Energy-Momentum Conservation for a Perfect Fluid(a) Derive the frame-independent expression (2.74b) for the perfect fluid stress energy tensor from its rest-frame components (2.74a).(b) Explain why the projection of ∇(vector) · T = 0 along the fluid 4-velocity, u(vector) . (∇(vector) · T ) = 0, should represent energy conservation as viewed by the fluid itself. Show that this equation reduces to

With the aid of Eq. (2.65), bring this into the form

where V is the 3-volume of some small fluid element as measured in the fluid’s local rest frame. What are the physical interpretations of the left- and right-hand sides of this equation, and how is it related to the first law of thermodynamics?

(c) Read the discussion in Ex. 2.10 about the tensor P = g + u(vector) ⊗ u(vector) that projects into the 3-space of the fluid’s rest frame. Explain why P_μαT^αβ_;β = 0 should represent the law of force balance (momentum conservation) as seen by the fluid. Show that this equation reduces to

where a(vector) = du(vector)/dτ is the fluid’s 4-acceleration. This equation is a relativistic version of Newton’s F = ma. Explain the physical meanings of the left- and righthand sides. Infer that ρ + P must be the fluid’s inertial mass per unit volume. It is also the enthalpy per unit volume, including the contribution of rest mass; see Ex. 5.5 and Box 13.2.

dp dr = -(p + P) .. (2.76a)