Relativistic Stellar Structure

Relativistic stellar structure is the bridge from a local material law to a global compact object. The central questions are deceptively simple: how does pressure balance gravity, which mass is conserved, where does a stellar family end, and when does an equilibrium cease to be stable? This chapter answers them for static spherical stars first, then shows exactly which assumptions must change for anisotropic or multifluid matter.

Enter this chapter

Read the six pages in this order when building the subject from first principles:

1. The Tolman–Oppenheimer–Volkoff Equations derives local hydrostatic balance, center regularity, surface location, and vacuum matching with $G$ explicit.
2. Mass–Radius Sequences turns one equilibrium into a controlled one-parameter computation and separates EOS uncertainty from numerical error.
3. Baryon Number and Binding Energy distinguishes conserved material content, proper energy, and gravitational mass.
4. Radial Stability and the Turning-Point Criterion states the mode and constrained-sequence hypotheses needed to interpret an extremum.
5. Anisotropic and Multifluid Stars introduces radial–tangential stress, separately moving currents, entrainment, and new mode families.
6. Exact Interiors and Physical Admissibility closes the chapter by testing analytic solutions against geometry, microphysics, matching, causality, and stability.

Readers who need only a dependable TOV calculation can follow pages 1–3 and then the admissibility checklist. Readers concerned with stability should add page 4 before interpreting extrema. Readers modeling superfluid cores or elastic/magnetic effective stresses should treat page 5 as a change of material model, not as a correction factor appended to a perfect fluid.

From local balance to a stellar family

For a static isotropic perfect fluid, the Einstein equation and $\nabla_\mu T^{\mu\nu}=0$ reduce to

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

These equations become predictive only after an EOS supplies $p(\epsilon)$ and the center and surface conditions are fixed. Tolman’s general static reduction and the neutron-core specialization by Oppenheimer and Volkoff are given in Tolman 1939, §2, Eqs. (2.2)–(2.4), pp. 365–366 and Oppenheimer and Volkoff 1939, §II, Eqs. (7)–(10), p. 376. Their notation and geometrized units must be translated as a complete package.

Varying central density generates a mass–radius sequence, but the curve is not self-interpreting. Its shape depends on the EOS domain, crust join, phase construction, and interpolation. Gravitational mass $M=m(R)$ is not baryonic mass, and comparing configurations at fixed $M$ differs from comparing them at fixed baryon number. Regular center series, exterior matching, thermodynamic identities, and convergence under solver and EOS-grid refinement provide independent checks.

Equilibrium is not stability

For a spherical perfect fluid with a specified adiabatic response, radial perturbations form a self-adjoint eigenvalue problem. The perturbative response must be specified even when the equilibrium follows a barotropic curve. Stability is controlled by the sign of the lowest $\omega^2$, not by the visual slope of a curve alone. Chandrasekhar’s variational formulation is the canonical starting point Chandrasekhar 1964, §VI, Eqs. (59)–(62), p. 424.

The turning-point method provides a powerful shortcut under more restrictive conditions. For uniformly rotating barotropic equilibria, a fixed-angular- momentum mass extremum or a fixed-baryon-number angular-momentum extremum can mark a change in axisymmetric secular stability. The exact constrained result is stated by Friedman, Ipser, and Sorkin 1988, theorem and corollary, pp. 723–724. Differential rotation, composition gradients, multifluid currents, magnetic flux, or a two-parameter family adds admissible directions that the simple criterion does not control.

This distinction organizes the whole chapter: an equilibrium equation answers whether a configuration exists; a sequence describes how configurations are related; a perturbation problem answers whether disturbances grow; and a formation calculation asks whether nature reaches the configuration.

Extensions and neighboring subjects

Anisotropy modifies radial balance by $2(p_t-p_r)/r$, while multifluid theory replaces one momentum parallel to one current with a master function and entrained momenta. Both require additional closure. A prescribed anisotropy profile or entrainment coefficient is an input whose regularity, dimensions, causal response, and microscopic interpretation must be stated.

The chapter stops at material equilibrium and radial response. Rotating and magnetized equilibria are developed later in this volume. Production discretizations and executable solvers belong in Computational Relativity. Generic horizon formation and trapped-region geometry belong in Gravitational Collapse and Black-Hole Dynamics. Exact interiors are used here as stellar tests; the broader classification of exact spacetimes remains in Exact Solutions of Einstein’s Equations.

Review the chapter

Two TOV integrations use the same EOS. Model A and model B have equal gravitational mass, but B has greater baryon number and lies beyond the first mass maximum. What can be concluded without a mode calculation?

Solution

At equal gravitational mass, B’s greater baryon number gives greater $M_{\rm b}$ and therefore a larger $E_{\rm bind}=M_{\rm b}-M$, provided the same reference baryon mass and composition convention are used. Neither the sign of that energy nor dynamical stability follows from the comparison. Otherwise even that comparison needs translation. Its position beyond the first maximum suggests a radial instability for an ordinary regular one-parameter nonrotating barotrope, but the plot alone is not sufficient. One must verify the EOS and family hypotheses and count the radial modes. No conclusion about nonlinear collapse, rotation-driven modes, or formation follows from the stated information.

References

• Chandrasekhar, Subrahmanyan. “The Dynamical Instability of Gaseous Masses Approaching the Schwarzschild Limit in General Relativity.” Astrophysical Journal 140 (1964): 417–433. DOI.
• Friedman, John L., James R. Ipser, and Rafael D. Sorkin. “Turning-Point Method for Axisymmetric Stability of Rotating Relativistic Stars.” Astrophysical Journal 325 (1988): 722–724. DOI.
• 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.