Deriving the Navier-Stokes equation from energy-momentum tensor

[PhyPsy]

The energy-momentum tensor for a perfect fluid is T^{ab}=(\rho_0+p)u^au^b-pg^{ab} (using the +--- Minkowski metric).

Using the conservation law \partial_bT^{ab}=0, I'm coming up with (\rho+\gamma^2p) [\frac{\partial\mathbb{u}}{{\partial}t}+ (\mathbb{u}\cdot\mathbb{\nabla})\mathbb{u}]= -{\nabla}p instead of \rho[\frac{\partial\mathbb{u}}{{\partial}t}+ (\mathbb{u}\cdot\mathbb{\nabla})\mathbb{u}]= -{\nabla}p (disregarding the body force part of the equation). Why is there a term for the part in brackets multiplied by \rho, but not for \gamma^2p?

 

[dextercioby]

Where did the gamma come from ? Can you post your derivation/scaned copy of it ?

 

[PhyPsy]

u^a=\gamma(1,\mathbf{u})
Keeping b constant and cycling a from x to z, I get \partial_tT^{at}= \partial_t[\gamma^2(\rho_0+p)\mathbf{u}]
\partial_xT^{ax}= \partial_x[\gamma^2(\rho_0+p)u_x\mathbf{u}]+\partial_xp
\partial_yT^{ay}= \partial_y[\gamma^2(\rho_0+p)u_y\mathbf{u}]+\partial_yp
\partial_zT^{az}= \partial_z[\gamma^2(\rho_0+p)u_z\mathbf{u}]+\partial_zp
\rho=\gamma^2\rho_0, so that is why you see a \gamma^2 coefficient for p but not for \rho. When I solve for \partial_bT^{tb}=0, I get \partial_t(\rho+\gamma^2p)+ \partial_x[(\rho+\gamma^2p)u_x]+ \partial_y[(\rho+\gamma^2p)u_y]+ \partial_z[(\rho+\gamma^2p)u_z]=0, so summing the four equations for T^{ab} above and simplifying using \partial_bT^{tb}=0, I get (\rho+\gamma^2p)\partial_t\mathbf{u}+ (\rho+\gamma^2p)u_x\partial_x\mathbf{u}+ (\rho+\gamma^2p)u_y\partial_y\mathbf{u}+ (\rho+\gamma^2p)u_z\partial_z\mathbf{u}+ {\nabla}p=\mathbf{0}
This simplifies to what I put in the first post: (\rho+\gamma^2p)[\partial_t\mathbf{u}+ (\mathbf{u}\cdot\mathbf{\nabla})\mathbf{u}]= -\mathbf{\nabla}p

 

[dextercioby]

There must be a specially relativistic correction to the Galilei invariant Euler equations. You may wish to check L&L <Fluid Mechanics>, pp. 505 to 508.

 

[PhyPsy]

It looks like the reason the \gamma^2p term is not in the Navier-Stokes equation is because a nonrelativistic limit is applied where p&lt;&lt;\rho, so \rho+\gamma^2p\approx\rho.