Editing Stress–energy tensor

{{Short description|Tensor describing energy momentum density in spacetime}}
{{Use American English|date=January 2019}}[[File:StressEnergyTensor contravariant.svg|right|236px|thumb|Contravariant components of the stress–energy tensor.]]
{{General relativity sidebar}}
{{Wikiversity|Gravitational 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 [[Volume-specific quantity|density]] and [[flux]] of [[energy]] and [[momentum]] in [[spacetime]], generalizing the [[Cauchy stress tensor|stress tensor]] of [[Newtonian physics]]. It is an attribute of [[matter]], [[radiation]], and non-gravitational [[force field (physics)|force field]]s. 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 ({{em|not}} exponents; see ''[[Tensor index notation]]'' and ''[[Einstein notation|Einstein summation notation]]''). If [[Cartesian coordinates]] in [[SI units]] are used, then the components of the position [[four-vector]] {{mvar|x}} are given by: {{math|{{bracket| ''x''{{sup|0}}, ''x''{{sup|1}}, ''x''{{sup|2}}, ''x''{{sup|3}} }}}}.  In traditional Cartesian coordinates these are instead customarily written {{math|{{bracket| ''t'', ''x'', ''y'', ''z'' }}}}, where {{math|''t''}} is coordinate time, and {{math|''x''}}, {{math|''y''}}, and {{math|''z''}} are coordinate distances.

The stress–energy tensor is defined as the [[tensor]] {{math|''T''<sup>''αβ''</sup>}} of order two that gives the [[flux]] of the {{mvar|α}}th component of the [[momentum]] [[vector (geometric)|vector]] across a surface with constant {{math|''x''<sup>''β''</sup>}} [[coordinate]]. In the theory of [[general relativity|relativity]], this momentum vector is taken as the [[four-momentum]]. In general relativity, the stress–energy tensor is symmetric,{{efn|
"All the stress–energy tensors explored above were symmetric. That they could not have been otherwise one sees as follows."
{{right| — [[Gravitation (book)|Misner, Thorne, and Wheeler]]<ref>
 {{cite book
  |last1=Misner  |first1=C.W. |author1-link=Charles W. Misner
  |last2=Thorne  |first2=K.S. |author2-link=Kip Thorne
  |last3=Wheeler |first3=J.A. |author3-link=John Archibald Wheeler
  |year=2017  |orig-year=1973
  |section=Symmetry of the stress–energy tensor 
  |title=[[Gravitation (book)|Gravitation]] |edition=reprint
  |publisher=Princeton University Press
  |place=Princeton, NJ
  |isbn=978-0-6911-7779-3 
  |at=section&nbsp;5.7, pp.&nbsp;141–142
 }}
 </ref> }}
}}
<math display=block> T^{\alpha \beta} = T^{\beta \alpha} .</math>

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]].

== Components ==
Because the stress–energy tensor is of order 2, its components can be displayed in {{math|4 × 4}} matrix form:
<math display=block>
T^{\mu\nu} = \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}\,,</math>
where the indices {{mvar|μ}} and {{mvar|ν}} take on the values 0, 1, 2, 3.

In the following, {{mvar|k}} and {{mvar|ℓ}} range from 1 through 3:

{{ordered list
 | list-style-type=lower-alpha
 | 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 [[proper frame|co-moving frame of reference]].<ref>{{cite book |first1=Charles W. |last1=Misner |last2=Thorne |first2=Kip S. |last3=Wheeler |first3=John A. |year=1973 |title=Gravitation |place=San&nbsp;Francisco, CA |publisher=W.H. Freeman and Company |isbn=0-7167-0334-3}}</ref> It has a direct physical interpretation. In the case of a perfect fluid this component is
<math display=block>T^{00} = \rho~,</math>
where <math display=inline>\rho</math> is the [[relativistic mass]] per unit volume, and for an electromagnetic field in otherwise empty space this component is
<math display=block>T^{00} = {1 \over c^2}\left(\frac{1}{2} \epsilon_0 E^2 + \frac{1}{2\mu_0} B^2 \right),</math>
where {{mvar|E}} and {{mvar|B}} are the electric and magnetic fields, respectively.<ref>{{cite book |last=d'Inverno |first=R.A. |year=1992 |title=Introducing Einstein's Relativity |place=New York, NY |publisher=Oxford University Press |isbn=978-0-19-859686-8}}</ref>
| The flux of relativistic mass across the {{mvar|x{{sup|k}}}} surface is equivalent to the {{mvar|k}}th component of linear [[momentum density]],
<math display=block>T^{0k} = T^{k0}~.</math>
| The components
<math display=block> T^{k\ell}</math>
represent flux of {{mvar|k}}th component of linear momentum across the {{mvar|x{{sup|ℓ}}}} surface. In particular,
<math display=block> T^{kk}</math>
(not summed) represents [[tensile stress|normal stress]] in the {{mvar|k}}th co-ordinate direction ({{math|''k'' {{=}} 1, 2, 3}}), which is called "[[pressure]]" when it is the same in every direction, {{mvar|k}}. The remaining components
<math display=block> T^{k\ell} \quad k \ne \ell </math>
represent [[shear stress]] (compare with the [[stress (physics)|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, {{math|''T''{{sup|''μν''}}}} of the stress–energy tensor. However, it is often convenient to work with the covariant form,
<math display=block>T_{\mu \nu} = T^{\alpha \beta} g_{\alpha \mu} g_{\beta \nu} ,</math>
or the mixed form,
<math display=block>T^\mu{}_\nu = T^{\mu \alpha} g_{\alpha \nu} .</math>

This article uses the spacelike [[Sign convention#Metric signature|sign convention]] {{math|(− + + +)}} for the metric signature.

== Conservation law ==

=== In special relativity ===
{{see also|Relativistic angular momentum|Four-momentum}}

The stress–energy tensor is the conserved [[Noether's theorem|Noether current]] associated with [[spacetime]] [[translation (physics)|translation]]s.

The divergence of the non-gravitational stress–energy is zero. In other words, non-gravitational energy and momentum are conserved,
<math display=block> 0 = T^{\mu \nu}{}_{;\nu}\ \equiv\ \nabla_\nu T^{\mu \nu}{} ~.</math>
When gravity is negligible and using a [[Cartesian coordinate system]] for spacetime, this may be expressed in terms of partial derivatives as 
<math display=block> 0 = T^{\mu \nu}{}_{,\nu}\ \equiv\ \partial_{\nu} T^{\mu \nu} ~.</math>

The integral form of the non-covariant formulation is
<math display=block> 0 = \int_{\partial N} T^{\mu \nu} \mathrm{d}^3 s_{\nu} </math>
where {{mvar|N}} is any compact four-dimensional region of spacetime; <math display=inline> \partial N </math> is its boundary, a three-dimensional hypersurface; and <math display=inline> \mathrm{d}^3 s_{\nu} </math> 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:
<math display=block> 0 = (x^{\alpha} T^{\mu \nu} - x^{\mu} T^{\alpha \nu})_{,\nu} \,.</math>

=== 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 [[Divergence#Generalizations|coordinate-free definition of the divergence]] is used which incorporates the [[covariant derivative]]
<math display=block>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}</math>
where <math display=inline>\Gamma^{\mu}{}_{\sigma \nu} </math> is the [[Christoffel symbol]], which is the gravitational [[force field (physics)|force field]].{{citation needed|date=December 2024}}

Consequently, if <math display=inline>\xi^{\mu}</math> is any [[Killing vector field]], then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as
<math display=block>0 = \nabla_\nu \left(\xi^{\mu} T^{\nu}{}_{\mu}\right) = \frac{1}{\sqrt{-g}} \partial_\nu \left(\sqrt{-g}\ \xi^{\mu} T_{\mu}^{\nu}\right) </math>

The integral form of this is
<math display=block>0 = \int_{\partial N} \xi^{\mu} T^{\nu}{}_{\mu} \sqrt{-g} \ \mathrm{d}^3 s_{\nu}\,.</math>

== 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.<ref>{{cite book |last1=Landau |first1=L.D. |last2=Lifshitz |first2=E.M. |title=The Classical Theory of Fields |date=2010 |publisher=Butterworth-Heinemann |isbn=978-0-7506-2768-9 |pages=84–85 |edition=4th}}</ref>

Given a Lagrangian density <math display=inline>\mathcal{L}</math> that is a function of a set of fields <math display=inline>\phi_{\alpha}</math> and their derivatives, but explicitly not of any of the spacetime coordinates, we can construct the [[#Canonical stress–energy tensor|canonical stress–energy tensor]] by looking at the total derivative with respect to one of the generalized coordinates of the system. So, with our condition
<math display=block>\frac{\partial \mathcal{L}}{\partial x^{\nu}} = 0</math>

By using the chain rule, we then have
<math display=block>\frac{d \mathcal{L}}{dx^{\nu}} = d_{\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}}</math>

Written in useful shorthand,
<math display=block>d_{\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}</math>

Then, we can use the Euler–Lagrange Equation:
<math display=block>\partial_{\mu}\left(\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi_{\alpha})}\right) = \frac{\partial\mathcal{L}}{\partial \phi_{\alpha}}</math>

And then use the fact that partial derivatives commute so that we now have
<math display=block>d_{\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}</math>

We can recognize the right hand side as a product rule. Writing it as the derivative of a product of functions tells us that
<math display=block>d_{\nu}\mathcal{L} = \partial_{\mu}\left[\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi_{\alpha})}\partial_{\nu}\phi_{\alpha}\right]</math>

Now, in flat space, one can write <math display=inline>d_{\nu}\mathcal{L} = \partial_{\mu}[\delta^{\mu}_{\nu}\mathcal{L}]</math>. Doing this and moving it to the other side of the equation tells us that
<math display=block>\partial_{\mu}\left[\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi_{\alpha})}\partial_{\nu}\phi_{\alpha}\right] - \partial_{\mu}\left(\delta^{\mu}_{\nu}\mathcal{L}\right) = 0</math>

And upon regrouping terms,
<math display=block>\partial_{\mu}\left[\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi_{\alpha})}\partial_{\nu}\phi_{\alpha} - \delta^{\mu}_{\nu}\mathcal{L}\right] = 0</math>

This is to say that the divergence of the tensor in the brackets is 0. Indeed, with this, we define the stress–energy tensor:
<math display=block>T^{\mu}{}_{\nu} \equiv \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi_{\alpha})}\partial_{\nu}\phi_{\alpha} - \delta^{\mu}_{\nu}\mathcal{L}</math>

By construction it has the property that
<math display=block>\partial_{\mu}T^{\mu}{}_{\nu} = 0</math>

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 <math display=inline>T^{0}{}_{0}</math> 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 <math display=inline>\partial_{\mu}T^{\mu}{}_{0} = 0</math>, we then have
<math display=block> \frac{\partial \mathcal{H}}{\partial t} + \nabla\cdot\left(\frac{\partial \mathcal{L}}{\partial\nabla \phi_{\alpha}}\dot{\phi}_{\alpha}\right) = 0</math>

We can then conclude that the terms of <math display=inline>\frac{\partial \mathcal{L}}{\partial\nabla \phi_{\alpha}}\dot{\phi}_{\alpha}</math> represent the energy flux density of the system.

=== Trace ===
The trace of the stress–energy tensor is defined to be {{tmath|1= T^{\mu}{}_{\mu} }}, so
<math display=block>T^{\mu}{}_{\mu} = \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi_{\alpha})}\partial_{\mu}\phi_{\alpha}-\delta^{\mu}_{\mu}\mathcal{L} .</math>

Since {{tmath|1= \delta^{\mu}_{\mu} = 4 }},
<math display=block>T^{\mu}{}_{\mu} = \frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi_{\alpha})}\partial_{\mu}\phi_{\alpha}-4\mathcal{L} .</math>

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

In general relativity, the [[partial derivatives]] used in special relativity are replaced by [[covariant derivative]]s. 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 energy|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 [[Stress–energy–momentum pseudotensor#Landau–Lifshitz pseudotensor|Landau–Lifshitz pseudotensor]] is a unique way to define the ''gravitational'' field energy and momentum densities. Any such [[Stress–energy–momentum pseudotensor|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|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
<math display=block>G_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu} ,</math>
where <math display="inline">G_{\mu \nu} = R_{\mu \nu} - \tfrac{1}{2} R\,g_{\mu \nu}</math> is the [[Einstein tensor]], <math display=inline>R_{\mu \nu}</math> is the [[Ricci tensor]], <math display=inline>R = g^{\alpha \beta}R_{\alpha \beta}</math> is the [[scalar curvature]], <math display=inline>g_{\mu \nu}\,</math> is the [[metric tensor (general relativity)|metric tensor]], {{math|Λ}} is the [[cosmological constant]] (negligible at the scale of a galaxy or smaller), and <math display=inline>\kappa = 8\pi G/c^4</math> is the [[Einstein gravitational constant]].

== Stress–energy in special situations ==

=== Isolated particle ===
In special relativity, the stress–energy of a non-interacting particle with rest mass {{mvar|m}} and trajectory <math display=inline> \mathbf{x}_\text{p}(t)</math> is:
<math display=block>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)) </math>
where <math display=inline>v^{\alpha}</math> is the velocity vector (which should not be confused with [[four-velocity]], since it is missing a <math display=inline>\gamma</math>)
<math display=block>v^{\alpha} = \left(1, \frac{d \mathbf{x}_\text{p}}{dt}(t) \right) \,,</math>
<math display=inline>\delta</math> is the [[Dirac delta function]] and <math display="inline"> E = \sqrt{p^2 c^2 + m^2 c^4} </math> is the [[energy]] of the particle.

Written in the language of classical physics, the stress–energy tensor would be (relativistic mass, momentum, the [[dyadic product]] of momentum and velocity) 
<math display=block>\left( \frac{E}{c^2} , \, \mathbf{p} , \, \mathbf{p} \, \mathbf{v} \right) \,.</math>

=== Stress–energy of a fluid in equilibrium ===
For a [[perfect fluid]] in [[thermodynamic equilibrium]], the stress–energy tensor takes on a particularly simple form
<math display=block>T^{\alpha \beta} \, = \left(\rho + {p \over c^2}\right)u^{\alpha}u^{\beta} + p g^{\alpha \beta}</math>
where <math display=inline>\rho</math> is the mass–energy density ([[kilogram]]s per cubic meter), <math display=inline>p</math> is the hydrostatic pressure ([[pascal (unit)|pascals]]), <math display=inline>u^{\alpha}</math> is the fluid's [[four-velocity]], and <math display=inline>g^{\alpha \beta}</math> is the matrix inverse of the [[metric tensor (general relativity)|metric tensor]]. Therefore, the trace is given by
<math display=block>T^{\alpha}{}_{\,\alpha} = g_{\alpha\beta} T^{\beta \alpha} = 3p - \rho c^2 \,.</math>

The [[four-velocity]] satisfies
<math display=block>u^{\alpha} u^{\beta} g_{\alpha \beta} = - c^2 \,.</math>

In an [[inertial frame of reference]] comoving with the fluid, better known as the fluid's [[proper frame]] of reference, the four-velocity is 
<math display=block>u^{\alpha} = (1, 0, 0, 0) \,,</math>

the matrix inverse of the metric tensor is simply
<math display=block>
g^{\alpha \beta} \, = \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)
</math>
and the stress–energy tensor is a diagonal matrix
<math display=block>
T^{\alpha \beta} = \left( \begin{matrix}
                   \rho & 0 & 0 & 0 \\
                   0 & p & 0 & 0 \\
                   0 & 0 & p & 0 \\
                   0 & 0 & 0 & p    
      \end{matrix} \right).
</math>

=== Electromagnetic stress–energy tensor ===
{{main|Electromagnetic stress–energy tensor}}
The Hilbert stress–energy tensor of a source-free electromagnetic field is
<math display=block> 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) </math>
where <math display=inline> F_{\mu \nu} </math> is the [[electromagnetic field tensor]].

=== Scalar field ===
{{main|Klein–Gordon equation}}
The stress–energy tensor for a complex scalar field <math display=inline>\phi </math> that satisfies the Klein–Gordon equation is
<math display=block>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} m c^2 \bar\phi \phi ,</math>
and when the metric is flat (Minkowski in Cartesian coordinates) its components work out to be:
<math display=block>\begin{align}
  T^{00} & = \frac{\hbar^2}{m c^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}{m c^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 + m c^2 \bar{\phi} \phi\right). 
\end{align}</math>

== Variant definitions of stress–energy ==
There are a number of inequivalent definitions<ref>{{cite journal |doi=10.1016/j.nuclphysb.2020.115240 |title=Noether and Hilbert (metric) energy–momentum tensors are not, in general, equivalent |year=2021 |last1=Baker |first1=M.R.  |last2=Kiriushcheva |first2=N. |last3=Kuzmin |first3=S.  |journal=Nuclear Physics B |volume=962 |issue=1 |pages=115240
 |arxiv=2011.10611 |bibcode=2021NuPhB.96215240B |s2cid=227127490 |url=https://doi.org/10.1016/j.nuclphysb.2020.115240 }}</ref> of non-gravitational stress–energy:

=== Hilbert stress–energy tensor ===
The Hilbert stress–energy tensor is defined as the [[functional derivative]]
<math display=block>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},
</math>
where <math display=inline>S_{\mathrm{matter}}</math> is the nongravitational part of the [[action (physics)|action]], <math display=inline>\mathcal{L}_{\mathrm{matter}}</math> is the nongravitational part of the [[Lagrangian (field theory)|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; for details see the section above on the stress–energy tensor in special relativity. 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 transformation]]s 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 pseudotensor.

=== Belinfante–Rosenfeld stress–energy tensor ===
{{main|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|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.

== See also ==
{{cols|colwidth=26em}}
* [[Electromagnetic stress–energy tensor]]
* [[Energy condition]]
* [[Energy density#Energy density of electric and magnetic fields|Energy density of electric and magnetic fields]]
* [[Maxwell stress tensor]]
* [[Poynting vector]]
* [[Ricci calculus]]
* [[Segre classification]]
{{colend}}

== Notes ==
{{notelist}}

== References ==
{{reflist|25em}}

== Further reading ==
* {{cite journal |first=Walter |last=Wyss |date=14 July 2005 |title=The energy–momentum tensor in classical field theory |journal=Universal Journal of Physics and Applications |series=''{{small|Old and New}} Concepts of Physics'' {{grey|[prior journal name]}} |volume=II |issue=3–4 |pages=295–310 |issn=2331-6543 |quote=...&nbsp;classical field theory and in particular in the role that a divergence term plays in a lagrangian&nbsp;... |url=https://www.hrpub.org/download/20040201/UJPA-18490185.pdf }}

== External links ==
* [https://web.archive.org/web/20060430094645/http://people.hofstra.edu/faculty/Stefan_Waner/diff_geom/Sec12.html Lecture, Stephan Waner]
* [https://web.archive.org/web/20140530175713/http://www.black-holes.org/numrel1.html Caltech Tutorial on Relativity] &mdash; A simple discussion of the relation between the stress–energy tensor of general relativity and the metric

{{Tensors}}

{{DEFAULTSORT:Stress-energy tensor}}
[[Category:Tensor physical quantities]]
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