The stress energy tensor for relativistic dust
$$
T_{\mu\nu} = \rho v_\mu v_\nu
$$
follows from the action
$$
S_M = -\int \rho c \sqrt{v_\mu v^\mu} \sqrt{ -g } d^4 x
= -\int c \sqrt{p_\mu p^\mu} d^4 x
$$
where $p^\mu=\rho v^\mu \sqrt{ -g }$ is the 4-momentum *density*. One uses the formula:
$$
T_{\mu\nu}
= - {2\over\sqrt{ -g }}{\delta S_M\over\delta g^{\mu\nu}}\quad\quad\quad\quad\quad\quad\quad(1)
$$
And the derivation is given for example in this question that I asked before (or in the Dirac book cited therein):
http://physics.stackexchange.com/questions/17604/lagrangian-for-relativistic-dust-derivation-questions

Question 1: is there any Lagrangian (or action) that would give the following stress energy tensor for a perfect fluid:
$$
T_{\mu\nu} = \left(\rho+{p\over c^2}\right) v_\mu v_\nu + p g_{\mu\nu}
$$
?

Question 2: What about Navier Stokes equations?

Question 3: If the answer is no to any of the questions above, can the equation (1) still be used as the definition of the stress energy tensor? Or should rather one use a definition that the stress energy tensor is whatever appears on the right hand side of the Einstein's equations (even if it can't be derived from an action)?

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