The Tolman–Oppenheimer–Volkoff Equations

A static relativistic star is not supported by the Newtonian equation $dp/dr=-Gm\rho/r^2$. Pressure itself gravitates, proper radial distance differs from coordinate distance, and the redshift of force must be included. The Tolman–Oppenheimer–Volkoff system packages those effects into two coupled ordinary differential equations. Given a physically admissible equation of state, it determines the star from its regular center to the surface and matches that surface to a vacuum exterior.

Required background. The Einstein Equation supplies the field equations and matter coupling; Killing Fields and Isometry Groups supplies the static and spherical symmetry reduction used here.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation fixes stress–energy signs; Relativistic Matter and Compact Objects orients the material model; Computational Relativity explains how differential-equation solutions are validated.

Static spherical geometry and matter

Assume a four-dimensional, asymptotically flat, static, spherically symmetric spacetime, zero cosmological constant, and an isotropic perfect fluid. Choose the areal radius $r$, so a symmetry sphere has area $4\pi r^2$, and write

$$ds^2=-e^{2\Phi(r)}dt^2 +\frac{dr^2}{1-2Gm(r)/r} +r^2d\Omega^2.$$

The fluid four-velocity is $u^\mu=e^{-\Phi}(\partial_t)^\mu$, and its stress–energy tensor is

$$T^{\mu\nu}=(\epsilon+p)u^\mu u^\nu+p g^{\mu\nu}.$$

Here $\epsilon$ is total rest-frame energy density, including rest mass, and $p$ is isotropic pressure. An equation of state closes the system, for example $p=p(\epsilon)$ for cold barotropic matter. The assumptions exclude rotation, anisotropic stress, heat flow, and time dependence; later pages relax some of them.

The mass function is defined geometrically by $g_{rr}^{-1}=1-2Gm/r$. The $tt$ Einstein equation then gives

$$\frac{dm}{dr}=4\pi r^2\epsilon.$$

Thus $m(r)$ is the gravitational mass inside the areal sphere, not the proper integral of $\epsilon$ over its curved spatial volume.

Hydrostatic balance from conservation

The radial projection of $\nabla_\mu T^{\mu\nu}=0$ gives

$$\frac{dp}{dr}=-(\epsilon+p)\frac{d\Phi}{dr}.$$

The $rr$ Einstein equation supplies the redshift gradient,

$$\frac{d\Phi}{dr} =\frac{G\left[m(r)+4\pi r^3p(r)\right]} {r^2\left[1-2Gm(r)/r\right]}.$$

Combining the two relations yields the TOV equation,

$$\frac{dp}{dr} =-\frac{G(\epsilon+p)\left(m+4\pi r^3p\right)} {r^2\left(1-2Gm/r\right)}.$$

To see where the metric equation enters, its radial component is

$$8\pi Gp=-\frac{2Gm}{r^3} +\frac{2}{r}\left(1-\frac{2Gm}{r}\right)\Phi'.$$

Solving for $\Phi'$ gives the expression above; conservation then supplies the pressure gradient. No independent equation for a gravitational force has been assumed.

The three relativistic corrections are visible: inertial density is $\epsilon+p$, active mass contains $4\pi r^3p$, and radial curvature contributes the factor $(1-2Gm/r)^{-1}$. This normalization follows directly from the original static reductions in Tolman 1939, §2, Eqs. (2.2)–(2.4), pp. 365–366 and Oppenheimer and Volkoff 1939, §II, Eqs. (7)–(10), p. 376. The latter use $G=c=1$, metric potentials $\nu=2\Phi$ and $\lambda=-\ln(1-2Gm/r)$, and an enclosed-mass variable $u=Gm$ after restoring $G$. Their energy density includes rest energy. Translating these variables and their opposite metric signature gives the complete system used here.

Center, surface, and exterior matching

Choose a central energy density $\epsilon_c$ and set $m(0)=0$. Regularity requires even expansions for the thermodynamic variables and

$$\begin{aligned} m(r)&=\frac{4\pi}{3}\epsilon_c r^3+O(r^5),\\ p(r)&=p_c-2\pi G(\epsilon_c+p_c) \left(\frac{\epsilon_c}{3}+p_c\right)r^2+O(r^4). \end{aligned}$$

The pressure coefficient follows by inserting the leading mass into TOV: $p'=-4\pi G(\epsilon_c+p_c)(\epsilon_c/3+p_c)r+O(r^3)$. It is negative for ordinary positive-density matter, as required for support. The expansion assumes a smooth central EOS branch; do not expand across a first-order density jump.

Starting an integration at a small positive radius with these series avoids the removable $0/0$ form at the center. The stellar surface $R$ is the first radius at which $p(R)=0$ for an isolated star with a finite pressure-free surface. An EOS extending into an infinite atmosphere needs a different outer-boundary prescription. The total mass is $M=m(R)$, and continuity of the first and second fundamental forms matches the interior to Schwarzschild vacuum. With the time coordinate normalized at infinity,

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

A density discontinuity at zero pressure can be allowed for a self-bound star; it is not automatically a surface layer. A nonzero surface pressure or an explicit membrane instead requires the appropriate junction stress. The denominator also demands $2Gm(r)\lt r$ for $0\lt r\leq R$ throughout a regular static configuration. The lapse can be reconstructed afterward from $\Phi(r)=\Phi(R)-H[p(r)]$, where $H(p)=\int_0^p dp'/[\epsilon(p')+p']$.

At a sharp internal phase boundary without a surface layer, $p$, $m$, and the induced geometry remain continuous while $\epsilon$ and $m'$ can jump. Integrate each EOS branch up to the transition pressure and restart on the other side. A constant-pressure Maxwell interval represents this interface; it is not a finite radial layer of constant pressure in an ordinary TOV star.

Limits and physical checks

In the weak-field, low-pressure regime, $p\ll\epsilon\simeq\rho$ and $2Gm/r\ll1$, the TOV equation reduces to $dp/dr=-Gm\rho/r^2$. Dimensional analysis provides another check: in units $c=1$, both $m$ and $4\pi r^3p$ have dimensions of mass, while the full right-hand side has dimensions of pressure per length.

Monotone pressure is not by itself enough for physical admissibility. The EOS must be thermodynamically stable, its sound characteristics must remain causal in its declared domain, and the solution must remain regular and matchable. The surface redshift and the equality of interior and exterior mass provide useful global consistency checks that are independent of the local ODE residuals. Conversely, the appearance of a maximum in a mass–radius sequence is not derived from the TOV equation alone; stability requires a perturbation analysis and carefully stated sequence constraints.

Check the Newtonian limit

Starting from the relativistic equation, show precisely which small parameters must be neglected to recover Newtonian hydrostatic equilibrium.

Solution

Define $\eta_p=p/\epsilon$, $\eta_m=4\pi r^3p/m$, and $\eta_C=2Gm/r$. When all three are small, expand $(\epsilon+p)=\epsilon(1+\eta_p)$, $m+4\pi r^3p=m(1+\eta_m)$, and $(1-\eta_C)^{-1}=1+O(\eta_C)$. Keeping only leading terms gives

$$\frac{dp}{dr}=-\frac{Gm\epsilon}{r^2}.$$

For matter whose total energy density is dominated by rest mass, $\epsilon\simeq\rho$, this is the Newtonian equation. Dropping compactness alone would be insufficient if pressure were still comparable to energy density.

References

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