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

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

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

• What is the scalar $v$ in the second inequality below Eq. 4? The magnitude of $\vec{v}$? Commented Jun 13, 2023 at 23:15
• Commented Oct 23, 2023 at 10:21

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 $$|\mathbf{u}|/c \rightarrow 0$$: