User:LeQuantum/gravitation

Stationary and axisymmetric metric in the co-rotating frame of a rotating body (angular velocity $\Omega$):

$\mathrm {d} s^{2}=e^{2\mu }\left(\mathrm {d} \rho ^{2}+\mathrm {d} \zeta ^{2}\right)+{\frac {W^{2}}{e^{2\nu }}}\left[\mathrm {d} \phi +\left(\Omega -\omega \right)\mathrm {d} t\right]^{2}-e^{2\nu }\mathrm {d} t^{2}$

4-velocity of a fluid in the co-rotating frame:

$u^{\alpha }\,=\left(0,0,0,{\frac {1}{e^{\nu }{\sqrt {1-v^{2}}}}}\right)$, with $v\equiv {\frac {W\left(\Omega -\omega \right)}{e^{2\nu }}}$

$u_{\beta }\,=u^{\alpha }g_{\alpha \beta }\,=\left(0,0,{\frac {Wv}{e^{\nu }{\sqrt {1-v^{2}}}}},-e^{\nu }{\sqrt {1-v^{2}}}\right)$

Stress-energy tensor of a fluid in the co-rotating frame:

$T_{\alpha \beta }\,=(\epsilon +p)u_{\alpha }u_{\beta }+p\,g_{\alpha \beta }$ :

$T_{\rho \rho }\,=T_{\zeta \zeta }\,=e^{2\mu }p$

$T_{\phi \phi }\,={\frac {W^{2}}{e^{2\nu }}}\left({\frac {p+v^{2}\epsilon }{1-v^{2}}}\right)$

$T_{\phi t}\,=-Wv\epsilon$

$T_{tt}\,=e^{2\nu }(1-v^{2})\epsilon$