The Ocean Roar Can Be Heard In the Shell Of The Conch

Should the theorem of the hippopotamus prove true through the avenue of science and the anthropologist world than within this post it is the continuing work that will eventually prove in theory the formation of the Earth’s core and the make-up wherefrom. 

 

First evolution and the path from the ocean to land animals; importance of time and the traveling biology into life form of new not complete.  As the gentle tack of evolution continues to explain our existence on this planet it is imperative that the work of past evolutionists is put to task.  I am not filling in the blanks as that would prove presumption, instead I have chosen to allow nature to take it’s natural course and place reliance on the obvious.

To be of interest in comprehension I know that each track that is found is exciting and full:  The trees and branches!

The elephant seal to the hippopotamus to the black rhino:  Explanation to be completed with more study on branches.  The passive nature to mention is simple, the Black rhinoceros is "Critically Endangered" (see below).  Is this the missing piece, and, is this where "Einstein's equation E = mc2 shows that energy and mass are interchangeable" (*see below) would prove applicable?  I believe that only part of the equation applies, specifically, MC2 leading down the prim rose path of:  Noted to: Special relativity (https://en.wikipedia.org/wiki/Special_relativity; see below).

Black rhinoceros

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Black rhinoceros or hook-lipped rhinoceros[1] Temporal range: Early Pleistocene - Recent 2.5–0 Ma PreЄ Є O S D C P T J K Pg N ↓
Male at Cincinnati Zoo
Female at the Taronga Zoo
Conservation status
Critically Endangered (IUCN 3.1)[2]
Scientific classification
Kingdom:	Animalia
Phylum:	Chordata
Class:	Mammalia
Order:	Perissodactyla
Family:	Rhinocerotidae
Genus:	Diceros Gray, 1821
Species:	D. bicornis
Binomial name
Diceros bicornis (Linnaeus, 1758)
Subspecies
Diceros bicornis bicornis † Diceros bicornis brucii † Diceros bicornis chobiensis Diceros bicornis ladoensis Diceros bicornis longipes † Diceros bicornis michaeli Diceros bicornis minor Diceros bicornis occidentalis
Historical black rhinoceros range (ca. 1700 A.D.).[3] Hatched: Possible historical range in West Africa.[4]
Current black rhinoceros range      Extant, resident      Extinct      Extant & Reintroduced (resident)      Extant & Assisted Colonisation (resident)
Synonyms
Rhinoceros bicornis Linnaeus, 1758

The black rhinoceros or hook-lipped rhinoceros (Diceros bicornis) is a species of rhinoceros, native to eastern and southern Africa including Angola, Botswana, Kenya, Malawi, Mozambique, Namibia, South Africa, Eswatini, Tanzania, Zambia, and Zimbabwe. Although the rhinoceros is referred to as black, its colours vary from brown to grey.
The other African rhinoceros is the white rhinoceros (Ceratotherium simum). The word "white" in the name "white rhinoceros" is often said to be a misinterpretation of the Afrikaans word wyd (Dutch wijd) meaning wide, referring to its square upper lip, as opposed to the pointed or hooked lip of the black rhinoceros. These species are now sometimes referred to as the square-lipped (for white) or hook-lipped (for black) rhinoceros.^[5]
The species overall is classified as critically endangered (even though the south-western black rhinoceros is classified as near threatened). Three subspecies have been declared extinct, including the western black rhinoceros, which was declared extinct by the International Union for Conservation of Nature (IUCN) in 2011.^[6][7]

Contents

• 1 Taxonomy
  • 1.1 Subspecies
• 2 Evolution
• 3 Description
• 4 Distribution
  • 4.1 Prehistorical range
  • 4.2 Historical and extant range
• 5 Behavior
  • 5.1 Diet
  • 5.2 Communication
  • 5.3 Reproduction
• 6 Conservation
• 7 Threats
• 8 References
• 9 Further reading
• 10 External links

Taxonomy

The species was first named Rhinoceros bicornis by Carl Linnaeus in the 10th edition of his Systema naturae in 1758. The name means "double-horned rhinoceros". There is some confusion about what exactly Linnaeus conceived under this name as this species was probably based upon the skull of a single-horned Indian rhinoceros (Rhinoceros unicornis), with a second horn artificially added by the collector. Such a skull is known to have existed and Linnaeus even mentioned India as origin of this species. However he also referred to reports from early travellers about a double-horned rhino in Africa and when it emerged that there is only one, single-horned species of rhino in India, Rhinoceros" bicornis was used to refer to the African rhinos (the white rhino only became recognised in 1812).^[8] In 1911 this was formally fixed and the Cape of Good Hope officially declared the type locality of the species.^[9]

Subspecies

The intraspecific variation in the black rhinoceros has been discussed by various authors and is not finally settled.^[10] The most accepted scheme considers seven or eight subspecies,^[3][11][12] of which three became extinct in historical times and one is on the brink of extinction:

• Southern black rhinoceros or Cape black rhinoceros (D. b. bicornis) – Extinct. Once abundant from the Cape of Good Hope to Transvaal, South Africa and probably into the south of Namibia, this was the largest subspecies. It became extinct due to excessive hunting and habitat destruction around 1850.^[13]
• North-eastern black rhinoceros (D. b. brucii) – Extinct. Formerly central Sudan, Eritrea, northern and southeastern Ethiopia, Djibouti and northern and southeastern Somalia. Primal populations in northern Somalia vanished during the early 20th century.
• Chobe black rhinoceros (D. b. chobiensis) – A local subspecies restricted to the Chobe Valley in southeastern Angola, Namibia (Zambezi Region) and northern Botswana. Nearly extinct, possibly only one surviving specimen in Botswana.^[12]
• Uganda black rhinoceros (D. b. ladoensis) – Former distribution from South Sudan, across Uganda into western Kenya and southwesternmost Ethiopia. Black rhinos are considered extinct across most of this area and its conservational status is unclear. Probably surviving in Kenyan reserves.
• Western black rhinoceros (D. b. longipes) – Extinct. Once lived in South Sudan, northern Central African Republic, southern Chad, northern Cameroon, northeastern Nigeria and south-eastern Niger. The range possibly stretched west to the Niger River in western Niger, though this is unconfirmed. The evidence from Liberia and Burkina Faso mainly rests upon the existence of indigenous names for the rhinoceros.^[4] A far greater former range in West Africa as proposed earlier^[14] is doubted by a 2004 study.^[4] The last known wild specimens lived in northern Cameroon. In 2006 an intensive survey across its putative range in Cameroon failed to locate any, leading to fears that it was extinct in the wild.^[6][15] On 10 November 2011 the IUCN declared the western black rhinoceros extinct.^[6]
• Eastern black rhinoceros (D. b. michaeli) – Had a historical distribution from South Sudan, Uganda, Ethiopia, down through Kenya into north-central Tanzania. Today, its range is limited primarily to Kenya and Tanzania.
• South-central black rhinoceros (D. b. minor) – Most widely distributed subspecies, characterised by a compact body, proportionally large head and prominent skin-folds. Ranged from north-eastern South Africa (KwaZulu-Natal) to northeastern Tanzania and southeastern Kenya. Preserved in reserves throughout most of its former range but probably extinct in eastern Angola, southern Democratic Republic of Congo and possibly Mozambique. Extinct but reintroduced in Malawi, Botswana, and Zambia.
• South-western black rhinoceros (D. b. occidentalis) – A small subspecies, adapted to survival in desert and semi-desert conditions. Originally distributed in north-western Namibia and southwestern Angola, today restricted to wildlife reserves in Namibia with sporadic sightings in Angola. These populations are often erroneously referred to D. b. bicornis or D. b. minor but represent a subspecies in their own right.^[12]

The most widely adopted alternative scheme only recognizes five subspecies or "eco-types", D. b. bicornis, D. b. brucii, D. b. longipes, D. b. michaeli, and D. b. minor.^[16] This concept is also used by the IUCN, listing three surviving subspecies and recognizing D. b. brucii and D. b. longipes as extinct. The most important difference to the above scheme is the inclusion of the extant southwestern subspecies from Namibia in D. b. bicornis instead of in its own subspecies, whereupon the nominal subspecies is not considered extinct.^[2]

Evolution

The rhinoceros originated in the Eocene about fifty million years ago alongside other members of Perissodactyla.^[17] Ancestors of the black and the white rhinoceros were present in Africa by the end of the Late Miocene about ten million years ago.^[17] The two species evolved from the common ancestral species Ceratotherium neumayri during this time. The clade comprising the genus Diceros is characterised by an increased adaptation to browsing. Between four and five million years ago, the black rhinoceros diverged from the white rhinoceros.^[17] After this split, the direct ancestor of Diceros bicornis, Diceros praecox was present in the Pliocene of East Africa (Ethiopia, Kenya, Tanzania). D. bicornis evolved from this species during the Late Pliocene – Early Pleistocene.^[18] With the oldest definitive record at the Pliocene-Pleistocene boundary c. 2.5 Ma at Koobi Fora, Kenya.^[19]

Description

Comparative illustration of black (top) and white rhinos (bottom)

An adult black rhinoceros stands 140–180 cm (55–71 in) high at the shoulder and is 3–3.75 m (9.8–12.3 ft) in length.^[20][21] An adult typically weighs from 800 to 1,400 kg (1,760 to 3,090 lb), however unusually large male specimens have been reported at up to 2,896 kg (6,385 lb).^[3][20] The females are smaller than the males. Two horns on the skull are made of keratin with the larger front horn typically 50 cm (20 in) long, exceptionally up to 140 cm (55 in).
The longest known black rhinoceros horn measured nearly 1.5 m (4.9 ft) in length.^[22] Sometimes, a third, smaller horn may develop.^[23] These horns are used for defense, intimidation, and digging up roots and breaking branches during feeding. The black rhino is smaller than the white rhino, and is close in size to the Javan rhino of Indonesia. It has a pointed and prehensile upper lip, which it uses to grasp leaves and twigs when feeding.^[22] The white rhinoceros has square lips used for eating grass. The black rhinoceros can also be distinguished from the white rhinoceros by its size, smaller skull, and ears; and by the position of the head, which is held higher than the white rhinoceros, since the black rhinoceros is a browser and not a grazer. This key differentiation is further illustrated by the shape of the two species mouths (lips): the "square" lip of the white rhinoceros is an adaptation for grazing, and the "hooked" lip of the black rhinoceros is an adaptation to help browsing.^{[citation needed]}

A black rhinoceros skull

Their thick-layered skin helps to protect the rhino from thorns and sharp grasses. Their skin harbors external parasites, such as mites and ticks, which may be eaten by oxpeckers and egrets.^[24] Such behaviour was originally thought to be an example of mutualism, but recent evidence suggests that oxpeckers may be parasites instead, feeding on rhino blood.^[25] It is commonly assumed that black rhinos have poor eyesight, relying more on hearing and smell. However, studies have shown that their eyesight is comparatively good, at about the level of a rabbit.^[26] Their ears have a relatively wide rotational range to detect sounds. An excellent sense of smell alerts rhinos to the presence of predators.

Distribution

Prehistorical range

As with many other components of the African large mammal fauna, black rhinos probably had a wider range in the northern part of the continent in prehistoric times than today. However this seems to have not been as extensive as that of the white rhino. Unquestionable fossil remains have not yet been found in this area and the abundant petroglyphs found across the Sahara desert are often too schematic to unambiguously decide whether they depict black or white rhinos. Petroglyphs from the Eastern Desert of southeastern Egypt relatively convincingly show the occurrence of black rhinos in these areas in prehistoric times.^[27]

Historical and extant range

The natural range of the black rhino included most of southern and eastern Africa, but it did not occur in the Congo Basin, the tropical rainforest areas along the Bight of Benin, the Ethiopian Highlands, and the Horn of Africa.^[3] Its former native occurrence in the extremely dry parts of the Kalahari desert of southwestern Botswana and northwestern South Africa is uncertain.^[28] In western Africa it was abundant in an area stretching east to west from Eritrea and Sudan through South Sudan to southeastern Niger, and especially around Lake Chad. Its occurrence further to the west is questionable, though often purported to in literature.^[4] Today it is totally restricted to protected nature reserves and has vanished from many countries in which it once thrived, especially in the west and north of its former range. The remaining populations are highly scattered. Some specimens have been relocated from their habitat to better protected locations, sometimes across national frontiers.^[2] The black rhino has been successfully reintroduced to Malawi since 1993, where it became extinct in 1990.^[29] Similarly it was reintroduced to Zambia (North Luangwa National Park) in 2008, where it had become extinct in 1998,^[30] and to Botswana (extinct in 1992, reintroduced in 2003).^[31]
In May 2017, 18 Eastern Black Rhinos were translocated from South Africa to the Akagera National Park in Rwanda. The park had around 50 rhinos in the 1970s but the numbers dwindled to zero by 2007. In September 2017, the birth of a calf raised the population to 19. The park has dedicated rhino monitoring teams to protect the animals from poaching.^[32][33]
In October 2017, The governments of Chad and South Africa reached an agreement to transfer six black rhinos from South Africa to Zakouma National Park in Chad. Once established, this will be the northernmost population of the species. The species was wiped out from Chad in the 1970s and is under severe pressure from poaching in South Africa. The agreement calls for South African experts to assess the habitat, local management capabilities, security and the infrastructure before the transfer can take place.^[34]

Behavior

An adult black rhinoceros with young grazing in Krefeld Zoo

Black rhino at Moringa waterhole, Etosha National Park

Black rhinoceros are generally thought to be solitary, with the only strong bond between a mother and her calf. In addition, males and females have a consort relationship during mating, also subadults and young adults frequently form loose associations with older individuals of either sex.^[35] They are not very territorial and often intersect other rhino territories. Home ranges vary depending on season and the availability of food and water. Generally they have smaller home ranges and larger density in habitats that have plenty of food and water available, and vice versa if resources are not readily available. Sex and age of an individual black rhino influence home range and size, with ranges of females larger than those of males, especially when accompanied by a calf.^[36] In the Serengeti home ranges are around 70 to 100 km² (27 to 39 sq mi), while in the Ngorongoro it is between 2.6 to 58.0 km² (1.0 to 22.4 sq mi).^[35] Black rhinos have also been observed to have a certain area they tend to visit and rest frequently called "houses" which are usually on a high ground level.^{[citation needed]} These "home" ranges can vary from 2.6 km² to 133 km² with smaller home ranges having more abundant resources than larger home ranges.^[37]
Black rhinoceros in captivity and reservations sleep patterns have been recently studied to show that males sleep longer on average than females by nearly double the time. Other factors that play a role in their sleeping patterns is the location of where they decide to sleep. Although they do not sleep any longer in captivity, they do sleep at different times due to their location in captivity, or section of the park.^[38]
The black rhino has a reputation for being extremely aggressive, and charges readily at perceived threats. They have even been observed to charge tree trunks and termite mounds.^{[citation needed]} Black rhinos will fight each other, and they have the highest rates of mortal combat recorded for any mammal: about 50% of males and 30% of females die from combat-related injuries.^[39] Adult rhinos normally have no natural predators, thanks to their imposing size as well as their thick skin and deadly horns.^[40] However, adult black rhinos have fallen prey to crocodiles in exceptional circumstances.^[41] Calves and, very seldom, small sub-adults may be preyed upon by lions as well.^[3]
Black rhinoceros follow the same trails that elephants use to get from foraging areas to water holes. They also use smaller trails when they are browsing. They are very fast and can get up to speeds of 55 kilometres per hour (34 mph) running on their toes.^[42][43]

Diet

Chewing on plants

The black rhinoceros is a herbivorous browser that eats leafy plants, branches, shoots, thorny wood bushes, and fruit.^[44] The optimum habitat seems to be one consisting of thick scrub and bushland, often with some woodland, which supports the highest densities. Their diet can reduce the amount of woody plants, which may benefit grazers (who focus on leaves and stems of grass), but not competing browsers (who focus on leaves, stems of trees, shrubs or herbs). It has been known to eat up to 220 species of plants. They have a significantly restricted diet with a preference for a few key plant species and a tendency to select leafy species in the dry season.^[45] The plant species they seem to be most attracted to when not in dry season are the woody plants. There are 18 species of woody plants known to the diet of the black rhinoceros, and 11 species that could possibly be a part of their diet too.^[46] Black rhinoceros also have a tendency to choose food based on quality over quantity, where researchers find more populations in areas where the food has better quality.^[47] In accordance with their feeding habit, adaptations of the chewing apparatus have been described for rhinos. The black rhinoceros has a twophased chewing activity with a cutting ectoloph and more grinding lophs on the lingual side. The black rhinoceros can also be considered a more challenging herbivore to feed in captivity compared to its grazing relatives.^[48] It can live up to 5 days without water during drought. Black rhinos live in several habitats including bushlands, Riverine woodland, marshes, and their least favorable, grasslands. Habitat preferences are shown in two ways, the amount of sign found in the different habitats, and the habitat content of home ranges and core areas. Habitat types are also identified based on the composition of dominant plant types in each area. Different subspecies live in different habitats including Vachellia and Senegalia savanna, Euclea bushlands, Albany thickets, and even desert.^[35] They browse for food in the morning and evening. They are selective browsers but, studies done in Kenya show that they do add the selection material with availability in order to satisfy their nutritional requirements.^[49] In the hottest part of the day they are most inactive- resting, sleeping, and wallowing in mud. Wallowing helps cool down body temperature during the day and protects against parasites. When black rhinos browse they use their lips to strip the branches of their leaves. Competition with elephants is causing the black rhinoceros to shift its diet. The black rhinoceros alters its selectivity with the absence of the elephant.^[50]
There is some variance in the exact chemical composition of rhinoceros horns. This variation is directly linked to diet and can be used as a means of rhino identification. Horn composition has helped scientists pinpoint the original location of individual rhinos, allowing for law enforcement to more accurately and more frequently identify and penalize poachers.^[51]

Communication

A male south-western black rhinoceros (D. b. occidentalis) in Etosha National Park, Namibia

Rhinos use several forms of communication. Due to their solitary nature, scent marking is often used to identify themselves to other black rhinos. Urine spraying occurs on trees and bushes, around water holes and feeding areas. Females urine spray more often when receptive for breeding. Defecation sometimes occurs in the same spot used by different rhinos, such as around feeding stations and watering tracks. Coming upon these spots, rhinos will smell to see who is in the area and add their own marking. When presented with adult feces, male and female rhinoceroses respond differently than when they are presented with subadult feces. The urine and feces of one black rhinoceros helps other black rhinoceroses to determine its age, sex, and identity.^[52] Less commonly they will rub their heads or horns against tree trunks to scent-mark.
The black rhino has powerful tube-shaped ears that can freely rotate in all directions. This highly developed sense of hearing allows black rhinos to detect sound over vast distances.^[53]

Reproduction

Mother and calf in Lewa, central Kenya

The adults are solitary in nature, coming together only for mating. Mating does not have a seasonal pattern but births tend to be towards the end of the rainy season in more arid environments.
When in season the females will mark dung piles. Males will follow females when they are in season; when she defecates he will scrape and spread the dung, making it more difficult for rival adult males to pick up her scent trail.
Courtship behaviors before mating include snorting and sparring with the horns among males. Another courtship behavior is called bluff and bluster, where the rhino will snort and swing its head from side to side aggressively before running away repeatedly. Breeding pairs stay together for 2–3 days and sometimes even weeks. They mate several times a day over this time and copulation lasts for a half-hour.
The gestation period for a black rhino is 15 months. The single calf weighs about 35–50 kilograms (80–110 lb) at birth, and can follow its mother around after just three days. Weaning occurs at around 2 years of age for the offspring. The mother and calf stay together for 2–3 years until the next calf is born; female calves may stay longer, forming small groups. The young are occasionally taken by hyenas and lions. Sexual maturity is reached from 5 to 7 years old for females, and 7 to 8 years for males. The life expectancy in natural conditions (without poaching pressure) is from 35 to 50 years.^[54]

Conservation

Black rhino in the Maasai Mara

For most of the 20th century the continental black rhino was the most numerous of all rhino species. Around 1900 there were probably several hundred thousand^[2] living in Africa. During the latter half of the 20th century their numbers were severely reduced from an estimated 70,000^[55] in the late 1960s to only 10,000 to 15,000 in 1981. In the early 1990s the number dipped below 2,500, and in 2004 it was reported that only 2,410 black rhinos remained. According to the International Rhino Foundation—housed in Yulee, Florida at White Oak Conservation, which breeds black rhinos^[56]—the total African population had recovered to 4,240 by 2008 (which suggests that the 2004 number was low).^[57] By 2019 the population of 5,500 was either steady or slowly increasing.^[58]
In 1992, nine rhinos were brought from Chete National Park, Zimbabwe to Australia via Cocos Island. After the natural deaths of the males in the group, four males were brought in from United States and have since adapted well to captivity and new climate.^[59] Calves and some subadults are preyed on by lions, but predation is rarely taken into account in managing the black rhinoceros.^{[citation needed]} This is a major flaw because predation should be considered when attributing cause to the poor performance of the black rhinoceros population.^[60] In 2002 only ten western black rhinos remained in Cameroon, and in 2006 intensive surveys across its putative range failed to locate any, leading to fears that this subspecies had become extinct.^[15] In 2011 the IUCN declared the western black rhino extinct.^[61] There was a conservation effort in which black rhinos were translocated, but their population did not improve, as they did not like to be in an unfamiliar habitat.
Under CITES Appendix I all international commercial trade of the black rhino horn is prohibited since 1977.^[37] China though having joined CITES since 8 April 1981 is the largest importer of black rhino horns.^{[62][citation needed]} However, this is a trade in which not only do the actors benefit, but so do the nation states ignoring them as well. Nonetheless, people continue to remove the rhino from its natural environment and allow for a dependence on human beings to save them from endangerment.^[63] Parks and reserves have been made for protecting the rhinos with armed guards keeping watch, but even still many poachers get through and harm the rhinos for their horns. Many have considered extracting rhino horns in order to deter poachers from slaughtering these animals or potentially bringing them to other breeding grounds such as the US and Australia.^[63] This method of extracting the horn, known as dehorning, consists of tranquilizing the rhino then sawing the horn almost completely off to decrease initiative for poaching, although the effectiveness of this in reducing poaching is not known and rhino mothers are known to use their horns to fend off predators.^[64]
The only rhino subspecies that has recovered somewhat from the brink of extinction is the southern white rhinoceros, whose numbers now are estimated around 14,500, up from fewer than 50 in the first decade of the 20th century.^[65] But there seems to be hope for the black rhinoceros in recovering their gametes from dead rhinos in captivity. This shows promising results for producing black rhinoceros embryos, which can be used for testing sperm in vitro.^[66]
A January 2014 auction for a permit to hunt a black rhinoceros in Namibia sold for $350,000 at a fundraiser hosted by the Dallas Safari Club. The auction drew considerable criticism as well as death threats directed towards members of the club and the man who purchased the permit.^[67] This permit was issued for 1 of 18 black rhinoceros specifically identified by Namibia's Ministry of Environment and Tourism as being past breeding age and considered a threat to younger rhinos. The $350,000 that the hunter paid for the permit was used by the Namibian government to fund anti-poaching efforts in the country.^[68]

Threats

Today, there are various threats posed to the black rhinoceros including habitat changes, illegal poaching, and competing species. Civil disturbances, such as war, have made mentionably negative effects on the black rhinoceros populations in since the 1960s in countries including, but not limited to, Chad, Cameroon, Rwanda, Mozambique, and Somalia.^[2] In the Addo Elephant National Park in South Africa, the African elephant Loxodonta africana is posing slight concern involving the black rhinoceroses who also inhabit the area. Both animals are browsers; however, the elephant's diet consists of a wider variety of foraging capacity, while the rhinoceros primarily sticks to dwarf shrubs. The black rhinoceros has been found to eat grass as well; however, the shortening of its range of available food could be potentially problematic.^[69]
Black rhinoceros face problems associated with the minerals they ingest. They have become adjusted to ingesting less iron in the wild due to their evolutionary progression, which poses a problem when placed in captivity. These rhinoceroses can overload on iron, which leads to build up in the lungs, liver, spleen and small intestine.^[70] Not only do these rhinoceros face threats being in the wild, but in captivity too. Black rhinoceros have become more susceptible to disease in captivity with high rates of mortality.^[66]
Illegal poaching for the international rhino horn trade is the main and most detrimental threat.^[2] The killing of these animals is not unique to modern-day society. The Chinese have maintained reliable documents of these happenings dating back to 1200 B.C.^[71] The ancient Chinese often hunted rhino horn for the making of wine cups, as well as the rhino's skin to manufacture imperial crowns, belts and armor for soldiers.^[71] A major market for rhino horn has historically been in the Middle East nations to make ornately carved handles for ceremonial daggers called jambiyas. Demand for these exploded in the 1970s, causing the black rhinoceros population to decline 96% between 1970 and 1992. The horn is also used in traditional Chinese medicine, and is said by herbalists to be able to revive comatose patients, facilitate exorcisms and various methods of detoxification,^[71] cure fevers, and aid male sexual stamina and fertility.^[72] It is also hunted for the superstitious belief that the horns allow direct access to Heaven due to their unique location and hollow nature.^[71] The purported effectiveness of the use of rhino horn in treating any illness has not been confirmed, or even suggested, by medical science. In June 2007, the first-ever documented case of the medicinal sale of black rhino horn in the United States (confirmed by genetic testing of the confiscated horn) occurred at a traditional Chinese medicine supply store in Portland, Oregon's Chinatown.^[72]

References

1. 
   

Grubb, P. (2005). "Order Perissodactyla". In Wilson, D.E.; Reeder, D.M (eds.). Mammal Species of the World: A Taxonomic and Geographic Reference (3rd ed.). Johns Hopkins University Press. pp. 635–636. ISBN 978-0-8018-8221-0. OCLC 62265494.

72. Patte, David (26 June 2007). "Portland Man Pleads Guilty to Selling Black Rhino Horn". U.S. Fish & Wildlife Service. Archived from the original on 8 August 2007. Retrieved 29 June 2007.

Further reading

• Emslie, R. & Brooks, M. (1999). African Rhino. Status Survey and Conservation Action Plan. IUCN/SSC African Rhino Specialist Group. IUCN, Gland, Switzerland and Cambridge, UK. ISBN 2-8317-0502-9.
• Rookmaaker, L. C. (2005). "Review of the European perception of the African rhinoceros". Journal of Zoology. 265 (4): 365–376. doi:10.1017/S0952836905006436. S2CID 86237288.

External links

Wikispecies has information related to Diceros bicornis

Wikimedia Commons has media related to Diceros bicornis.

• Black rhinoceros at Curlie
• Black Rhino Info & Black Rhino Pictures on the Rhino Resource Center website.
• "Black Rhinoceros" (PDF). Zoological Parks Board of New South Wales. Archived from the original (PDF) on 28 September 2007. Retrieved 9 October 2007.
• www.savetherhino.org
• WildLifeNow Website for the Tony Fitzjohn/George Adamson African Wildlife Preservation Trust, supporting the Mkomazi Game Reserve and Mkomazi Rhino Sanctuary in Tanzania
• U.S. Fish & Wildlife Service Species Profile
• Sebakwe Black Rhino Trust dedicated to black rhino conservation in Zimbabwe
• A Radiolab interview with the man that won the auction to hunt a black rhino

v t e Extant Perissodactyla (Odd-toed ungulates) species by suborder
Kingdom Animalia Phylum Chordata Class Mammalia Infraclass Eutheria Superorder Laurasiatheria
Hippomorpha Equidae (Horse family) Equus (including Zebras) Subgenus Equus: Wild horse (E. ferus) Domestic horse (E. ferus caballus) Subgenus Asinus: African wild ass (E. africanus) Donkey (E. africanus asinus) Onager (E. hemionus) Kiang (E. kiang) Subgenus Dolichohippus: Grévy's zebra (E. grevyi) Subgenus Hippotigris: Plains zebra (E. quagga) Mountain zebra (E. zebra)
Ceratomorpha Rhinocerotidae (Rhinoceroses) Rhinoceros Indian rhinoceros (R. unicornis) Javan rhinoceros (R. sondaicus) Dicerorhinus Sumatran rhinoceros (D. sumatrensis) Ceratotherium White rhinoceros (C. simum) Diceros Black rhinoceros (D. bicornis) Tapiridae (Tapirs) Tapirus Baird's tapir (T. bairdii) Little black tapir (T. kabomani) Malayan tapir (T. indicus) Mountain tapir (T. pinchaque) South American tapir (T. terrestris)
Category

Taxon identifiers	Wikidata: Q95036 Wikispecies: Diceros bicornis ARKive: diceros-bicornis ECOS: 1234 EoL: 311501 EPPO: DKRSBI Fossilworks: 197764 GBIF: 5220111 iNaturalist: 43350 IRMNG: 10227003 ITIS: 625003 IUCN: 6557 MSW: 14100059 NCBI: 9805 Species+: 6656 TSA: 5971

Authority control	BNF: cb13328971f (data) GND: 4182388-6 LCCN: sh85014609

Categories:

• IUCN Red List critically endangered species
• Rhinoceroses
• EDGE species
• Fauna of East Africa
• Mammals of Southern Africa
• Mammals of Sub-Saharan Africa
• Critically endangered fauna of Africa
• Mammals described in 1758
• Species endangered by use in wearables
• Species endangered by human consumption for medicinal or magical purposes

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Missing link (human evolution)

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For other uses, see Missing link.

A symbolic portrayal of human evolution, wrongly implying that evolution is linear and progressive.

"Missing link" is an unscientific term for a transitional fossil. It is often used in popular science and in the media for any new transitional form. The term originated to describe the hypothetical intermediate form in the evolutionary series of anthropoid ancestors to anatomically modern humans (hominization). The term was influenced by the pre-Darwinian evolutionary theory of the Great Chain of Being and the now-outdated notion (orthogenesis) that simple organisms are more primitive than complex organisms.

Phylogenetic tree of hominid evolution

The term "missing link" has fallen out of favor with biologists because it implies the evolutionary process is a linear phenomenon and that forms originate consecutively in a chain. Instead, last common ancestor is preferred since this does not have the connotation of linear evolution, as evolution is a branching process.
In addition to implying a linear evolution, the term also implies that a particular fossil has not yet been found. Many of the famous discoveries in human evolution are often termed "missing links". For example, there were the Peking Man and the Java Man, despite the fact that these fossils are not missing. Transitional forms that have not been discovered are also termed missing links; however, there is no singular missing link. The scarcity of transitional fossils can be attributed to the incompleteness of the fossil record.

Contents

• 1 Historical origins
• 2 Historical beliefs about the missing link
• 3 Famous "missing links" in human evolution
• 4 Portrayals in media
• 5 References

Historical origins

The term "missing link" was influenced by the 18th century Enlightenment thinkers such as Alexander Pope and Jean-Jacques Rousseau who thought of humans as links in the Great Chain of Being. The Great Chain of Being is a hierarchical structure of all matter and life. Influenced by Aristotle's theory of higher and lower animals, the Great Chain of Being was created during the Medieval period in Europe and was strongly influenced by religious thought.^[1] God was at the top of the chain followed by man and then animals. It was during the 18th century that the set nature of species and their immutable place in the great chain was questioned. The dual nature of the chain, divided yet united, had always allowed for seeing creation as essentially one continuous whole, with the potential for overlap between the links.^[2] Radical thinkers like Jean-Baptiste Lamarck saw a progression of life forms from the simplest creatures striving towards complexity and perfection, a schema accepted by zoologists like Henri de Blainville.^[3] The very idea of an ordering of organisms, even if supposedly fixed, laid the basis for the idea of transmutation of species, for example Charles Darwin's theory of evolution.^[4]
The earliest publication that explicitly uses the term “missing link” was in 1844 in Vestiges of the Natural History of Creation by Robert Chambers, which uses the term in an evolutionary context relating to gaps in the fossil record.^[5] Charles Lyell employed the term a few years later in 1851 in his third edition of Elements of Geology to as a metaphor for the missing gaps in the continuity of the geological column.^[6] The first time it was used as a name for transitional types between different taxa was in 1863, in Lyell's Geological Evidences of the Antiquity of Man.^[7] "Missing link" later became a name for transitional fossils, particularly those seen as bridging the gulf between man and animal. Subsequently, Charles Darwin, Thomas Henry Huxley, and Ernst Haeckel used it in their works with this meaning.

Historical beliefs about the missing link

Haeckel's Chain of the Animal Ancestors of Man

Jean-Baptiste Lamarck envisioned that life is generated in the form of the simplest creatures constantly, and then strive towards complexity and perfection (i.e. humans) through a series of lower forms. In his view, lower animals were simply newcomers on the evolutionary scene. After Darwin's On the Origin of Species, the idea of "lower animals" representing earlier stages in evolution lingered, as demonstrated in Ernst Haeckel's figure of the human pedigree. While the vertebrates were then seen as forming a sort of evolutionary sequence, the various classes were distinct, the undiscovered intermediate forms being called "missing links."
Haeckel claimed that human evolution occurred in 24 stages and that the 23rd stage was a theoretical missing link he named Pithecanthropus alalus ("ape-man lacking speech").^[8] Haeckel claimed the origin of humanity was to be found in Asia. He theorized that the missing link was to be found on the lost continent of Lemuria located in the Indian Ocean. He believed that Lemuria was the home of the first humans and that Asia was the home of many of the earliest primates; he thus supported that Asia was the cradle of hominid evolution. Haeckel argued that humans were closely related to the primates of Southeast Asia and rejected Darwin's hypothesis of human origins in Africa.^[1][9]
The search for a fossil that connected man and ape was unproductive until the Dutch paleontologist Eugene Dubois went to Indonesia. Between 1886 and 1895 Dubois discovered remains that he later described as "an intermediate species between humans and monkeys". He named the hominin Pithecanthropus erectus (erect ape-man), which has now been reclassified as Homo erectus. In the media, the Java Man was hailed as the missing link. For instance, the headline of the Philadelphia Enquirer on February 3, 1895, was "The Missing Link: A Dutch Surgeon in Java Unearths the Needed Specimen".^[10]

Famous "missing links" in human evolution

Further information: List of human evolution fossils, Pithecanthropus, Archaic humans, and Homo habilis

Java Man, the original "missing link" found in Java

Among the famous fossil finds credited as the "missing link" in human evolution are:

• Java Man (Homo erectus): Discovered by Eugene Dubois in 1891 in Indonesia. Originally named Pithecanthropus erectus.
• Piltdown Man: A set of bones found in 1912 thought to be the "missing link" between ape and man. Eventually revealed to be a hoax.
• Taung Child (Australopithecus africanus): Discovered by Raymond Dart in 1924 in South Africa.
• Homo habilis (described in 1964) has features intermediate between Australopithecus and Homo erectus, and its classification in Homo rather than Australopithecus has been questioned.^[11]
• Lucy (Australopithecus afarensis): Discovered in 1974 by Donald Johanson in Ethiopia
• Australopithecus sediba: A series of skeletons discovered in South Africa between 2008-2010

Portrayals in media

• The Missing Link in Monsters vs. Aliens (2009)
• Missing Link (1988)
• The Missing Link (1927)
• The Missing Link (1980)
• Missing Link (2019)

References

1. 
   

11. Wood and Richmond; Richmond, BG (2000). "Human evolution: taxonomy and paleobiology". Journal of Anatomy. 197 (Pt 1): 19–60. doi:10.1046/j.1469-7580.2000.19710019.x. PMC 1468107. PMID 10999270. p. 41: "A recent reassessment of cladistic and functional evidence concluded that there are few, if any, grounds for retaining H. habilis in Homo, and recommended that the material be transferred (or, for some, returned) to Australopithecus (Wood & Collard, 1999)."

Categories:

• Human evolution
• Gaps in the fossil record

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Mass–energy equivalence

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"E=MC2" and "E=mc2" redirect here. For other uses, see E=MC2 (disambiguation).

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In physics, mass–energy equivalence is the principle that anything having mass has an equivalent amount of energy and vice versa, with these fundamental quantities directly relating to one another by Albert Einstein's famous formula:^[1]

This formula states that the equivalent energy (E) can be calculated as the mass (m) multiplied by the speed of light (c = ~3×10⁸ m/s) squared. Similarly, anything having energy exhibits a corresponding mass m given by its energy E divided by the speed of light squared c². Because the speed of light is a large number in everyday units, the formula implies that even an everyday object at rest with a modest amount of mass has a very large amount of energy intrinsically. Chemical reactions, nuclear reactions, and other energy transformations may cause a system to lose some of its energy content to the environment (and thus some corresponding mass), releasing it as the radiant energy of light or as thermal energy for example.
Mass–energy equivalence arose originally from special relativity as a paradox described by Henri Poincaré.^[2] Einstein proposed it on 21 November 1905, in the paper Does the inertia of a body depend upon its energy-content?, one of his Annus Mirabilis (Miraculous Year) papers.^[3] Einstein was the first to propose that the equivalence of mass and energy is a general principle and a consequence of the symmetries of space and time.
A consequence of the mass–energy equivalence is that if a body is stationary, it still has some internal or intrinsic energy, called its rest energy, corresponding to its rest mass. When the body is in motion, its total energy is greater than its rest energy, and equivalently its total mass (also called relativistic mass in this context) is greater than its rest mass. This rest mass is also called the intrinsic or invariant mass because it remains the same regardless of this motion, even for the extreme speeds or gravity considered in special and general relativity.
The mass–energy formula also serves to convert units of mass to units of energy (and vice versa), no matter what system of measurement units is used.

Contents

• 1 Nomenclature
• 2 Conservation of mass and energy
  • 2.1 Fast-moving objects and systems of objects
• 3 Applicability of the strict formula
• 4 Meanings of the strict formula
  • 4.1 Binding energy and the "mass defect"
  • 4.2 Massless particles
  • 4.3 Massless particles contribute rest mass and invariant mass to systems
  • 4.4 Relation to gravity
• 5 Application to nuclear physics
• 6 Practical examples
• 7 Efficiency
• 8 Background
  • 8.1 Mass–velocity relationship
  • 8.2 Relativistic mass
  • 8.3 Low speed expansion
• 9 History
  • 9.1 Newton: matter and light
  • 9.2 Swedenborg: matter composed of "pure and total motion"
  • 9.3 Electromagnetic mass
  • 9.4 Radiation pressure and inertia
  • 9.5 Einstein: mass–energy equivalence
    • 9.5.1 The first derivation by Einstein (1905)
    • 9.5.2 Alternative version
    • 9.5.3 Relativistic center-of-mass theorem (1906)
  • 9.6 Others
  • 9.7 Radioactivity and nuclear energy
• 10 See also
• 11 References
• 12 External links

Nomenclature

The formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.^[4][5] In "Does the inertia of a body depend upon its energy content?" (1905), Einstein used V to mean the speed of light in a vacuum and L to mean the energy lost by a body in the form of radiation.^[3] Consequently, the equation E = mc² was not originally written as a formula but as a sentence in German saying that "if a body gives off the energy L in the form of radiation, its mass diminishes by L/V²." A remark placed above it informed that the equation was approximated by neglecting "magnitudes of fourth and higher orders" of a series expansion.^[6]
In May 1907, Einstein explained that the expression for energy ε of a moving mass point assumes the simplest form, when its expression for the state of rest is chosen to be ε₀ = μV² (where μ is the mass), which is in agreement with the "principle of the equivalence of mass and energy". In addition, Einstein used the formula μ = E₀/V², with E₀ being the energy of a system of mass points, to describe the energy and mass increase of that system when the velocity of the differently moving mass points is increased.^[7]
In June 1907, Max Planck rewrote Einstein's mass–energy relationship as M = E₀ + pV₀/c², where p is the pressure and V the volume to express the relation between mass, its latent energy, and thermodynamic energy within the body.^[8] Subsequently, in October 1907, this was rewritten as M₀ = E₀/c² and given a quantum interpretation by Johannes Stark, who assumed its validity and correctness (Gültigkeit).^[9]
In December 1907, Einstein expressed the equivalence in the form M = μ + E₀/c² and concluded: "A mass μ is equivalent, as regards inertia, to a quantity of energy μc². [...] It appears far more natural to consider every inertial mass as a store of energy."^[10][11]
In 1909, Gilbert N. Lewis and Richard C. Tolman used two variations of the formula: m = E/c² and m₀ = E₀/c², with E being the relativistic energy (the energy of an object when the object is moving), E₀ is the rest energy (the energy when not moving), m is the relativistic mass (the rest mass and the extra mass gained when moving), and m₀ is the rest mass (the mass when not moving).^[12] The same relations in different notation were used by Hendrik Lorentz in 1913 (published 1914), though he placed the energy on the left-hand side: ε = Mc² and ε₀ = mc², with ε being the total energy (rest energy plus kinetic energy) of a moving material point, ε₀ its rest energy, M the relativistic mass, and m the invariant (or rest) mass.^[13]
In 1911, Max von Laue gave a more comprehensive proof of M₀ = E₀/c² from the stress–energy tensor,^[14] which was later (1918) generalized by Felix Klein.^[15]
Einstein returned to the topic once again after World War II and this time he wrote E = mc² in the title of his article^[16] intended as an explanation for a general reader by analogy.^[17]

Conservation of mass and energy

Main articles: Conservation of energy and Conservation of mass

Mass and energy can be seen as two names (and two measurement units) for the same underlying, conserved physical quantity.^[18] Thus, the laws of conservation of energy and conservation of (total) mass are equivalent and both hold true.^[19] Einstein elaborated in a 1946 essay that "the principle of the conservation of mass [...] proved inadequate in the face of the special theory of relativity. It was therefore merged with the energy conservation principle—just as, about 60 years before, the principle of the conservation of mechanical energy had been combined with the principle of the conservation of heat [thermal energy]. We might say that the principle of the conservation of energy, having previously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass—and holds the field alone."^[20]
If the conservation of mass law is interpreted as conservation of rest mass, it does not hold true in special relativity. The rest energy (equivalently, rest mass) of a particle can be converted, not "to energy" (it already is energy (mass)), but rather to other forms of energy (mass) that require motion, such as kinetic energy, thermal energy, or radiant energy. Similarly, kinetic or radiant energy can be converted to other kinds of particles that have rest energy (rest mass). In the transformation process, neither the total amount of mass nor the total amount of energy changes, since both properties are connected via a simple constant.^[21][22] This view requires that if either energy or (total) mass disappears from a system, it is always found that both have simply moved to another place, where they are both measurable as an increase of both energy and mass that corresponds to the loss in the first system.

Fast-moving objects and systems of objects

When an object is pushed in the direction of motion, it gains momentum and energy, but when the object is already traveling near the speed of light, it cannot move much faster, no matter how much energy it absorbs. Its momentum and energy continue to increase without bounds, whereas its speed approaches (but never reaches) a constant value—the speed of light. This implies that in relativity the momentum of an object cannot be a constant times the velocity, nor can the kinetic energy be a constant times the square of the velocity.
A property called the relativistic mass is defined as the ratio of the momentum of an object to its velocity.^[23] Relativistic mass depends on the motion of the object, so that different observers in relative motion see different values for it. If the object is moving slowly, the relativistic mass is nearly equal to the rest mass and both are nearly equal to the usual Newtonian mass. If the object is moving quickly, the relativistic mass is greater than the rest mass by an amount equal to the mass associated with the kinetic energy of the object. As the object approaches the speed of light, the relativistic mass grows infinitely, because the kinetic energy grows infinitely and this energy is associated with mass.
The relativistic mass is always equal to the total energy (rest energy plus kinetic energy) divided by c².^[24] Because the relativistic mass is exactly proportional to the energy, relativistic mass and relativistic energy are nearly synonyms; the only difference between them is the units. If length and time are measured in natural units, the speed of light is equal to 1, and even this difference disappears. Then mass and energy have the same units and are always equal, so it is redundant to speak about relativistic mass, because it is just another name for the energy. This is why physicists usually reserve the useful short word "mass" to mean rest mass, or invariant mass, and not relativistic mass.
The relativistic mass of a moving object is larger than the relativistic mass of an object that is not moving, because a moving object has extra kinetic energy. The rest mass of an object is defined as the mass of an object when it is at rest, so that the rest mass is always the same, independent of the motion of the observer: it is the same in all inertial frames.
For things and systems made up of many parts, like an atomic nucleus, planet, or star, the relativistic mass is the sum of the relativistic masses (or energies) of the parts, because energies are additive in isolated systems. This is not true in open systems, however, if energy is subtracted. For example, if a system is bound by attractive forces, and the energy gained due to the forces of attraction in excess of the work done is removed from the system, then mass is lost with this removed energy. For example, the mass of an atomic nucleus is less than the total mass of the protons and neutrons that make it up, but this is only true after this energy from binding has been removed in the form of a gamma ray (which in this system, carries away the mass of the energy of binding). This mass decrease is also equivalent to the energy required to break up the nucleus into individual protons and neutrons (in this case, work and mass would need to be supplied). Similarly, the mass of the solar system is slightly less than the sum of the individual masses of the sun and planets.
For a system of particles going off in different directions, the invariant mass of the system is the analog of the rest mass, and is the same for all observers, even those in relative motion. It is defined as the total energy (divided by c²) in the center of mass frame (where by definition, the system total momentum is zero). A simple example of an object with moving parts but zero total momentum is a container of gas. In this case, the mass of the container is given by its total energy (including the kinetic energy of the gas molecules), since the system total energy and invariant mass are the same in any reference frame where the momentum is zero, and such a reference frame is also the only frame in which the object can be weighed. In a similar way, the theory of special relativity posits that the thermal energy in all objects (including solids) contributes to their total masses and weights, even though this energy is present as the kinetic and potential energies of the atoms in the object, and it (in a similar way to the gas) is not seen in the rest masses of the atoms that make up the object.
In a similar manner, even photons (light quanta), if trapped in a container space (as a photon gas or thermal radiation), would contribute a mass associated with their energy to the container. Such an extra mass, in theory, could be weighed in the same way as any other type of rest mass. This is true in special relativity theory, even though individually photons have no rest mass. The property that trapped energy in any form adds weighable mass to systems that have no net momentum is one of the characteristic and notable consequences of relativity. It has no counterpart in classical Newtonian physics, in which radiation, light, heat, and kinetic energy never exhibit weighable mass under any circumstances.
Just as the relativistic mass of an isolated system is conserved through time, so also is its invariant mass.This property allows the conservation of all types of mass in systems, and also conservation of all types of mass in reactions where matter is destroyed (annihilated), leaving behind the energy that was associated with it (which is now in non-material form, rather than material form). Matter may appear and disappear in various reactions, but mass and energy are both unchanged in this process.

Applicability of the strict formula

As is noted above, two different definitions of mass have been used in special relativity, and also two different definitions of energy. The simple equation is not generally applicable to all these types of mass and energy, except in the special case that the total additive momentum is zero for the system under consideration. In such a case, which is always guaranteed when observing the system from either its center of mass frame or its center of momentum frame, is always true for any type of mass and energy that are chosen. Thus, for example, in the center of mass frame, the total energy of an object or system is equal to its rest mass times , a useful equality. This is the relationship used for the container of gas in the previous example. It is not true in other reference frames where the center of mass is in motion. In these systems or for such an object, its total energy depends on both its rest (or invariant) mass, and its (total) momentum.^[25]
In inertial reference frames other than the rest frame or center of mass frame, the equation remains true if the energy is the relativistic energy and the mass is the relativistic mass. It is also correct if the energy is the rest or invariant energy (also the minimum energy), and the mass is the rest mass, or the invariant mass. However, connection of the total or relativistic energy () with the rest or invariant mass () requires consideration of the system's total momentum, in systems and reference frames where the total momentum (of magnitude p) has a non-zero value. The formula then required to connect the two different kinds of mass and energy, is the extended version of Einstein's equation, called the relativistic energy–momentum relation:^[26]

or

Here the term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. This equation reduces to when the momentum term is zero. For photons where , the equation reduces to .

Meanings of the strict formula

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The mass–energy equivalence formula was displayed on Taipei 101 during the event of the World Year of Physics 2005.

Mass–energy equivalence states that any object has a certain energy, even when it is stationary. In Newtonian mechanics, a motionless body has no kinetic energy, and it may or may not have other amounts of internal stored energy, like chemical energy or thermal energy, in addition to any potential energy it may have from its position in a field of force. In Newtonian mechanics, all of these energies are much smaller than the mass of the object times the speed of light squared.
In relativity, all the energy that moves with an object (that is, all the energy present in the object's rest frame) contributes to the total mass of the body, which measures how much it resists acceleration. Each bit of potential and kinetic energy makes a proportional contribution to the mass. As noted above, even if a box of ideal mirrors "contains" light, then the individually massless photons still contribute to the total mass of the box, by the amount of their energy divided by c².^[27]
In relativity, removing energy is removing mass, and for an observer in the center of mass frame, the formula m = E/c² indicates how much mass is lost when energy is removed. In a nuclear reaction, the mass of the atoms that come out is less than the mass of the atoms that go in, and the difference in mass shows up as heat and light with the same relativistic mass as the difference (and also the same invariant mass in the center of mass frame of the system). In this case, the E in the formula is the energy released and removed, and the mass m is how much the mass decreases. In the same way, when any sort of energy is added to an isolated system, the increase in the mass is equal to the added energy divided by c². For example, when water is heated it gains about 1.11×10⁻¹⁷ kg of mass for every joule of heat added to the water.
An object moves with different speed in different frames, depending on the motion of the observer, so the kinetic energy in both Newtonian mechanics and relativity is frame dependent. This means that the amount of relativistic energy, and therefore the amount of relativistic mass, that an object is measured to have depends on the observer. The rest mass is defined as the mass that an object has when it is not moving (or when an inertial frame is chosen such that it is not moving). The term also applies to the invariant mass of systems when the system as a whole is not "moving" (has no net momentum). The rest and invariant masses are the smallest possible value of the mass of the object or system. They also are conserved quantities, so long as the system is isolated. Because of the way they are calculated, the effects of moving observers are subtracted, so these quantities do not change with the motion of the observer.
The rest mass is almost never additive: the rest mass of an object is not the sum of the rest masses of its parts. The rest mass of an object is the total energy of all the parts, including kinetic energy, as measured by an observer that sees the center of the mass of the object to be standing still. The rest mass adds up only if the parts are standing still and do not attract or repel, so that they do not have any extra kinetic or potential energy. The other possibility is that they have a positive kinetic energy and a negative potential energy that exactly cancels.

Binding energy and the "mass defect"

Main articles: Binding energy and Mass defect

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Whenever any type of energy is removed from a system, the mass associated with the energy is also removed, and the system therefore loses mass. This mass defect in the system may be simply calculated as Δm = ΔE/c², and this was the form of the equation historically first presented by Einstein in 1905. However, use of this formula in such circumstances has led to the false idea that mass has been "converted" to energy. This may be particularly the case when the energy (and mass) removed from the system is associated with the binding energy of the system. In such cases, the binding energy is observed as a "mass defect" or deficit in the new system.
The fact that the released energy is not easily weighed in many such cases, may cause its mass to be neglected as though it no longer existed. This circumstance has encouraged the false idea of conversion of mass to energy, rather than the correct idea that the binding energy of such systems is relatively large, and exhibits a measurable mass, which is removed when the binding energy is removed.^{[citation needed]}.
The difference between the rest mass of a bound system and of the unbound parts is the binding energy of the system, if this energy has been removed after binding. For example, a water molecule weighs a little less than two free hydrogen atoms and an oxygen atom. The minuscule mass difference is the energy needed to split the molecule into three individual atoms (divided by c²), which was given off as heat when the molecule formed (this heat had mass). Likewise, a stick of dynamite in theory weighs a little bit more than the fragments after the explosion, but this is true only so long as the fragments are cooled and the heat removed. In this case the mass difference is the energy/heat that is released when the dynamite explodes, and when this heat escapes, the mass associated with it escapes, only to be deposited in the surroundings, which absorb the heat (so that total mass is conserved).
Such a change in mass may only happen when the system is open, and the energy and mass escapes. Thus, if a stick of dynamite is blown up in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light would still be equal to the original mass of the chamber and dynamite. If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation.^[22] Thus, a 21.5 kiloton (9×10¹³ joule) nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents of the box would be heated to millions of degrees without changing total mass and weight. If then, however, a transparent window (passing only electromagnetic radiation) were opened in such an ideal box after the explosion, and a beam of X-rays and other lower-energy light allowed to escape the box, it would eventually be found to weigh one gram less than it had before the explosion. This weight loss and mass loss would happen as the box was cooled by this process, to room temperature. However, any surrounding mass that absorbed the X-rays (and other "heat") would gain this gram of mass from the resulting heating, so the mass "loss" would represent merely its relocation. Thus, no mass (or, in the case of a nuclear bomb, no matter) would be "converted" to energy in such a process. Mass and energy, as always, would both be separately conserved.

Massless particles

Massless particles have zero rest mass. Their relativistic mass is simply their relativistic energy, divided by c², or m_rel = E/c².^[28][29] The energy for photons is E = hf, where h is Planck's constant and f is the photon frequency. This frequency and thus the relativistic energy are frame-dependent.
If an observer runs away from a photon in the direction the photon travels from a source, and it catches up with the observer—when the photon catches up, the observer sees it as having less energy than it had at the source. The faster the observer is traveling with regard to the source when the photon catches up, the less energy the photon has. As an observer approaches the speed of light with regard to the source, the photon looks redder and redder, by relativistic Doppler effect (the Doppler shift is the relativistic formula), and the energy of a very long-wavelength photon approaches zero. This is because the photon is massless—the rest mass of a photon is zero.

Massless particles contribute rest mass and invariant mass to systems

Two photons moving in different directions cannot both be made to have arbitrarily small total energy by changing frames, or by moving toward or away from them. The reason is that in a two-photon system, the energy of one photon is decreased by chasing after it, but the energy of the other increases with the same shift in observer motion. Two photons not moving in the same direction comprise an inertial frame where the combined energy is smallest, but not zero. This is called the center of mass frame or the center of momentum frame; these terms are almost synonyms (the center of mass frame is the special case of a center of momentum frame where the center of mass is put at the origin). The most that chasing a pair of photons can accomplish to decrease their energy is to put the observer in a frame where the photons have equal energy and are moving directly away from each other. In this frame, the observer is now moving in the same direction and speed as the center of mass of the two photons. The total momentum of the photons is now zero, since their momenta are equal and opposite. In this frame the two photons, as a system, have a mass equal to their total energy divided by c². This mass is called the invariant mass of the pair of photons together. It is the smallest mass and energy the system may be seen to have, by any observer. It is only the invariant mass of a two-photon system that can be used to make a single particle with the same rest mass.
If the photons are formed by the collision of a particle and an antiparticle, the invariant mass is the same as the total energy of the particle and antiparticle (their rest energy plus the kinetic energy), in the center of mass frame, where they automatically move in equal and opposite directions (since they have equal momentum in this frame). If the photons are formed by the disintegration of a single particle with a well-defined rest mass, like the neutral pion, the invariant mass of the photons is equal to rest mass of the pion. In this case, the center of mass frame for the pion is just the frame where the pion is at rest, and the center of mass does not change after it disintegrates into two photons. After the two photons are formed, their center of mass is still moving the same way the pion did, and their total energy in this frame adds up to the mass energy of the pion. Thus, by calculating the invariant mass of pairs of photons in a particle detector, pairs can be identified that were probably produced by pion disintegration.
A similar calculation illustrates that the invariant mass of systems is conserved, even when massive particles (particles with rest mass) within the system are converted to massless particles (such as photons). In such cases, the photons contribute invariant mass to the system, even though they individually have no invariant mass or rest mass. Thus, an electron and positron (each of which has rest mass) may undergo annihilation with each other to produce two photons, each of which is massless (has no rest mass). However, in such circumstances, no system mass is lost. Instead, the system of both photons moving away from each other has an invariant mass, which acts like a rest mass for any system in which the photons are trapped, or that can be weighed. Thus, not only the quantity of relativistic mass, but also the quantity of invariant mass does not change in transformations between "matter" (electrons and positrons) and energy (photons).

Relation to gravity

In physics, there are two distinct concepts of mass: the gravitational mass and the inertial mass. The gravitational mass is the quantity that determines the strength of the gravitational field generated by an object, as well as the gravitational force acting on the object when it is immersed in a gravitational field produced by other bodies. The inertial mass, on the other hand, quantifies how much an object accelerates if a given force is applied to it. The mass–energy equivalence in special relativity refers to the inertial mass. However, already in the context of Newton gravity, the Weak Equivalence Principle is postulated: the gravitational and the inertial mass of every object are the same. Thus, the mass–energy equivalence, combined with the Weak Equivalence Principle, results in the prediction that all forms of energy contribute to the gravitational field generated by an object. This observation is one of the pillars of the general theory of relativity.
The above prediction, that all forms of energy interact gravitationally, has been subject to experimental tests. The first observation testing this prediction was made in 1919.^[30] During a solar eclipse, Arthur Eddington observed that the light from stars passing close to the Sun was bent. The effect is due to the gravitational attraction of light by the Sun. The observation confirmed that the energy carried by light indeed is equivalent to a gravitational mass. Another seminal experiment, the Pound–Rebka experiment, was performed in 1960.^[31] In this test a beam of light was emitted from the top of a tower and detected at the bottom. The frequency of the light detected was higher than the light emitted. This result confirms that the energy of photons increases when they fall in the gravitational field of the Earth. The energy, and therefore the gravitational mass, of photons is proportional to their frequency as stated by the Planck's relation.

Application to nuclear physics

Task Force One, the world's first nuclear-powered task force. Enterprise, Long Beach and Bainbridge in formation in the Mediterranean, 18 June 1964. Enterprise crew members are spelling out Einstein's mass–energy equivalence formula E = mc² on the flight deck.

Max Planck pointed out that the mass–energy equivalence formula implied^[how?] that bound systems would have a mass less than the sum of their constituents, once the binding energy had been allowed to escape. However, Planck was thinking about chemical reactions, where the binding energy is too small to measure. Einstein suggested that radioactive materials such as radium would provide a test of the theory, but even though a large amount of energy is released per atom in radium, due to the half-life of the substance (1602 years), only a small fraction of radium atoms decay over an experimentally measurable period of time.
Once the nucleus was discovered, experimenters realized that the very high binding energies of the atomic nuclei should allow calculation of their binding energies, simply from mass differences. But it was not until the discovery of the neutron in 1932, and the measurement of the neutron mass, that this calculation could actually be performed (see nuclear binding energy for example calculation). A little while later, the Cockcroft–Walton accelerator produced the first transmutation reaction (⁷
₃Li + ¹
₁p → 2 ⁴
₂He), verifying Einstein's formula to an accuracy of ±0.5%.^{[citation needed]} In 2005, Rainville et al. published a direct test of the energy-equivalence of mass lost in the binding energy of a neutron to atoms of particular isotopes of silicon and sulfur, by comparing the mass lost to the energy of the emitted gamma ray associated with the neutron capture. The binding mass-loss agreed with the gamma ray energy to a precision of ±0.00004%, the most accurate test of E = mc² to date.^[32]
The mass–energy equivalence formula was used in the understanding of nuclear fission reactions, and implies the great amount of energy that can be released by a nuclear fission chain reaction, used in both nuclear weapons and nuclear power. By measuring the mass of different atomic nuclei and subtracting from that number the total mass of the protons and neutrons as they would weigh separately, one gets the exact binding energy available in an atomic nucleus. This is used to calculate the energy released in any nuclear reaction, as the difference in the total mass of the nuclei that enter and exit the reaction.

Practical examples

Einstein used the CGS system of units (centimeters, grams, seconds, dynes, and ergs), but the formula is independent of the system of units. In natural units, the numerical value of the speed of light is set to equal 1, and the formula expresses an equality of numerical values: E = m. In the SI system (expressing the ratio E/m in joules per kilogram using the value of c in meters per second):^[33]

E/m = c² = (299792458 m/s)² = 89875517873681764 J/kg (≈ 9.0 × 10¹⁶ joules per kilogram).

So the energy equivalent of one kilogram of mass is

• 89.9 petajoules
• 25.0 billion kilowatt-hours (≈ 25,000 GW·h)
• 21.5 trillion kilocalories (≈ 21 Pcal)^[34]
• 85.2 trillion BTUs^[34]
• 0.0852 quads

or the energy released by combustion of the following:

• 21 500 kilotons of TNT-equivalent energy (≈ 21 Mt)^[34]
• 2630000000 litres or 695000000 US gallons of automotive gasoline

Any time energy is released, the process can be evaluated from an E = mc² perspective. For instance, the "Gadget"-style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling. The electromagnetic radiation and kinetic energy (thermal and blast energy) released in this explosion carried the missing one gram of mass.^[35] This occurs because nuclear binding energy is released whenever elements with more than 62 nucleons fission.^{[citation needed]}
Another example is hydroelectric generation. The electrical energy produced by Grand Coulee Dam's turbines every 3.7 hours represents one gram of mass. This mass passes to electrical devices (such as lights in cities) powered by the generators, where it appears as a gram of heat and light.^[36] Turbine designers look at their equations in terms of pressure, torque, and RPM. However, Einstein's equations show that all energy has mass, and thus the electrical energy produced by a dam's generators, and the resulting heat and light, all retain their mass—which is equivalent to the energy. The potential energy—and equivalent mass—represented by the waters of the Columbia River as it descends to the Pacific Ocean would be converted to heat due to viscous friction and the turbulence of white water rapids and waterfalls were it not for the dam and its generators. This heat would remain as mass on site at the water, were it not for the equipment that converted some of this potential and kinetic energy into electrical energy, which can move from place to place (taking mass with it).^{[citation needed]}
Whenever energy is added to a system, the system gains mass, as shown when the equation is rearranged:

• A spring's mass increases whenever it is put into compression or tension. Its added mass arises from the added potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring.
• Raising the temperature of an object (increasing its heat energy) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of platinum/iridium. If its temperature is allowed to change by 1 °C, its mass changes by 1.5 picograms (1 pg = 1×10⁻¹² g).^[37]
• A spinning ball weighs more than a ball that is not spinning. Its increase of mass is exactly the equivalent of the mass of energy of rotation, which is itself the sum of the kinetic energies of all the moving parts of the ball. For example, the Earth itself is more massive due to its rotation, than it would be with no rotation. This rotational energy (2.14×10²⁹ J) represents 2.38 billion metric tons of added mass.^[38]

Note that no net mass or energy is really created or lost in any of these examples and scenarios. Mass/energy simply moves from one place to another. These are some examples of the transfer of energy and mass in accordance with the principle of mass–energy conservation.^{[citation needed]}

Efficiency

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Although mass cannot be converted to energy,^[22] in some reactions matter particles (which contain a form of rest energy) can be destroyed and the energy released can be converted to other types of energy that are more usable and obvious as forms of energy—such as light and energy of motion (heat, etc.). However, the total amount of energy and mass does not change in such a transformation. Even when particles are not destroyed, a certain fraction of the ill-defined "matter" in ordinary objects can be destroyed, and its associated energy liberated and made available as the more dramatic energies of light and heat, even though no identifiable real particles are destroyed, and even though (again) the total energy is unchanged (as also the total mass). Such conversions between types of energy (resting to active energy) happen in nuclear weapons, in which the protons and neutrons in atomic nuclei lose a small fraction of their average mass, but this mass loss is not due to the destruction of any protons or neutrons (or even, in general, lighter particles like electrons). Also the mass is not destroyed, but simply removed from the system in the form of heat and light from the reaction.
In nuclear reactions, typically only a small fraction of the total mass–energy of the bomb converts into the mass–energy of heat, light, radiation, and motion—which are "active" forms that can be used. When an atom fissions, it loses only about 0.1% of its mass (which escapes from the system and does not disappear), and additionally, in a bomb or reactor not all the atoms can fission. In a modern fission-based atomic bomb, the efficiency is only about 40%, so only 40% of the fissionable atoms actually fission, and only about 0.03% of the fissile core mass appears as energy in the end. In nuclear fusion, more of the mass is released as usable energy, roughly 0.3%. But in a fusion bomb, the bomb mass is partly casing and non-reacting components, so that in practicality, again (coincidentally) no more than about 0.03% of the total mass of the entire weapon is released as usable energy (which, again, retains the "missing" mass). See nuclear weapon yield for practical details of this ratio in modern nuclear weapons.
In theory, it should be possible to destroy matter and convert all of the rest-energy associated with matter into heat and light (which would of course have the same mass), but none of the theoretically known methods are practical. One way to convert all the energy within matter into usable energy is to annihilate matter with antimatter. But antimatter is rare in our universe, and must be made first. Due to inefficient mechanisms of production, making antimatter always requires far more usable energy than would be released when it was annihilated.
Since most of the mass of ordinary objects resides in protons and neutrons, converting all the energy of ordinary matter into more useful energy requires that the protons and neutrons be converted to lighter particles, or particles with no rest-mass at all. In the Standard Model of particle physics, the number of protons plus neutrons is nearly exactly conserved. Still, Gerard 't Hooft showed that there is a process that converts protons and neutrons to antielectrons and neutrinos.^[39] This is the weak SU(2) instanton proposed by Belavin Polyakov Schwarz and Tyupkin.^[40] This process, can in principle destroy matter and convert all the energy of matter into neutrinos and usable energy, but it is normally extraordinarily slow. Later it became clear that this process happens at a fast rate at very high temperatures,^[41] since then, instanton-like configurations are copiously produced from thermal fluctuations. The temperature required is so high that it would only have been reached shortly after the Big Bang.
Many extensions of the standard model contain magnetic monopoles, and in some models of grand unification, these monopoles catalyze proton decay, a process known as the Callan-Rubakov effect.^[42] This process would be an efficient mass–energy conversion at ordinary temperatures, but it requires making monopoles and anti-monopoles first. The energy required to produce monopoles is believed to be enormous, but magnetic charge is conserved, so that the lightest monopole is stable. All these properties are deduced in theoretical models—magnetic monopoles have never been observed, nor have they been produced in any experiment so far.
A third known method of total matter–energy "conversion" (which again in practice only means conversion of one type of energy into a different type of energy), is using gravity, specifically black holes. Stephen Hawking theorized^[43] that black holes radiate thermally with no regard to how they are formed. So, it is theoretically possible to throw matter into a black hole and use the emitted heat to generate power. According to the theory of Hawking radiation, however, the black hole used radiates at a higher rate the smaller it is, producing usable powers at only small black hole masses, where usable may for example be something greater than the local background radiation. It is also worth noting that the ambient irradiated power would change with the mass of the black hole, increasing as the mass of the black hole decreases, or decreasing as the mass increases, at a rate where power is proportional to the inverse square of the mass. In a "practical" scenario, mass and energy could be dumped into the black hole to regulate this growth, or keep its size, and thus power output, near constant. This could result from the fact that mass and energy are lost from the hole with its thermal radiation.

Background

Mass–velocity relationship

In developing special relativity, Einstein found that the kinetic energy of a moving body is

with v the velocity, m₀ the rest mass, and γ the Lorentz factor.
He included the second term on the right to make sure that for small velocities the energy would be the same as in classical mechanics, thus satisfying the correspondence principle:

Without this second term, there would be an additional contribution in the energy when the particle is not moving.
Einstein found that the total momentum of a moving particle is:

It is this quantity that is conserved in collisions. The ratio of the momentum to the velocity is the relativistic mass, m.

And the relativistic mass and the relativistic kinetic energy are related by the formula:

Einstein wanted to omit the unnatural second term on the right-hand side, whose only purpose is to make the energy at rest zero, and to declare that the particle has a total energy, which obeys:

which is a sum of the rest energy m₀c² and the kinetic energy. This total energy is mathematically more elegant, and fits better with the momentum in relativity. But to come to this conclusion, Einstein needed to think carefully about collisions. This expression for the energy implied that matter at rest has a huge amount of energy, and it is not clear whether this energy is physically real, or just a mathematical artifact with no physical meaning.
In a collision process where all the rest-masses are the same at the beginning as at the end, either expression for the energy is conserved. The two expressions only differ by a constant that is the same at the beginning and at the end of the collision. Still, by analyzing the situation where particles are thrown off a heavy central particle, it is easy to see that the inertia of the central particle is reduced by the total energy emitted. This allowed Einstein to conclude that the inertia of a heavy particle is increased or diminished according to the energy it absorbs or emits.

Relativistic mass

Main article: Mass in special relativity

After Einstein first made his proposal, it became clear that the word mass can have two different meanings. Some denote the relativistic mass with an explicit index:

This mass is the ratio of momentum to velocity, and it is also the relativistic energy divided by c² (it is not Lorentz-invariant, in contrast to ). The equation E = m_relc² holds for moving objects. When the velocity is small, the relativistic mass and the rest mass are almost exactly the same.

• E = mc² either means E = m₀c² for an object at rest, or E = m_relc² when the object is moving.

Also Einstein (following Hendrik Lorentz and Max Abraham) used velocity- and direction-dependent mass concepts (longitudinal and transverse mass) in his 1905 electrodynamics paper and in another paper in 1906.^[44][45] However, in his first paper on E = mc² (1905), he treated m as what would now be called the rest mass.^[3] Some claim that (in later years) he did not like the idea of "relativistic mass".^[46]  When modern physicists say "mass", they are usually talking about rest mass, since if they meant "relativistic mass", they would just say "energy".
Considerable debate has ensued over the use of the concept "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. For example, one view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. A perspective that avoids this debate, due to Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection.^[47][48]

Low speed expansion

We can rewrite the expression E = γm₀c² as a Taylor series:

For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because v/c is small. For low speeds we can ignore all but the first two terms:

The total energy is a sum of the rest energy and the Newtonian kinetic energy.
The classical energy equation ignores both the m₀c² part, and the high-speed corrections. This is appropriate, because all the high-order corrections are small. Since only changes in energy affect the behavior of objects, whether we include the m₀c² part makes no difference, since it is constant. For the same reason, it is possible to subtract the rest energy from the total energy in relativity. By considering the emission of energy in different frames, Einstein could show that the rest energy has a real physical meaning.
The higher-order terms are extra corrections to Newtonian mechanics, and become important at higher speeds. The Newtonian equation is only a low-speed approximation, but an extraordinarily good one. All of the calculations used in putting astronauts on the moon, for example, could have been done using Newton's equations without any of the higher-order corrections.^{[citation needed]} The total mass energy equivalence should also include the rotational and vibrational kinetic energies as well as the linear kinetic energy at low speeds.

History

While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass. But nearly all previous authors thought that the energy that contributes to mass comes only from electromagnetic fields.^{[49][50][51][52]}

Newton: matter and light

In 1717 Isaac Newton speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks, where he asks:

Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?

Swedenborg: matter composed of "pure and total motion"

In 1734 the Swedish scientist and theologian Emanuel Swedenborg in his Principia theorized that all matter is ultimately composed of dimensionless points of "pure and total motion". He described this motion as being without force, direction or speed, but having the potential for force, direction and speed everywhere within it.^[53][54]

Electromagnetic mass

Main article: Electromagnetic mass

There were many attempts in the 19th and the beginning of the 20th century—like those of J. J. Thomson (1881), Oliver Heaviside (1888), and George Frederick Charles Searle (1897), Wilhelm Wien (1900), Max Abraham (1902), Hendrik Antoon Lorentz (1904) — to understand how the mass of a charged object depends on the electrostatic field.^[49][50] This concept was called electromagnetic mass, and was considered as being dependent on velocity and direction as well. Lorentz (1904) gave the following expressions for longitudinal and transverse electromagnetic mass:

,

where

Radiation pressure and inertia

Main article: Electromagnetic mass § Inertia of energy and radiation paradoxes

Another way of deriving some sort of electromagnetic mass was based on the concept of radiation pressure. In 1900, Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass^[2]

By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes.^[52]
Friedrich Hasenöhrl showed in 1904, that electromagnetic cavity radiation contributes the "apparent mass"

to the cavity's mass. He argued that this implies mass dependence on temperature as well.^[55]

Einstein: mass–energy equivalence

Albert Einstein did not formulate exactly the formula E = mc² in his 1905 Annus Mirabilis paper "Does the Inertia of an object Depend Upon Its Energy Content?";^[3] rather, the paper states that if a body gives off the energy L in the form of radiation, its mass diminishes by L/c². (Here, "radiation" means electromagnetic radiation, or light, and mass means the ordinary Newtonian mass of a slow-moving object.) This formulation relates only a change Δm in mass to a change L in energy without requiring the absolute relationship.
Objects with zero mass presumably have zero energy, so the extension that all mass is proportional to energy is obvious from this result. In 1905, even the hypothesis that changes in energy are accompanied by changes in mass was untested. Not until the discovery of the first type of antimatter (the positron in 1932) was it found that all of the mass of pairs of resting particles could be converted to radiation.

The first derivation by Einstein (1905)

Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct expression for the kinetic energy of particles:

.

Now the question remained open as to which formulation applies to bodies at rest. This was tackled by Einstein in his paper "Does the inertia of a body depend upon its energy content?", where he used a body emitting two light pulses in opposite directions, having energies of E₀ before and E₁ after the emission as seen in its rest frame. As seen from a moving frame, this becomes H₀ and H₁. Einstein obtained:

then he argued that H − E can only differ from the kinetic energy K by an additive constant, which gives

Neglecting effects higher than third order in v/c after a Taylor series expansion of the right side of this gives:

Einstein concluded that the emission reduces the body's mass by E/c², and that the mass of a body is a measure of its energy content.
The correctness of Einstein's 1905 derivation of E = mc² was criticized by Max Planck (1907), who argued that it is only valid to first approximation. Another criticism was formulated by Herbert Ives (1952) and Max Jammer (1961), asserting that Einstein's derivation is based on begging the question.^[4][56] On the other hand, John Stachel and Roberto Torretti (1982) argued that Ives' criticism was wrong, and that Einstein's derivation was correct.^[57] Hans Ohanian (2008) agreed with Stachel/Torretti's criticism of Ives, though he argued that Einstein's derivation was wrong for other reasons.^[58] For a recent review, see Hecht (2011).^[5]

Alternative version

An alternative version of Einstein's thought experiment was proposed by Fritz Rohrlich (1990), who based his reasoning on the Doppler effect.^[59] Like Einstein, he considered a body at rest with mass M. If the body is examined in a frame moving with nonrelativistic velocity v, it is no longer at rest and in the moving frame it has momentum P = Mv. Then he supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy E/2. In its rest frame, the object remains at rest after the emission since the two beams are equal in strength and carry opposite momentum.
However, if the same process is considered in a frame that moves with velocity v to the left, the pulse moving to the left is redshifted, while the pulse moving to the right is blue shifted. The blue light carries more momentum than the red light, so that the momentum of the light in the moving frame is not balanced: the light is carrying some net momentum to the right.
The object has not changed its velocity before or after the emission. Yet in this frame it has lost some right-momentum to the light. The only way it could have lost momentum is by losing mass. This also solves Poincaré's radiation paradox, discussed above.
The velocity is small, so the right-moving light is blueshifted by an amount equal to the nonrelativistic Doppler shift factor 1 − v/c. The momentum of the light is its energy divided by c, and it is increased by a factor of v/c. So the right-moving light is carrying an extra momentum ΔP given by:

The left-moving light carries a little less momentum, by the same amount ΔP. So the total right-momentum in both light pulses is twice ΔP. This is the right-momentum that the object lost.

The momentum of the object in the moving frame after the emission is reduced to this amount:

So the change in the object's mass is equal to the total energy lost divided by c². Since any emission of energy can be carried out by a two step process, where first the energy is emitted as light and then the light is converted to some other form of energy, any emission of energy is accompanied by a loss of mass. Similarly, by considering absorption, a gain in energy is accompanied by a gain in mass.

Relativistic center-of-mass theorem (1906)

Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for the center-of-mass theorem to hold. On this occasion, Einstein referred to Poincaré's 1900 paper and wrote:^[60]

Although the merely formal considerations, which we will need for the proof, are already mostly contained in a work by H. Poincaré², for the sake of clarity I will not rely on that work.^[61]

In Einstein's more physical, as opposed to formal or mathematical, point of view, there was no need for fictitious masses. He could avoid the perpetuum mobile problem because, on the basis of the mass–energy equivalence, he could show that the transport of inertia that accompanies the emission and absorption of radiation solves the problem. Poincaré's rejection of the principle of action–reaction can be avoided through Einstein's E = mc², because mass conservation appears as a special case of the energy conservation law.

Others

During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various ether theories.^[62] In 1873 Nikolay Umov pointed out a relation between mass and energy for ether in the form of Е = kmc², where 0.5 ≤ k ≤ 1.^[63] The writings of Samuel Tolver Preston,^[64][65] and a 1903 paper by Olinto De Pretto,^[66][67] presented a mass–energy relation. Bartocci (1999) observed that there were only three degrees of separation linking De Pretto to Einstein, concluding that Einstein was probably aware of De Pretto's work.^[68]
Preston and De Pretto, following Le Sage, imagined that the universe was filled with an ether of tiny particles that always move at speed c. Each of these particles has a kinetic energy of mc² up to a small numerical factor. The nonrelativistic kinetic energy formula did not always include the traditional factor of 1/2, since Leibniz introduced kinetic energy without it, and the 1/2 is largely conventional in prerelativistic physics.^[69] By assuming that every particle has a mass that is the sum of the masses of the ether particles, the authors concluded that all matter contains an amount of kinetic energy either given by E = mc² or 2E = mc² depending on the convention. A particle ether was usually considered unacceptably speculative science at the time,^[70] and since these authors did not formulate relativity, their reasoning is completely different from that of Einstein, who used relativity to change frames.
Independently, Gustave Le Bon in 1905 speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative philosophy of physics.^[71][72]

Radioactivity and nuclear energy

It was quickly noted after the discovery of radioactivity in 1897, that the total energy due to radioactive processes is about one million times greater than that involved in any known molecular change. However, it raised the question where this energy is coming from. After eliminating the idea of absorption and emission of some sort of Lesagian ether particles, the existence of a huge amount of latent energy, stored within matter, was proposed by Ernest Rutherford and Frederick Soddy in 1903. Rutherford also suggested that this internal energy is stored within normal matter as well. He went on to speculate in 1904:^[73][74]

If it were ever found possible to control at will the rate of disintegration of the radio-elements, an enormous amount of energy could be obtained from a small quantity of matter.

Einstein's equation is in no way an explanation of the large energies released in radioactive decay (this comes from the powerful nuclear forces involved; forces that were still unknown in 1905). In any case, the enormous energy released from radioactive decay (which had been measured by Rutherford) was much more easily measured than the (still small) change in the gross mass of materials as a result. Einstein's equation, by theory, can give these energies by measuring mass differences before and after reactions, but in practice, these mass differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive decay energies with a calorimeter was thought possibly likely to allow measurement of changes in mass difference, as a check on Einstein's equation itself. Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which releases enough energy (the quantitative amount known roughly by 1905) to possibly be "weighed," when missing from the system (having been given off as heat). However, radioactivity seemed to proceed at its own unalterable (and quite slow, for radioactives known then) pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. Rutherford was reported in 1933 to have declared that this energy could not be exploited efficiently: "Anyone who expects a source of power from the transformation of the atom is talking moonshine."^[75]

The popular connection between Einstein, E = mc², and the atomic bomb was prominently indicated on the cover of Time magazine in July 1946 by the writing of the equation on the mushroom cloud.

This situation changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single nuclides and their reactions to be calculated directly, and compared with the sum of masses for the particles that made up their composition. In 1933, the energy released from the reaction of lithium-7 plus protons giving rise to 2 alpha particles (as noted above by Rutherford), allowed Einstein's equation to be tested to an error of ±0.5%. However, scientists still did not see such reactions as a practical source of power, due to the energy cost of accelerating reaction particles.
After the very public demonstration of huge energies released from nuclear fission after the atomic bombings of Hiroshima and Nagasaki in 1945, the equation E = mc² became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured as early as page 2 of the Smyth Report, the official 1945 release by the US government on the development of the atomic bomb, and by 1946 the equation was linked closely enough with Einstein's work that the cover of Time magazine prominently featured a picture of Einstein next to an image of a mushroom cloud emblazoned with the equation.^[76] Einstein himself had only a minor role in the Manhattan Project: he had cosigned a letter to the U.S. President in 1939 urging funding for research into atomic energy, warning that an atomic bomb was theoretically possible. The letter persuaded Roosevelt to devote a significant portion of the wartime budget to atomic research. Without a security clearance, Einstein's only scientific contribution was an analysis of an isotope separation method in theoretical terms. It was inconsequential, on account of Einstein not being given sufficient information (for security reasons) to fully work on the problem.^[77]
While E = mc² is useful for understanding the amount of energy potentially released in a fission reaction, it was not strictly necessary to develop the weapon, once the fission process was known, and its energy measured at 200 MeV (which was directly possible, using a quantitative Geiger counter, at that time). As the physicist and Manhattan Project participant Robert Serber put it: "Somehow the popular notion took hold long ago that Einstein's theory of relativity, in particular his famous equation E = mc², plays some essential role in the theory of fission. Albert Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of relativity is not required in discussing fission. The theory of fission is what physicists call a non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly."^[78] However the association between E = mc² and nuclear energy has since stuck, and because of this association, and its simple expression of the ideas of Albert Einstein himself, it has become "the world's most famous equation".^[1]
While Serber's view of the strict lack of need to use mass–energy equivalence in designing the atomic bomb is correct, it does not take into account the pivotal role this relationship played in making the fundamental leap to the initial hypothesis that large atoms were energetically allowed to split into approximately equal parts (before this energy was in fact measured). In late 1938, Lise Meitner and Otto Robert Frisch—while on a winter walk during which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission—directly used Einstein's equation to help them understand the quantitative energetics of the reaction that overcame the "surface tension-like" forces that hold the nucleus together, and allowed the fission fragments to separate to a configuration from which their charges could force them into an energetic fission. To do this, they used packing fraction, or nuclear binding energy values for elements, which Meitner had memorized. These, together with use of E = mc² allowed them to realize on the spot that the basic fission process was energetically possible:

...We walked up and down in the snow, I on skis and she on foot. ...and gradually the idea took shape... explained by Bohr's idea that the nucleus is like a liquid drop; such a drop might elongate and divide itself... We knew there were strong forces that would resist, ..just as surface tension. But nuclei differed from ordinary drops. At this point we both sat down on a tree trunk and started to calculate on scraps of paper. ...the Uranium nucleus might indeed be a very wobbly, unstable drop, ready to divide itself... But, ...when the two drops separated they would be driven apart by electrical repulsion, about 200 MeV in all. Fortunately Lise Meitner remembered how to compute the masses of nuclei... and worked out that the two nuclei formed... would be lighter by about one-fifth the mass of a proton. Now whenever mass disappears energy is created, according to Einstein's formula E = mc², and... the mass was just equivalent to 200 MeV; it all fitted!^[79][80]

See also

• Physics portal

• Energy density
• Index of energy articles
• Index of wave articles
• Outline of energy

References

1. 
   

Bodanis, David (2009). E=mc^2: A Biography of the World's Most Famous Equation (illustrated ed.). Bloomsbury Publishing. ISBN 978-0-8027-1821-1.

80. Sime, Ruth (1996), Lise Meitner: A Life in Physics, California Studies in the History of Science, 13, Berkeley: University of California Press, pp. 236–237, ISBN 0-520-20860-9

External links

Wikisource has original text related to this article: Relativity: The Special and General Theory

Wikimedia Commons has media related to Einstein formula.

• Einstein on the Inertia of Energy – MathPages
• Lasky, Ronald C. (April 23, 2007), What is the significance of E = mc²? And what does it mean?, Scientific American
• Mass and Energy – Conversations About Science with Theoretical Physicist Matt Strassler
• Ask an Astrophysicist | Energy–Matter Conversion, NASA, 1997
• The Equivalence of Mass and Energy – Entry in the Stanford Encyclopedia of Philosophy
• Gail Wilson (May 2014) Scientists discover how to turn light into matter after 80-year quest Imperial College
• Merrifield, Michael; Copeland, Ed; Bowley, Roger. "E=mc² – Mass–Energy Equivalence". Sixty Symbols. Brady Haran for the University of Nottingham.
• Hecht, Eugene (2009). "Einstein on mass and energy". American Journal of Physics. 77 (9): 799–806. Bibcode:2009AmJPh..77..799H. CiteSeerX 10.1.1.205.7995. doi:10.1119/1.3160671. Early on, Einstein embraced the idea of a speed-dependent mass but changed his mind in 1906 and thereafter carefully avoided that notion entirely. He shunned, and explicitly rejected, what later came to be known as 'relativistic mass'. ... He consistently related the rest energy of a system to its invariant inertial mass.

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Conversions used: 1956 International (Steam) Table (IT) values where one calorie ≡ 4.1868 J and one BTU ≡ 1055.05585262 J. Weapons designers' conversion value of one gram TNT ≡ 1000 calories used.

The 6.2 kg core comprised 0.8% gallium by weight. Also, about 20% of the Gadget's yield was due to fast fissioning in its natural uranium tamper. This resulted in 4.1 moles of plutonium fissioning with 180 MeV per atom actually contributing prompt kinetic energy to the explosion. Note too that the term "Gadget"-style is used here instead of "Fat Man" because this general design of bomb was very rapidly upgraded to a more efficient one requiring only 5 kg of the plutonium–gallium alloy.^{[citation needed]}

Assuming the dam is generating at its peak capacity of 6,809 MW.^{[citation needed]}

Assuming a 90/10 alloy of Pt/Ir by weight, a C_p of 25.9 for Pt and 25.1 for Ir, a Pt-dominated average C_p of 25.8, 5.134 moles of metal, and 132 J⋅K⁻¹ for the prototype. A variation of ±1.5 picograms is of course, much smaller than the actual uncertainty in the mass of the international prototype, which is ±2 micrograms.

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(*) 

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Special relativity

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For history and motivation, see History of special relativity.

Special relativity
Principle of relativity Theory of relativity Doubly special relativity de Sitter invariant special relativity General relativity
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Alternative formulations of special relativity
v t e

In physics, special relativity (also known as the special theory of relativity) is the generally accepted and experimentally confirmed physical theory regarding the relationship between space and time. In Albert Einstein's original treatment, it is based on two postulates:

1. the laws of physics are invariant (i.e., identical) in all inertial frames of reference (i.e., non-accelerating frames of reference); and
2. the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer.

Some of the work of Albert Einstein in special relativity is built on the earlier work by Hendrik Lorentz.
Special relativity was originally proposed by Albert Einstein in a paper published on 26 September 1905 titled "On the Electrodynamics of Moving Bodies".^{[p 1]} The incompatibility of Newtonian mechanics with Maxwell's equations of electromagnetism and, experimentally, the Michelson-Morley null result (and subsequent similar experiments) demonstrated that the historically hypothesized luminiferous aether did not exist. This led to Einstein's development of special relativity, which corrects mechanics to handle situations involving all motions and especially those at a speed close to that of light (known as relativistic velocities). Today, special relativity is proven to be the most accurate model of motion at any speed when gravitational effects are negligible. Even so, the Newtonian model is still valid as a simple and accurate approximation at low velocities (relative to the speed of light), for example, everyday motions on Earth.
Special relativity has a wide range of consequences. These have been experimentally verified,^[1] and include length contraction, time dilation, relativistic mass, a universal speed limit, mass–energy equivalence, the speed of causality and relativity of simultaneity. It has, for example, replaced the conventional notion of an absolute universal time with the notion of a time that is dependent on reference frame and spatial position. Rather than an invariant time interval between two events, there is an invariant spacetime interval. Combined with other laws of physics, the two postulates of special relativity predict the equivalence of mass and energy, as expressed in the mass–energy equivalence formula E = mc² (c is the speed of light in a vacuum).^[2][3]
A defining feature of special relativity is the replacement of the Galilean transformations of Newtonian mechanics with the Lorentz transformations. Time and space cannot be defined separately from each other (as was earlier thought to be the case). Rather, space and time are interwoven into a single continuum known as "spacetime". Events that occur at the same time for one observer can occur at different times for another.
Until Einstein developed general relativity, introducing a curved spacetime to incorporate gravity, the phrase "special relativity" was not used. A translation sometimes used is "restricted relativity"; "special" really means "special case".^{[p 2][p 3][p 4][note 1]}
The theory is "special" in that it only applies in the special case where the spacetime is "flat", that is, the curvature of spacetime, described by the energy-momentum tensor and causing gravity, is negligible.^{[4][note 2]} In order to correctly accommodate gravity, Einstein formulated general relativity in 1915. Special relativity, contrary to some historical descriptions, does accommodate accelerations as well as accelerating frames of reference.^[5][6]
Just as Galilean relativity is now accepted to be an approximation of special relativity that is valid for low speeds, special relativity is considered an approximation of general relativity that is valid for weak gravitational fields, that is, at a sufficiently small scale (e.g., for tidal forces) and in conditions of free fall. General relativity, however, incorporates noneuclidean geometry in order to represent gravitational effects as the geometric curvature of spacetime. Special relativity is restricted to the flat spacetime known as Minkowski space. As long as the universe can be modeled as a pseudo-Riemannian manifold, a Lorentz-invariant frame that abides by special relativity can be defined for a sufficiently small neighborhood of each point in this curved spacetime.
Galileo Galilei had already postulated that there is no absolute and well-defined state of rest (no privileged reference frames), a principle now called Galileo's principle of relativity. Einstein extended this principle so that it accounted for the constant speed of light,^[7] a phenomenon that had been observed in the Michelson–Morley experiment. He also postulated that it holds for all the laws of physics, including both the laws of mechanics and of electrodynamics.^[8]

Albert Einstein around 1905, the year his "Annus Mirabilis papers" were published. These included Zur Elektrodynamik bewegter Körper, the paper founding special relativity.

Contents

• 1 Traditional "two postulates" approach to special relativity
• 2 Principle of relativity
  • 2.1 Reference frames and relative motion
  • 2.2 Standard configuration
  • 2.3 Lack of an absolute reference frame
  • 2.4 Relativity without the second postulate
• 3 Lorentz invariance as the essential core of special relativity
  • 3.1 Alternative approaches to special relativity
  • 3.2 Lorentz transformation and its inverse
  • 3.3 Graphical representation of the Lorentz transformation
• 4 Consequences derived from the Lorentz transformation
  • 4.1 Invariant interval
  • 4.2 Relativity of simultaneity
  • 4.3 Time dilation
  • 4.4 Length contraction
  • 4.5 Lorentz transformation of velocities
  • 4.6 Thomas rotation
  • 4.7 Causality and prohibition of motion faster than light
• 5 Optical effects
  • 5.1 Dragging effects
  • 5.2 Relativistic aberration of light
  • 5.3 Relativistic Doppler effect
  • 5.4 Measurement versus visual appearance
• 6 Dynamics
  • 6.1 Equivalence of mass and energy
  • 6.2 How far can one travel from the Earth?
• 7 Relativity and unifying electromagnetism
• 8 Theories of relativity and quantum mechanics
• 9 Status
• 10 Technical discussion of spacetime
  • 10.1 Geometry of spacetime
  • 10.2 Physics in spacetime
• 11 See also
• 12 Primary sources
• 13 References
• 14 Notes
  • 14.1 Textbooks
  • 14.2 Journal articles
• 15 External links
  • 15.1 Original works
  • 15.2 Special relativity for a general audience (no mathematical knowledge required)
  • 15.3 Special relativity explained (using simple or more advanced mathematics)
  • 15.4 Visualization

Traditional "two postulates" approach to special relativity

Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results ... How, then, could such a universal principle be found?

— Albert Einstein: Autobiographical Notes^{[p 5]}

Einstein discerned two fundamental propositions that seemed to be the most assured, regardless of the exact validity of the (then) known laws of either mechanics or electrodynamics. These propositions were the constancy of the speed of light in a vacuum and the independence of physical laws (especially the constancy of the speed of light) from the choice of inertial system. In his initial presentation of special relativity in 1905 he expressed these postulates as:^{[p 1]}

• The Principle of Relativity – the laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.^{[p 1]}
• The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" (from the preface).^{[p 1]} That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source.

The constancy of the speed of light was motivated by Maxwell's theory of electromagnetism and the lack of evidence for the luminiferous ether. There is conflicting evidence on the extent to which Einstein was influenced by the null result of the Michelson–Morley experiment.^[9][10] In any case, the null result of the Michelson–Morley experiment helped the notion of the constancy of the speed of light gain widespread and rapid acceptance.
The derivation of special relativity depends not only on these two explicit postulates, but also on several tacit assumptions (made in almost all theories of physics), including the isotropy and homogeneity of space and the independence of measuring rods and clocks from their past history.^{[p 6]}
Following Einstein's original presentation of special relativity in 1905, many different sets of postulates have been proposed in various alternative derivations.^[11] However, the most common set of postulates remains those employed by Einstein in his original paper. A more mathematical statement of the Principle of Relativity made later by Einstein, which introduces the concept of simplicity not mentioned above is:

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K′ moving in uniform translation relatively to K.^[12]

Henri Poincaré provided the mathematical framework for relativity theory by proving that Lorentz transformations are a subset of his Poincaré group of symmetry transformations. Einstein later derived these transformations from his axioms.
Many of Einstein's papers present derivations of the Lorentz transformation based upon these two principles.^{[p 7]}

Principle of relativity

Main article: Principle of relativity

Reference frames and relative motion

Figure 2-1. The primed system is in motion relative to the unprimed system with constant velocity v only along the x-axis, from the perspective of an observer stationary in the unprimed system. By the principle of relativity, an observer stationary in the primed system will view a likewise construction except that the velocity they record will be −v. The changing of the speed of propagation of interaction from infinite in non-relativistic mechanics to a finite value will require a modification of the transformation equations mapping events in one frame to another.

Reference frames play a crucial role in relativity theory. The term reference frame as used here is an observational perspective in space which is not undergoing any change in motion (acceleration), from which a position can be measured along 3 spatial axes (so, at rest or constant velocity). In addition, a reference frame has the ability to determine measurements of the time of events using a 'clock' (any reference device with uniform periodicity).
An event is an occurrence that can be assigned a single unique moment and location in space relative to a reference frame: it is a "point" in spacetime. Since the speed of light is constant in relativity irrespective of reference frame, pulses of light can be used to unambiguously measure distances and refer back the times that events occurred to the clock, even though light takes time to reach the clock after the event has transpired.
For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four spacetime coordinates: The time of occurrence and its 3-dimensional spatial location define a reference point. Let's call this reference frame S.
In relativity theory, we often want to calculate the coordinates of an event from differing reference frames. The equations that relate measurements made in different frames are called transformation equations.

Standard configuration

To gain insight in how the spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in a standard configuration.^[13]:107 With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2‑1, two Galilean reference frames (i.e., conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime" or "S dash") belongs to a second observer O′.

• The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′.
• Frame S′ moves, for simplicity, in a single direction: the x-direction of frame S with a constant velocity v as measured in frame S.
• The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.

Since there is no absolute reference frame in relativity theory, a concept of 'moving' doesn't strictly exist, as everything may be moving with respect to some other reference frame. Instead, any two frames that move at the same speed in the same direction are said to be comoving. Therefore, S and S′ are not comoving.

Lack of an absolute reference frame

The principle of relativity, which states that physical laws have the same form in each inertial reference frame, dates back to Galileo, and was incorporated into Newtonian physics. However, in the late 19th century, the existence of electromagnetic waves led some physicists to suggest that the universe was filled with a substance that they called "aether", which, they postulated, would act as the medium through which these waves, or vibrations, propagated (in many respects similar to the way sound propagates through air). The aether was thought to be an absolute reference frame against which all speeds could be measured, and could be considered fixed and motionless relative to Earth or some other fixed reference point. The aether supposedly possessed some wonderful properties: it was sufficiently elastic to support electromagnetic waves, and those waves could interact with matter, yet it offered no resistance to bodies passing through it (its one property was that it allowed electromagnetic waves to propagate). The results of various experiments, including the Michelson–Morley experiment in 1887 (subsequently verified with more accurate and innovative experiments), led to the theory of special relativity, by showing that the aether did not exist.^[14] Einstein's solution was to discard the notion of an aether and the absolute state of rest. In relativity, any reference frame moving with uniform motion will observe the same laws of physics. In particular, the speed of light in vacuum is always measured to be c, even when measured by multiple systems that are moving at different (but constant) velocities.

Relativity without the second postulate

From the principle of relativity alone without assuming the constancy of the speed of light (i.e., using the isotropy of space and the symmetry implied by the principle of special relativity) it can be shown that the spacetime transformations between inertial frames are either Euclidean, Galilean, or Lorentzian. In the Lorentzian case, one can then obtain relativistic interval conservation and a certain finite limiting speed. Experiments suggest that this speed is the speed of light in vacuum.^{[p 8][15]}

Lorentz invariance as the essential core of special relativity

Main article: Lorentz transformation

Alternative approaches to special relativity

Main article: Derivations of the Lorentz transformations

Einstein consistently based the derivation of Lorentz invariance (the essential core of special relativity) on just the two basic principles of relativity and light-speed invariance. He wrote:

The insight fundamental for the special theory of relativity is this: The assumptions relativity and light speed invariance are compatible if relations of a new type ("Lorentz transformation") are postulated for the conversion of coordinates and times of events ... The universal principle of the special theory of relativity is contained in the postulate: The laws of physics are invariant with respect to Lorentz transformations (for the transition from one inertial system to any other arbitrarily chosen inertial system). This is a restricting principle for natural laws ...^{[p 5]}

Thus many modern treatments of special relativity base it on the single postulate of universal Lorentz covariance, or, equivalently, on the single postulate of Minkowski spacetime.^{[p 9][p 10]}
Rather than considering universal Lorentz covariance to be a derived principle, this article considers it to be the fundamental postulate of special relativity. The traditional two-postulate approach to special relativity is presented in innumerable college textbooks and popular presentations.^[16] Textbooks starting with the single postulate of Minkowski spacetime include those by Taylor and Wheeler^[17] and by Callahan.^[18] This is also the approach followed by the Wikipedia articles Spacetime and Minkowski diagram.

Lorentz transformation and its inverse

Define an event to have spacetime coordinates (t,x,y,z) in system S and (t′,x′,y′,z′) in a reference frame moving at a velocity v with respect to that frame, S′. Then the Lorentz transformation specifies that these coordinates are related in the following way:

where

is the Lorentz factor and c is the speed of light in vacuum, and the velocity v of S′, relative to S, is parallel to the x-axis. For simplicity, the y and z coordinates are unaffected; only the x and t coordinates are transformed. These Lorentz transformations form a one-parameter group of linear mappings, that parameter being called rapidity.
Solving the four transformation equations above for the unprimed coordinates yields the inverse Lorentz transformation:

Enforcing this inverse Lorentz transformation to coincide with the Lorentz transformation from the primed to the unprimed system, shows the unprimed frame as moving with the velocity v′ = −v, as measured in the primed frame.
There is nothing special about the x-axis. The transformation can apply to the y- or z-axis, or indeed in any direction parallel to the motion (which are warped by the γ factor) and perpendicular; see the article Lorentz transformation for details.
A quantity invariant under Lorentz transformations is known as a Lorentz scalar.
Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates (x₁, t₁) and (x′₁, t′₁), another event has coordinates (x₂, t₂) and (x′₂, t′₂), and the differences are defined as

Eq. 1:   

Eq. 2:   

we get

Eq. 3:   

Eq. 4:   

If we take differentials instead of taking differences, we get

Eq. 5:   

Eq. 6:   

Graphical representation of the Lorentz transformation

Figure 3-1. Drawing a Minkowski spacetime diagram to illustrate a Lorentz transformation.

Spacetime diagrams (Minkowski diagrams) are an extremely useful aid to visualizing how coordinates transform between different reference frames. Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario.^[15]
To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S', in standard configuration, as shown in Fig. 2‑1.^{[15][19]:155–199}
Fig. 3‑1a. Draw the and axes of frame S. The axis is horizontal and the (actually ) axis is vertical, which is the opposite of the usual convention in kinematics. The axis is scaled by a factor of so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45° diagonal lines represent the worldlines of two photons passing through the origin at time The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, and have been plotted on this graph so that their coordinates may be compared in the S and S' frames.
Fig. 3‑1b. Draw the and axes of frame S'. The axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, Both the and axes are tilted from the unprimed axes by an angle where The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that when
Fig. 3‑1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that coordinates of in the primed coordinate system transform to in the unprimed coordinate system. Likewise, coordinates of in the primed coordinate system transform to in the unprimed system. Draw gridlines parallel with the axis through points as measured in the unprimed frame, where is an integer. Likewise, draw gridlines parallel with the axis through as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between units equals times the spacing between units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as
Fig. 3‑1d. Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time still plot as 45° diagonal lines. The primed coordinates of and are related to the unprimed coordinates through the Lorentz transformations and could be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure, we observe that the two timelike-separated events that had different x-coordinates in the unprimed frame are now at the same position in space.
While the unprimed frame is drawn with space and time axes that meet at right angles, the primed frame is drawn with axes that meet at acute or obtuse angles. The frames are actually equivalent. The asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a Cartesian plane. By analogy, planar maps of the world are unavoidably distorted, but with experience and intuition, one learns to mentally account for these distortions.

Consequences derived from the Lorentz transformation

See also: Twin paradox and Relativistic mechanics

The consequences of special relativity can be derived from the Lorentz transformation equations.^[20] These transformations, and hence special relativity, lead to different physical predictions than those of Newtonian mechanics at all relative velocities, and most pronounced when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything most humans encounter that some of the effects predicted by relativity are initially counterintuitive.

Invariant interval

In Galilean relativity, length ()^{[note 3]} and temporal separation between two events () are independent invariants, the values of which do not change when observed from different frames of reference.^{[note 4][note 5]}
In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as :

^{[note 6]}

The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.
The form of being the difference of the squared time lapse and the squared spatial distance, demonstrates a fundamental discrepancy between Euclidean and spacetime distances.^{[note 7]} The invariance of this interval is a property of the general Lorentz transform (also called the Poincaré transformation), making it an isometry of spacetime. The general Lorentz transform extends the standard Lorentz transform (which deals with translations without rotation, that is, Lorentz boosts, in the x-direction) with all other translations, reflections, and rotations between any Cartesian inertial frame.^[24]:33–34
In the analysis of simplified scenarios, such as spacetime diagrams, a reduced-dimensionality form of the invariant interval is often employed:

Demonstrating that the interval is invariant is straightforward for the reduced-dimensionality case and with frames in standard configuration:^[15]

The value of is hence independent of the frame in which it is measured.
In considering the physical significance of , there are three cases to note:^{[15][25]:25–39}

• Δs² > 0: In this case, the two events are separated by more time than space, and they are hence said to be timelike separated. This implies that and given the Lorentz transformation it is evident that there exists a less than for which (in particular, ). In other words, given two events that are timelike separated, it is possible to find a frame in which the two events happen at the same place. In this frame, the separation in time, is called the proper time.
• Δs² < 0: In this case, the two events are separated by more space than time, and they are hence said to be spacelike separated. This implies that and given the Lorentz transformation there exists a less than for which (in particular, ). In other words, given two events that are spacelike separated, it is possible to find a frame in which the two events happen at the same time. In this frame, the separation in space, is called the proper distance, or proper length. For values of greater than and less than the sign of changes, meaning that the temporal order of spacelike-separated events changes depending on the frame in which the events are viewed. The temporal order of timelike-separated events, however, is absolute, since the only way that could be greater than would be if
• Δs² = 0: In this case, the two events are said to be lightlike separated. This implies that and this relationship is frame independent due to the invariance of From this, we observe that the speed of light is in every inertial frame. In other words, starting from the assumption of universal Lorentz covariance, the constant speed of light is a derived result, rather than a postulate as in the two-postulates formulation of the special theory.

Relativity of simultaneity

See also: Relativity of simultaneity and Ladder paradox

Figure 4-1. The three events (A, B, C) are simultaneous in the reference frame of some observer O. In a reference frame moving at v = 0.3c, as measured by O, the events occur in the order C, B, A. In a reference frame moving at v = -0.5c with respect to O, the events occur in the order A, B, C. The white lines, the lines of simultaneity, move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on it. They are the locus of all events occurring at the same time in the respective frame. The gray area is the light cone with respect to the origin of all considered frames.

Consider two events happening in two different locations that occur simultaneously in the reference frame of one inertial observer. They may occur non-simultaneously in the reference frame of another inertial observer (lack of absolute simultaneity).
From Equation 3 (the forward Lorentz transformation in terms of coordinate differences)

It is clear that the two events that are simultaneous in frame S (satisfying Δt = 0), are not necessarily simultaneous in another inertial frame S′ (satisfying Δt′ = 0). Only if these events are additionally co-local in frame S (satisfying Δx = 0), will they be simultaneous in another frame S′.
The Sagnac effect can be considered a manifestation of the relativity of simultaneity.^[26] Since relativity of simultaneity is a first order effect in ,^[15] instruments based on the Sagnac effect for their operation, such as ring laser gyroscopes and fiber optic gyroscopes, are capable of extreme levels of sensitivity.^{[p 14]}

Time dilation

See also: Time dilation

The time lapse between two events is not invariant from one observer to another, but is dependent on the relative speeds of the observers' reference frames (e.g., the twin paradox which concerns a twin who flies off in a spaceship traveling near the speed of light and returns to discover that the non-traveling twin sibling has aged much more, the paradox being that at constant velocity we are unable to discern which twin is non-traveling and which twin travels).
Suppose a clock is at rest in the unprimed system S. The location of the clock on two different ticks is then characterized by Δx = 0. To find the relation between the times between these ticks as measured in both systems, Equation 3 can be used to find:

    for events satisfying    

This shows that the time (Δt′) between the two ticks as seen in the frame in which the clock is moving (S′), is longer than the time (Δt) between these ticks as measured in the rest frame of the clock (S). Time dilation explains a number of physical phenomena; for example, the lifetime of high speed muons created by the collision of cosmic rays with particles in the Earth's outer atmosphere and moving towards the surface is greater than the lifetime of slowly moving muons, created and decaying in a laboratory.^[27]

Length contraction

See also: Lorentz contraction

The dimensions (e.g., length) of an object as measured by one observer may be smaller than the results of measurements of the same object made by another observer (e.g., the ladder paradox involves a long ladder traveling near the speed of light and being contained within a smaller garage).
Similarly, suppose a measuring rod is at rest and aligned along the x-axis in the unprimed system S. In this system, the length of this rod is written as Δx. To measure the length of this rod in the system S′, in which the rod is moving, the distances x′ to the end points of the rod must be measured simultaneously in that system S′. In other words, the measurement is characterized by Δt′ = 0, which can be combined with Equation 3 to find the relation between the lengths Δx and Δx′:

    for events satisfying    

This shows that the length (Δx′) of the rod as measured in the frame in which it is moving (S′), is shorter than its length (Δx) in its own rest frame (S).
Time dilation and length contraction are not merely appearances. Time dilation is explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system (called "co-local" events). These time intervals (which can be, and are, actually measured experimentally by relevant observers) are different in another coordinate system moving with respect to the first, unless the events, in addition to being co-local, are also simultaneous. Similarly, length contraction relates to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system.

Lorentz transformation of velocities

See also: Velocity-addition formula

Consider two frames S and S′ in standard configuration. A particle in S moves in the x direction with velocity vector What is its velocity in frame S′ ?
We can write

Eq. 7:   

Eq. 8:   

Substituting expressions for and from Equation 5 into Equation 8, followed by straightforward mathematical manipulations and back-substitution from Equation 7 yields the Lorentz transformation of the speed to :

Eq. 9:   

The inverse relation is obtained by interchanging the primed and unprimed symbols and replacing with

Eq. 10:   

For not aligned along the x-axis, we write:^[8]:47–49

Eq. 11:   

Eq. 12:   

The forward and inverse transformations for this case are:

Eq. 13:       

Eq. 14:       

Equation 10 and Equation 14 can be interpreted as giving the resultant of the two velocities and and they replace the formula which is valid in Galilean relativity. Interpreted in such a fashion, they are commonly referred to as the relativistic velocity addition (or composition) formulas, valid for the three axes of S and S′ being aligned with each other (although not necessarily in standard configuration).^[8]:47–49
We note the following points:

• If an object (e.g., a photon) were moving at the speed of light in one frame (i.e., u = ±c or u′ = ±c), then it would also be moving at the speed of light in any other frame, moving at |v| < c.
• The resultant speed of two velocities with magnitude less than c is always a velocity with magnitude less than c.
• If both |u| and |v| (and then also |u′| and |v′|) are small with respect to the speed of light (that is, e.g., |u/c| ≪ 1), then the intuitive Galilean transformations are recovered from the transformation equations for special relativity
• Attaching a frame to a photon (riding a light beam like Einstein considers) requires special treatment of the transformations.

There is nothing special about the x direction in the standard configuration. The above formalism applies to any direction; and three orthogonal directions allow dealing with all directions in space by decomposing the velocity vectors to their components in these directions. See Velocity-addition formula for details.

Thomas rotation

See also: Thomas rotation

Figure 4-2. Thomas-Wigner rotation

The composition of two non-collinear Lorentz boosts (i.e., two non-collinear Lorentz transformations, neither of which involve rotation) results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation.
Thomas rotation results from the relativity of simultaneity. In Fig. 4‑2a, a rod of length in its rest frame (i.e., having a proper length of ) rises vertically along the y‑axis in the ground frame.
In Fig. 4‑2b, the same rod is observed from the frame of a rocket moving at speed to the right. If we imagine two clocks situated at the left and right ends of the rod that are synchronized in the frame of the rod, relativity of simultaneity causes the observer in the rocket frame to observe (not see) the clock at the right end of the rod as being advanced in time by and the rod is correspondingly observed as tilted.^[25]:98–99
Unlike second-order relativistic effects such as length contraction or time dilation, this effect becomes quite significant even at fairly low velocities. For example, this can be seen in the spin of moving particles, where Thomas precession is a relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope, relating the angular velocity of the spin of a particle following a curvilinear orbit to the angular velocity of the orbital motion.^{[25]:169–174}
Thomas rotation provides the resolution to the well-known "meter stick and hole paradox".^{[p 15][25]:98–99}

Causality and prohibition of motion faster than light

See also: Causality (physics) and Tachyonic antitelephone

Figure 4-3. Light cone

In Fig. 4‑3, the time interval between the events A (the "cause") and B (the "effect") is 'time-like'; that is, there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames accessible by a Lorentz transformation. It is possible for matter (or information) to travel (below light speed) from the location of A, starting at the time of A, to the location of B, arriving at the time of B, so there can be a causal relationship (with A the cause and B the effect).
The interval AC in the diagram is 'space-like'; that is, there is a frame of reference in which events A and C occur simultaneously, separated only in space. There are also frames in which A precedes C (as shown) and frames in which C precedes A. However, there are no frames accessible by a Lorentz transformation, in which events A and C occur at the same location. If it were possible for a cause-and-effect relationship to exist between events A and C, then paradoxes of causality would result.
For example, if signals could be sent faster than light, then signals could be sent into the sender's past (observer B in the diagrams).^{[28][p 16]} A variety of causal paradoxes could then be constructed.

Figure 4-4. Causality violation by the use of fictitious
"instantaneous communicators"

Consider the spacetime diagrams in Fig. 4‑4. A and B stand alongside a railroad track, when a high speed train passes by, with C riding in the last car of the train and D riding in the leading car. The world lines of A and B are vertical (ct), distinguishing the stationary position of these observers on the ground, while the world lines of C and D are tilted forwards (ct′), reflecting the rapid motion of the observers C and D stationary in their train, as observed from the ground.

1. Fig. 4‑4a. The event of "B passing a message to D", as the leading car passes by, is at the origin of D's frame. D sends the message along the train to C in the rear car, using a fictitious "instantaneous communicator". The worldline of this message is the fat red arrow along the axis, which is a line of simultaneity in the primed frames of C and D. In the (unprimed) ground frame the signal arrives earlier than it was sent.
2. Fig. 4‑4b. The event of "C passing the message to A", who is standing by the railroad tracks, is at the origin of their frames. Now A sends the message along the tracks to B via an "instantaneous communicator". The worldline of this message is the blue fat arrow, along the axis, which is a line of simultaneity for the frames of A and B. As seen from the spacetime diagram, B will receive the message before having sent it out, a violation of causality.^[29]

It is not necessary for signals to be instantaneous to violate causality. Even if the signal from D to C were slightly shallower than the axis (and the signal from A to B slightly steeper than the axis), it would still be possible for B to receive his message before he had sent it. By increasing the speed of the train to near light speeds, the and axes can be squeezed very close to the dashed line representing the speed of light. With this modified setup, it can be demonstrated that even signals only slightly faster than the speed of light will result in causality violation.^[30]
Therefore, if causality is to be preserved, one of the consequences of special relativity is that no information signal or material object can travel faster than light in vacuum.
This is not to say that all faster than light speeds are impossible. Various trivial situations can be described where some "things" (not actual matter or energy) move faster than light.^[31] For example, the location where the beam of a search light hits the bottom of a cloud can move faster than light when the search light is turned rapidly (although this does not violate causality or any other relativistic phenomenon).^[32][33]

Optical effects

Dragging effects

Main article: Fizeau experiment

Figure 5-1. Highly simplified diagram of Fizeau's 1851 experiment.

In 1850, Hippolyte Fizeau and Léon Foucault independently established that light travels more slowly in water than in air, thus validating a prediction of Fresnel's wave theory of light and invalidating the corresponding prediction of Newton's corpuscular theory.^[34] The speed of light was measured in still water. What would be the speed of light in flowing water?
In 1851, Fizeau conducted an experiment to answer this question, a simplified representation of which is illustrated in Fig. 5‑1. A beam of light is divided by a beam splitter, and the split beams are passed in opposite directions through a tube of flowing water. They are recombined to form interference fringes, indicating a difference in optical path length, that an observer can view. The experiment demonstrated that dragging of the light by the flowing water caused displacement of the fringes, showing that the motion of the water had affected the speed of the light.
According to the theories prevailing at the time, light traveling through a moving medium would be a simple sum of its speed through the medium plus the speed of the medium. Contrary to expectation, Fizeau found that although light appeared to be dragged by the water, the magnitude of the dragging was much lower than expected. If is the speed of light in still water, and is the speed of the water, and is the water-bourne speed of light in the lab frame with the flow of water adding to or subtracting from the speed of light, then

Fizeau's results, although consistent with Fresnel's earlier hypothesis of partial aether dragging, were extremely disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since depends on wavelength, the aether must be capable of sustaining different motions at the same time.^{[note 8]} A variety of theoretical explanations were proposed to explain Fresnel's dragging coefficient that were completely at odds with each other. Even before the Michelson–Morley experiment, Fizeau's experimental results were among a number of observations that created a critical situation in explaining the optics of moving bodies.^[35]
From the point of view of special relativity, Fizeau's result is nothing but an approximation to Equation 10, the relativistic formula for composition of velocities.^[24]

Relativistic aberration of light

Main articles: Aberration of light and Light-time correction

Figure 5-2. Illustration of stellar aberration

Because of the finite speed of light, if the relative motions of a source and receiver include a transverse component, then the direction from which light arrives at the receiver will be displaced from the geometric position in space of the source relative to the receiver. The classical calculation of the displacement takes two forms and makes different predictions depending on whether the receiver, the source, or both are in motion with respect to the medium. (1) If the receiver is in motion, the displacement would be the consequence of the aberration of light. The incident angle of the beam relative to the receiver would be calculable from the vector sum of the receiver's motions and the velocity of the incident light.^[36] (2) If the source is in motion, the displacement would be the consequence of light-time correction. The displacement of the apparent position of the source from its geometric position would be the result of the source's motion during the time that its light takes to reach the receiver.^[37]
The classical explanation failed experimental test. Since the aberration angle depends on the relationship between the velocity of the receiver and the speed of the incident light, passage of the incident light through a refractive medium should change the aberration angle. In 1810, Arago used this expected phenomenon in a failed attempt to measure the speed of light,^[38] and in 1870, George Airy tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope.^[39] A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag,^[40] but was incompatible with the results of the Michelson–Morley experiment, which apparently demanded complete aether-drag.^[41]
Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig. 5‑2, these include^[24]:57–60

  OR     OR  

Relativistic Doppler effect

Main article: Relativistic Doppler effect

Relativistic longitudinal Doppler effect

The classical Doppler effect depends on whether the source, receiver, or both are in motion with respect to the medium. The relativistic Doppler effect is independent of any medium. Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a time dilation term, and that is the treatment described here.^[42][43]
Assume the receiver and the source are moving away from each other with a relative speed as measured by an observer on the receiver or the source (The sign convention adopted here is that is negative if the receiver and the source are moving towards each other). Assume that the source is stationary in the medium. Then

where is the speed of sound.
For light, and with the receiver moving at relativistic speeds, clocks on the receiver are time dilated relative to clocks at the source. The receiver will measure the received frequency to be

where

   and

   is the Lorentz factor.

An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source.^[44][15]

Transverse Doppler effect

Figure 5-3. Transverse Doppler effect for two scenarios: (a) receiver moving in a circle around the source; (b) source moving in a circle around the receiver.

The transverse Doppler effect is one of the main novel predictions of the special theory of relativity.
Classically, one might expect that if source and receiver are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver.
Special relativity predicts otherwise. Fig. 5‑3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments.^[15] In Fig. 5‑3a, the receiver observes light from the source as being blueshifted by a factor of . In Fig. 5‑3b, the light is redshifted by the same factor.

Measurement versus visual appearance

Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact of Doppler shift, nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer.
Scientists make a fundamental distinction between measurement or observation on the one hand, versus visual appearance, or what one sees. The measured shape of an object is a hypothetical snapshot of all of the object's points as they exist at a single moment in time. The visual appearance of an object, however, is affected by the varying lengths of time that light takes to travel from different points on the object to one's eye.

Figure 5-4. Comparison of the measured length contraction of a cube versus its visual appearance.

For many years, the distinction between the two had not been generally appreciated, and it had generally been thought that a length contracted object passing by an observer would in fact actually be seen as length contracted. In 1959, James Terrell and Roger Penrose independently pointed out that differential time lag effects in signals reaching the observer from the different parts of a moving object result in a fast moving object's visual appearance being quite different from its measured shape. For example, a receding object would appear contracted, an approaching object would appear elongated, and a passing object would have a skew appearance that has been likened to a rotation.^{[p 19][p 20][45][46]} A sphere in motion retains the appearance of a sphere, although images on the surface of the sphere will appear distorted.^[47]

Figure 5-5. Galaxy M87 streams out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.

Fig. 5‑4 illustrates a cube viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. This illusion has come to be known as Terrell rotation or the Terrell–Penrose effect.^{[note 9]}
Another example where visual appearance is at odds with measurement comes from the observation of apparent superluminal motion in various radio galaxies, BL Lac objects, quasars, and other astronomical objects that eject relativistic-speed jets of matter at narrow angles with respect to the viewer. An apparent optical illusion results giving the appearance of faster than light travel.^[48][49][50] In Fig. 5‑5, galaxy M87 streams out a high-speed jet of subatomic particles almost directly towards us, but Penrose–Terrell rotation causes the jet to appear to be moving laterally in the same manner that the appearance of the cube in Fig. 5‑4 has been stretched out.^[51]

Dynamics

Section Consequences derived from the Lorentz transformation dealt strictly with kinematics, the study of the motion of points, bodies, and systems of bodies without considering the forces that caused the motion. This section discusses masses, forces, energy and so forth, and as such requires consideration of physical effects beyond those encompassed by the Lorentz transformation itself.

Equivalence of mass and energy

Main article: Mass–energy equivalence

As an object's speed approaches the speed of light from an observer's point of view, its relativistic mass increases thereby making it more and more difficult to accelerate it from within the observer's frame of reference.
The energy content of an object at rest with mass m equals mc². Conservation of energy implies that, in any reaction, a decrease of the sum of the masses of particles must be accompanied by an increase in kinetic energies of the particles after the reaction. Similarly, the mass of an object can be increased by taking in kinetic energies.
In addition to the papers referenced above—which give derivations of the Lorentz transformation and describe the foundations of special relativity—Einstein also wrote at least four papers giving heuristic arguments for the equivalence (and transmutability) of mass and energy, for E = mc².
Mass–energy equivalence is a consequence of special relativity. The energy and momentum, which are separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is (E/c, 0, 0, 0): it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes (E/c, Ev/c², 0, 0). The momentum is equal to the energy multiplied by the velocity divided by c². As such, the Newtonian mass of an object, which is the ratio of the momentum to the velocity for slow velocities, is equal to E/c².
The energy and momentum are properties of matter and radiation, and it is impossible to deduce that they form a four-vector just from the two basic postulates of special relativity by themselves, because these don't talk about matter or radiation, they only talk about space and time. The derivation therefore requires some additional physical reasoning. In his 1905 paper, Einstein used the additional principles that Newtonian mechanics should hold for slow velocities, so that there is one energy scalar and one three-vector momentum at slow velocities, and that the conservation law for energy and momentum is exactly true in relativity. Furthermore, he assumed that the energy of light is transformed by the same Doppler-shift factor as its frequency, which he had previously shown to be true based on Maxwell's equations.^{[p 1]} The first of Einstein's papers on this subject was "Does the Inertia of a Body Depend upon its Energy Content?" in 1905.^{[p 21]} Although Einstein's argument in this paper is nearly universally accepted by physicists as correct, even self-evident, many authors over the years have suggested that it is wrong.^[52] Other authors suggest that the argument was merely inconclusive because it relied on some implicit assumptions.^[53]
Einstein acknowledged the controversy over his derivation in his 1907 survey paper on special relativity. There he notes that it is problematic to rely on Maxwell's equations for the heuristic mass–energy argument. The argument in his 1905 paper can be carried out with the emission of any massless particles, but the Maxwell equations are implicitly used to make it obvious that the emission of light in particular can be achieved only by doing work. To emit electromagnetic waves, all you have to do is shake a charged particle, and this is clearly doing work, so that the emission is of energy.^{[p 22][note 10]}

How far can one travel from the Earth?

See also: Space travel using constant acceleration

Since one can not travel faster than light, one might conclude that a human can never travel farther from Earth than 40 light years if the traveler is active between the ages of 20 and 60. One would easily think that a traveler would never be able to reach more than the very few solar systems which exist within the limit of 20–40 light years from the earth. But that would be a mistaken conclusion. Because of time dilation, a hypothetical spaceship can travel thousands of light years during the pilot's 40 active years. If a spaceship could be built that accelerates at a constant 1g, it will, after a little less than a year, be travelling at almost the speed of light as seen from Earth. This is described by:

where v(t) is the velocity at a time t, a is the acceleration of 1g and t is the time as measured by people on Earth.^{[p 23]} Therefore, after one year of accelerating at 9.81 m/s², the spaceship will be travelling at v = 0.77c relative to Earth. Time dilation will increase the travellers life span as seen from the reference frame of the Earth to 2.7 years, but his lifespan measured by a clock travelling with him will not change. During his journey, people on Earth will experience more time than he does. A 5-year round trip for him will take 6.5 Earth years and cover a distance of over 6 light-years. A 20-year round trip for him (5 years accelerating, 5 decelerating, twice each) will land him back on Earth having travelled for 335 Earth years and a distance of 331 light years.^[54] A full 40-year trip at 1g will appear on Earth to last 58,000 years and cover a distance of 55,000 light years. A 40-year trip at 1.1g will take 148,000 Earth years and cover about 140,000 light years. A one-way 28 year (14 years accelerating, 14 decelerating as measured with the astronaut's clock) trip at 1g acceleration could reach 2,000,000 light-years to the Andromeda Galaxy.^[54] This same time dilation is why a muon travelling close to c is observed to travel much farther than c times its half-life (when at rest).^[55]

Relativity and unifying electromagnetism

Main articles: Classical electromagnetism and special relativity and Covariant formulation of classical electromagnetism

Theoretical investigation in classical electromagnetism led to the discovery of wave propagation. Equations generalizing the electromagnetic effects found that finite propagation speed of the E and B fields required certain behaviors on charged particles. The general study of moving charges forms the Liénard–Wiechert potential, which is a step towards special relativity.
The Lorentz transformation of the electric field of a moving charge into a non-moving observer's reference frame results in the appearance of a mathematical term commonly called the magnetic field. Conversely, the magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. Maxwell's equations are thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.
Maxwell's equations in the 3D form are already consistent with the physical content of special relativity, although they are easier to manipulate in a manifestly covariant form, that is, in the language of tensor calculus.^[56]

Theories of relativity and quantum mechanics

Special relativity can be combined with quantum mechanics to form relativistic quantum mechanics and quantum electrodynamics. It is an unsolved problem in physics how general relativity and quantum mechanics can be unified; quantum gravity and a "theory of everything", which require a unification including general relativity too, are active and ongoing areas in theoretical research.
The early Bohr–Sommerfeld atomic model explained the fine structure of alkali metal atoms using both special relativity and the preliminary knowledge on quantum mechanics of the time.^[57]
In 1928, Paul Dirac constructed an influential relativistic wave equation, now known as the Dirac equation in his honour,^{[p 24]} that is fully compatible both with special relativity and with the final version of quantum theory existing after 1926. This equation not only describe the intrinsic angular momentum of the electrons called spin, it also led to the prediction of the antiparticle of the electron (the positron),^{[p 24][p 25]} and fine structure could only be fully explained with special relativity. It was the first foundation of relativistic quantum mechanics.
On the other hand, the existence of antiparticles leads to the conclusion that relativistic quantum mechanics is not enough for a more accurate and complete theory of particle interactions. Instead, a theory of particles interpreted as quantized fields, called quantum field theory, becomes necessary; in which particles can be created and destroyed throughout space and time.

Status

Main articles: Tests of special relativity and Criticism of relativity theory

Special relativity in its Minkowski spacetime is accurate only when the absolute value of the gravitational potential is much less than c² in the region of interest.^[58] In a strong gravitational field, one must use general relativity. General relativity becomes special relativity at the limit of a weak field. At very small scales, such as at the Planck length and below, quantum effects must be taken into consideration resulting in quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is experimentally tested to extremely high degree of accuracy (10⁻²⁰)^[59] and thus accepted by the physics community. Experimental results which appear to contradict it are not reproducible and are thus widely believed to be due to experimental errors.
Special relativity is mathematically self-consistent, and it is an organic part of all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields).
Newtonian mechanics mathematically follows from special relativity at small velocities (compared to the speed of light) – thus Newtonian mechanics can be considered as a special relativity of slow moving bodies. See classical mechanics for a more detailed discussion.
Several experiments predating Einstein's 1905 paper are now interpreted as evidence for relativity. Of these it is known Einstein was aware of the Fizeau experiment before 1905,^[60] and historians have concluded that Einstein was at least aware of the Michelson–Morley experiment as early as 1899 despite claims he made in his later years that it played no role in his development of the theory.^[10]

• The Fizeau experiment (1851, repeated by Michelson and Morley in 1886) measured the speed of light in moving media, with results that are consistent with relativistic addition of colinear velocities.
• The famous Michelson–Morley experiment (1881, 1887) gave further support to the postulate that detecting an absolute reference velocity was not achievable. It should be stated here that, contrary to many alternative claims, it said little about the invariance of the speed of light with respect to the source and observer's velocity, as both source and observer were travelling together at the same velocity at all times.
• The Trouton–Noble experiment (1903) showed that the torque on a capacitor is independent of position and inertial reference frame.
• The Experiments of Rayleigh and Brace (1902, 1904) showed that length contraction doesn't lead to birefringence for a co-moving observer, in accordance with the relativity principle.

Particle accelerators routinely accelerate and measure the properties of particles moving at near the speed of light, where their behavior is completely consistent with relativity theory and inconsistent with the earlier Newtonian mechanics. These machines would simply not work if they were not engineered according to relativistic principles. In addition, a considerable number of modern experiments have been conducted to test special relativity. Some examples:

• Tests of relativistic energy and momentum – testing the limiting speed of particles
• Ives–Stilwell experiment – testing relativistic Doppler effect and time dilation
• Experimental testing of time dilation – relativistic effects on a fast-moving particle's half-life
• Kennedy–Thorndike experiment – time dilation in accordance with Lorentz transformations
• Hughes–Drever experiment – testing isotropy of space and mass
• Modern searches for Lorentz violation – various modern tests
• Experiments to test emission theory demonstrated that the speed of light is independent of the speed of the emitter.
• Experiments to test the aether drag hypothesis – no "aether flow obstruction".

Technical discussion of spacetime

Main article: Minkowski space

Geometry of spacetime

Comparison between flat Euclidean space and Minkowski space

See also: line element

Figure 10-1. Orthogonality and rotation of coordinate systems compared between left: Euclidean space through circular angle φ, right: in Minkowski spacetime through hyperbolic angle φ (red lines labelled c denote the worldlines of a light signal, a vector is orthogonal to itself if it lies on this line).^[61]

Special relativity uses a 'flat' 4-dimensional Minkowski space – an example of a spacetime. Minkowski spacetime appears to be very similar to the standard 3-dimensional Euclidean space, but there is a crucial difference with respect to time.
In 3D space, the differential of distance (line element) ds is defined by

where dx = (dx₁, dx₂, dx₃) are the differentials of the three spatial dimensions. In Minkowski geometry, there is an extra dimension with coordinate X⁰ derived from time, such that the distance differential fulfills

where dX = (dX₀, dX₁, dX₂, dX₃) are the differentials of the four spacetime dimensions. This suggests a deep theoretical insight: special relativity is simply a rotational symmetry of our spacetime, analogous to the rotational symmetry of Euclidean space (see Fig. 10‑1).^[62] Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski metric. Basically, special relativity can be stated as the invariance of any spacetime interval (that is the 4D distance between any two events) when viewed from any inertial reference frame. All equations and effects of special relativity can be derived from this rotational symmetry (the Poincaré group) of Minkowski spacetime.
The actual form of ds above depends on the metric and on the choices for the X⁰ coordinate. To make the time coordinate look like the space coordinates, it can be treated as imaginary: X₀ = ict (this is called a Wick rotation). According to Misner, Thorne and Wheeler (1971, §2.3), ultimately the deeper understanding of both special and general relativity will come from the study of the Minkowski metric (described below) and to take X⁰ = ct, rather than a "disguised" Euclidean metric using ict as the time coordinate.
Some authors use X⁰ = t, with factors of c elsewhere to compensate; for instance, spatial coordinates are divided by c or factors of c^±2 are included in the metric tensor.^[63] These numerous conventions can be superseded by using natural units where c = 1. Then space and time have equivalent units, and no factors of c appear anywhere.

3D spacetime

Figure 10-2. Three-dimensional dual-cone.

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3D space

we see that the null geodesics lie along a dual-cone (see Fig. 10‑2) defined by the equation;

or simply

which is the equation of a circle of radius c dt.

4D spacetime

If we extend this to three spatial dimensions, the null geodesics are the 4-dimensional cone:

so

Figure 10-3. Concentric spheres, illustrating in 3-space the null geodesics of a 4-dimensional cone in spacetime.

As illustrated in Fig. 10‑3, the null geodesics can be visualized as a set of continuous concentric spheres with radii = c dt.
This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am receiving is X years old", we are looking down this line of sight: a null geodesic. We are looking at an event a distance away and a time d/c in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the Fig. 10‑2 represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)
The cone in the −t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.
The geometry of Minkowski space can be depicted using Minkowski diagrams, which are useful also in understanding many of the thought-experiments in special relativity.
Note that, in 4d spacetime, the concept of the center of mass becomes more complicated, see center of mass (relativistic).

Physics in spacetime

Transformations of physical quantities between reference frames

Above, the Lorentz transformation for the time coordinate and three space coordinates illustrates that they are intertwined. This is true more generally: certain pairs of "timelike" and "spacelike" quantities naturally combine on equal footing under the same Lorentz transformation.
The Lorentz transformation in standard configuration above, that is, for a boost in the x direction, can be recast into matrix form as follows:

In Newtonian mechanics, quantities which have magnitude and direction are mathematically described as 3d vectors in Euclidean space, and in general they are parametrized by time. In special relativity, this notion is extended by adding the appropriate timelike quantity to a spacelike vector quantity, and we have 4d vectors, or "four vectors", in Minkowski spacetime. The components of vectors are written using tensor index notation, as this has numerous advantages. The notation makes it clear the equations are manifestly covariant under the Poincaré group, thus bypassing the tedious calculations to check this fact. In constructing such equations, we often find that equations previously thought to be unrelated are, in fact, closely connected being part of the same tensor equation. Recognizing other physical quantities as tensors simplifies their transformation laws. Throughout, upper indices (superscripts) are contravariant indices rather than exponents except when they indicate a square (this should be clear from the context), and lower indices (subscripts) are covariant indices. For simplicity and consistency with the earlier equations, Cartesian coordinates will be used.
The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ct and spacelike component x = (x, y, z), in a contravariant position four vector with components:

where we define X⁰ = ct so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally.^[64][65][66] Now the transformation of the contravariant components of the position 4-vector can be compactly written as:

where there is an implied summation on from 0 to 3, and is a matrix.
More generally, all contravariant components of a four-vector transform from one frame to another frame by a Lorentz transformation:

Examples of other 4-vectors include the four-velocity defined as the derivative of the position 4-vector with respect to proper time:

where the Lorentz factor is:

The relativistic energy and relativistic momentum of an object are respectively the timelike and spacelike components of a contravariant four momentum vector:

where m is the invariant mass.
The four-acceleration is the proper time derivative of 4-velocity:

The transformation rules for three-dimensional velocities and accelerations are very awkward; even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of four-velocity and four-acceleration are simpler by means of the Lorentz transformation matrix.
The four-gradient of a scalar field φ transforms covariantly rather than contravariantly:

which is the transpose of:

only in Cartesian coordinates. It's the covariant derivative which transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates.
More generally, the covariant components of a 4-vector transform according to the inverse Lorentz transformation:

where is the reciprocal matrix of .
The postulates of special relativity constrain the exact form the Lorentz transformation matrices take.
More generally, most physical quantities are best described as (components of) tensors. So to transform from one frame to another, we use the well-known tensor transformation law^[67]

where is the reciprocal matrix of . All tensors transform by this rule.
An example of a four dimensional second order antisymmetric tensor is the relativistic angular momentum, which has six components: three are the classical angular momentum, and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time is the relativistic torque, also second order antisymmetric tensor.
The electromagnetic field tensor is another second order antisymmetric tensor field, with six components: three for the electric field and another three for the magnetic field. There is also the stress–energy tensor for the electromagnetic field, namely the electromagnetic stress–energy tensor.

Metric

The metric tensor allows one to define the inner product of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the Minkowski metric η has components (valid in any inertial reference frame) which can be arranged in a 4 × 4 matrix:

which is equal to its reciprocal, , in those frames. Throughout we use the signs as above, different authors use different conventions – see Minkowski metric alternative signs.
The Poincaré group is the most general group of transformations which preserves the Minkowski metric:

and this is the physical symmetry underlying special relativity.
The metric can be used for raising and lowering indices on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector T with another 4-vector S is:

Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no Λ appears in its trivial transformation. The magnitude of the 4-vector T is the positive square root of the inner product with itself:

One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants:

similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one doesn't need to perform Lorentz transformations to determine the invariants.

Relativistic kinematics and invariance

The coordinate differentials transform also contravariantly:

so the squared length of the differential of the position four-vector dX^μ constructed using

is an invariant. Notice that when the line element dX² is negative that √−dX² is the differential of proper time, while when dX² is positive, √dX² is differential of the proper distance.
The 4-velocity U^μ has an invariant form:

which means all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. Differentiating the above equation by τ produces:

So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal.

Relativistic dynamics and invariance

The invariant magnitude of the momentum 4-vector generates the energy–momentum relation:

We can work out what this invariant is by first arguing that, since it is a scalar, it doesn't matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.

We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.
The rest energy is related to the mass according to the celebrated equation discussed above:

The mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.
To use Newton's third law of motion, both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector among its components.
If a particle is not traveling at c, one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the four-force. It is the rate of change of the above energy momentum four-vector with respect to proper time. The covariant version of the four-force is:

In the rest frame of the object, the time component of the four force is zero unless the "invariant mass" of the object is changing (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times c. In general, though, the components of the four force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, that is, dp/dt while the four force is defined by the rate of change of momentum with respect to proper time, that is, dp/dτ.
In a continuous medium, the 3D density of force combines with the density of power to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/c times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.

See also

• Physics portal

People: Hendrik Lorentz | Henri Poincaré | Albert Einstein | Max Planck | Hermann Minkowski | Max von Laue | Arnold Sommerfeld | Max Born | Gustav Herglotz | Richard C. Tolman

Relativity: Theory of relativity | History of special relativity | Principle of relativity | Doubly special relativity | General relativity | Frame of reference | Inertial frame of reference | Lorentz transformations | Bondi k-calculus | Einstein synchronisation | Rietdijk–Putnam argument | Special relativity (alternative formulations) | Criticism of relativity theory | Relativity priority dispute

Physics: Einstein's thought experiments | Newtonian Mechanics | spacetime | speed of light | simultaneity | center of mass (relativistic) | physical cosmology | Doppler effect | relativistic Euler equations | Aether drag hypothesis | Lorentz ether theory | Moving magnet and conductor problem | Shape waves | Relativistic heat conduction | Relativistic disk | Thomas precession | Born rigidity | Born coordinates

Mathematics: Derivations of the Lorentz transformations | Minkowski space | four-vector | world line | light cone | Lorentz group | Poincaré group | geometry | tensors | split-complex number | Relativity in the APS formalism

Philosophy: actualism | conventionalism | formalism

Paradoxes: Twin paradox | Ehrenfest paradox | Ladder paradox | Bell's spaceship paradox | Velocity composition paradox | Lighthouse paradox

Primary sources

1. 
   

Albert Einstein (1905) "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17: 891; English translation On the Electrodynamics of Moving Bodies by George Barker Jeffery and Wilfrid Perrett (1923); Another English translation On the Electrodynamics of Moving Bodies by Megh Nad Saha (1920).

25. C.D. Anderson (1933). "The Positive Electron". Phys. Rev. 43 (6): 491–494. Bibcode:1933PhRv...43..491A. doi:10.1103/PhysRev.43.491.

References

1. 
   

67. Sean M. Carroll (2004). Spacetime and Geometry: An Introduction to General Relativity. Addison Wesley. p. 22. ISBN 978-0-8053-8732-2.

Notes

1. 
   

10. In a letter to Carl Seelig in 1955, Einstein wrote "I had already previously found that Maxwell's theory did not account for the micro-structure of radiation and could therefore have no general validity.", Einstein letter to Carl Seelig, 1955.

Textbooks

• Einstein, Albert (1920). Relativity: The Special and General Theory.
• Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. ISBN 1-56731-136-9
• Logunov, Anatoly A. (2005) Henri Poincaré and the Relativity Theory (transl. from Russian by G. Pontocorvo and V. O. Soloviev, edited by V. A. Petrov) Nauka, Moscow.
• Charles Misner, Kip Thorne, and John Archibald Wheeler (1971) Gravitation. W. H. Freeman & Co. ISBN 0-7167-0334-3
• Post, E.J., 1997 (1962) Formal Structure of Electromagnetics: General Covariance and Electromagnetics. Dover Publications.
• Wolfgang Rindler (1991). Introduction to Special Relativity (2nd ed.), Oxford University Press. ISBN 978-0-19-853952-0; ISBN 0-19-853952-5
• Harvey R. Brown (2005). Physical relativity: space–time structure from a dynamical perspective, Oxford University Press, ISBN 0-19-927583-1; ISBN 978-0-19-927583-0
• Qadir, Asghar (1989). Relativity: An Introduction to the Special Theory. Singapore: World Scientific Publications. p. 128. Bibcode:1989rist.book.....Q. ISBN 978-9971-5-0612-4.
• French, A. P. (1968). Special Relativity (M.I.T. Introductory Physics) (1st ed.). W. W. Norton & Company. ISBN 978-0393097931.
• Silberstein, Ludwik (1914) The Theory of Relativity.
• Lawrence Sklar (1977). Space, Time and Spacetime. University of California Press. ISBN 978-0-520-03174-6.
• Lawrence Sklar (1992). Philosophy of Physics. Westview Press. ISBN 978-0-8133-0625-4.
• Sergey Stepanov (2018). Relativistic World. De Gruyter. ISBN 9783110515879.
• Taylor, Edwin, and John Archibald Wheeler (1992) Spacetime Physics (2nd ed.). W.H. Freeman & Co. ISBN 0-7167-2327-1
• Tipler, Paul, and Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman & Co. ISBN 0-7167-4345-0

Journal articles

• Alvager, T.; Farley, F. J. M.; Kjellman, J.; Wallin, L.; et al. (1964). "Test of the Second Postulate of Special Relativity in the GeV region". Physics Letters. 12 (3): 260–262. Bibcode:1964PhL....12..260A. doi:10.1016/0031-9163(64)91095-9.
• Darrigol, Olivier (2004). "The Mystery of the Poincaré–Einstein Connection". Isis. 95 (4): 614–26. doi:10.1086/430652. PMID 16011297.
• Wolf, Peter; Petit, Gerard (1997). "Satellite test of Special Relativity using the Global Positioning System". Physical Review A. 56 (6): 4405–09. Bibcode:1997PhRvA..56.4405W. doi:10.1103/PhysRevA.56.4405.
• Special Relativity Scholarpedia
• Rindler, Wolfgang. "Special relativity: Kinematics". Scholarpedia. 6 (2): 8520. doi:10.4249/scholarpedia.8520.

External links

Wikisource has original text related to this article: Relativity: The Special and General Theory

Wikisource has original works on the topic: Relativity

Wikibooks has a book on the topic of: Special Relativity

Wikiversity has learning resources about Special Relativity

Look up special relativity in Wiktionary, the free dictionary.

Original works

• Zur Elektrodynamik bewegter Körper Einstein's original work in German, Annalen der Physik, Bern 1905
• On the Electrodynamics of Moving Bodies English Translation as published in the 1923 book The Principle of Relativity.

Special relativity for a general audience (no mathematical knowledge required)

• Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.
• Einstein Online Introduction to relativity theory, from the Max Planck Institute for Gravitational Physics.
• Audio: Cain/Gay (2006) – Astronomy Cast. Einstein's Theory of Special Relativity

Special relativity explained (using simple or more advanced mathematics)

• Greg Egan's Foundations.
• The Hogg Notes on Special Relativity A good introduction to special relativity at the undergraduate level, using calculus.
• Relativity Calculator: Special Relativity – An algebraic and integral calculus derivation for E = mc².
• MathPages – Reflections on Relativity A complete online book on relativity with an extensive bibliography.
• Special Relativity An introduction to special relativity at the undergraduate level.
• Relativity: the Special and General Theory at Project Gutenberg, by Albert Einstein
• Special Relativity Lecture Notes is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University.
• Understanding Special Relativity The theory of special relativity in an easily understandable way.
• An Introduction to the Special Theory of Relativity (1964) by Robert Katz, "an introduction ... that is accessible to any student who has had an introduction to general physics and some slight acquaintance with the calculus" (130 pp; pdf format).
• Lecture Notes on Special Relativity by J D Cresser Department of Physics Macquarie University.
• SpecialRelativity.net - An overview with visualizations and minimal mathematics.

Visualization

• Raytracing Special Relativity Software visualizing several scenarios under the influence of special relativity.
• Real Time Relativity The Australian National University. Relativistic visual effects experienced through an interactive program.
• Spacetime travel A variety of visualizations of relativistic effects, from relativistic motion to black holes.
• Through Einstein's Eyes The Australian National University. Relativistic visual effects explained with movies and images.
• Warp Special Relativity Simulator A computer program to show the effects of traveling close to the speed of light.
• Animation clip on YouTube visualizing the Lorentz transformation.
• Original interactive FLASH Animations from John de Pillis illustrating Lorentz and Galilean frames, Train and Tunnel Paradox, the Twin Paradox, Wave Propagation, Clock Synchronization, etc.
• lightspeed An OpenGL-based program developed to illustrate the effects of special relativity on the appearance of moving objects.
• Animation showing the stars near Earth, as seen from a spacecraft accelerating rapidly to light speed.

v t e Branches of physics

v t e Albert Einstein

v t e Relativity

v t e Tests of special relativity

v t e Tensors

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The number of works is vast, see as example:
Sidney Coleman; Sheldon L. Glashow (1997). "Cosmic Ray and Neutrino Tests of Special Relativity". Physics Letters B. 405 (3–4): 249–252. arXiv:hep-ph/9703240. Bibcode:1997PhLB..405..249C. doi:10.1016/S0370-2693(97)00638-2.
An overview can be found on this page

John D. Norton, John D. (2004). "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905". Archive for History of Exact Sciences. 59 (1): 45–105. Bibcode:2004AHES...59...45N. doi:10.1007/s00407-004-0085-6.

J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 58. ISBN 978-0-7167-0344-0.

J.R. Forshaw; A.G. Smith (2009). Dynamics and Relativity. Wiley. p. 247. ISBN 978-0-470-01460-8.

R. Penrose (2007). The Road to Reality. Vintage books. ISBN 978-0-679-77631-4.

Jean-Bernard Zuber & Claude Itzykson, Quantum Field Theory, pg 5, ISBN 0-07-032071-3

Charles W. Misner, Kip S. Thorne & John A. Wheeler, Gravitation, pg 51, ISBN 0-7167-0344-0

George Sterman, An Introduction to Quantum Field Theory, pg 4 , ISBN 0-521-31132-2

Einstein himself, in The Foundations of the General Theory of Relativity, Ann. Phys. 49 (1916), writes "The word "special" is meant to intimate that the principle is restricted to the case ...". See p. 111 of The Principle of Relativity, A. Einstein, H. A. Lorentz, H. Weyl, H. Minkowski, Dover reprint of 1923 translation by Methuen and Company.]

Wald, General Relativity, p. 60: "... the special theory of relativity asserts that spacetime is the manifold ℝ⁴ with a flat metric of Lorentz signature defined on it. Conversely, the entire content of special relativity ... is contained in this statement ..."

In a spacetime setting, the length of a rigid object is the spatial distance between the ends of the object measured at the same time.

The results of the Michelson–Morley experiment led George Francis FitzGerald and Hendrik Lorentz independently to propose the phenomenon of length contraction. Lorentz believed that length contraction represented a physical contraction of the atoms making up an object. He envisioned no fundamental change in the nature of space and time.^[21]:62–68
     Lorentz expected that length contraction would result in compressive strains in an object that should result in measurable effects. Such effects would include optical effects in transparent media, such as optical rotation^{[p 11]} and induction of double refraction,^{[p 12]} and the induction of torques on charged condensers moving at an angle with respect to the aether.^{[p 12]} Lorentz was perplexed by experiments such as the Trouton–Noble experiment and the experiments of Rayleigh and Brace which failed to validate his theoretical expectations.^[21]

For mathematical consistency, Lorentz proposed a new time variable, the "local time", called that because it depended on the position of a moving body, following the relation .^{[p 13]} Lorentz considered local time not to be "real"; rather, it represented an ad hoc change of variable.^[22]:51,80
     Impressed by Lorentz's "most ingenious idea", Poincaré saw more in local time than a mere mathematical trick. It represented the actual time that would be shown on a moving observer's clocks. On the other hand, Poincaré did not consider this measured time to be the "true time" that would be exhibited by clocks at rest in the aether. Poincaré made no attempt to redefine the concepts of space and time. To Poincaré, Lorentz transformation described the apparent states of the field for a moving observer. True states remained those defined with respect to the ether.^[23]

This concept is counterintuitive at least for the fact that, in contrast to usual concepts of distance, it may assume negative values (is not positive definite for non-coinciding events), and that the square-denotation is misleading. This negative square lead to, now not broadly used, concepts of imaginary time. It is immediate that the negative of is also an invariant, generated by a variant of the metric signature of spacetime.

The invariance of Δs² under standard Lorentz transformation in analogous to the invariance of squared distances Δr² under rotations in Euclidean space. Although space and time have an equal footing in relativity, the minus sign in front of the spatial terms marks space and time as being of essentially different character. They are not the same. Because it treats time differently than it treats the 3 spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.

The refractive index dependence of the presumed partial aether-drag was eventually confirmed by Pieter Zeeman in 1914–1915, long after special relativity had been accepted by the mainstream. Using a scaled-up version of Michelson's apparatus connected directly to Amsterdam's main water conduit, Zeeman was able to perform extended measurements using monochromatic light ranging from violet (4358 Å) through red (6870 Å).^{[p 17][p 18]}

Even though it has been many decades since Terrell and Penrose published their observations, popular writings continue to conflate measurement versus appearance. For example, Michio Kaku wrote in Einstein's Cosmos (W. W. Norton & Company, 2004. p. 65): "... imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length."

 

 

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