Editing Hydrostatic equilibrium (section)

=== Derivation from general relativity ===
By plugging the [[energy–momentum tensor]] for a [[perfect fluid]]
<math display="block">T^{\mu\nu} = \left(\rho c^{2} + P\right) u^\mu u^\nu + P g^{\mu\nu}</math>
into the [[Einstein field equations]]
<math display="block">R_{\mu\nu} = \frac{8\pi G}{c^4} \left(T_{\mu\nu} - \frac{1}{2} g_{\mu\nu} T\right)</math>
and using the conservation condition
<math display="block">\nabla_\mu T^{\mu\nu} = 0</math>
one can derive the [[Tolman–Oppenheimer–Volkoff equation]] for the structure of a static, spherically symmetric relativistic star in isotropic coordinates:
<math display="block">\frac{dP}{dr} = -\frac{G M(r)\rho(r)}{r^2} \left(1+\frac{P(r)}{\rho(r)c^2}\right) \left(1+\frac{4\pi r^3 P(r)}{M(r) c^2}\right) \left(1 - \frac{2GM(r)}{r c^2}\right)^{-1}</math>
In practice, ''Ρ'' and ''ρ'' are related by an equation of state of the form ''f''(''Ρ'',''ρ'')&nbsp;=&nbsp;0, with ''f'' specific to makeup of the star. ''M''(''r'') is a foliation of spheres weighted by the mass density ''ρ''(''r''), with the largest sphere having radius ''r'':
<math display="block">M(r) = 4\pi \int_0^r dr' \, r'^2 \rho(r').</math>
Per standard procedure in taking the nonrelativistic limit, we let {{nowrap|''c'' → ∞}}, so that the factor
<math display="block">\left(1+\frac{P(r)}{\rho(r)c^2}\right) \left(1+\frac{4\pi r^3P(r)}{M(r)c^2}\right) \left(1-\frac{2GM(r)}{r c^2} \right)^{-1} \rightarrow 1</math>
Therefore, in the nonrelativistic limit the Tolman–Oppenheimer–Volkoff equation reduces to Newton's hydrostatic equilibrium:
<math display="block">\frac{dP}{dr} = -\frac{GM(r)\rho(r)}{r^2} = -g(r)\,\rho(r)\longrightarrow dP = - \rho(h)\,g(h)\, dh</math>
(we have made the trivial notation change ''h''&nbsp;=&nbsp;''r'' and have used ''f''(''Ρ'',''ρ'')&nbsp;=&nbsp;0 to express ''ρ'' in terms of ''P'').<ref>{{cite book|last1=Zee|first1=A.|title=Einstein gravity in a nutshell|date=2013|publisher=Princeton University Press | location=Princeton | isbn=9780691145587|pages=451–454}}</ref> A similar equation can be computed for rotating, axially symmetric stars, which in its gauge independent form reads:
<math display="block">\frac{\partial_i P}{P+\rho} - \partial_i \ln u^t + u_t u^\varphi\partial_i\frac{u_\varphi}{u_t}=0</math> 
Unlike the TOV equilibrium equation, these are two equations (for instance, if as usual when treating stars, one chooses spherical coordinates as basis coordinates <math>(t,r,\theta,\varphi)</math>, the index ''i'' runs for the coordinates ''r'' and <math>\theta</math>).