Acute radiation syndrome

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https://en.wikipedia.org/wiki/Acute_radiation_syndrome

 

Acute radiation syndrome
Other names	Radiation poisoning, radiation sickness, radiation toxicity
Radiation causes cellular degradation by autophagy.
Specialty	Critical care medicine
Symptoms	Early: Nausea, vomiting, skin burns, loss of appetite Later: Infections, bleeding, dehydration, confusion
Complications	Cancer
Usual onset	Within days
Types	Bone marrow syndrome, gastrointestinal syndrome, neurovascular syndrome
Causes	Large amounts of ionizing radiation over a short period of time
Diagnostic method	Based on history of exposure and symptoms
Treatment	Supportive care (blood transfusions, antibiotics, colony stimulating factors, stem cell transplant)
Prognosis	Depends on the exposure dose
Frequency	Rare

Acute radiation syndrome (ARS), also known as radiation sickness or radiation poisoning, is a collection of health effects that are caused by being exposed to high amounts of ionizing radiation in a short period of time. Symptoms can start within an hour of exposure, and can last for several months. Early symptoms are usually nausea, vomiting and loss of appetite. In the following hours or weeks, initial symptoms may appear to improve, before the development of additional symptoms, after which either recovery or death follow.

ARS involves a total dose of greater than 0.7 Gy (70 rad), that generally occurs from a source outside the body, delivered within a few minutes. Sources of such radiation can occur accidentally or intentionally. They may involve nuclear reactors, cyclotrons, certain devices used in cancer therapy, nuclear weapons, or radiological weapons. It is generally divided into three types: bone marrow, gastrointestinal, and neurovascular syndrome, with bone marrow syndrome occurring at 0.7 to 10 Gy, and neurovascular syndrome occurring at doses that exceed 50 Gy. The cells that are most affected are generally those that are rapidly dividing. At high doses, this causes DNA damage that may be irreparable. Diagnosis is based on a history of exposure and symptoms. Repeated complete blood counts (CBCs) can indicate the severity of exposure.

Treatment of ARS is generally supportive care. This may include blood transfusions, antibiotics, colony-stimulating factors, or stem cell transplant. Radioactive material remaining on the skin or in the stomach should be removed. If radioiodine was inhaled or ingested, potassium iodide is recommended. Complications such as leukemia and other cancers among those who survive are managed as usual. Short-term outcomes depend on the dose exposure.

ARS is generally rare. A single event can affect a large number of people, as happened in the atomic bombings of Hiroshima and Nagasaki and the Chernobyl nuclear power plant disaster. ARS differs from chronic radiation syndrome, which occurs following prolonged exposures to relatively low doses of radiation.

Signs and symptoms

See also: Effects of nuclear explosions on human health

Radiation sickness

Classically, ARS is divided into three main presentations: hematopoietic, gastrointestinal, and neurovascular. These syndromes may be preceded by a prodrome. The speed of symptom onset is related to radiation exposure, with greater doses resulting in a shorter delay in symptom onset. These presentations presume whole-body exposure, and many of them are markers that are invalid if the entire body has not been exposed. Each syndrome requires that the tissue showing the syndrome itself be exposed (e.g., gastrointestinal syndrome is not seen if the stomach and intestines are not exposed to radiation). Some areas affected are:

1. Hematopoietic. This syndrome is marked by a drop in the number of blood cells, called aplastic anemia. This may result in infections, due to a low number of white blood cells, bleeding, due to a lack of platelets, and anemia, due to too few red blood cells in circulation. These changes can be detected by blood tests after receiving a whole-body acute dose as low as 0.25 grays (25 rad), though they might never be felt by the patient if the dose is below 1 gray (100 rad). Conventional trauma and burns resulting from a bomb blast are complicated by the poor wound healing caused by hematopoietic syndrome, increasing mortality.
2. Gastrointestinal. This syndrome often follows absorbed doses of 6–30 grays (600–3,000 rad).^[3] The signs and symptoms of this form of radiation injury include nausea, vomiting, loss of appetite, and abdominal pain. Vomiting in this time-frame is a marker for whole body exposures that are in the fatal range above 4 grays (400 rad). Without exotic treatment such as bone marrow transplant, death with this dose is common, due generally more to infection than gastrointestinal dysfunction.
3. Neurovascular. This syndrome typically occurs at absorbed doses greater than 30 grays (3,000 rad), though it may occur at doses as low as 10 grays (1,000 rad). It presents with neurological symptoms such as dizziness, headache, or decreased level of consciousness, occurring within minutes to a few hours, with an absence of vomiting, and is almost always fatal, even with aggressive intensive care.

Early symptoms of ARS typically include nausea, vomiting, headaches, fatigue, fever, and a short period of skin reddening. These symptoms may occur at radiation doses as low as 0.35 grays (35 rad). These symptoms are common to many illnesses, and may not, by themselves, indicate acute radiation sickness.

Dose effects

Phase	Symptom	Whole-body absorbed dose (Gy)
		1–2 Gy	2–6 Gy	6–8 Gy	8–30 Gy	> 30 Gy
Immediate	Nausea and vomiting	5–50%	50–100%	75–100%	90–100%	100%
	Time of onset	2–6 h	1–2 h	10–60 min	< 10 min	Minutes
	Duration	< 24 h	24–48 h	< 48 h	< 48 h	— (patients die in < 48 h)
	Diarrhea	None	None to mild (< 10%)	Heavy (> 10%)	Heavy (> 95%)	Heavy (100%)
	Time of onset	—	3–8 h	1–3 h	< 1 h	< 1 h
	Headache	Slight	Mild to moderate (50%)	Moderate (80%)	Severe (80–90%)	Severe (100%)
	Time of onset	—	4–24 h	3–4 h	1–2 h	< 1 h
	Fever	None	Moderate increase (10–100%)	Moderate to severe (100%)	Severe (100%)	Severe (100%)
	Time of onset	—	1–3 h	< 1 h	< 1 h	< 1 h
	CNS function	No impairment	Cognitive impairment 6–20 h	Cognitive impairment > 24 h	Rapid incapacitation	Seizures, tremor, ataxia, lethargy
Latent period		28–31 days	7–28 days	< 7 days	None	None
Illness		Mild to moderate Leukopenia Fatigue Weakness	Moderate to severe Leukopenia Purpura Hemorrhage Infections Alopecia after 3 Gy	Severe leukopenia High fever Diarrhea Vomiting Dizziness and disorientation Hypotension Electrolyte disturbance	Nausea Vomiting Severe diarrhea High fever Electrolyte disturbance Shock	— (patients die in < 48h)
Mortality	Without care	0–5%	5–95%	95–100%	100%	100%
	With care	0–5%	5–50%	50–100%	99–100%	100%
	Death	6–8 weeks	4–6 weeks	2–4 weeks	2 days – 2 weeks	1–2 days

A similar table and description of symptoms (given in rems, where 100 rem = 1 Sv), derived from data from the effects on humans subjected to the atomic bombings of Hiroshima and Nagasaki, the indigenous peoples of the Marshall Islands subjected to the Castle Bravo thermonuclear bomb, animal studies and lab experiment accidents, have been compiled by the U.S. Department of Defense.

A person who was less than 1 mile (1.6 km) from the atomic bomb Little Boy's hypocenter at Hiroshima, Japan, was found to have absorbed about 9.46 grays (Gy) of ionizing radiation. The doses at the hypocenters of the Hiroshima and Nagasaki atomic bombings were 240 and 290 Gy, respectively.

Skin changes

Main article: Radiation burn

Harry K. Daghlian's hand 9 days after he had manually stopped a prompt critical fission reaction during an accident with what later obtained the nickname the demon core. He received a dose of 5.1 Sv, or 3.1 Gy. He died 16 days after this photo was taken.

Cutaneous radiation syndrome (CRS) refers to the skin symptoms of radiation exposure. Within a few hours after irradiation, a transient and inconsistent redness (associated with itching) can occur. Then, a latent phase may occur and last from a few days up to several weeks, when intense reddening, blistering, and ulceration of the irradiated site is visible. In most cases, healing occurs by regenerative means; however, very large skin doses can cause permanent hair loss, damaged sebaceous and sweat glands, atrophy, fibrosis (mostly keloids), decreased or increased skin pigmentation, and ulceration or necrosis of the exposed tissue.

As seen at Chernobyl, when skin is irradiated with high energy beta particles, moist desquamation (peeling of skin) and similar early effects can heal, only to be followed by the collapse of the dermal vascular system after two months, resulting in the loss of the full thickness of the exposed skin. Another example of skin loss caused by high-level exposure of radiation is during the 1999 Tokaimura nuclear accident, where technician Hisashi Ouchi had lost a majority of his skin due to the high amounts of radiation he absorbed during the irradiation. This effect had been demonstrated previously with pig skin using high energy beta sources at the Churchill Hospital Research Institute, in Oxford.

Cause

Both dose and dose rate contribute to the severity of acute radiation syndrome. The effects of dose fractionation or rest periods before repeated exposure also shift the LD50 dose upwards.

Comparison of Radiation Doses – includes the amount detected on the trip from Earth to Mars by the RAD on the MSL (2011–2013).

ARS is caused by exposure to a large dose of ionizing radiation (> ~0.1 Gy) over a short period of time (> ~0.1 Gy/h). Alpha and beta radiation have low penetrating power and are unlikely to affect vital internal organs from outside the body. Any type of ionizing radiation can cause burns, but alpha and beta radiation can only do so if radioactive contamination or nuclear fallout is deposited on the individual's skin or clothing.

Gamma and neutron radiation can travel much greater distances and penetrate the body easily, so whole-body irradiation generally causes ARS before skin effects are evident. Local gamma irradiation can cause skin effects without any sickness. In the early twentieth century, radiographers would commonly calibrate their machines by irradiating their own hands and measuring the time to onset of erythema.

Accidental

Main article: Nuclear and radiation accidents and incidents

Accidental exposure may be the result of a criticality or radiotherapy accident. There have been numerous criticality accidents dating back to atomic testing during World War II, while computer-controlled radiation therapy machines such as Therac-25 played a major part in radiotherapy accidents. The latter of the two is caused by the failure of equipment software used to monitor the radiational dose given. Human error has played a large part in accidental exposure incidents, including some of the criticality accidents, and larger scale events such as the Chernobyl disaster. Other events have to do with orphan sources, in which radioactive material is unknowingly kept, sold, or stolen. The Goiânia accident is an example, where a forgotten radioactive source was taken from a hospital, resulting in the deaths of 4 people from ARS. Theft and attempted theft of radioactive material by clueless thieves has also led to lethal exposure in at least one incident.

Exposure may also come from routine spaceflight and solar flares that result in radiation effects on earth in the form of solar storms. During spaceflight, astronauts are exposed to both galactic cosmic radiation (GCR) and solar particle event (SPE) radiation. The exposure particularly occurs during flights beyond low Earth orbit (LEO). Evidence indicates past SPE radiation levels that would have been lethal for unprotected astronauts. GCR levels that might lead to acute radiation poisoning are less well understood. The latter cause is rarer, with an event possibly occurring during the solar storm of 1859.

Intentional

Main articles: Atomic bombings of Hiroshima and Nagasaki and Crimes involving radioactive substances

Intentional exposure is controversial as it involves the use of nuclear weapons, human experiments, or is given to a victim in an act of murder. The intentional atomic bombings of Hiroshima and Nagasaki resulted in tens of thousands of casualties; the survivors of these bombings are known today as Hibakusha. Nuclear weapons emit large amounts of thermal radiation as visible, infrared, and ultraviolet light, to which the atmosphere is largely transparent. This event is also known as "Flash", where radiant heat and light are bombarded into any given victim's exposed skin, causing radiation burns. Death is highly likely, and radiation poisoning is almost certain if one is caught in the open with no terrain or building masking-effects within a radius of 0–3 km from a 1 megaton airburst. The 50% chance of death from the blast extends out to ~8 km from a 1 megaton atmospheric explosion.

Scientific testing on humans within the United States occurred extensively throughout the atomic age. Experiments took place on a range of subjects including, but not limited to; the disabled, children, soldiers, and incarcerated persons, with the level of understanding and consent given by subjects varying from complete to none. Since 1997 there have been requirements for patients to give informed consent, and to be notified if experiments were classified. Across the world, the Soviet nuclear program involved human experiments on a large scale, which is still kept secret by the Russian government and the Rosatom agency. The human experiments that fall under intentional ARS exclude those that involved long term exposure. Criminal activity has involved murder and attempted murder carried out through abrupt victim contact with a radioactive substance such as polonium or plutonium.

Pathophysiology

The most commonly used predictor of ARS is the whole-body absorbed dose. Several related quantities, such as the equivalent dose, effective dose, and committed dose, are used to gauge long-term stochastic biological effects such as cancer incidence, but they are not designed to evaluate ARS. To help avoid confusion between these quantities, absorbed dose is measured in units of grays (in SI, unit symbol Gy) or rad (in CGS), while the others are measured in sieverts (in SI, unit symbol Sv) or rem (in CGS). 1 rad = 0.01 Gy and 1 rem = 0.01 Sv.

In most of the acute exposure scenarios that lead to radiation sickness, the bulk of the radiation is external whole-body gamma, in which case the absorbed, equivalent, and effective doses are all equal. There are exceptions, such as the Therac-25 accidents and the 1958 Cecil Kelley criticality accident, where the absorbed doses in Gy or rad are the only useful quantities, because of the targeted nature of the exposure to the body.

Radiotherapy treatments are typically prescribed in terms of the local absorbed dose, which might be 60 Gy or higher. The dose is fractionated to about 2 Gy per day for curative treatment, which allows normal tissues to undergo repair, allowing them to tolerate a higher dose than would otherwise be expected. The dose to the targeted tissue mass must be averaged over the entire body mass, most of which receives negligible radiation, to arrive at a whole-body absorbed dose that can be compared to the table above.

DNA damage

Exposure to high doses of radiation causes DNA damage, later creating serious and even lethal chromosomal aberrations if left unrepaired. Ionizing radiation can produce reactive oxygen species, and does directly damage cells by causing localized ionization events. The former is very damaging to DNA, while the latter events create clusters of DNA damage. This damage includes loss of nucleobases and breakage of the sugar-phosphate backbone that binds to the nucleobases. The DNA organization at the level of histones, nucleosomes, and chromatin also affects its susceptibility to radiation damage. Clustered damage, defined as at least two lesions within a helical turn, is especially harmful. While DNA damage happens frequently and naturally in the cell from endogenous sources, clustered damage is a unique effect of radiation exposure. Clustered damage takes longer to repair than isolated breakages, and is less likely to be repaired at all. Larger radiation doses are more prone to cause tighter clustering of damage, and closely localized damage is increasingly less likely to be repaired.

Somatic mutations cannot be passed down from parent to offspring, but these mutations can propagate in cell lines within an organism. Radiation damage can also cause chromosome and chromatid aberrations, and their effects depend on in which stage of the mitotic cycle the cell is when the irradiation occurs. If the cell is in interphase, while it is still a single strand of chromatin, the damage will be replicated during the S1 phase of the cell cycle, and there will be a break on both chromosome arms; the damage then will be apparent in both daughter cells. If the irradiation occurs after replication, only one arm will bear the damage; this damage will be apparent in only one daughter cell. A damaged chromosome may cyclize, binding to another chromosome, or to itself.

Diagnosis

Diagnosis is typically made based on a history of significant radiation exposure and suitable clinical findings. An absolute lymphocyte count can give a rough estimate of radiation exposure. Time from exposure to vomiting can also give estimates of exposure levels if they are less than 10 Gy (1000 rad).

Prevention

A guiding principle of radiation safety is as low as reasonably achievable (ALARA). This means try to avoid exposure as much as possible and includes the three components of time, distance, and shielding.

Time

The longer that humans are subjected to radiation the larger the dose will be. The advice in the nuclear war manual entitled Nuclear War Survival Skills published by Cresson Kearny in the U.S. was that if one needed to leave the shelter then this should be done as rapidly as possible to minimize exposure.

In chapter 12, he states that "[q]uickly putting or dumping wastes outside is not hazardous once fallout is no longer being deposited. For example, assume the shelter is in an area of heavy fallout and the dose rate outside is 400 roentgen (R) per hour, enough to give a potentially fatal dose in about an hour to a person exposed in the open. If a person needs to be exposed for only 10 seconds to dump a bucket, in this 1/360 of an hour he will receive a dose of only about 1 R. Under war conditions, an additional 1-R dose is of little concern." In peacetime, radiation workers are taught to work as quickly as possible when performing a task that exposes them to radiation. For instance, the recovery of a radioactive source should be done as quickly as possible.

Shielding

See also: Radiation protection

Matter attenuates radiation in most cases, so placing any mass (e.g., lead, dirt, sandbags, vehicles, water, even air) between humans and the source will reduce the radiation dose. This is not always the case, however; care should be taken when constructing shielding for a specific purpose. For example, although high atomic number materials are very effective in shielding photons, using them to shield beta particles may cause higher radiation exposure due to the production of bremsstrahlung x-rays, and hence low atomic number materials are recommended. Also, using material with a high neutron activation cross section to shield neutrons will result in the shielding material itself becoming radioactive and hence more dangerous than if it were not present.

There are many types of shielding strategies that can be used to reduce the effects of radiation exposure. Internal contamination protective equipment such as respirators are used to prevent internal deposition as a result of inhalation and ingestion of radioactive material. Dermal protective equipment, which protects against external contamination, provides shielding to prevent radioactive material from being deposited on external structures. While these protective measures do provide a barrier from radioactive material deposition, they do not shield from externally penetrating gamma radiation. This leaves anyone exposed to penetrating gamma rays at high risk of ARS.

Naturally, shielding the entire body from high energy gamma radiation is optimal, but the required mass to provide adequate attenuation makes functional movement nearly impossible. In the event of a radiation catastrophe, medical and security personnel need mobile protection equipment in order to safely assist in containment, evacuation, and many other necessary public safety objectives.

Research has been done exploring the feasibility of partial body shielding, a radiation protection strategy that provides adequate attenuation to only the most radio-sensitive organs and tissues inside the body. Irreversible stem cell damage in the bone marrow is the first life-threatening effect of intense radiation exposure and therefore one of the most important bodily elements to protect. Due to the regenerative property of hematopoietic stem cells, it is only necessary to protect enough bone marrow to repopulate the exposed areas of the body with the shielded supply. This concept allows for the development of lightweight mobile radiation protection equipment, which provides adequate protection, deferring the onset of ARS to much higher exposure doses. One example of such equipment is the 360 gamma, a radiation protection belt that applies selective shielding to protect the bone marrow stored in the pelvic area as well as other radio sensitive organs in the abdominal region without hindering functional mobility.

Reduction of incorporation

Where radioactive contamination is present, an elastomeric respirator, dust mask, or good hygiene practices may offer protection, depending on the nature of the contaminant. Potassium iodide (KI) tablets can reduce the risk of cancer in some situations due to slower uptake of ambient radioiodine. Although this does not protect any organ other than the thyroid gland, their effectiveness is still highly dependent on the time of ingestion, which would protect the gland for the duration of a twenty-four-hour period. They do not prevent ARS as they provide no shielding from other environmental radionuclides.

Fractionation of dose

Main article: Dose fractionation

If an intentional dose is broken up into a number of smaller doses, with time allowed for recovery between irradiations, the same total dose causes less cell death. Even without interruptions, a reduction in dose rate below 0.1 Gy/h also tends to reduce cell death. This technique is routinely used in radiotherapy.

The human body contains many types of cells and a human can be killed by the loss of a single type of cells in a vital organ. For many short term radiation deaths (3–30 days), the loss of two important types of cells that are constantly being regenerated causes death. The loss of cells forming blood cells (bone marrow) and the cells in the digestive system (microvilli, which form part of the wall of the intestines) is fatal.

Management

Effect of medical care on acute radiation syndrome

Treatment usually involves supportive care with possible symptomatic measures employed. The former involves the possible use of antibiotics, blood products, colony stimulating factors, and stem cell transplant.

Antimicrobials

Main article: Treatment of infections after exposure to ionizing radiation

There is a direct relationship between the degree of the neutropenia that emerges after exposure to radiation and the increased risk of developing infection. Since there are no controlled studies of therapeutic intervention in humans, most of the current recommendations are based on animal research.

The treatment of established or suspected infection following exposure to radiation (characterized by neutropenia and fever) is similar to the one used for other febrile neutropenic patients. However, important differences between the two conditions exist. Individuals that develop neutropenia after exposure to radiation are also susceptible to irradiation damage in other tissues, such as the gastrointestinal tract, lungs and central nervous system. These patients may require therapeutic interventions not needed in other types of neutropenic patients. The response of irradiated animals to antimicrobial therapy can be unpredictable, as was evident in experimental studies where metronidazole and pefloxacin therapies were detrimental.

Antimicrobials that reduce the number of the strict anaerobic component of the gut flora (i.e., metronidazole) generally should not be given because they may enhance systemic infection by aerobic or facultative bacteria, thus facilitating mortality after irradiation.

An empirical regimen of antimicrobials should be chosen based on the pattern of bacterial susceptibility and nosocomial infections in the affected area and medical center and the degree of neutropenia. Broad-spectrum empirical therapy (see below for choices) with high doses of one or more antibiotics should be initiated at the onset of fever. These antimicrobials should be directed at the eradication of Gram-negative aerobic bacilli (i.e., Enterobacteriaceae, Pseudomonas) that account for more than three quarters of the isolates causing sepsis. Because aerobic and facultative Gram-positive bacteria (mostly alpha-hemolytic streptococci) cause sepsis in about a quarter of the victims, coverage for these organisms may also be needed.

A standardized management plan for people with neutropenia and fever should be devised. Empirical regimens contain antibiotics broadly active against Gram-negative aerobic bacteria (quinolones: i.e., ciprofloxacin, levofloxacin, a third- or fourth-generation cephalosporin with pseudomonal coverage: e.g., cefepime, ceftazidime, or an aminoglycoside: i.e. gentamicin, amikacin).

Prognosis

See also: Radiation-induced cancer

The prognosis for ARS is dependent on the exposure dose, with anything above 8 Gy being almost always lethal, even with medical care. Radiation burns from lower-level exposures usually manifest after 2 months, while reactions from the burns occur months to years after radiation treatment. Complications from ARS include an increased risk of developing radiation-induced cancer later in life. According to the controversial but commonly applied linear no-threshold model, any exposure to ionizing radiation, even at doses too low to produce any symptoms of radiation sickness, can induce cancer due to cellular and genetic damage. The probability of developing cancer is a linear function with respect to the effective radiation dose. Radiation cancer may occur after ionizing radiation exposure following a latent period averaging 20 to 40 years.

History

Acute effects of ionizing radiation were first observed when Wilhelm Röntgen intentionally subjected his fingers to X-rays in 1895. He published his observations concerning the burns that developed that eventually healed, and misattributed them to ozone. Röntgen believed the free radical produced in air by X-rays from the ozone was the cause, but other free radicals produced within the body are now understood to be more important. David Walsh first established the symptoms of radiation sickness in 1897.

Ingestion of radioactive materials caused many radiation-induced cancers in the 1930s, but no one was exposed to high enough doses at high enough rates to bring on ARS.

The atomic bombings of Hiroshima and Nagasaki resulted in high acute doses of radiation to a large number of Japanese people, allowing for greater insight into its symptoms and dangers. Red Cross Hospital Surgeon Terufumi Sasaki led intensive research into the syndrome in the weeks and months following the Hiroshima and Nagasaki bombings. Sasaki and his team were able to monitor the effects of radiation in patients of varying proximities to the blast itself, leading to the establishment of three recorded stages of the syndrome. Within 25–30 days of the explosion, Sasaki noticed a sharp drop in white blood cell count and established this drop, along with symptoms of fever, as prognostic standards for ARS. Actress Midori Naka, who was present during the atomic bombing of Hiroshima, was the first incident of radiation poisoning to be extensively studied. Her death on 24 August 1945 was the first death ever to be officially certified as a result of ARS (or "Atomic bomb disease").

There are two major databases that track radiation accidents: The American ORISE REAC/TS and the European IRSN ACCIRAD. REAC/TS shows 417 accidents occurring between 1944 and 2000, causing about 3000 cases of ARS, of which 127 were fatal. ACCIRAD lists 580 accidents with 180 ARS fatalities for an almost identical period. The two deliberate bombings are not included in either database, nor are any possible radiation-induced cancers from low doses. The detailed accounting is difficult because of confounding factors. ARS may be accompanied by conventional injuries such as steam burns, or may occur in someone with a pre-existing condition undergoing radiotherapy. There may be multiple causes for death, and the contribution from radiation may be unclear. Some documents may incorrectly refer to radiation-induced cancers as radiation poisoning, or may count all overexposed individuals as survivors without mentioning if they had any symptoms of ARS.

Notable cases

The following table includes only those known for their attempted survival with ARS. These cases exclude chronic radiation syndrome such as Albert Stevens, in which radiation is exposed to a given subject over a long duration. The table also necessarily excludes cases where the individual was exposed to so much radiation that death occurred before medical assistance or dose estimations could be made, such as an attempted cobalt-60 thief who reportedly died 30 minutes after exposure. The result column represents the time of exposure to the time of death attributed to the short and long term effects attributed to initial exposure. As ARS is measured by a whole-body absorbed dose, the exposure column only includes units of gray (Gy).

Date	Name	Exposure (Gy or Sv)	Incident/accident	Result
August 21, 1945	Harry Daghlian	3.1 Gy	Harry Daghlian criticality accident	Death in 25 days
May 21, 1946	Louis Slotin	11 Gy	Slotin criticality accident	Death in 9 days
	Alvin C. Graves	1.9 Gy		Death in 19 years
December 30, 1958	Cecil Kelley	36 Gy	Cecil Kelley criticality accident	Death in 38 hours
July 24, 1964	Robert Peabody	~100 Gy	Robert Peabody criticality accident	Death in 49 hours
April 26, 1986	Aleksandr Akimov	15 Gy	Chernobyl disaster	Death in 14 days
September 30, 1999	Hisashi Ouchi	17 Sv	Tokaimura nuclear accident	Death in 83 days
December 2, 2001	Patient "1-DN"	3.6 Gy	Lia radiological accident	Death in 893 days

Other animals

Thousands of scientific experiments have been performed to study ARS in animals. There is a simple guide for predicting survival and death in mammals, including humans, following the acute effects of inhaling radioactive particles.

Elliptical galaxy

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Elliptical_galaxy

The giant elliptical galaxy ESO 325-G004

An elliptical galaxy is a type of galaxy with an approximately ellipsoidal shape and a smooth, nearly featureless image. They are one of the four main classes of galaxy described by Edwin Hubble in his Hubble sequence and 1936 work The Realm of the Nebulae, along with spiral and lenticular galaxies. Elliptical (E) galaxies are, together with lenticular galaxies (S0) with their large-scale disks, and ES galaxies with their intermediate scale disks, a subset of the "early-type" galaxy population.

Most elliptical galaxies are composed of older, low-mass stars, with a sparse interstellar medium, and they tend to be surrounded by large numbers of globular clusters. Star formation activity in elliptical galaxies is typically minimal; they may, however, undergo brief periods of star formation when merging with other galaxies. Elliptical galaxies are believed to make up approximately 10–15% of galaxies in the Virgo Supercluster, and they are not the dominant type of galaxy in the universe overall. They are preferentially found close to the centers of galaxy clusters.

Elliptical galaxies range in size from dwarf ellipticals with tens of millions of stars, to supergiants of over one hundred trillion stars that dominate their galaxy clusters. Originally, Edwin Hubble hypothesized that elliptical galaxies evolved into spiral galaxies, which was later discovered to be false, although the accretion of gas and smaller galaxies may build a disk around a pre-existing ellipsoidal structure. Stars found inside of elliptical galaxies are on average much older than stars found in spiral galaxies.

Examples

• 3C 244.1
• M49 (NGC 4472)
• M59 (NGC 4621)
• M60 (NGC 4649)
• M87 (NGC 4486), whose supermassive black hole was the first black hole to be imaged by the Event Horizon Telescope.
• M89 (NGC 4552)
• M105 (NGC 3379)
• NGC 4697, part of the NGC 4697 Group
• ESO 383-76, one of the largest galaxies known.
• IC 1101, the central galaxy of Abell 2029
• Hercules A, supergiant elliptical galaxy
• Maffei 1, the closest giant elliptical galaxy
• CGCG 049-033, known for having the longest galactic jet discovered
• Centaurus A (NGC 5128), an elliptical/lenticular radio galaxy with peculiar morphology and unusual dust lanes
• NeVe 1, the source of the Ophiuchus Supercluster eruption, the most powerful astronomical event known
• M86 (4406) an elliptical or lenticular galaxy in the constellation Virgo

General characteristics

Elliptical galaxy IC 2006

Elliptical galaxies are characterized by several properties that make them distinct from other classes of galaxy. They are spherical or ovoid masses of stars, starved of star-making gases. Furthermore, there is very little interstellar matter (neither gas nor dust), which results in low rates of star formation, few open star clusters, and few young stars; rather elliptical galaxies are dominated by old stellar populations, giving them red colors. Large elliptical galaxies typically have an extensive system of globular clusters. They generally have two distinct populations of globular clusters: one that is redder and metal-rich, and another that is bluer and metal-poor.

The dynamical properties of elliptical galaxies and the bulges of disk galaxies are similar, suggesting that they may be formed by the same physical processes, although this remains controversial. The luminosity profiles of both elliptical galaxies and bulges are well fit by Sersic's law, and a range of scaling relations between the elliptical galaxies' structural parameters unify the population.

Every massive elliptical galaxy contains a supermassive black hole at its center. Observations of 46 elliptical galaxies, 20 classical bulges, and 22 pseudobulges show that each contain a black hole at the center. The mass of the black hole is tightly correlated with the mass of the galaxy, evidenced through correlations such as the M–sigma relation which relates the velocity dispersion of the surrounding stars to the mass of the black hole at the center.

Elliptical galaxies are preferentially found in galaxy clusters and in compact groups of galaxies.

Unlike flat spiral galaxies with organization and structure, elliptical galaxies are more three-dimensional, without much structure, and their stars are in somewhat random orbits around the center.

Sizes and shapes

Hercules A, a supergiant elliptical galaxy and also a radio galaxy. The radio lobes shown here in pink are over a million light-years across.

The largest galaxies are supergiant ellipticals, or type-cD galaxies. Elliptical galaxies vary greatly in both size and mass with diameters ranging from 3,000 light years to more than 700,000 light years, and masses from 10⁵ to nearly 10¹³ solar masses. This range is much broader for this galaxy type than for any other. The smallest, the dwarf elliptical galaxies, may be no larger than a typical globular cluster, but contain a considerable amount of dark matter not present in clusters. Most of these small galaxies may not be related to other ellipticals.

The brilliant central object is the supergiant elliptical galaxy SDSS J142347.87+240442.4, the dominant member of the galaxy cluster MACS J1423.8+2404. It has a diameter of 380,000 light-years. Note the gravitational lensing.

The Hubble classification of elliptical galaxies contains an integer that describes how elongated the galaxy image is. The classification is determined by the ratio of the major (a) to the minor (b) axes of the galaxy's isophotes:

${\displaystyle 10\times \left(1-\frac{b}{a}\right)}$

Thus for a spherical galaxy with a equal to b, the number is 0, and the Hubble type is E0. While the limit in the literature is about E7, it has been known since 1966 that the E4 to E7 galaxies are misclassified lenticular galaxies with disks inclined at different angles to our line of sight. This has been confirmed through spectral observations revealing the rotation of their stellar disks. Hubble recognized that his shape classification depends both on the intrinsic shape of the galaxy, as well as the angle with which the galaxy is observed. Hence, some galaxies with Hubble type E0 are actually elongated.

It is sometimes said that there are two physical types of ellipticals: the giant ellipticals with slightly "boxy"-shaped isophotes, whose shapes result from random motion which is greater in some directions than in others (anisotropic random motion); and the "disky" normal and dwarf ellipticals, which contain disks. This is, however, an abuse of the nomenclature, as there are two types of early-type galaxy, those with disks and those without. Given the existence of ES galaxies with intermediate-scale disks, it is reasonable to expect that there is a continuity from E to ES, and onto the S0 galaxies with their large-scale stellar disks that dominate the light at large radii.

Dwarf spheroidal galaxies appear to be a distinct class: their properties are more similar to those of irregulars and late spiral-type galaxies.

At the large end of the elliptical spectrum, there is further division, beyond Hubble's classification. Beyond gE giant ellipticals, lies D-galaxies and cD-galaxies. These are similar to their smaller brethren, but more diffuse, with large haloes that may as much belong to the galaxy cluster within which they reside than the centrally-located giant galaxy.

NGC 3597 is the product of a collision between two galaxies. It is evolving into a giant elliptical galaxy.

Star formation

In recent years, evidence has shown that a reasonable proportion (~25%) of early-type (E, ES and S0) galaxies have residual gas reservoirs and low level star-formation.

Herschel Space Observatory researchers have speculated that the central black holes in elliptical galaxies keep the gas from cooling enough for star formation.

Mathematics of general relativity

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Mathematics_of_general_relativity

 

General relativity
${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }}$

When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity.

Note: General relativity articles using tensors will use the abstract index notation.

Tensors

The principle of general covariance was one of the central principles in the development of general relativity. It states that the laws of physics should take the same mathematical form in all reference frames. The term 'general covariance' was used in the early formulation of general relativity, but the principle is now often referred to as 'diffeomorphism covariance'.

Diffeomorphism covariance is not the defining feature of general relativity, and controversies remain regarding its present status in general relativity. However, the invariance property of physical laws implied in the principle, coupled with the fact that the theory is essentially geometrical in character (making use of non-Euclidean geometries), suggested that general relativity be formulated using the language of tensors. This will be discussed further below.

Spacetime as a manifold

Main articles: Spacetime and Spacetime topology

Most modern approaches to mathematical general relativity begin with the concept of a manifold. More precisely, the basic physical construct representing gravitation — a curved spacetime — is modelled by a four-dimensional, smooth, connected, Lorentzian manifold. Other physical descriptors are represented by various tensors, discussed below.

The rationale for choosing a manifold as the fundamental mathematical structure is to reflect desirable physical properties. For example, in the theory of manifolds, each point is contained in a (by no means unique) coordinate chart, and this chart can be thought of as representing the 'local spacetime' around the observer (represented by the point). The principle of local Lorentz covariance, which states that the laws of special relativity hold locally about each point of spacetime, lends further support to the choice of a manifold structure for representing spacetime, as locally around a point on a general manifold, the region 'looks like', or approximates very closely Minkowski space (flat spacetime).

The idea of coordinate charts as 'local observers who can perform measurements in their vicinity' also makes good physical sense, as this is how one actually collects physical data - locally. For cosmological problems, a coordinate chart may be quite large.

Local versus global structure

An important distinction in physics is the difference between local and global structures. Measurements in physics are performed in a relatively small region of spacetime and this is one reason for studying the local structure of spacetime in general relativity, whereas determining the global spacetime structure is important, especially in cosmological problems.

An important problem in general relativity is to tell when two spacetimes are 'the same', at least locally. This problem has its roots in manifold theory where determining if two Riemannian manifolds of the same dimension are locally isometric ('locally the same'). This latter problem has been solved and its adaptation for general relativity is called the Cartan–Karlhede algorithm.

Tensors in general relativity

Further information: Tensor

One of the profound consequences of relativity theory was the abolition of privileged reference frames. The description of physical phenomena should not depend upon who does the measuring - one reference frame should be as good as any other. Special relativity demonstrated that no inertial reference frame was preferential to any other inertial reference frame, but preferred inertial reference frames over noninertial reference frames. General relativity eliminated preference for inertial reference frames by showing that there is no preferred reference frame (inertial or not) for describing nature.

Any observer can make measurements and the precise numerical quantities obtained only depend on the coordinate system used. This suggested a way of formulating relativity using 'invariant structures', those that are independent of the coordinate system (represented by the observer) used, yet still have an independent existence. The most suitable mathematical structure seemed to be a tensor. For example, when measuring the electric and magnetic fields produced by an accelerating charge, the values of the fields will depend on the coordinate system used, but the fields are regarded as having an independent existence, this independence represented by the electromagnetic tensor .

Mathematically, tensors are generalised linear operators - multilinear maps. As such, the ideas of linear algebra are employed to study tensors.

At each point ${\displaystyle p}$ of a manifold, the tangent and cotangent spaces to the manifold at that point may be constructed. Vectors (sometimes referred to as contravariant vectors) are defined as elements of the tangent space and covectors (sometimes termed covariant vectors, but more commonly dual vectors or one-forms) are elements of the cotangent space.

At ${\displaystyle p}$, these two vector spaces may be used to construct type ${\displaystyle (r,s)}$ tensors, which are real-valued multilinear maps acting on the direct sum of ${\displaystyle r}$ copies of the cotangent space with ${\displaystyle s}$ copies of the tangent space. The set of all such multilinear maps forms a vector space, called the tensor product space of type ${\displaystyle (r,s)}$ at ${\displaystyle p}$ and denoted by ${\displaystyle (T_p{)}_s^{r}M.}$ If the tangent space is n-dimensional, it can be shown that ${\displaystyle \dim (T_p{)}_s^{r}M=n^{r+s}.}$

In the general relativity literature, it is conventional to use the component syntax for tensors.

A type ${\displaystyle (r,s)}$ tensor may be written as

$${\displaystyle T={T^{a_1\dots a_r}}_{b_1\dots b_s}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{a_1}}\otimes \cdots \otimes \frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{a_r}}\otimes d{x}^{b_1}\otimes \cdots \otimes d{x}^{b_s}}$$ where $\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{a_i}}$ is a basis for the i-th tangent space and ${\displaystyle d{x}^{b_j}}$ a basis for the j-th cotangent space.

As spacetime is assumed to be four-dimensional, each index on a tensor can be one of four values. Hence, the total number of elements a tensor possesses equals 4^R, where R is the count of the number of covariant ${\displaystyle (b_i)}$ and contravariant ${\displaystyle (a_i)}$ indices on the tensor, ${\displaystyle r+s}$ (a number called the rank of the tensor).

Symmetric and antisymmetric tensors

Main articles: Antisymmetric tensor and Symmetric tensor

Some physical quantities are represented by tensors not all of whose components are independent. Important examples of such tensors include symmetric and antisymmetric tensors. Antisymmetric tensors are commonly used to represent rotations (for example, the vorticity tensor).

Although a generic rank R tensor in 4 dimensions has 4^R components, constraints on the tensor such as symmetry or antisymmetry serve to reduce the number of distinct components. For example, a symmetric rank two tensor ${\displaystyle T}$ satisfies ${\displaystyle T_{\alpha \beta }=T_{\beta \alpha }}$ and possesses 10 independent components, whereas an antisymmetric (skew-symmetric) rank two tensor ${\displaystyle P}$ satisfies ${\displaystyle P_{\alpha \beta }=-P_{\beta \alpha }}$ and has 6 independent components. For ranks greater than two, the symmetric or antisymmetric index pairs must be explicitly identified.

Antisymmetric tensors of rank 2 play important roles in relativity theory. The set of all such tensors - often called bivectors - forms a vector space of dimension 6, sometimes called bivector space.

The metric tensor

Main article: Metric tensor (general relativity)

The metric tensor is a central object in general relativity that describes the local geometry of spacetime (as a result of solving the Einstein field equations). Using the weak-field approximation, the metric tensor can also be thought of as representing the 'gravitational potential'. The metric tensor is often just called 'the metric'.

The metric is a symmetric tensor and is an important mathematical tool. As well as being used to raise and lower tensor indices, it also generates the connections which are used to construct the geodesic equations of motion and the Riemann curvature tensor.

A convenient means of expressing the metric tensor in combination with the incremental intervals of coordinate distance that it relates to is through the line element: $${\displaystyle d{s}^2=g_{ab}d{x}^{a}d{x}^b}$$

This way of expressing the metric was used by the pioneers of differential geometry. While some relativists consider the notation to be somewhat old-fashioned, many readily switch between this and the alternative notation: $${\displaystyle g=g_{ab}d{x}^a\otimes d{x}^b}$$

The metric tensor is commonly written as a 4×4 matrix. This matrix is symmetric and thus has 10 independent components.

Invariants

One of the central features of GR is the idea of invariance of physical laws. This invariance can be described in many ways, for example, in terms of local Lorentz covariance, the general principle of relativity, or diffeomorphism covariance.

A more explicit description can be given using tensors. The crucial feature of tensors used in this approach is the fact that (once a metric is given) the operation of contracting a tensor of rank R over all R indices gives a number - an invariant - that is independent of the coordinate chart one uses to perform the contraction. Physically, this means that if the invariant is calculated by any two observers, they will get the same number, thus suggesting that the invariant has some independent significance. Some important invariants in relativity include:

• The Ricci scalar: ${\displaystyle R=R^{\alpha \beta }{g}_{\alpha \beta }}$
• The Kretschmann scalar: ${\displaystyle K=R^{abcd}{R}_{abcd}}$

Other examples of invariants in relativity include the electromagnetic invariants, and various other curvature invariants, some of the latter finding application in the study of gravitational entropy and the Weyl curvature hypothesis.

Tensor classifications

The classification of tensors is a purely mathematical problem. In GR, however, certain tensors that have a physical interpretation can be classified with the different forms of the tensor usually corresponding to some physics. Examples of tensor classifications useful in general relativity include the Segre classification of the energy–momentum tensor and the Petrov classification of the Weyl tensor. There are various methods of classifying these tensors, some of which use tensor invariants.

Tensor fields in general relativity

Main article: Tensor field

Tensor fields on a manifold are maps which attach a tensor to each point of the manifold. This notion can be made more precise by introducing the idea of a fibre bundle, which in the present context means to collect together all the tensors at all points of the manifold, thus 'bundling' them all into one grand object called the tensor bundle. A tensor field is then defined as a map from the manifold to the tensor bundle, each point ${\displaystyle p}$ being associated with a tensor at ${\displaystyle p}$.

The notion of a tensor field is of major importance in GR. For example, the geometry around a star is described by a metric tensor at each point, so at each point of the spacetime the value of the metric should be given to solve for the paths of material particles. Another example is the values of the electric and magnetic fields (given by the electromagnetic field tensor) and the metric at each point around a charged black hole to determine the motion of a charged particle in such a field.

Vector fields are contravariant rank one tensor fields. Important vector fields in relativity include the four-velocity, ${\displaystyle U^a={\dot{x}}^a}$, which is the coordinate distance travelled per unit of proper time, the four-acceleration ${\displaystyle A^a={\ddot{x}}^a}$ and the four-current ${\displaystyle J^a}$ describing the charge and current densities. Other physically important tensor fields in relativity include the following:

• The stress–energy tensor ${\displaystyle T^{ab}}$, a symmetric rank-two tensor.
• The electromagnetic field tensor ${\displaystyle F^{ab}}$, a rank-two antisymmetric tensor.

Although the word 'tensor' refers to an object at a point, it is common practice to refer to tensor fields on a spacetime (or a region of it) as just 'tensors'.

At each point of a spacetime on which a metric is defined, the metric can be reduced to the Minkowski form using Sylvester's law of inertia.

Tensorial derivatives

Before the advent of general relativity, changes in physical processes were generally described by partial derivatives, for example, in describing changes in electromagnetic fields (see Maxwell's equations). Even in special relativity, the partial derivative is still sufficient to describe such changes. However, in general relativity, it is found that derivatives which are also tensors must be used. The derivatives have some common features including that they are derivatives along integral curves of vector fields.

The problem in defining derivatives on manifolds that are not flat is that there is no natural way to compare vectors at different points. An extra structure on a general manifold is required to define derivatives. Below are described two important derivatives that can be defined by imposing an additional structure on the manifold in each case.

Affine connections

Main article: Affine connection

The curvature of a spacetime can be characterised by taking a vector at some point and parallel transporting it along a curve on the spacetime. An affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction.

By definition, an affine connection is a bilinear map ${\displaystyle \mathrm{\Gamma }(TM)\times \mathrm{\Gamma }(TM)\to \mathrm{\Gamma }(TM)}$, where ${\displaystyle \mathrm{\Gamma }(TM)}$ is a space of all vector fields on the spacetime. This bilinear map can be described in terms of a set of connection coefficients (also known as Christoffel symbols) specifying what happens to components of basis vectors under infinitesimal parallel transport: $${\displaystyle {\mathrm{\nabla }}_{e_i}{e}_j={\mathrm{\Gamma }}_{ji}^k{e}_k}$$

Despite their appearance, the connection coefficients are not the components of a tensor.

Generally speaking, there are ${\displaystyle D^3}$ independent connection coefficients at each point of spacetime. The connection is called symmetric or torsion-free, if ${\displaystyle {\mathrm{\Gamma }}_{ji}^k={\mathrm{\Gamma }}_{ij}^k}$. A symmetric connection has at most $\frac{1}{2}{D}^2(D+1)$ unique coefficients.

For any curve ${\displaystyle \gamma }$ and two points ${\displaystyle A=\gamma (0)}$ and ${\displaystyle B=\gamma (t)}$ on this curve, an affine connection gives rise to a map of vectors in the tangent space at ${\displaystyle A}$ into vectors in the tangent space at ${\displaystyle B}$: $${\displaystyle X(t)={\mathrm{\Pi }}_{0,t,\gamma }X(0)}$$ and ${\displaystyle X(t)}$ can be computed component-wise by solving the differential equation $${\displaystyle \frac{d}{dt}{X}^i(t)={\mathrm{\nabla }}_{C(t)}{X}^i(t)={\mathrm{\Gamma }}_{jk}^i{X}^j(t)C^k(t)}$$ where ${\displaystyle C^j(t)}$ is the vector tangent to the curve at the point ${\displaystyle \gamma (t)}$.

An important affine connection in general relativity is the Levi-Civita connection, which is a symmetric connection obtained from parallel transporting a tangent vector along a curve whilst keeping the inner product of that vector constant along the curve. The resulting connection coefficients (Christoffel symbols) can be calculated directly from the metric. For this reason, this type of connection is often called a metric connection.

The covariant derivative

Main article: Covariant derivative

Let ${\displaystyle X}$ be a point, ${\displaystyle \overrightarrow{A}}$ a vector located at ${\displaystyle X}$, and ${\displaystyle \overrightarrow{B}}$ a vector field. The idea of differentiating ${\displaystyle \overrightarrow{B}}$ at ${\displaystyle X}$ along the direction of ${\displaystyle \overrightarrow{A}}$ in a physically meaningful way can be made sense of by choosing an affine connection and a parameterized smooth curve ${\displaystyle \gamma (t)}$ such that ${\displaystyle X=\gamma (0)}$ and $\overrightarrow{A}=\frac{d}{dt}\gamma (0)$. The formula $${\displaystyle {\mathrm{\nabla }}_{\overrightarrow{A}}\overrightarrow{B}(X)=\underset{\epsilon \to 0}{lim}\frac{1}{\epsilon }\left[{\mathrm{\Pi }}_{(\epsilon ,0,\gamma )}\overrightarrow{B}(\gamma [\epsilon ])-\overrightarrow{B}(X)\right]}$$ for a covariant derivative of ${\displaystyle \overrightarrow{B}}$ along ${\displaystyle \overrightarrow{A}}$ associated with connection ${\displaystyle \mathrm{\Pi }}$ turns out to give curve-independent results and can be used as a "physical definition" of a covariant derivative.

It can be expressed using connection coefficients: $${\displaystyle {\mathrm{\nabla }}_{\overrightarrow{Y}}\overrightarrow{X}=X^a{}_{;b}{Y}^b\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^a}=(X^a{}_{,b}+{\mathrm{\Gamma }}_{bc}^a{X}^c)Y^b\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^a}}$$

The expression in brackets, called a covariant derivative of ${\displaystyle X}$ (with respect to the connection) and denoted by ${\displaystyle \mathrm{\nabla }\overrightarrow{X}}$, is more often used in calculations: $${\displaystyle \mathrm{\nabla }\overrightarrow{X}=X^a{}_{;b}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^a}\otimes d{x}^b=(X^a{}_{,b}+{\mathrm{\Gamma }}_{bc}^a{X}^c)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^a}\otimes d{x}^b}$$

A covariant derivative of ${\displaystyle X}$ can thus be viewed as a differential operator acting on a vector field sending it to a type (1, 1) tensor (increasing the covariant index by 1) and can be generalised to act on type ${\displaystyle (r,s)}$ tensor fields sending them to type ${\displaystyle (r,s+1)}$ tensor fields. Notions of parallel transport can then be defined similarly as for the case of vector fields. By definition, a covariant derivative of a scalar field is equal to the regular derivative of the field.

In the literature, there are three common methods of denoting covariant differentiation: $${\displaystyle D_a{T}_{d\dots e}^{b\dots c}={\mathrm{\nabla }}_a{T}_{d\dots e}^{b\dots c}=T_{d\dots e;a}^{b\dots c}}$$

Many standard properties of regular partial derivatives also apply to covariant derivatives: $${\displaystyle \begin{array}{rl}{\mathrm{\nabla }}_a(X^b+Y^b)& ={\mathrm{\nabla }}_a{X}^b+{\mathrm{\nabla }}_a{Y}^b\\ {\mathrm{\nabla }}_a(c{X}^b)& =c{\mathrm{\nabla }}_a{X}^b& & c\text{ a constant}\\ {\mathrm{\nabla }}_a(X^b{Y}^c)& =Y^c({\mathrm{\nabla }}_a{X}^b)+X^b({\mathrm{\nabla }}_a{Y}^c)\\ {\mathrm{\nabla }}_a(f(x)X^b)& =f{\mathrm{\nabla }}_a{X}^b+X^b{\mathrm{\nabla }}_{a}f=f{\mathrm{\nabla }}_a{X}^b+X^b\frac{\mathrm{\partial }f}{\mathrm{\partial }{x}^a}\end{array}}$$

In general relativity, one usually refers to "the" covariant derivative, which is the one associated with Levi-Civita affine connection. By definition, Levi-Civita connection preserves the metric under parallel transport, therefore, the covariant derivative gives zero when acting on a metric tensor (as well as its inverse). It means that we can take the (inverse) metric tensor in and out of the derivative and use it to raise and lower indices: $${\displaystyle {\mathrm{\nabla }}_a{T}^b={\mathrm{\nabla }}_a(T_c{g}^{bc})=g^{bc}{\mathrm{\nabla }}_a{T}_c}$$

The Lie derivative

Main articles: Lie derivative and Spacetime symmetries

Another important tensorial derivative is the Lie derivative. Unlike the covariant derivative, the Lie derivative is independent of the metric, although in general relativity one usually uses an expression that seemingly depends on the metric through the affine connection. Whereas the covariant derivative required an affine connection to allow comparison between vectors at different points, the Lie derivative uses a congruence from a vector field to achieve the same purpose. The idea of Lie dragging a function along a congruence leads to a definition of the Lie derivative, where the dragged function is compared with the value of the original function at a given point. The Lie derivative can be defined for type ${\displaystyle (r,s)}$ tensor fields and in this respect can be viewed as a map that sends a type ${\displaystyle (r,s)}$ to a type ${\displaystyle (r,s)}$ tensor.

The Lie derivative is usually denoted by ${\displaystyle {\mathcal{L}}_X}$, where ${\displaystyle X}$ is the vector field along whose congruence the Lie derivative is taken.

The Lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field. The Lie derivative of a scalar is just the directional derivative: $${\displaystyle {\mathcal{L}}_X\varphi =X^a{\mathrm{\nabla }}_a\varphi =X^a\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{x}^a}}$$

Higher rank objects pick up additional terms when the Lie derivative is taken. For example, the Lie derivative of a type (0, 2) tensor is $${\displaystyle {\mathcal{L}}_X{T}_{ab}=X^c{\mathrm{\nabla }}_c{T}_{ab}+({\mathrm{\nabla }}_a{X}^c)T_{cb}+({\mathrm{\nabla }}_b{X}^c)T_{ac}=X^c{T}_{ab,c}+X_{,a}^c{T}_{cb}+X_{,b}^c{T}_{ac}}$$

More generally, $${\displaystyle \begin{array}{rl}{\mathcal{L}}_X& T^{a_1\dots a_r}{}_{b_1\dots b_s}=X^c({\mathrm{\nabla }}_c{T}^{a_1\dots a_r}{}_{b_1\dots b_s})-\\ & ({\mathrm{\nabla }}_c{X}^{a_1})T^{c\dots a_r}{}_{b_1\dots b_s}-\cdots -({\mathrm{\nabla }}_c{X}^{a_r})T^{a_1\dots a_{r-1}c}{}_{b_1\dots b_s}+\\ & ({\mathrm{\nabla }}_{b_1}{X}^c)T^{a_1\dots a_r}{}_{c\dots b_s}+\cdots +({\mathrm{\nabla }}_{b_s}{X}^c)T^{a_1\dots a_r}{}_{b_1\dots b_{s-1}c}\end{array}}$$

In fact in the above expression, one can replace the covariant derivative ${\displaystyle {\mathrm{\nabla }}_a}$ with any torsion free connection ${\displaystyle {\tilde{\mathrm{\nabla }}}_a}$ or locally, with the coordinate dependent derivative ${\displaystyle {\mathrm{\partial }}_a}$, showing that the Lie derivative is independent of the metric. The covariant derivative is convenient however because it commutes with raising and lowering indices.

One of the main uses of the Lie derivative in general relativity is in the study of spacetime symmetries where tensors or other geometrical objects are preserved. In particular, Killing symmetry (symmetry of the metric tensor under Lie dragging) occurs very often in the study of spacetimes. Using the formula above, we can write down the condition that must be satisfied for a vector field to generate a Killing symmetry: $${\displaystyle \begin{array}{rl}{\mathcal{L}}_X{g}_{ab}=0& ⟺{\mathrm{\nabla }}_a{X}_b+{\mathrm{\nabla }}_b{X}_a=0\\ & ⟺X^c{g}_{ab,c}+X_{,a}^c{g}_{bc}+X_{,b}^c{g}_{ac}=0\end{array}}$$

The Riemann curvature tensor

Main article: Riemann tensor (general relativity)

A crucial feature of general relativity is the concept of a curved manifold. A useful way of measuring the curvature of a manifold is with an object called the Riemann (curvature) tensor.

This tensor measures curvature by use of an affine connection by considering the effect of parallel transporting a vector between two points along two curves. The discrepancy between the results of these two parallel transport routes is essentially quantified by the Riemann tensor.

This property of the Riemann tensor can be used to describe how initially parallel geodesics diverge. This is expressed by the equation of geodesic deviation and means that the tidal forces experienced in a gravitational field are a result of the curvature of spacetime.

Using the above procedure, the Riemann tensor is defined as a type (1, 3) tensor and when fully written out explicitly contains the Christoffel symbols and their first partial derivatives. The Riemann tensor has 20 independent components. The vanishing of all these components over a region indicates that the spacetime is flat in that region. From the viewpoint of geodesic deviation, this means that initially parallel geodesics in that region of spacetime will stay parallel.

The Riemann tensor has a number of properties sometimes referred to as the symmetries of the Riemann tensor. Of particular relevance to general relativity are the algebraic and differential Bianchi identities.

The connection and curvature of any Riemannian manifold are closely related, the theory of holonomy groups, which are formed by taking linear maps defined by parallel transport around curves on the manifold, providing a description of this relationship.

What the Riemann tensor allows us to do is tell, mathematically, whether a space is flat or, if curved, how much curvature takes place in any given region. In order to derive the Riemann curvature tensor we must first recall the definition of the covariant derivative of a tensor with one and two indices;

1. $${\displaystyle {\mathrm{\nabla }}_{\mu }{V}_{\nu }={\mathrm{\partial }}_{\mu }{V}_{\nu }-{\mathrm{\Gamma }}^{\rho }{}_{\mu \nu }{V}_{\rho }}$$
2. $${\displaystyle {\mathrm{\nabla }}_{\sigma }[V_{\mu \nu }]={\mathrm{\partial }}_{\sigma }{V}_{\mu \nu }-{\mathrm{\Gamma }}^{\rho }{}_{\mu \sigma }{V}_{\rho \nu }-{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }{V}_{\mu \rho }}$$

For the formation of the Riemann tensor, the covariant derivative is taken twice with the respect to a tensor of rank one. The equation is set up as follows; $${\displaystyle \begin{array}{rl}{\mathrm{\nabla }}_{\sigma ;\mu }{V}_{\nu }& ={\mathrm{\nabla }}_{\sigma }[{\mathrm{\nabla }}_{\mu }{V}_{\nu }]\\ & ={\mathrm{\partial }}_{\sigma }[{\mathrm{\nabla }}_{\mu }{V}_{\nu }]-{\mathrm{\Gamma }}^{\rho }{}_{\mu \sigma }[{\mathrm{\nabla }}_{\rho }{V}_{\nu }]-{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }[{\mathrm{\nabla }}_{\mu }{V}_{\rho }]\\ & ={\mathrm{\partial }}_{\sigma }[{\mathrm{\partial }}_{\mu }{V}_{\nu }-{\mathrm{\Gamma }}^{\alpha }{}_{\nu \mu }{V}_{\alpha }]-{\mathrm{\Gamma }}^{\rho }{}_{\mu \sigma }[{\mathrm{\partial }}_{\rho }{V}_{\nu }-{\mathrm{\Gamma }}^{\alpha }{}_{\nu \rho }{V}_{\alpha }]-{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }[{\mathrm{\partial }}_{\mu }{V}_{\rho }-{\mathrm{\Gamma }}^{\alpha }{}_{\rho \mu }{V}_{\alpha }]\\ & ={\mathrm{\partial }}_{\sigma }{\mathrm{\partial }}_{\mu }{V}_{\nu }-{\mathrm{\partial }}_{\sigma }({\mathrm{\Gamma }}^{\alpha }{}_{\nu \mu }{V}_{\alpha })-{\mathrm{\Gamma }}^{\rho }{}_{\mu \sigma }{\mathrm{\partial }}_{\rho }{V}_{\nu }+{\mathrm{\Gamma }}^{\rho }{}_{\mu \sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\nu \rho }{V}_{\alpha }-{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }{\mathrm{\partial }}_{\mu }{V}_{\rho }+{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\rho \mu }{V}_{\alpha }\\ & ={\mathrm{\partial }}_{\sigma }{\mathrm{\partial }}_{\mu }{V}_{\nu }-{\mathrm{\partial }}_{\sigma }({\mathrm{\Gamma }}^{\alpha }{}_{\nu \mu })V_{\alpha }-{\mathrm{\Gamma }}^{\alpha }{}_{\nu \mu }{\mathrm{\partial }}_{\sigma }(V_{\alpha })-{\mathrm{\Gamma }}^{\rho }{}_{\mu \sigma }{\mathrm{\partial }}_{\rho }{V}_{\nu }+{\mathrm{\Gamma }}^{\rho }{}_{\mu \sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\nu \rho }{V}_{\alpha }-{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }{\mathrm{\partial }}_{\mu }{V}_{\rho }+{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\rho \mu }{V}_{\alpha }\end{array}}$$

Similarly we have: $${\displaystyle {\mathrm{\nabla }}_{\mu ;\sigma }{V}_{\nu }={\mathrm{\partial }}_{\mu }{\mathrm{\partial }}_{\sigma }{V}_{\nu }-{\mathrm{\partial }}_{\mu }({\mathrm{\Gamma }}^{\alpha }{}_{\nu \sigma })V_{\alpha }-{\mathrm{\Gamma }}^{\alpha }{}_{\nu \sigma }{\mathrm{\partial }}_{\mu }(V_{\alpha })-{\mathrm{\Gamma }}^{\rho }{}_{\sigma \mu }{\mathrm{\partial }}_{\rho }{V}_{\nu }+{\mathrm{\Gamma }}^{\rho }{}_{\sigma \mu }{\mathrm{\Gamma }}^{\alpha }{}_{\nu \rho }{V}_{\alpha }-{\mathrm{\Gamma }}^{\rho }{}_{\nu \mu }{\mathrm{\partial }}_{\sigma }{V}_{\rho }+{\mathrm{\Gamma }}^{\rho }{}_{\nu \mu }{\mathrm{\Gamma }}^{\alpha }{}_{\rho \sigma }{V}_{\alpha }}$$

Subtracting the two equations, swapping dummy indices and using the symmetry of Christoffel symbols leaves: $${\displaystyle {\mathrm{\nabla }}_{\sigma ;\mu }{V}_{\nu }-{\mathrm{\nabla }}_{\mu ;\sigma }{V}_{\nu }=({\mathrm{\partial }}_{\mu }{\mathrm{\Gamma }}^{\alpha }{}_{\nu \sigma }-{\mathrm{\partial }}_{\sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\nu \mu }+{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\rho \mu }-{\mathrm{\Gamma }}^{\rho }{}_{\nu \mu }{\mathrm{\Gamma }}^{\alpha }{}_{\rho \sigma })V_{\alpha }}$$ or $${\displaystyle R_{\nu \mu \sigma }^{\alpha }{V}_{\alpha }=({\mathrm{\partial }}_{\mu }{\mathrm{\Gamma }}^{\alpha }{}_{\nu \sigma }b-{\mathrm{\partial }}_{\sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\nu \mu }+{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\rho \mu }-{\mathrm{\Gamma }}^{\rho }{}_{\nu \mu }{\mathrm{\Gamma }}^{\alpha }{}_{\rho \sigma })V_{\alpha }}$$

Finally the Riemann curvature tensor is written as $${\displaystyle R_{\nu \mu \sigma }^{\alpha }={\mathrm{\partial }}_{\mu }{\mathrm{\Gamma }}^{\alpha }{}_{\nu \sigma }-{\mathrm{\partial }}_{\sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\nu \mu }+{\mathrm{\Gamma }}^{\rho }{}_{\nu \sigma }{\mathrm{\Gamma }}^{\alpha }{}_{\rho \mu }-{\mathrm{\Gamma }}^{\rho }{}_{\nu \mu }{\mathrm{\Gamma }}^{\alpha }{}_{\rho \sigma }}$$

You can contract indices to make the tensor covariant simply by multiplying by the metric, which will be useful when working with Einstein's field equations, $${\displaystyle g_{\alpha \lambda }{R}_{\nu \mu \sigma }^{\lambda }=R_{\alpha \nu \mu \sigma }}$$ and by further decomposition, $${\displaystyle g^{\alpha \mu }{R}_{\alpha \nu \mu \sigma }=R_{\nu \sigma }}$$

This tensor is called the Ricci tensor which can also be derived by setting ${\displaystyle \alpha }$ and ${\displaystyle \mu }$ in the Riemann tensor to the same indice and summing over them. Then the curvature scalar can be found by going one step further, $${\displaystyle g^{\nu \sigma }{R}_{\nu \sigma }=R}$$

So now we have 3 different objects,

1. the Riemann curvature tensor: ${\displaystyle R_{\nu \mu \sigma }^{\alpha }}$ or ${\displaystyle R_{\alpha \nu \mu \sigma }}$
2. the Ricci tensor: ${\displaystyle R_{\nu \sigma }}$
3. the scalar curvature: ${\displaystyle R}$

all of which are useful in calculating solutions to Einstein's field equations.

The energy–momentum tensor

Main article: Energy-momentum tensor (general relativity)

The sources of any gravitational field (matter and energy) is represented in relativity by a type (0, 2) symmetric tensor called the energy–momentum tensor. It is closely related to the Ricci tensor. Being a second rank tensor in four dimensions, the energy–momentum tensor may be viewed as a 4 by 4 matrix. The various admissible matrix types, called Jordan forms cannot all occur, as the energy conditions that the energy–momentum tensor is forced to satisfy rule out certain forms.

Energy conservation

In special and general relativity, there is a local law for the conservation of energy–momentum. It can be succinctly expressed by the tensor equation: $${\displaystyle T^{ab}{}_{;b}=0}$$

This illustrates the rule of thumb that 'partial derivatives go to covariant derivatives'.

The Einstein field equations

Main article: Einstein field equations

See also: Solutions of the Einstein field equations

The Einstein field equations (EFE) are the core of general relativity theory. The EFE describe how mass and energy (as represented in the stress–energy tensor) are related to the curvature of space-time (as represented in the Einstein tensor). In abstract index notation, the EFE reads as follows: $${\displaystyle G_{ab}+\mathrm{\Lambda }{g}_{ab}=\frac{8\pi G}{c^4}{T}_{ab}}$$ where ${\displaystyle G_{ab}}$ is the Einstein tensor, ${\displaystyle \mathrm{\Lambda }}$ is the cosmological constant, ${\displaystyle g_{ab}}$ is the metric tensor, ${\displaystyle c}$ is the speed of light in vacuum and ${\displaystyle G}$ is the gravitational constant, which comes from Newton's law of universal gravitation.

The solutions of the EFE are metric tensors. The EFE, being non-linear differential equations for the metric, are often difficult to solve. There are a number of strategies used to solve them. For example, one strategy is to start with an ansatz (or an educated guess) of the final metric, and refine it until it is specific enough to support a coordinate system but still general enough to yield a set of simultaneous differential equations with unknowns that can be solved for. Metric tensors resulting from cases where the resultant differential equations can be solved exactly for a physically reasonable distribution of energy–momentum are called exact solutions. Examples of important exact solutions include the Schwarzschild solution and the Friedman-Lemaître-Robertson–Walker solution.

The EIH approximation plus other references (e.g. Geroch and Jang, 1975 - 'Motion of a body in general relativity', JMP, Vol. 16 Issue 1).

The geodesic equations

Main article: Geodesic (general relativity)

Once the EFE are solved to obtain a metric, it remains to determine the motion of inertial objects in the spacetime. In general relativity, it is assumed that inertial motion occurs along timelike and null geodesics of spacetime as parameterized by proper time. Geodesics are curves that parallel transport their own tangent vector ${\displaystyle \overrightarrow{U}}$; i.e., ${\displaystyle {\mathrm{\nabla }}_{\overrightarrow{U}}\overrightarrow{U}=0}$. This condition, the geodesic equation, can be written in terms of a coordinate system ${\displaystyle x^a}$ with the tangent vector ${\displaystyle U^a=\frac{d{x}^a}{d\tau }}$: $${\displaystyle {\ddot{x}}^a+{{\mathrm{\Gamma }}^a}_{bc}{\dot{x}}^b{\dot{x}}^c=0}$$ where ${\displaystyle \dot{}}$ denotes the derivative by proper time, ${\displaystyle d/d\tau }$, with τ parametrising proper time along the curve and making manifest the presence of the Christoffel symbols.

A principal feature of general relativity is to determine the paths of particles and radiation in gravitational fields. This is accomplished by solving the geodesic equations.

The EFE relate the total matter (energy) distribution to the curvature of spacetime. Their nonlinearity leads to a problem in determining the precise motion of matter in the resultant spacetime. For example, in a system composed of one planet orbiting a star, the motion of the planet is determined by solving the field equations with the energy–momentum tensor the sum of that for the planet and the star. The gravitational field of the planet affects the total spacetime geometry and hence the motion of objects. It is therefore reasonable to suppose that the field equations can be used to derive the geodesic equations.

When the energy–momentum tensor for a system is that of dust, it may be shown by using the local conservation law for the energy–momentum tensor that the geodesic equations are satisfied exactly.

Lagrangian formulation

Main article: Variational methods in general relativity

The issue of deriving the equations of motion or the field equations in any physical theory is considered by many researchers to be appealing. A fairly universal way of performing these derivations is by using the techniques of variational calculus, the main objects used in this being Lagrangians.

Many consider this approach to be an elegant way of constructing a theory, others as merely a formal way of expressing a theory (usually, the Lagrangian construction is performed after the theory has been developed).

Mathematical techniques for analysing spacetimes

Having outlined the basic mathematical structures used in formulating the theory, some important mathematical techniques that are employed in investigating spacetimes will now be discussed.

Frame fields

Main article: Frame fields in general relativity

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A frame field is an orthonormal set of 4 vector fields (1 timelike, 3 spacelike) defined on a spacetime. Each frame field can be thought of as representing an observer in the spacetime moving along the integral curves of the timelike vector field. Every tensor quantity can be expressed in terms of a frame field, in particular, the metric tensor takes on a particularly convenient form. When allied with coframe fields, frame fields provide a powerful tool for analysing spacetimes and physically interpreting the mathematical results.

Symmetry vector fields

Main article: Spacetime symmetries

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Some modern techniques in analysing spacetimes rely heavily on using spacetime symmetries, which are infinitesimally generated by vector fields (usually defined locally) on a spacetime that preserve some feature of the spacetime. The most common type of such symmetry vector fields include Killing vector fields (which preserve the metric structure) and their generalisations called generalised Killing vector fields. Symmetry vector fields find extensive application in the study of exact solutions in general relativity and the set of all such vector fields usually forms a finite-dimensional Lie algebra.

The Cauchy problem

Main article: Initial value formulation (general relativity)

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The Cauchy problem (sometimes called the initial value problem) is the attempt at finding a solution to a differential equation given initial conditions. In the context of general relativity, it means the problem of finding solutions to Einstein's field equations - a system of hyperbolic partial differential equations - given some initial data on a hypersurface. Studying the Cauchy problem allows one to formulate the concept of causality in general relativity, as well as 'parametrising' solutions of the field equations. Ideally, one desires global solutions, but usually local solutions are the best that can be hoped for. Typically, solving this initial value problem requires selection of particular coordinate conditions.

Spinor formalism

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Spinors find several important applications in relativity. Their use as a method of analysing spacetimes using tetrads, in particular, in the Newman–Penrose formalism is important.

Another appealing feature of spinors in general relativity is the condensed way in which some tensor equations may be written using the spinor formalism. For example, in classifying the Weyl tensor, determining the various Petrov types becomes much easier when compared with the tensorial counterpart.

Regge calculus

Main article: Regge calculus

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Regge calculus is a formalism which chops up a Lorentzian manifold into discrete 'chunks' (four-dimensional simplicial blocks) and the block edge lengths are taken as the basic variables. A discrete version of the Einstein–Hilbert action is obtained by considering so-called deficit angles of these blocks, a zero deficit angle corresponding to no curvature. This novel idea finds application in approximation methods in numerical relativity and quantum gravity, the latter using a generalisation of Regge calculus.

Singularity theorems

Main article: Penrose–Hawking singularity theorems

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In general relativity, it was noted that, under fairly generic conditions, gravitational collapse will inevitably result in a so-called singularity. A singularity is a point where the solutions to the equations become infinite, indicating that the theory has been probed at inappropriate ranges.

Numerical relativity

Main article: Numerical relativity

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Numerical relativity is the sub-field of general relativity which seeks to solve Einstein's equations through the use of numerical methods. Finite difference, finite element and pseudo-spectral methods are used to approximate the solution to the partial differential equations which arise. Novel techniques developed by numerical relativity include the excision method and the puncture method for dealing with the singularities arising in black hole spacetimes. Common research topics include black holes and neutron stars.

Perturbation methods

Main article: Perturbation methods in general relativity

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The nonlinearity of the Einstein field equations often leads one to consider approximation methods in solving them. For example, an important approach is to linearise the field equations. Techniques from perturbation theory find ample application in such areas.