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): 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)?