Past exam of the general relativity and cosmology course of the University of Oxford / 2017 / 1 / b / Solution

Use metric signature $(-,+,+,+)$ and a bulk four-velocity normalized by $u^{\alpha }{u}_{\alpha }=-c^2$. A perfect fluid in general relativity is isotropic in its local rest frame, with no rest-frame energy flux or anisotropic stress. In an orthonormal frame in spacetime comoving with that fluid,Here $\rho$ is rest-frame energy density divided by $c^2$, including random kinetic energy, and $p$ is isotropic pressure. For a particle gas, $p=\frac{1}{3}\int \mathbf{p}\cdot \mathbf{v}dn$ and $\rho c^2=\int Edn$, with $dn$ the rest-frame particle distribution. Isotropy makes the off-diagonal spatial stresses and net energy flux vanish.

Introduce the spatial projector of a timelike congruenceThe only isotropic decomposition with the preceding components isSince both sides are tensors, equality in the comoving orthonormal frame in spacetime establishesCoordinate-independent definitions of the two scalar fields areThus $\rho$ is not, in general, only the sum of the particle rest masses per volume.

The microscopic averaging notation needs a normalization qualification. The normalization of an averaged four-velocity matters because a literal average of normalized microscopic four-velocities is not normally a normalized bulk four-velocity. For example, six equally weighted streams with the same ordinary speed $v$, directed along the positive and negative coordinate axes, have an isotropic velocity distribution andThe physical rest-frame bulk four-velocity is $(c,0,0,0)$, not this unnormalized average. Even taking the source's schematic kinetic expression literally with prefactor ${\rho }_s$ givesSubstituting $u=⟨v⟩$ and the same ${\rho }_s$ into the claimed fluid expression would instead increase $T^{00}$ by $p({\gamma }^2-1)$. The standard result therefore requires a normalized bulk velocity and a carefully defined rest-frame total energy density; it does not follow with both microscopic identifications interpreted literally. More generally the bulk velocity can be defined as the future-directed timelike energy eigenvector of the stress-energy tensor.

The equation ${\nabla }_{\alpha }{T}^{\alpha \beta }=0$ is local stress-energy conservation. In a weak gravitational field and the nonrelativistic limit its leading energy component gives mass conservation,while its spatial components give momentum conservation, or the inviscid Euler equationsThe next energy order gives conservation of internal plus kinetic energy, including pressure work and gravitational work. These are local conservation laws, not an assertion that arbitrary curved spacetimes possess a global conserved total energy.