Theory

The TOV Equations

The structure of a static, spherically symmetric relativistic star is governed by the Tolman-Oppenheimer-Volkoff equations:

\[\begin{aligned} \frac{dP}{dr} &= -(\epsilon + P) \frac{m + 4\pi r^3 P}{r(r - 2m)} \\ \frac{dm}{dr} &= 4\pi r^2 \epsilon \\ \frac{d\nu}{dr} &= -\frac{1}{\epsilon + P} \frac{dP}{dr} \end{aligned}\]

where:

• \[P(r)\]

  is the pressure,

• \[\epsilon(r)\]

  is the total energy density,

• \[m(r)\]

  is the enclosed gravitational mass,

• \[\nu(r)\]

  is the metric potential defined by $g_{tt} = -e^{2\nu}$.

Metric Matching

At the stellar surface ($R$), the interior metric must match the exterior Schwarzschild metric:

\[e^{2\nu(R)} = 1 - \frac{2M}{R}\]

Our solver automatically shifts the interior potential $\nu(r)$ to ensure this condition is met.

Baryonic Mass

The total baryonic mass $M_b$ (rest mass) is calculated by integrating the proper volume element weighted by the rest-mass density $\rho_0$:

\[\frac{dM_b}{dr} = \frac{4\pi r^2 \rho_0}{\sqrt{1 - 2m/r}}\]

This quantity is conserved during evolution and is critical for binding energy calculations ($E_b = M_b - M$).