Balance a spherical relativistic star

Use metric signature $(-,+,+,+)$ and geometric units $G=c=1$ in the structural equations. The coordinate r is areal radius, so a sphere has area $4\pi r^2$. The function m is enclosed gravitational mass expressed as a length; physical mass is recovered as $c^2m/G$. Energy density epsilon includes rest energy and has the same geometric units as pressure, inverse length squared. It is not merely the baryon rest density.

Assume a nonrotating isotropic perfect fluid in hydrostatic equilibrium with an appropriate barotropic relation between pressure and total energy density. Mature cold stars often use a neutrino-transparent beta-equilibrated equation of state. Hot newborn remnants require entropy and composition information instead. Large rotational, magnetic, or elastic anisotropies need extended equations, so a spherical TOV calculation is a defined baseline rather than a universal compact-object simulator.

Choose a central pressure and use the equation of state to obtain its corresponding energy density. Regularity requires enclosed mass to vanish as radius cubed and the central pressure gradient to vanish. Direct substitution of r equal to zero into the differential formula creates an apparent singularity, so numerical integrations start at a small radius using a regular expansion. The expansion coefficients must use the same equation of state and units as the subsequent integration.

Integrate outward until pressure reaches the prescribed surface value, normally zero for an isolated ideal star. The corresponding areal radius is R and mass is M. An ordinary stellar exterior is Schwarzschild after matching the surface metric and pressure conditions. Self-bound matter can retain a finite energy density at zero surface pressure, so vanishing pressure and vanishing density are not interchangeable boundary conditions. Crust matching also matters for radii and tidal observables.

In the weak-field limit, pressure is small compared with energy density, the pressure contribution to enclosed gravitational attraction is small, and $2m/r$ is small. Restoring units then recovers Newtonian hydrostatic balance with mass density approximately epsilon divided by c squared. Neutron-star interiors can violate all three small-correction conditions. A Newtonian estimate that ignores pressure gravity can therefore underestimate the magnitude of the required pressure gradient.

Varying central pressure generates an equilibrium mass–radius sequence for the selected equation of state. For a simple one-parameter cold nonrotating sequence, the first maximum-mass turning point is associated with loss of radial stability under the relevant perturbation assumptions. Extra branches or sharp phase transitions require a fuller mode analysis. Stability is a property of perturbations and constraints, not a conclusion obtained from drawing a smooth mass–radius curve or checking causality alone.