Nonrelativistic limit of perfect fluid equations $\partial^a T_{ab} = 0$

Question

On page 62 of his book “General Relativity”, Robert Wald writes the stress-energy tensor $T_{ab}$ for a perfect fluid in the form

$$T_{ab} = \rho u_au_b + P(\eta_{ab} + u_au_b), \tag{1}$$

and then says that the equation of motion, when the fluid is not subject to external forces, is

$$\partial^a T_{ab}=0. \tag{2}$$

He projects this equation parallel and perpendicular to $u_b$ to find

$$u^a\partial_a\rho + (\rho + P)\partial^au_a = 0, \tag{3}$$ $$(P +\rho) u^a\partial_a u_b + (\eta_{ab} + u_au_b)\partial^a P = 0. \tag{4}$$

Finally, he states that in the nonrelativistic limit $P\ll\rho$, $u = (1,\vec{v})$ and $v \frac{dP}{dt} \ll |\vec{\nabla}P|$. Why is this?

And why, under these conditions, do equations $(3)$ and $(4)$ become

$$\frac{\partial \rho}{\partial t} + \vec{\nabla}\cdot(\rho\vec{v})=0, \tag{5}$$ $$\rho\left[\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot\vec{\nabla})\vec{v}\right] = -\vec{\nabla} P?\tag{6}$$

Answer 1

I don't have the book you mentioned, but if you want to get the classical limit of the equations, without doing any projection, I'd suggest to write the 4 components of the divergence of the energy-momentum tensor in an orthonormal basis, and then evaluate them in the limit $|\mathbf{u}|/c \rightarrow 0$:

• the time component should give you the mass continuity equation,
• the spatial components should give you the momentum equation.

Now I'm from the smartphone, but if you need a more detailed derivation, and nobody provides it, I'll try my best as soon as I'm in front of a computer