The Stress–Energy Tensor

The right-hand side of the Einstein field equations is the stress–energy tensor (equivalently, the energy–momentum tensor) $T_{\mu \nu }$ — the source of gravity. Where Newtonian gravity is sourced by mass density alone, general relativity is sourced by the full density and flux of energy and momentum, including pressure, stress, and the energy of fields. This page builds $T_{\mu \nu }$, catalogues its standard forms, and derives the conservation law ${\nabla }_{\mu }{T}^{\mu \nu }=0$ that ties it to the geometry.

We use $c=1$, keep $G$ explicit, and adopt the mostly-plus signature $(-,+,+,+)$.

Definition and components

$T_{\mu \nu }$ is a symmetric $(0,2)$ tensor field whose components measure the flux of the $\mu$-component of four-momentum across a surface of constant $x^{\nu }$. In an orthonormal frame the components have direct physical readings:

$$T_{\mu \nu }=\left(\begin{array}{cc}{T}_{00}& T_{0j}\\ T_{i0}& T_{ij}\end{array}\right)=\left(\begin{array}{cc}\text{energy density}& \text{energy flux}\\ \text{momentum density}& \text{momentum flux (stress)}\end{array}\right).$$

• $T_{00}=\rho$ — energy density (mass density in the rest frame, $c=1$);
• $T_{0i}$ — energy flux in direction $i$; equal (symmetry) to the momentum density;
• $T_{ij}$ — stress: the $i$-momentum flowing in the $j$-direction. Diagonal entries are pressure, off-diagonal are shear stress.

Symmetry $T_{\mu \nu }=T_{\nu \mu }$ encodes the equality of momentum density and energy flux, and (in flat space) the conservation of angular momentum. This is the same object that appears in special-relativistic field theory; GR simply lets it curve spacetime and evaluates it on the curved metric.

Canonical forms

Dust (pressureless matter)

A cloud of non-interacting particles with rest-mass density $\rho$ and common four-velocity $u^{\mu }$ ($u^{\mu }{u}_{\mu }=-1$):

$$T^{\mu \nu }=\rho u^{\mu }{u}^{\nu }.$$

The simplest source — galaxies on cosmological scales, or any cold, collisionless matter. Conservation ${\nabla }_{\mu }{T}^{\mu \nu }=0$ splits into continuity (${\nabla }_{\mu }(\rho u^{\mu })=0$) plus the geodesic equation $u^{\mu }{\nabla }_{\mu }{u}^{\nu }=0$: dust free-falls, recovering the claim that matter follows geodesics as a consequence of the field equations rather than a separate postulate.

Perfect fluid

Matter with isotropic pressure $p$ and no shear or heat conduction, density $\rho$, four-velocity $u^{\mu }$:

$$\boxed{T^{\mu \nu }=(\rho +p)u^{\mu }{u}^{\nu }+p{g}^{\mu \nu }.}$$

In the fluid rest frame $T^{\mu }{}_{\nu }=\mathrm{diag}(-\rho ,p,p,p)$. This is the workhorse source: stars, and the cosmic fluid of FLRW cosmology. Its equation of state $p=w\rho$ classifies the content:

Conservation gives the relativistic Euler equation; in the Newtonian limit it reduces to ${\partial }_t\rho +\nabla \cdot (\rho \mathbf{v})=0$ and $\rho \dot{\mathbf{v}}=-\nabla p-\rho \nabla \Phi$.

Electromagnetic field

The covariant Maxwell field $F_{\mu \nu }$ on a curved background sources gravity through

$$T^{\mu \nu }=F^{\mu \alpha }{F}^{\nu }{}_{\alpha }-\frac{1}{4}{g}^{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta },$$

which is traceless ($T^{\mu }{}_{\mu }=0$) — a general feature of massless/conformally invariant fields. Its energy density is $\frac{1}{2}(E^2+B^2)$ and its momentum density the Poynting vector, now curving spacetime. Radiation ($w=1/3$) shares this tracelessness.

Scalar field

A scalar field $\varphi$ with potential $V(\varphi )$ — central to inflation and dark-energy models — has

$$T_{\mu \nu }={\nabla }_{\mu }\varphi {\nabla }_{\nu }\varphi -g_{\mu \nu }(\frac{1}{2}{\nabla }^{\alpha }\varphi {\nabla }_{\alpha }\varphi +V(\varphi )),$$

which behaves as a perfect fluid with $\rho =\frac{1}{2}{\dot{\varphi }}^2+V$ and $p=\frac{1}{2}{\dot{\varphi }}^2-V$; a slowly rolling field (${\dot{\varphi }}^2\ll V$) gives $p\approx -\rho$, mimicking a cosmological constant.

Where $T_{\mu \nu }$ comes from: the variational definition

The forms above are not guessed independently. The systematic source is the matter action $S_{\text{matter}}[\varphi ,g]$: the stress–energy tensor is its response to varying the metric,

$$\boxed{T_{\mu \nu }=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\text{matter}}}{\delta g^{\mu \nu }}.}$$

This is the definition that makes the Einstein–Hilbert variation produce $G_{\mu \nu }=8\pi G{T}_{\mu \nu }$ with a consistent right-hand side, and it automatically yields a symmetric, generally-covariant tensor. (The naive Noether/canonical tensor from spacetime-translation symmetry is generally neither symmetric nor gauge-invariant; the metric-variation — "Hilbert" — tensor is the correct gravitational source, and the Belinfante–Rosenfeld procedure reconciles the two.) The variational route is developed further in the planned foundations/action.md.

Conservation and its geometric meaning

Because the Einstein tensor is divergence-free by the contracted Bianchi identity, the field equations force

$$\boxed{{\nabla }_{\mu }{T}^{\mu \nu }=0.}$$

This is local energy–momentum conservation. Two cautions distinguish it from the flat-space law:

• It is $\nabla$, not $\partial$. Expanded, ${\nabla }_{\mu }{T}^{\mu \nu }={\partial }_{\mu }{T}^{\mu \nu }+{\Gamma }^{\mu }{}_{\mu \lambda }{T}^{\lambda \nu }+{\Gamma }^{\nu }{}_{\mu \lambda }{T}^{\mu \lambda }=0$. The $\Gamma$ terms describe the exchange of energy and momentum between matter and the gravitational field — e.g. a photon climbing out of a well loses energy to the field (redshift).
• No global conserved charge, in general. Unlike SR, ${\nabla }_{\mu }{T}^{\mu \nu }=0$ does not integrate to a conserved total four-momentum on a curved spacetime, because $T_{\mu \nu }$ omits the energy of gravity itself, and gravitational energy has no local, tensorial density (it can be transformed away pointwise by the EEP). Globally conserved quantities are recovered only when the spacetime has symmetries (Killing vectors) or suitable asymptotic structure (ADM/Bondi mass).

Killing vectors and conserved quantities

When a spacetime has a continuous symmetry — a Killing vector ${\xi }^{\mu }$ satisfying ${\nabla }_{(\mu }{\xi }_{\nu )}=0$ — the current $J^{\mu }=T^{\mu \nu }{\xi }_{\nu }$ is covariantly conserved, ${\nabla }_{\mu }{J}^{\mu }=0$, and yields a genuine conserved charge. A timelike Killing vector gives conserved energy (as in the static Schwarzschild exterior); rotational Killing vectors give conserved angular momentum (as for Kerr). This is the curved-space shadow of Noether's theorem and the practical route to constants of motion along geodesics.

Summary

• $T_{\mu \nu }$ is the symmetric source of gravity: $T_{00}$ energy density, $T_{0i}$ energy flux = momentum density, $T_{ij}$ stress (pressure + shear).
• Canonical forms: dust $\rho u^{\mu }{u}^{\nu }$, perfect fluid $(\rho +p)u^{\mu }{u}^{\nu }+p{g}^{\mu \nu }$ (with equation of state $p=w\rho$), electromagnetic (traceless), scalar field.
• The correct, symmetric, covariant source is the Hilbert tensor $T_{\mu \nu }=-\frac{2}{\sqrt{-g}}\delta S_{\text{matter}}/\delta g^{\mu \nu }$.
• The Bianchi identity forces local conservation ${\nabla }_{\mu }{T}^{\mu \nu }=0$; the $\Gamma$ terms exchange energy with gravity, and global charges exist only given Killing symmetries or asymptotic structure.

Next

Not every symmetric tensor is a physically reasonable source. The energy conditions impose positivity constraints on $T_{\mu \nu }$ — that energy densities are non-negative and pressures not too negative — which underpin the singularity theorems and the positive-mass results.