Relativistic energy and the stress tensor

Throughout this module use signature $(+,-,-,-)$ and $x^0=ct$. A massive observer has four-velocity $u^\mu=dx^\mu/d\tau$ with $u^\mu u_\mu=c^2$. The proper time $\tau$ is read by a clock moving along that observer’s trajectory. For a particle of rest mass $m$, four-momentum is $p^\mu=mu^\mu$. Its squared norm is invariant even though its energy and spatial momentum change between inertial frames.

An observer with four-velocity $w^\mu$ measures energy $E=p_\mu w^\mu$. In that observer’s inertial rest frame this becomes $E=cp^0$. Thus “the particle’s energy” always implies an observer, whereas its rest mass does not. For a massless particle no rest frame or timelike four-velocity exists; momentum remains meaningful and has zero norm. These distinctions prevent the misleading claim that all energy is simply rest mass multiplied by a universal constant.

In a local Cartesian inertial frame, $T^{00}$ is energy per volume. The components $cT^{0i}$ describe energy flux, while $T^{ij}$ describe flux of the ith momentum component across a surface normal to direction j. For a symmetric matter stress tensor, $T^{i0}/c$ is momentum density. The factors of c are necessary because all tensor components here have the units of energy density. Forgetting them can produce dimensionally plausible but physically incorrect conservation equations.

Consider a gas at rest on average. Its total momentum density can vanish while molecules still strike a wall and transfer momentum. Isotropy makes the spatial stress diagonal with pressure p. An ideal perfect fluid assumes no viscosity or heat conduction in its own rest frame. It is a useful macroscopic model, not a universal description of plasma turbulence, a solid under shear, or matter far from local equilibrium.

In flat spacetime an isolated matter system obeys $\partial_\mu T^{\mu\nu}=0$. Integrating its time component over a region relates changing energy inside to energy flowing through the boundary. Curved spacetime replaces the partial derivative by the metric-compatible covariant derivative. This expresses local energy-momentum accounting in freely falling frames; it does not automatically produce one globally conserved energy for an arbitrary expanding universe.

A global conserved energy requires additional structure, often a timelike Killing symmetry and suitable boundary conditions. Gravity also cannot generally be assigned a coordinate-independent local stress tensor on the same footing as matter. Later we use asymptotic or approximate gravitational energy measures when appropriate. For now, the essential lesson is that the matter source contains energy density, momentum transport, and stress together; specifying only a mass density throws away relativistic information.