stressenergy

The stress-energy tensor, also called the stress-energy–momentum tensor, is a rank-2 tensor T^{μν} that encodes the density and flux of energy and momentum in spacetime. It unifies energy density, momentum density, and stresses (pressures and shear) of matter and fields at a point.

In special relativity, with flat spacetime, components have intuitive meaning: T^{00} is energy density, T^{0i} is energy flux or momentum density, and T^{ij} describe stresses. In general relativity, T^{μν} is the source of spacetime curvature in Einstein's field equations G_{μν} = 8πG/c^4 T^{μν}, and it obeys ∇_μ T^{μν} = 0.

T^{μν} can be derived from a Lagrangian via variation with respect to the metric: T^{μν} = -2/√(-g) δS/δg_{μν}. For flat spacetime and translational symmetry, Noether's theorem yields the same conserved quantity. In many theories, T^{μν} is symmetric (often after improvements).

Common examples: a perfect fluid has T^{μν} = (ρ + p) u^μ u^ν + p g^{μν}, where ρ is rest energy density, p pressure, and u^μ the four-velocity. The electromagnetic field has T^{μν} = F^{μαν} F^{ν}{}_α − 1/4 g^{μν} F^{αβ}F_{αβ}. A real scalar field gives T^{μν} = ∂^μ φ ∂^ν φ − 1/2 g^{μν} ∂_α φ ∂^α φ.

Conservation and applications: ∇_μ T^{μν} = 0 expresses local energy-momentum conservation, yielding hydrodynamic and gravitational dynamics. The trace and energy conditions constrain possible T^{μν}; for conformal theories the trace can vanish, and energy conditions demand nonnegative energy density for all timelike observers.