rium and carefully considers the interplay of Newton Theory, General Relativity and Quantum Mechanics.

6.2 The Stress Energy Tensor of a Perfect Fluid

In agreement with the discussion presented in our historical outline we come to the conclusion that in order to write down and solve the relevant Einstein field equations an essential ingredient is provided by the general form of the stress-energy tensor Tμν for a perfect fluid. This is what we derive in the present section. To this effect we make a step back and we consider the same problem in the context of Special Relativity. As we shall see the final result can be immediately promoted to General Relativity by invoking general covariance.

A perfect fluid can be idealized as a system of N → ∞ identical, non-interacting point particles of mass m. At time t the nth particle is characterized by its energy and tri-momentum:

and also by its position x(n)(t). In (6.2.1) the symbol γ denotes the Lorentz factor:

1 − vc22

The stress-energy tensor of the nth particle is given by:3

By summing on all the particles contained in the system we obtain its stress-energy tensor:

3Let us recall that the various components of the stress-energy tensor have the following physical meaning:

T00 is the energy density

Tk0 is the flux of energy in the kth direction

T0i is the density of ith component of tri-momentum

Tki is the flux of ith momentum component in the direction k

Next using the relation P μ = Ec dxdtμ we can rewrite (6.2.4) as follows:

In a reference frame where the perfect fluid is at rest it must be homogeneous and isotropic. This corresponds to imposing the following conditions on the spatial part of the stress-energy tensor:

where p = p(x) is a scalar function. Indeed the only symmetric 2-tensor that is invariant with respect to the SO(3) rotation group is the Kronecker delta. Looking

at (6.2.5) we easily calculate the physical dimensions of the stress-energy tensor T ij . We have:

Hence by dimensional analysis we conclude that the scalar function p(x) appearing in (6.2.6) is to be interpreted as the pressure of the fluid in its rest frame. Also by comparison of (6.2.6) with (6.2.5) we obtain:

that has the following obvious physical interpretation. Since the total energy of the

nth particle is given by E(n) = m2c4 + P2n the pressure is due to the fraction of the total energy not due to rest masses, namely the kinetic energy.

In the rest frame there is neither energy or nor momentum flux so that T 0i = T i0 = 0. On the other hand the T 00 component is given by:

Indeed the energy density in the space-time point x = (x0, x) is the sum of the energies of all the particles that happen to be in x at time x0. Summarizing, in the rest frame of the fluid, the stress-energy tensor takes the following form

To go back to a generic frame where the fluid has tri-velocity v it suffices to make a special Lorentz transformation with velocity v, namely:

Explicitly performing the transformation (6.2.12) and recalling that in Special Relativity the tetra-velocity is given by:

vi

we conclude that in the new generic frame the stress-energy tensor of a perfect fluid is given by the following simple formula:

where ημν = diag(+, −, −, −) is the standard Minkowski metric in the mostly minus convention.

Equation (6.2.15) gives the stress-energy tensor of a perfect fluid in Special Relativity. Its analogue in General Relativity is easily obtained: it just suffices to replace the flat metric ημν with a generic one gμν (x) obtaining

which is the starting point for all discussions of General Relativity in presence of matter.