The Stress-Energy Tensor

Core Idea

The stress-energy tensor T_μν is the source term in Einstein's field equations — it encodes the density and flux of energy and momentum for all non-gravitational matter and fields. T_{00} is the energy density, T_{0i} is the momentum density (or equivalently the energy flux), and T_{ij} is the stress (momentum flux, including pressure and shear). For a perfect fluid, T_μν = (ρ + p/c²)u_μ u_ν + p g_μν, where ρ is the energy density, p is the pressure, and u^μ is the four-velocity. The local conservation law ∇^μ T_μν = 0 generalizes energy-momentum conservation to curved spacetime and is automatically enforced by the Einstein equations through the contracted Bianchi identity. The stress-energy tensor is symmetric (T_μν = T_νμ), which ensures conservation of angular momentum.

Explainer

The stress-energy tensor is the object that tells spacetime how to curve. In Newtonian gravity, the source of the gravitational field is mass density ρ, a single scalar. In general relativity, a single scalar is insufficient: the source must be a symmetric (0,2) tensor T_μν to match the Einstein tensor on the other side of the field equations. This is because energy, momentum, and stress all contribute to gravity in relativity. A moving object has more energy (and therefore more gravitational effect) than a stationary one. Pressure contributes to the gravitational field. Stresses — internal forces within a material — contribute. All of this information is packaged into T_μν.

The physical interpretation of the components is clearest in a local inertial frame. T_{00} is the energy density (including rest-mass energy). T_{0i} = T_{i0} is the momentum density in the i-direction, which is equivalently the flux of energy in the i-direction — this equivalence is a consequence of the symmetry T_μν = T_νμ and is a relativistic identity (energy flow carries momentum). T_{ij} is the flux of i-momentum in the j-direction, which is the stress tensor from continuum mechanics: the diagonal components T_{ii} are the pressures (normal stress), and the off-diagonal components T_{ij} (i ≠ j) are the shear stresses.

The most important special case is the perfect fluid: a fluid with no viscosity or heat conduction, characterized entirely by its energy density ρ, pressure p, and four-velocity u^μ. Its stress-energy tensor is T_μν = (ρ + p/c²)u_μ u_ν + p g_μν. In the fluid's rest frame, this reduces to T_{00} = ρ, T_{ij} = p δ_{ij}, with no momentum density or shear stress. The perfect fluid model describes the matter content in most cosmological models, the interior of stars, and many other astrophysical situations. For dust (pressureless matter), p = 0 and T_μν = ρ u_μ u_ν. For radiation, p = ρc²/3, which follows from the tracelessness of the electromagnetic stress-energy tensor.

The conservation law ∇^μ T_μν = 0 is the curved-spacetime generalization of energy-momentum conservation. In flat spacetime with Cartesian coordinates, it reduces to ∂^μ T_μν = 0, which can be integrated over a spatial volume to give global conservation of energy and momentum. In curved spacetime, the covariant divergence cannot generally be converted to a global conservation law because there is no coordinate-invariant way to compare vectors (including momentum vectors) at different points. Global energy conservation is recovered only in spacetimes with special symmetries — specifically, those possessing a timelike Killing vector field. In an expanding universe, for example, the energy of photons decreases as they redshift, and there is no compensating increase elsewhere. This is not a violation of ∇^μ T_μν = 0 — the local law is satisfied everywhere — but rather a reflection of the fact that total energy is not a meaningful concept in a general curved spacetime.