60 General Relativity
matter are compactly described by thestress–energy tensor푇휇휈with the following
components.
- The time–time component푇 00 is the energy density휌푐^2 , which includes the mass
as well as internal and kinetic energies. - The diagonal space–space components푇iiare the pressure components in the
푖direction푝푖, or the momentum components per unit area. - The time–space components푐푇 0 푖are the energy flow components per unit area
in the푖direction. - The space–time components푐푇푖 0 are the momentum densities in the푖direction.
- The nondiagonal space–space components푇ijare the shear components of the
pressure푝푖in the푗direction.
It is important to note that the stress–energy tensor is of rank 2 and is symmet-
ric, thus it has ten independent components in four-space. However, a comoving
observer in the Robertson–Walker space-time following the motion of the fluid sees
no time–space or space–time components. Moreover, we can invoke the cosmological
principle to neglect the anisotropic nondiagonal space–space components. Thus the
stress–energy tensor can be cast into purely diagonal form:
푇휇휇=(푝+휌푐^2 )푣휇푣휇
푐^2
−푝푔휇휇. (3.30)
In particular, the time–time component푇 00 is 휌푐^2. The conservation of energy
and three-momentum, or equivalently the conservation of four-momentum, can be
written
D푇휇휈
D푥휈= 0. (3.31)
Thus the stress–energy tensor is divergence free.
Taking푇휇휈to describe relativistic matter, one has to pay attention to its Lorentz
transformation properties, which differ from the classical case. Under Lorentz trans-
formations the different components of a tensor do not remain unchanged, but
become dependent on each other. Thus the physics embodied by푇휇휈also differs: the
gravitational field does not depend on mass densities alone, but also on pressure. All
the components of푇휇휈are therefore responsible for warping the space-time.
The stress-energy tensor푇휇휈is the sum of the stress-energy tensors for the vari-
ous components of energy, baryons, radiation, neutrinos, dark matter and possible
other forms. Einstein’s formula [Equation (3.29)] expresses that the energy densities,
pressures and shears embodied by the stress-energy tensor determine the geometry
of space-time, which, in turn, determines the motion of matter.
Energy of Gravity Waves. The electromagnetic field which is described by Maxwell’s
field equation has a source, the electric charge. In contrast, the gravitational field푔휇휈