Astrophysics

From Wikipedia, the free encyclopedia

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

Astrophysics is a science that employs the methods and principles of physics in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the heavenly bodies, rather than their positions or motions in space–what they are, rather than where they are." Among the subjects studied are the Sun, other stars, galaxies, extrasolar planets, the interstellar medium and the cosmic microwave background. Emissions from these objects are examined across all parts of the electromagnetic spectrum, and the properties examined include luminosity, density, temperature, and chemical composition. Because astrophysics is a very broad subject, astrophysicists apply concepts and methods from many disciplines of physics, including classical mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics.

In practice, modern astronomical research often involves a substantial amount of work in the realms of theoretical and observational physics. Some areas of study for astrophysicists include their attempts to determine the properties of dark matter, dark energy, black holes, and other celestial bodies; and the origin and ultimate fate of the universe. Topics also studied by theoretical astrophysicists include Solar System formation and evolution; stellar dynamics and evolution; galaxy formation and evolution; magnetohydrodynamics; large-scale structure of matter in the universe; origin of cosmic rays; general relativity, special relativity, quantum and physical cosmology, including string cosmology and astroparticle physics.

History

Early 1900s comparison of elemental, solar, and stellar spectra

Astronomy is an ancient science, long separated from the study of terrestrial physics. In the Aristotelian worldview, bodies in the sky appeared to be unchanging spheres whose only motion was uniform motion in a circle, while the earthly world was the realm which underwent growth and decay and in which natural motion was in a straight line and ended when the moving object reached its goal. Consequently, it was held that the celestial region was made of a fundamentally different kind of matter from that found in the terrestrial sphere; either Fire as maintained by Plato, or Aether as maintained by Aristotle. During the 17th century, natural philosophers such as Galileo, Descartes, and Newton began to maintain that the celestial and terrestrial regions were made of similar kinds of material and were subject to the same natural laws. Their challenge was that the tools had not yet been invented with which to prove these assertions.

For much of the nineteenth century, astronomical research was focused on the routine work of measuring the positions and computing the motions of astronomical objects. A new astronomy, soon to be called astrophysics, began to emerge when William Hyde Wollaston and Joseph von Fraunhofer independently discovered that, when decomposing the light from the Sun, a multitude of dark lines (regions where there was less or no light) were observed in the spectrum. By 1860 the physicist, Gustav Kirchhoff, and the chemist, Robert Bunsen, had demonstrated that the dark lines in the solar spectrum corresponded to bright lines in the spectra of known gases, specific lines corresponding to unique chemical elements. Kirchhoff deduced that the dark lines in the solar spectrum are caused by absorption by chemical elements in the Solar atmosphere. In this way it was proved that the chemical elements found in the Sun and stars were also found on Earth.

Among those who extended the study of solar and stellar spectra was Norman Lockyer, who in 1868 detected radiant, as well as dark, lines in solar spectra. Working with chemist Edward Frankland to investigate the spectra of elements at various temperatures and pressures, he could not associate a yellow line in the solar spectrum with any known elements. He thus claimed the line represented a new element, which was called helium, after the Greek Helios, the Sun personified.

In 1885, Edward C. Pickering undertook an ambitious program of stellar spectral classification at Harvard College Observatory, in which a team of woman computers, notably Williamina Fleming, Antonia Maury, and Annie Jump Cannon, classified the spectra recorded on photographic plates. By 1890, a catalog of over 10,000 stars had been prepared that grouped them into thirteen spectral types. Following Pickering's vision, by 1924 Cannon expanded the catalog to nine volumes and over a quarter of a million stars, developing the Harvard Classification Scheme which was accepted for worldwide use in 1922.

In 1895, George Ellery Hale and James E. Keeler, along with a group of ten associate editors from Europe and the United States, established The Astrophysical Journal: An International Review of Spectroscopy and Astronomical Physics. It was intended that the journal would fill the gap between journals in astronomy and physics, providing a venue for publication of articles on astronomical applications of the spectroscope; on laboratory research closely allied to astronomical physics, including wavelength determinations of metallic and gaseous spectra and experiments on radiation and absorption; on theories of the Sun, Moon, planets, comets, meteors, and nebulae; and on instrumentation for telescopes and laboratories.

Around 1920, following the discovery of the Hertzsprung–Russell diagram still used as the basis for classifying stars and their evolution, Arthur Eddington anticipated the discovery and mechanism of nuclear fusion processes in stars, in his paper The Internal Constitution of the Stars. At that time, the source of stellar energy was a complete mystery; Eddington correctly speculated that the source was fusion of hydrogen into helium, liberating enormous energy according to Einstein's equation E = mc². This was a particularly remarkable development since at that time fusion and thermonuclear energy, and even that stars are largely composed of hydrogen (see metallicity), had not yet been discovered.

In 1925 Cecilia Helena Payne (later Cecilia Payne-Gaposchkin) wrote an influential doctoral dissertation at Radcliffe College, in which she applied ionization theory to stellar atmospheres to relate the spectral classes to the temperature of stars. Most significantly, she discovered that hydrogen and helium were the principal components of stars. Despite Eddington's suggestion, this discovery was so unexpected that her dissertation readers convinced her to modify the conclusion before publication. However, later research confirmed her discovery.

By the end of the 20th century, studies of astronomical spectra had expanded to cover wavelengths extending from radio waves through optical, x-ray, and gamma wavelengths. In the 21st century it further expanded to include observations based on gravitational waves.

Observational astrophysics

Supernova remnant LMC N 63A imaged in x-ray (blue), optical (green) and radio (red) wavelengths. The X-ray glow is from material heated to about ten million degrees Celsius by a shock wave generated by the supernova explosion.

Observational astronomy is a division of the astronomical science that is concerned with recording and interpreting data, in contrast with theoretical astrophysics, which is mainly concerned with finding out the measurable implications of physical models. It is the practice of observing celestial objects by using telescopes and other astronomical apparatus.

The majority of astrophysical observations are made using the electromagnetic spectrum.

• Radio astronomy studies radiation with a wavelength greater than a few millimeters. Example areas of study are radio waves, usually emitted by cold objects such as interstellar gas and dust clouds; the cosmic microwave background radiation which is the redshifted light from the Big Bang; pulsars, which were first detected at microwave frequencies. The study of these waves requires very large radio telescopes.
• Infrared astronomy studies radiation with a wavelength that is too long to be visible to the naked eye but is shorter than radio waves. Infrared observations are usually made with telescopes similar to the familiar optical telescopes. Objects colder than stars (such as planets) are normally studied at infrared frequencies.
• Optical astronomy was the earliest kind of astronomy. Telescopes paired with a charge-coupled device or spectroscopes are the most common instruments used. The Earth's atmosphere interferes somewhat with optical observations, so adaptive optics and space telescopes are used to obtain the highest possible image quality. In this wavelength range, stars are highly visible, and many chemical spectra can be observed to study the chemical composition of stars, galaxies, and nebulae.
• Ultraviolet, X-ray and gamma ray astronomy study very energetic processes such as binary pulsars, black holes, magnetars, and many others. These kinds of radiation do not penetrate the Earth's atmosphere well. There are two methods in use to observe this part of the electromagnetic spectrum—space-based telescopes and ground-based imaging air Cherenkov telescopes (IACT). Examples of Observatories of the first type are RXTE, the Chandra X-ray Observatory and the Compton Gamma Ray Observatory. Examples of IACTs are the High Energy Stereoscopic System (H.E.S.S.) and the MAGIC telescope.

Other than electromagnetic radiation, few things may be observed from the Earth that originate from great distances. A few gravitational wave observatories have been constructed, but gravitational waves are extremely difficult to detect. Neutrino observatories have also been built, primarily to study our Sun. Cosmic rays consisting of very high-energy particles can be observed hitting the Earth's atmosphere.

Observations can also vary in their time scale. Most optical observations take minutes to hours, so phenomena that change faster than this cannot readily be observed. However, historical data on some objects is available, spanning centuries or millennia. On the other hand, radio observations may look at events on a millisecond timescale (millisecond pulsars) or combine years of data (pulsar deceleration studies). The information obtained from these different timescales is very different.

The study of our very own Sun has a special place in observational astrophysics. Due to the tremendous distance of all other stars, the Sun can be observed in a kind of detail unparalleled by any other star. Our understanding of our own Sun serves as a guide to our understanding of other stars.

The topic of how stars change, or stellar evolution, is often modeled by placing the varieties of star types in their respective positions on the Hertzsprung–Russell diagram, which can be viewed as representing the state of a stellar object, from birth to destruction.

Theoretical astrophysics

See also: Theoretical astronomy

Theoretical astrophysicists use a wide variety of tools which include analytical models (for example, polytropes to approximate the behaviors of a star) and computational numerical simulations. Each has some advantages. Analytical models of a process are generally better for giving insight into the heart of what is going on. Numerical models can reveal the existence of phenomena and effects that would otherwise not be seen.

Theorists in astrophysics endeavor to create theoretical models and figure out the observational consequences of those models. This helps allow observers to look for data that can refute a model or help in choosing between several alternate or conflicting models.

Theorists also try to generate or modify models to take into account new data. In the case of an inconsistency, the general tendency is to try to make minimal modifications to the model to fit the data. In some cases, a large amount of inconsistent data over time may lead to total abandonment of a model.

Topics studied by theoretical astrophysicists include stellar dynamics and evolution; galaxy formation and evolution; magnetohydrodynamics; large-scale structure of matter in the universe; origin of cosmic rays; general relativity and physical cosmology, including string cosmology and astroparticle physics. Astrophysical relativity serves as a tool to gauge the properties of large-scale structures for which gravitation plays a significant role in physical phenomena investigated and as the basis for black hole (astro)physics and the study of gravitational waves.

Some widely accepted and studied theories and models in astrophysics, now included in the Lambda-CDM model, are the Big Bang, cosmic inflation, dark matter, dark energy and fundamental theories of physics.

Popularization

The roots of astrophysics can be found in the seventeenth century emergence of a unified physics, in which the same laws applied to the celestial and terrestrial realms. There were scientists who were qualified in both physics and astronomy who laid the firm foundation for the current science of astrophysics. In modern times, students continue to be drawn to astrophysics due to its popularization by the Royal Astronomical Society and notable educators such as prominent professors Lawrence Krauss, Subrahmanyan Chandrasekhar, Stephen Hawking, Hubert Reeves, Carl Sagan, Neil deGrasse Tyson and Patrick Moore. The efforts of the early, late, and present scientists continue to attract young people to study the history and science of astrophysics.

Stress–energy tensor

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor 

 

Contravariant components of the stress–energy tensor.

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

Definition

The stress–energy tensor involves the use of superscripted variables (not exponents; see tensor index notation and Einstein summation notation). If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: x⁰ = t, x¹ = x, x² = y, and x³ = z, where t is time in seconds, and x, y, and z are distances in meters.

The stress–energy tensor is defined as the tensor T^αβ of order two that gives the flux of the αth component of the momentum vector across a surface with constant x^β coordinate. In the theory of relativity, this momentum vector is taken as the four-momentum. In general relativity, the stress–energy tensor is symmetric,

${\displaystyle T^{\alpha \beta }=T^{\beta \alpha }.}$

In some alternative theories like Einstein–Cartan theory, the stress–energy tensor may not be perfectly symmetric because of a nonzero spin tensor, which geometrically corresponds to a nonzero torsion tensor.

The components of the stress-energy tensor

Because the stress–energy tensor is of order 2, its components can be displayed in 4 × 4 matrix form:

${\displaystyle (T^{\mu \nu })_{\mu ,\nu =0,1,2,3}={\begin{pmatrix}T^{00}&T^{01}&T^{02}&T^{03}\\T^{10}&T^{11}&T^{12}&T^{13}\\T^{20}&T^{21}&T^{22}&T^{23}\\T^{30}&T^{31}&T^{32}&T^{33}\end{pmatrix}}.}$

In the following, k and ℓ range from 1 through 3:

1. The time–time component is the density of relativistic mass, i.e., the energy density divided by the speed of light squared, while being in the co-moving frame of reference. It has a direct physical interpretation. In the case of a perfect fluid this component is

   ${\displaystyle T^{00}=\rho ~,}$

   where ${\displaystyle \rho }$ is the relativistic mass per unit volume, and for an electromagnetic field in otherwise empty space this component is

   ${\displaystyle T^{00}={1 \over c^{2}}\left({\frac {1}{2}}\epsilon _{0}E^{2}+{\frac {1}{2\mu _{0}}}B^{2}\right),}$

   where E and B are the electric and magnetic fields, respectively.
2. The flux of relativistic mass across the x^k surface is equivalent to the density of the kth component of linear momentum,

   ${\displaystyle T^{0k}=T^{k0}~.}$

3. The components
   ${\displaystyle T^{k\ell }}$
   represent flux of kth component of linear momentum across the x^ℓ surface. In particular,

   ${\displaystyle T^{kk}}$

   (not summed) represents normal stress in the kth co-ordinate direction (k = 1, 2, 3), which is called "pressure" when it is the same in every direction, k. The remaining components

   ${\displaystyle T^{k\ell }\quad k\neq \ell }$

   represent shear stress (compare with the stress tensor).

In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress–energy tensor in the proper frame of reference. In other words, the stress energy tensor in engineering differs from the relativistic stress–energy tensor by a momentum-convective term.

Covariant and mixed forms

Most of this article works with the contravariant form, T^μν of the stress–energy tensor. However, it is often necessary to work with the covariant form,

${\displaystyle T_{\mu \nu }=T^{\alpha \beta }g_{\alpha \mu }g_{\beta \nu },}$

or the mixed form,

${\displaystyle T^{\mu }{}_{\nu }=T^{\mu \alpha }g_{\alpha \nu },}$

or as a mixed tensor density

${\displaystyle {\mathfrak {T}}^{\mu }{}_{\nu }=T^{\mu }{}_{\nu }{\sqrt {-g}}\,.}$

This article uses the spacelike sign convention (−+++) for the metric signature.

Conservation law

In special relativity

See also: Relativistic angular momentum and Four-momentum

The stress–energy tensor is the conserved Noether current associated with spacetime translations.

The divergence of the non-gravitational stress–energy is zero. In other words, non-gravitational energy and momentum are conserved,

${\displaystyle 0=T^{\mu \nu }{}_{;\nu }=\nabla _{\nu }T^{\mu \nu }{}.\!}$

When gravity is negligible and using a Cartesian coordinate system for spacetime, this may be expressed in terms of partial derivatives as

${\displaystyle 0=T^{\mu \nu }{}_{,\nu }=\partial _{\nu }T^{\mu \nu }.\!}$

The integral form of this is

${\displaystyle 0=\int _{\partial N}T^{\mu \nu }\mathrm {d} ^{3}s_{\nu }\!}$

where N is any compact four-dimensional region of spacetime; ${\displaystyle \partial N}$ is its boundary, a three-dimensional hypersurface; and ${\displaystyle \mathrm {d} ^{3}s_{\nu }}$ is an element of the boundary regarded as the outward pointing normal.

In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that angular momentum is also conserved:

${\displaystyle 0=(x^{\alpha }T^{\mu \nu }-x^{\mu }T^{\alpha \nu })_{,\nu }.\!}$

In general relativity

When gravity is non-negligible or when using arbitrary coordinate systems, the divergence of the stress–energy still vanishes. But in this case, a coordinate-free definition of the divergence is used which incorporates the covariant derivative

${\displaystyle 0=\operatorname {div} T=T^{\mu \nu }{}_{;\nu }=\nabla _{\nu }T^{\mu \nu }=T^{\mu \nu }{}_{,\nu }+\Gamma ^{\mu }{}_{\sigma \nu }T^{\sigma \nu }+\Gamma ^{\nu }{}_{\sigma \nu }T^{\mu \sigma }}$

where ${\displaystyle \Gamma ^{\mu }{}_{\sigma \nu }}$ is the Christoffel symbol which is the gravitational force field.

Consequently, if ${\displaystyle \xi ^{\mu }}$ is any Killing vector field, then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as

${\displaystyle 0=\nabla _{\nu }\left(\xi ^{\mu }T_{\mu }^{\nu }\right)={\frac {1}{\sqrt {-g}}}\partial _{\nu }\left({\sqrt {-g}}\ \xi ^{\mu }T_{\mu }^{\nu }\right)}$

The integral form of this is

${\displaystyle 0=\int _{\partial N}{\sqrt {-g}}\ \xi ^{\mu }T_{\mu }^{\nu }\ \mathrm {d} ^{3}s_{\nu }=\int _{\partial N}\xi ^{\mu }{\mathfrak {T}}_{\mu }^{\nu }\ \mathrm {d} ^{3}s_{\nu }}$

In special relativity

In special relativity, the stress–energy tensor contains information about the energy and momentum densities of a given system, in addition to the momentum and energy flux densities.

Given a Lagrangian Density ${\displaystyle {\mathcal {L}}}$ that is a function of a set of fields ${\displaystyle \phi _{\alpha }}$ and their derivatives, but explicitly not of any of the spacetime coordinates, we can construct the tensor by looking at the total derivative with respect to one of the generalized coordinates of the system. So, with our condition

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial x_{\nu }}}=0}$

By using the chain rule, we then have

${\displaystyle {\frac {d{\mathcal {L}}}{dx_{\nu }}}=\partial ^{\nu }{\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}{\frac {\partial (\partial _{\mu }\phi _{\alpha })}{\partial x_{\nu }}}+{\frac {\partial {\mathcal {L}}}{\partial \phi _{\alpha }}}{\frac {\partial \phi _{\alpha }}{\partial x_{\nu }}}}$

Written in useful shorthand,

${\displaystyle \partial ^{\nu }{\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial ^{\nu }\partial _{\mu }\phi _{\alpha }+{\frac {\partial {\mathcal {L}}}{\partial \phi _{\alpha }}}\partial ^{\nu }\phi _{\alpha }}$

Then, we can use the Euler–Lagrange Equation:

${\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\right)={\frac {\partial {\mathcal {L}}}{\partial \phi _{\alpha }}}}$

And then use the fact that partial derivatives commute so that we now have

${\displaystyle \partial ^{\nu }{\mathcal {L}}={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\mu }\partial ^{\nu }\phi _{\alpha }+\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\right)\partial ^{\nu }\phi _{\alpha }}$

We can recognize the right hand side as a product rule. Writing it as the derivative of a product of functions tells us that

${\displaystyle \partial ^{\nu }{\mathcal {L}}=\partial _{\mu }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial ^{\nu }\phi _{\alpha }\right]}$

Now, in flat space, one can write ${\displaystyle \partial ^{\nu }{\mathcal {L}}=\partial _{\mu }g^{\mu \nu }{\mathcal {L}}}$. Doing this and moving it to the other side of the equation tells us that

${\displaystyle \partial _{\mu }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial ^{\nu }\phi _{\alpha }\right]-\partial _{\mu }\left(g^{\mu \nu }{\mathcal {L}}\right)=0}$

And upon regrouping terms,

${\displaystyle \partial _{\mu }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial ^{\nu }\phi _{\alpha }-g^{\mu \nu }{\mathcal {L}}\right]=0}$

This is to say that the divergence of the tensor in the brackets is 0. Indeed, with this, we define the stress–energy tensor:

${\displaystyle T^{\mu \nu }\equiv {\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial ^{\nu }\phi _{\alpha }-g^{\mu \nu }{\mathcal {L}}}$

By construction it has the property that

${\displaystyle \partial _{\mu }T^{\mu \nu }=0}$

Note that this divergenceless property of this tensor is equivalent to four continuity equations. That is, fields have at least four sets of quantities that obey the continuity equation. As an example, it can be seen that ${\displaystyle T_{0}^{0}}$ is the energy density of the system and that it is thus possible to obtain the Hamiltonian density from the stress–energy tensor.

Indeed, since this is the case, observing that ${\displaystyle \partial _{\mu }T^{\mu 0}=0}$, we then have

${\displaystyle {\frac {\partial {\mathcal {H}}}{\partial t}}+\nabla \cdot \left({\frac {\partial {\mathcal {L}}}{\partial \nabla \phi _{\alpha }}}{\dot {\phi }}_{\alpha }\right)=0}$

We can then conclude that the terms of ${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial \nabla \phi _{\alpha }}}{\dot {\phi }}_{\alpha }}$ represent the energy flux density of the system.

Trace

Note that the trace of the stress–energy tensor is defined to be ${\displaystyle T_{\mu }^{\mu }}$, where

${\displaystyle T_{\mu }^{\mu }=T^{\mu \nu }g_{\nu \mu }.}$

When we use the formula for the stress–energy tensor found above,

${\displaystyle T_{\mu }^{\mu }={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}g_{\mu \nu }\partial ^{\nu }\phi _{\alpha }-g_{\mu \nu }g^{\mu \nu }{\mathcal {L}}.}$

Using the raising and lowering properties of the metric and that ${\displaystyle g^{\mu \nu }g_{\mu \alpha }=\delta _{\alpha }^{\nu }}$,

${\displaystyle T_{\mu }^{\mu }={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\mu }\phi _{\alpha }-\delta _{\mu }^{\mu }{\mathcal {L}}.}$

Since ${\displaystyle \delta _{\mu }^{\mu }=4}$,

${\displaystyle T_{\mu }^{\mu }={\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{\alpha })}}\partial _{\mu }\phi _{\alpha }-4{\mathcal {L}}.}$

In general relativity

In general relativity, the symmetric stress–energy tensor acts as the source of spacetime curvature, and is the current density associated with gauge transformations of gravity which are general curvilinear coordinate transformations. (If there is torsion, then the tensor is no longer symmetric. This corresponds to the case with a nonzero spin tensor in Einstein–Cartan gravity theory.)

In general relativity, the partial derivatives used in special relativity are replaced by covariant derivatives. What this means is that the continuity equation no longer implies that the non-gravitational energy and momentum expressed by the tensor are absolutely conserved, i.e. the gravitational field can do work on matter and vice versa. In the classical limit of Newtonian gravity, this has a simple interpretation: kinetic energy is being exchanged with gravitational potential energy, which is not included in the tensor, and momentum is being transferred through the field to other bodies. In general relativity the Landau–Lifshitz pseudotensor is a unique way to define the gravitational field energy and momentum densities. Any such stress–energy pseudotensor can be made to vanish locally by a coordinate transformation.

In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energy–momentum vector in a general curved spacetime.

Einstein field equations

Main article: Einstein field equations

In general relativity, the stress-energy tensor is studied in the context of the Einstein field equations which are often written as

${\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu },}$

where ${\displaystyle R_{\mu \nu }}$ is the Ricci tensor, ${\displaystyle R}$ is the Ricci scalar (the tensor contraction of the Ricci tensor), ${\displaystyle g_{\mu \nu }\,}$ is the metric tensor, Λ is the cosmological constant (negligible at the scale of a galaxy or smaller), and ${\displaystyle G}$ is the universal gravitational constant.

Stress–energy in special situations

Isolated particle

In special relativity, the stress–energy of a non-interacting particle with rest mass m and trajectory ${\displaystyle \mathbf {x} _{\text{p}}(t)}$ is:

${\displaystyle T^{\alpha \beta }(\mathbf {x} ,t)={\frac {m\,v^{\alpha }(t)v^{\beta }(t)}{\sqrt {1-(v/c)^{2}}}}\;\,\delta \left(\mathbf {x} -\mathbf {x} _{\text{p}}(t)\right)={\frac {E}{c^{2}}}\;v^{\alpha }(t)v^{\beta }(t)\;\,\delta (\mathbf {x} -\mathbf {x} _{\text{p}}(t))}$

where ${\displaystyle \left(v^{\alpha }\right)_{\alpha =0,1,2,3}\!}$ is the velocity vector (which should not be confused with four-velocity, since it is missing a ${\displaystyle \gamma }$)

${\displaystyle \left(v^{\alpha }\right)_{\alpha =0,1,2,3}=\left(1,{\frac {d\mathbf {x} _{\text{p}}}{dt}}(t)\right)\,,}$

${\displaystyle \delta }$ is the Dirac delta function and ${\displaystyle E={\sqrt {p^{2}c^{2}+m^{2}c^{4}}}}$ is the energy of the particle.

Written in language of classical physics, the stress-energy tensor would be (relativistic mass, momentum, the dyad product of momentum and velocity)

${\displaystyle \left({\frac {E}{c^{2}}},\,\mathbf {p} ,\,\mathbf {p} \,\mathbf {v} \right)}$.

Stress–energy of a fluid in equilibrium

For a perfect fluid in thermodynamic equilibrium, the stress–energy tensor takes on a particularly simple form

${\displaystyle T^{\alpha \beta }\,=\left(\rho +{p \over c^{2}}\right)u^{\alpha }u^{\beta }+pg^{\alpha \beta }}$

where ${\displaystyle \rho }$ is the mass–energy density (kilograms per cubic meter), ${\displaystyle p}$ is the hydrostatic pressure (pascals), ${\displaystyle u^{\alpha }}$ is the fluid's four velocity, and ${\displaystyle g^{\alpha \beta }}$ is the reciprocal of the metric tensor. Therefore, the trace is given by

${\displaystyle T_{\,\alpha }^{\alpha }=g_{\alpha \beta }T^{\beta \alpha }=3p-\rho c^{2}\,.}$

The four-velocity satisfies

${\displaystyle u^{\alpha }u^{\beta }g_{\alpha \beta }=-c^{2}\,.}$

In an inertial frame of reference comoving with the fluid, better known as the fluid's proper frame of reference, the four velocity is

${\displaystyle (u^{\alpha })_{\alpha =0,1,2,3}=(1,0,0,0)\,,}$

the reciprocal of the metric tensor is simply

${\displaystyle (g^{\alpha \beta })_{\alpha ,\beta =0,1,2,3}\,=\left({\begin{matrix}-{\frac {1}{c^{2}}}&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)\,}$

and the stress–energy tensor is a diagonal matrix

${\displaystyle (T^{\alpha \beta })_{\alpha ,\beta =0,1,2,3}=\left({\begin{matrix}\rho &0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right).}$

Electromagnetic stress–energy tensor

Main article: Electromagnetic stress–energy tensor

The Hilbert stress–energy tensor of a source-free electromagnetic field is

${\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left(F^{\mu \alpha }g_{\alpha \beta }F^{\nu \beta }-{\frac {1}{4}}g^{\mu \nu }F_{\delta \gamma }F^{\delta \gamma }\right)}$

where ${\displaystyle F_{\mu \nu }}$ is the electromagnetic field tensor.

Scalar field

Main article: Klein–Gordon equation

The stress–energy tensor for a complex scalar field ${\displaystyle \phi }$ which satisfies the Klein–Gordon equation is

${\displaystyle T^{\mu \nu }={\frac {\hbar ^{2}}{m}}\left(g^{\mu \alpha }g^{\nu \beta }+g^{\mu \beta }g^{\nu \alpha }-g^{\mu \nu }g^{\alpha \beta }\right)\partial _{\alpha }{\bar {\phi }}\partial _{\beta }\phi -g^{\mu \nu }mc^{2}{\bar {\phi }}\phi ,}$

and when the metric is flat (Minkowski in Cartesian coordinates) its components work out to be:

${\displaystyle {\begin{aligned}T^{00}&={\frac {\hbar ^{2}}{mc^{4}}}\left(\partial _{0}{\bar {\phi }}\partial _{0}\phi +c^{2}\partial _{k}{\bar {\phi }}\partial _{k}\phi \right)+m{\bar {\phi }}\phi ,\\T^{0i}=T^{i0}&=-{\frac {\hbar ^{2}}{mc^{2}}}\left(\partial _{0}{\bar {\phi }}\partial _{i}\phi +\partial _{i}{\bar {\phi }}\partial _{0}\phi \right),\ \mathrm {and} \\T^{ij}&={\frac {\hbar ^{2}}{m}}\left(\partial _{i}{\bar {\phi }}\partial _{j}\phi +\partial _{j}{\bar {\phi }}\partial _{i}\phi \right)-\delta _{ij}\left({\frac {\hbar ^{2}}{m}}\eta ^{\alpha \beta }\partial _{\alpha }{\bar {\phi }}\partial _{\beta }\phi +mc^{2}{\bar {\phi }}\phi \right).\end{aligned}}}$

Variant definitions of stress–energy

There are a number of inequivalent definitions of non-gravitational stress–energy:

Hilbert stress–energy tensor

The Hilbert stress–energy tensor is defined as the functional derivative

${\displaystyle T_{\mu \nu }={\frac {-2}{\sqrt {-g}}}{\frac {\delta S_{\mathrm {matter} }}{\delta g^{\mu \nu }}}={\frac {-2}{\sqrt {-g}}}{\frac {\partial \left({\sqrt {-g}}{\mathcal {L}}_{\mathrm {matter} }\right)}{\partial g^{\mu \nu }}}=-2{\frac {\partial {\mathcal {L}}_{\mathrm {matter} }}{\partial g^{\mu \nu }}}+g_{\mu \nu }{\mathcal {L}}_{\mathrm {matter} },}$

where ${\displaystyle S_{\mathrm {matter} }}$ is the nongravitational part of the action, ${\displaystyle {\mathcal {L}}_{\mathrm {matter} }}$ is the nongravitational part of the Lagrangian density, and the Euler-Lagrange equation has been used. This is symmetric and gauge-invariant. See Einstein–Hilbert action for more information.

Canonical stress–energy tensor

Noether's theorem implies that there is a conserved current associated with translations through space and time. This is called the canonical stress–energy tensor. Generally, this is not symmetric and if we have some gauge theory, it may not be gauge invariant because space-dependent gauge transformations do not commute with spatial translations.

In general relativity, the translations are with respect to the coordinate system and as such, do not transform covariantly. See the section below on the gravitational stress–energy pseudo-tensor.

Belinfante–Rosenfeld stress–energy tensor

Main article: Belinfante–Rosenfeld stress–energy tensor

In the presence of spin or other intrinsic angular momentum, the canonical Noether stress energy tensor fails to be symmetric. The Belinfante–Rosenfeld stress energy tensor is constructed from the canonical stress–energy tensor and the spin current in such a way as to be symmetric and still conserved. In general relativity, this modified tensor agrees with the Hilbert stress–energy tensor.

Gravitational stress–energy

Main article: Stress–energy–momentum pseudotensor

By the equivalence principle gravitational stress–energy will always vanish locally at any chosen point in some chosen frame, therefore gravitational stress–energy cannot be expressed as a non-zero tensor; instead we have to use a pseudotensor.

In general relativity, there are many possible distinct definitions of the gravitational stress–energy–momentum pseudotensor. These include the Einstein pseudotensor and the Landau–Lifshitz pseudotensor. The Landau–Lifshitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.