The Stress-Energy Tensor: What Matter Tells Space
Einstein’s field equations relate geometry (curvature) to matter (energy and momentum). The geometry side uses the Ricci tensor and scalar. The matter side uses the stress-energy tensor $T^{\mu\nu}$ — a symmetric rank-(2,0) tensor that encodes everything about how matter and energy are distributed and flowing at each point in spacetime.
What each component means
$T^{\mu\nu}$ is a $4 \times 4$ symmetric matrix (10 independent components). In a local inertial frame:
| $\nu = 0$ (time) | $\nu = 1$ (x) | $\nu = 2$ (y) | $\nu = 3$ (z) | |
|---|---|---|---|---|
| $\mu = 0$ | Energy density $\rho c^2$ | Energy flux in $x$ | Energy flux in $y$ | Energy flux in $z$ |
| $\mu = 1$ | Momentum density in $x$ | Pressure / stress $_{xx}$ | Shear stress $_{xy}$ | Shear stress $_{xz}$ |
| $\mu = 2$ | Momentum density in $y$ | Shear stress $_{yx}$ | Pressure / stress $_{yy}$ | Shear stress $_{yz}$ |
| $\mu = 3$ | Momentum density in $z$ | Shear stress $_{zx}$ | Shear stress $_{zy}$ | Pressure / stress $_{zz}$ |
Hover over each cell to see what it represents physically. Toggle “Perfect Fluid” to see how most astrophysical situations simplify to just two numbers.
Key components
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$T^{00}$: Energy density — the amount of energy per unit volume. For non-relativistic matter, this is approximately $\rho c^2$ where $\rho$ is mass density. This is the dominant component in most situations and the one that corresponds to Newton’s gravitational source.
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$T^{0i} = T^{i0}$: Energy flux = momentum density. These are equal (a deep consequence of relativity): the flow of energy in direction $i$ equals the density of momentum in direction $i$.
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$T^{ij}$ (spatial components): Stress. The diagonal elements are pressure (force per area perpendicular to the surface). The off-diagonal elements are shear stress.
Perfect fluids
Most astrophysical matter (stars, gas, the universe’s contents) is well-modelled as a perfect fluid: no viscosity, no heat conduction. The stress-energy tensor simplifies dramatically:
$$T^{\mu\nu} = (\rho + p/c^2) u^\mu u^\nu + p \, g^{\mu\nu}$$
where $\rho$ is mass-energy density, $p$ is pressure, and $u^\mu$ is the fluid’s four-velocity. Two numbers instead of ten.
Conservation
The stress-energy tensor satisfies a conservation law:
$$\nabla_\mu T^{\mu\nu} = 0$$
This is four equations (one for each $\nu$). The $\nu = 0$ equation is conservation of energy. The $\nu = 1,2,3$ equations are conservation of momentum. In flat spacetime, these reduce to the familiar conservation laws. In curved spacetime, the covariant derivative $\nabla_\mu$ includes gravitational effects — energy and momentum are “conserved” in the curved-space sense.
What curves spacetime
In Newtonian gravity, only mass creates gravity. In GR, everything in $T^{\mu\nu}$ curves spacetime: energy, momentum, pressure, stress. Pressure gravitates. This is why neutron stars — where pressure is enormous — are more curved than you’d expect from their mass alone, and why the interior of a collapsing star can’t stop gravitational collapse once it passes a critical point.