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:

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


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


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


 * $$ T_{\phi t} \, = -W v \epsilon $$


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