Stationary and axisymmetric metric in the co-rotating frame of a rotating body (angular velocity Ω {\displaystyle \Omega } ):
d s 2 = e 2 μ ( d ρ 2 + d ζ 2 ) + W 2 e 2 ν [ d ϕ + ( Ω − ω ) d t ] 2 − e 2 ν d t 2 {\displaystyle \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 α = ( 0 , 0 , 0 , 1 e ν 1 − v 2 ) {\displaystyle u^{\alpha }\,=\left(0,0,0,{\frac {1}{e^{\nu }{\sqrt {1-v^{2}}}}}\right)} , with v ≡ W ( Ω − ω ) e 2 ν {\displaystyle v\equiv {\frac {W\left(\Omega -\omega \right)}{e^{2\nu }}}}
u β = u α g α β = ( 0 , 0 , W v e ν 1 − v 2 , − e ν 1 − v 2 ) {\displaystyle 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 α β = ( ϵ + p ) u α u β + p g α β {\displaystyle T_{\alpha \beta }\,=(\epsilon +p)u_{\alpha }u_{\beta }+p\,g_{\alpha \beta }} :
):
![{\displaystyle \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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79ad743fb18d9da7e807fcad092cca402fc9a643)
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