Relativistic Stellar Equilibria

Relativistic stellar equilibria provide both astrophysical initial data and unusually discriminating numerical benchmarks. Spherical models reduce to ordinary differential equations, while rotating stars require coupled elliptic fields and a free surface. In either case, a solver residual is insufficient: mass, baryon number, virial identities, surface location, stationarity under evolution, and mode frequencies test different parts of the implementation. This page focuses on constructing and verifying equilibria, leaving equation-of-state physics and stellar interpretation to Relativistic Matter and Compact Objects.

Required background. Hypersurfaces and Foliations supplies the spatial metric and extrinsic geometry on which the stellar data are posed.

Helpful background. Formulations of General Relativity supplies the constraint and stationary field equations, Ordinary Differential Equations supplies shooting and tolerance control, and Relativistic Matter and Compact Objects supplies fluid closure, stability, and stellar observables.

Spherical equilibrium as a reference problem

For a barotropic perfect fluid, a nonrotating star obeys the Tolman–Oppenheimer–Volkoff system. In this section use the sustained geometrized convention $G=c=1$: $m=G M_{\rm phys}$, $\epsilon=G\epsilon_{\rm phys}$, and $p=G p_{\rm phys}$, while $r$ remains a length. Then

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

Specify the equation of state, central density, regular center expansion, surface condition $p(R)=0$, and exterior matching. Integrating until pressure first crosses zero and interpolating the crossing gives the radius; taking the nearest grid point introduces a first-order surface error. Restore the reported physical mass as $M_{\rm phys}=m(R)/G$ and the compactness as $2m(R)/R=2G M_{\rm phys}/R$. The input record must preserve these conversions and the definitions of rest-mass density, total energy density, enthalpy, and baryon mass.

A TOV sequence is a benchmark because an independent one-dimensional ODE integration can be compared with the multidimensional initial-data code. Report gravitational mass, baryon mass, radius, central lapse, and constraint norms. Paschalidis and Stergioulas 2017, §2.7 including §2.7.8 and Table 2 reviews equilibrium construction and code comparisons.

Start at a regular center

The apparent $0/0$ at $r=0$ is a coordinate limit, not a stellar singularity. For a smooth central EOS state, substitute $m=m_3r^3+O(r^5)$ and $p=p_c+p_2r^2+O(r^4)$ into the TOV equations. Matching coefficients gives

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

Start at a small positive $r_0$ with these values and reduce $r_0$ independently of the integrator tolerance. Setting $m(r_0)=0$ discards the central ball’s mass and creates a separate starting error. The derivation assumes a regular center, positive $\epsilon_c+p_c$, and a single smooth EOS branch near the center; a phase transition elsewhere must be resolved separately.

If $\rho$ is the geometrized rest-mass density, the baryon rest mass is

$$m_b=4\pi\int_0^R \rho(r)\frac{r^2\,dr}{\sqrt{1-2m(r)/r}}.$$

The metric factor is essential: $m(R)$ integrates total energy with the TOV coordinate measure, whereas $m_b$ counts rest mass with proper spatial volume. Restore the physical baryon mass as $M_b=m_b/G$. For a cold polytrope $p=K\rho^\Gamma$, the total energy density is $\epsilon=\rho+p/(\Gamma-1)$; feeding only $\rho$ into $dm/dr$ solves the wrong relativistic star. See Baumgarte and Shapiro 2010, §1.3, Eqs. (1.77)–(1.84), pp. 15–16, and the discussion following Eq. (1.86), p. 17.

Rotating stars and free surfaces

Uniform rotation introduces centrifugal flattening and coupled metric potentials. Common schemes expand angular dependence spectrally or solve on surface-fitted domains while iterating matter and geometry. The free boundary should align with a domain interface when possible; otherwise the density’s limited differentiability reduces spectral convergence and creates Gibbs oscillations.

Inputs include central enthalpy or density, angular velocity or axis ratio, rotation law, EOS version, atmosphere policy if any, domain maps, and asymptotic boundary treatment. Track ADM mass, baryon mass, angular momentum, equatorial and polar circumferential radii, and the dimensionless spin. Virial indicators GRV2 and GRV3 provide global consistency checks derived differently from the field-equation residual. Paschalidis and Stergioulas 2017, §§2.7.8–2.7.9 and Table 2 describes these virial checks and numerical code comparisons.

Compare a selected uniformly rotating model with an independent published or independently implemented equilibrium at the same EOS, central density, and rotation parameter. Coordinate radii are not comparable until coordinate conventions match; circumferential radii and global charges are safer. Stergioulas and Friedman provide a classic cross-code comparison with global quantities Stergioulas and Friedman 1995, Table 1.

Evolution and convergence tests

Run at least three radial and angular resolutions and move the outer boundary. For TOV data, compare with the independent ODE solution and monitor the Hamiltonian constraint. Evolve the star using the same EOS implementation but an independently discretized hydrodynamics code. A stable model should remain stationary up to truncation-induced pulsations; central density, baryon mass, and metric quantities should converge toward constants.

Measure the dominant radial oscillation frequencies and compare them across resolution or with a perturbative calculation. This probes derivatives of the EOS and the coupled evolution, not only the initial elliptic solve. For rotating stars, monitor angular momentum and surface shape. The validity domain excludes unstable branches unless deliberate collapse or migration is the test; a drifting unstable star is not a failure of equilibrium construction by itself.

Convergence can change order at the surface. Report both bulk norms and surface-sensitive quantities rather than fitting one global rate. Refine the EOS interpolation and spatial grid independently so interpolation error cannot masquerade as discretization convergence.

Environment and failure criteria

Pin the EOS table or analytic formula, unit conversions, central parameter, rotation law, domain maps, quadrature, elliptic and ODE solver versions, precision, surface finder, atmosphere policy, evolution gauge, and code revision. A clean rerun interface might be stellar-equilibrium --config tov-polytrope.toml --output run/, emitting profiles, charges, virial indicators, constraints, refinement sweeps, evolution time series, and environment metadata. This is a specification, not a claim that a code accompanies the page.

Reject a model if mass and radius fail to converge, the first law or EOS range is violated, GRV2 or GRV3 stalls above truncation error, baryon mass drifts without accounted boundary flux, or an independent TOV solution disagrees. Other failure modes include surface interpolation at the wrong pressure branch, inconsistent energy-density definitions, atmosphere accretion, and apparent stationarity caused by excessive numerical dissipation.

Diagnostic

A rotating-star solver reaches an elliptic residual of $10^{-12}$ and GRV2 decreases spectrally. GRV3 stalls at $10^{-4}$, while angular momentum changes by the same fraction when the outer boundary moves. Is the model verified to $10^{-8}$?

Solution

No. The algebraic residual and one virial indicator do not control the outer-domain error detected by GRV3 and angular momentum. Extend or compactify the exterior, repeat the resolution and boundary sequence, and compare an independently generated model at identical EOS and rotation parameters. The supported accuracy is no better than the largest converged diagnostic.

References

• Baumgarte, T. W., and Shapiro, S. L. (2010). Numerical Relativity: Solving Einstein’s Equations on the Computer. Cambridge University Press. DOI.
• Paschalidis, V., and Stergioulas, N. (2017). “Rotating stars in relativity.” Living Reviews in Relativity 20, 7. arXiv.
• Stergioulas, N., and Friedman, J. L. (1995). “Comparing models of rapidly rotating relativistic stars constructed by two numerical methods.” Astrophysical Journal 444, 306–311. arXiv.