Stress-energy tensor

The stress-energy tensor describes local energy density, momentum density, and stress. Compatibility with the Einstein field equations requires covariant conservation ${\nabla }^{\mu }{T}_{\mu \nu }=0$.

Adding ${\partial }_{\rho }{B}^{\rho \mu \nu }$ with $B^{\rho \mu \nu }=-B^{\mu \rho \nu }$ to a stress-energy tensor preserves its divergence identically, since commuting partial derivatives annihilates the antisymmetric pair. The total four-momentum changes only by a spatial boundary term. Improvements can make local densities symmetric or gauge-invariant; equality of integrated charges requires the chosen boundary conditions. Equation-of-motion terms can also relate representatives on shell.

A stress-energy superpotential is an antisymmetric tensor $B^{\rho \mu \nu }=-B^{\mu \rho \nu }$ whose divergence supplies a stress-energy tensor improvement. In the Maxwell field, the potential-dependent expression $B^{\rho \mu }{}_{\nu }=F^{\mu \rho }{A}_{\nu }$ produces the difference between canonical and gauge-invariant stress-energy tensors on shell. Antisymmetry ensures identically conserved improvement terms.

A pressureless fluid with proper number density $n_0$, rest mass $m$ and four-velocity $u$ has tensor $m{n}_0{u}^{\mu }{u}^{\nu }$. In its local rest frame there is energy density but no spatial stress. Particle-number conservation and the Lorentz force make its divergence the electromagnetic force density.

Momentum density is momentum per unit physical volume. In a local orthonormal tetrad of the measuring observer it is $q^{\hat{i}}=T^{\hat{0}\hat{i}}$. A perfect fluid in general relativity moving at a small velocity relative to that observer has $q^{\hat{i}}=(\rho +P)v^{\hat{i}}$ to first order. For isotropic background photons, $P=\rho /3$, so the factor is $4\rho /3$ rather than $\rho$. Integrating the kinetic stress-energy tensor gives $q^{\hat{i}}=\int f{p}^{\hat{i}}{d}^{3}p$.

In a local orthonormal tetrad, anisotropic stress is the trace-free spatial part of the stress-energy tensor, conventionally denoted ${\pi }^{ij}$. Some linear cosmological perturbation theory conventions instead use ${\Pi }^{ij}=-{\pi }^{ij}$ and write $T^{ij}=P{\delta }^{ij}-{\Pi }^{ij}$. The sign must be specified before using a photon Euler equation. The photon quadrupole determines the linear photon anisotropic stress.

For action $S=\int \sqrt{-g}[-\frac{1}{2}(\nabla \Phi {)}^2-\xi R{\Phi }^2]d^{4}x$, metric variation defines $\delta S=\frac{1}{2}\int \sqrt{-g}{T}^{ab}\delta g_{ab}{d}^{4}x$ and gives the displayed tensor. The contracted Bianchi identity and the scalar derivative commutator yield ${\nabla }^a{T}_{ab}=(\square \Phi -2\xi R\Phi ){\nabla }_b\Phi$, so the stress-energy tensor is conserved on the field equation.

Local energy-momentum conservation is the covariant equation ${\nabla }_{\mu }{T}^{\mu \nu }=0$. Its normal and spatial projections give energy and momentum evolution equations in a spacetime foliation.

A Noether identity is an off-shell differential relation among Euler-Lagrange expressions implied by a local gauge symmetry. Diffeomorphism invariance relates the divergence of the stress-energy tensor to the matter field equations.

For a diffeomorphism-invariant pure metric action, write $\delta S=-\int \sqrt{-g}{E}^{ab}\delta g_{ab}$. Substitute $\delta g_{ab}={\nabla }_a{\xi }_b+{\nabla }_b{\xi }_a$ for an arbitrary compactly supported vector field and integrate by parts. Symmetry of $E^{ab}$ gives ${\nabla }_a{E}^{ab}=0$ without using field equations. This Noether identity concerns the gravitational variational tensor alone.

Simultaneously varying a metric tensor and a scalar field by the Lie derivative along a compactly supported vector field leaves a covariant matter action invariant. Integration by parts gives the displayed Noether identity. It implies stress-energy conservation when the scalar equation holds without needing to impose the metric field equation.

A relativistic perfect fluid has energy density $\rho$, isotropic pressure $p$ and unit timelike four-velocity $u$. Projecting ${\nabla }_a{T}^{ab}=0$ along and orthogonally to $u$ gives its energy and Euler equations.

The timelike projection of stress-energy conservation is $u^a{\nabla }_a\rho +(\rho +p){\nabla }_a{u}^a=0$. It is the local adiabatic first law for a perfect fluid.

The spatial projection of stress-energy conservation is $(\rho +p)u^b{\nabla }_b{u}^a=-(g^{ab}+u^a{u}^b){\nabla }_{b}p$.

The Tolman–Oppenheimer–Volkoff equation governs a static, spherically symmetric perfect fluid in general relativity. In geometrized units, if $m(r)$ is the mass inside areal radius $r$, then

The ratio of enclosed gravitational mass to areal radius in units $G=c=1$. In a spherical star the local value is $m(r)/r$, and the surface value is $M/R$. Bounds on this ratio constrain relativistic stellar models; it is unrelated to topological compactness.

For a smooth centre with density ${\rho }_c$ and pressure $p_c$, the Tolman–Oppenheimer–Volkoff equation gives $m(r)=4\pi {\rho }_c{r}^3/3+O(r^5)$ and $p(r)=p_c-2\pi ({\rho }_c+p_c)({\rho }_c/3+p_c)r^2+O(r^4)$. A fixed barotropic stellar equation of state leaves one physical central parameter; the additive lapse constant fixes time normalization.

The interior Schwarzschild metric is the exact static solution for a spherical fluid of constant density. If the star has mass $M$ and radius $R$, its pressure is

The Interior Schwarzschild metric has central pressureIt diverges when $M/R=4/9$, the limiting compactness in Buchdahl's theorem.

The dominant energy condition for the Interior Schwarzschild metric reduces to its largest pressure satisfying $p_c\le \rho$. Substitution of the exact central pressure gives $\sqrt{1-2M/R}\ge 1/2$, hence $M/R\le 3/8$. Equality has finite central pressure and is allowed by the non-strict condition.

Buchdahl's theorem gives $2M/R\le 8/9$ for a static spherical perfect-fluid star whose density is nonincreasing outwards, subject to its regularity assumptions. Equality would require divergent central pressure, so every regular such star satisfies $M/R<4/9$.

Let ${\rho }_0>0$ be a density threshold and let $a,\mu$ be the radius and enclosed mass where a monotonically decreasing stellar density reaches that threshold. Then $\mu \ge 4\pi {\rho }_0{a}^3/3$. A local compactness bound $\mu /a<4/9$ implies $a<(3\pi {\rho }_0{)}^{-1/2}$ and $\mu <4/(9\sqrt{3\pi {\rho }_0})$, independently of the equation of state above the threshold. A total-mass bound also requires suitable control of the known low-density envelope.

The null energy condition requires $T_{\mu \nu }{k}^{\mu }{k}^{\nu }\ge 0$ for every null vector $k$. Through the Einstein field equations this is equivalent to $R_{\mu \nu }{k}^{\mu }{k}^{\nu }\ge 0$ when the cosmological term is absent.

In a planar warped spacetime, contraction of the Ricci tensor with a null vector in an orthonormal frame gives $R_{ab}{k}^a{k}^b=-2(\log A{)}^{\prime \prime }(k^3{)}^2$. The Einstein field equations identify this with $8\pi T_{ab}{k}^a{k}^b$; metric terms, including a cosmological constant, vanish in a null contraction. Thus the null energy condition is equivalent to concavity of $\log A$ in the proper transverse coordinate $z$. Null directions confined to a planar slice saturate the condition.

The null convergence condition requires $R_{ab}{k}^a{k}^b\ge 0$ for every null vector. Through the Einstein field equations, it follows from the null energy condition, since all terms proportional to the metric tensor vanish on a null vector.

The averaged null energy condition requiresalong every complete affinely parametrized null geodesic. It is weaker than the pointwise null energy condition.

For the curvature-coupled scalar equation considered here, contraction with an affine null tangent and integration by parts gives, for $\xi <0$,when $\Phi {\Phi }^{\prime }/(1-\xi {\Phi }^2)$ vanishes at the endpoints.

For ${\nabla }_{\alpha }{\nabla }^{\alpha }\varphi =m^2\varphi$, considerCovariant conservation for all solutions holds exactly when

For a real scalar satisfying ${\nabla }_a{\nabla }^a\psi -{\mu }^2\psi =0$,is symmetric and covariantly conserved.

The weak energy condition requiresfor every timelike vector $X$, so every observer measures nonnegative energy density.

For the general scalar ansatz in Conserved Klein-Gordon scalar stress-energy ansatz, work in a unit timelike frame. ThenThe weak energy condition for arbitrary local field data is therefore equivalent to

The strong energy condition requires $(T_{ab}-T{g}_{ab}/2)u^a{u}^b\ge 0$ for every timelike vector $u$ in four spacetime dimensions. Through the Einstein field equations, it implies the timelike convergence condition $R_{ab}{u}^a{u}^b\ge 0$ when the cosmological constant vanishes.

The dominant energy condition requires the energy flux measured by every future-directed timelike observer to be future-directed and causal. Equivalently, $T_{ab}{u}^a{v}^b\ge 0$ for every pair of future-directed causal vectors $u$ and $v$ under the $(-,+,+,+)$ convention.

For a scalar field with $X=-q^2/2$, Lagrangian $F(X)$ and stress-energy tensor $T_{ab}=F\prime q_a{q}_b+F{g}_{ab}$, the inequalities $A=F\prime \ge 0$ and $Y=XF\prime -F\ge 0$ imply the dominant energy condition for any gradient type. In a local Lorentz frame, $\rho =A(q_0^2+{\sum }_i{q}_i^2)/2+Y\ge 0$ and ${\rho }^2-{\sum }_i{T}_{0i}^2=A^2{X}^2+A(q_0^2+{\sum }_i{q}_i^2)Y+Y^2\ge 0$.