Static Spherical Fluid Geometry

A static spherical perfect fluid is regular only when geometry and matter approach a smooth rotationally invariant limit at the center and join consistently to an exterior at the surface. The useful variables are the areal radius $r$, the mass function $m(r)$, the redshift potential $\Phi(r)$, and the orthonormal density and pressure. In these variables Einstein’s equation becomes a first-order system whose singular-looking denominators reveal both genuine compactness limits and removable central behavior.

Required background. Hilbert Stress–Energy from Metric Variation supplies the stress tensor and conservation law.

Helpful background. The Einstein Equation fixes the gravitational equations; The Riemann Curvature Tensor supplies invariant checks; Exact Solutions of Einstein’s Equations places the solution family; GR Formulations develops initial-value methods; and Continuum Kinematics and Balance Laws supplies perfect-fluid kinematics.

This page uses geometrized units $G=c=1$. The function $m(r)$ is a length, $m_{\mathrm{geom}}=G M_{\mathrm{phys}}$, so the metric factor $1-2m/r$ restores to $1-2G M_{\mathrm{phys}}(r)/r$.

Static spherical field equations

On a region where the static Killing field is timelike, choose areal radius and write

$$ds^2=-e^{2\Phi(r)}dt^2+\frac{dr^2}{1-2m(r)/r}+r^2(d\theta^2+\sin^2\theta\,d\phi^2).$$

For a perfect fluid at rest in this chart,

$$u^a=e^{-\Phi}(\partial_t)^a, \qquad T_{ab}=(\rho+p)u_a u_b+p g_{ab}.$$

Thus $T^{\hat a}{}_{\hat b}=\operatorname{diag}(-\rho,p,p,p)$ in the comoving orthonormal frame. With $G_{ab}=8\pi T_{ab}$, the $tt$ and $rr$ equations give

$$\frac{dm}{dr}=4\pi r^2\rho, \qquad \frac{d\Phi}{dr}=\frac{m+4\pi r^3p}{r(r-2m)}.$$

The radial component of $\nabla_aT^{ab}=0$ gives hydrostatic balance,

$$\frac{dp}{dr}=-(\rho+p)\frac{m+4\pi r^3p}{r(r-2m)}.$$

This is the Tolman–Oppenheimer–Volkoff equation. It is not an additional independent field equation: the contracted Bianchi identity makes it follow from Einstein’s equation once the source is conserved. The classic relativistic equilibrium construction appears in Oppenheimer and Volkoff 1939, pp. 374–381; Tolman’s complementary exact-solution analysis is Tolman 1939, pp. 364–373.

The quantity $m(r)$ is geometrically defined by

$$1-\frac{2m}{r}=g^{ab}(\nabla_a r)(\nabla_b r).$$

It is therefore invariant under reparameterizations of the radial coordinate, provided $r$ continues to denote the areal radius. Calling an arbitrary coefficient in another gauge “the mass” without checking this identity is unsafe.

Regular center and zero-pressure surface

Smooth spherical scalars are even functions of proper radial distance near the center. If $\rho_c$ and $p_c$ are finite, then

$$\rho(r)=\rho_c+O(r^2), \qquad p(r)=p_c+O(r^2), \qquad m(r)=\frac{4\pi}{3}\rho_c r^3+O(r^5).$$

Consequently $2m/r=O(r^2)$ and

$$\Phi'(r)=4\pi\left(p_c+\frac{\rho_c}{3}\right)r+O(r^3), \qquad p'(r)=-4\pi(\rho_c+p_c) \left(p_c+\frac{\rho_c}{3}\right)r+O(r^3).$$

These expansions are the practical regular-center conditions. A solution with finite $m(0)\ne0$, a linear term in $p$, or divergent orthonormal stress eigenvalues does not describe a smooth center even if its coordinate components look finite.

A finite body ends at the first radius $R$ for which $p(R)=0$. If no thin shell is present and the exterior is asymptotically flat vacuum, the exterior is Schwarzschild with mass $M=m(R)$. Continuity of the time metric fixes only a multiplicative normalization of $t$:

$$e^{2\Phi(R)}=1-\frac{2M}{R}.$$

The density may jump at $R$. What must vanish for a perfect-fluid-to-vacuum junction without a surface layer is the normal pressure, not necessarily $\rho(R)$. The full geometric statement is developed on Fluid–Vacuum Matching.

Interpretation and checks

Three diagnostics should accompany every claimed static fluid solution.

First, differentiate the proposed $m(r)$ and verify $m'=4\pi r^2\rho$. This catches many normalization errors without computing the full Einstein tensor. Second, compute one unused angular field equation; it must follow from the displayed system and conservation. Third, inspect invariant matter and curvature scalars, for example

$$T=-\rho+3p, \qquad R=-8\pi T=8\pi(\rho-3p).$$

A pole in $p$ is therefore normally an invariant curvature singularity, not a poor choice of static coordinates. By contrast, a pole in $g_{rr}$ at $r=2m$ needs separate analysis: it may be a coordinate horizon in vacuum, but a static material configuration cannot be continued through it while retaining the same timelike fluid velocity.

The equations also display their own boundary of applicability. They assume isotropic pressure, a hypersurface-orthogonal timelike Killing field, and an areal-radius region with $r>2m(r)$. Rotation, heat flow, elasticity, anisotropy, time dependence, and trapped regions require different reductions. Nor does an algebraically admissible $p(\rho)$ establish dynamical stability or causal microphysics.

Common pitfalls

Treating coordinate regularity as central regularity. Finite $g_{tt}$ is insufficient. Require $m=O(r^3)$, even scalar profiles, and finite curvature invariants in a local Cartesian frame.

Counting conservation twice. The TOV equation is essential, but it is linked to Einstein’s equations by the Bianchi identity. Solve a nonredundant set and use the remaining component as a check.

Exercises

Assume $\rho\ge0$, $p\ge0$, and $r>2m$ inside a static perfect-fluid body. Show that $\Phi'(r)\ge0$ and $p'(r)\le0$.

Solution

The mass equation and a regular center give $m(r)=4\pi\int_0^r\rho(s)s^2ds\ge0$. Every factor in

$$\Phi'=\frac{m+4\pi r^3p}{r(r-2m)}$$

is then nonnegative, with a positive denominator, so $\Phi'\ge0$. Since $\rho+p\ge0$, conservation gives $p'=-(\rho+p)\Phi'\le0$. Thus pressure cannot rise outward under these assumptions; violating monotonicity signals either a relaxed assumption or an incorrect solution.

References

• J. Robert Oppenheimer and George M. Volkoff, “On Massive Neutron Cores,” Physical Review 55, 374–381 (1939). DOI.
• Richard C. Tolman, “Static Solutions of Einstein’s Field Equations for Spheres of Fluid,” Physical Review 55, 364–373 (1939). DOI.