Radial Stability and the Turning-Point Criterion

An extremum on a stellar sequence is useful only after the sequence and the allowed perturbations have been specified. For a nonrotating cold barotrope, the first maximum of gravitational mass usually marks the fundamental radial mode passing through zero frequency. For uniformly rotating stars, the turning-point method gives a related but narrower statement about axisymmetric secular stability along constrained sequences. Neither result says that every mass maximum of every plotted family is a universal dynamical boundary.

Required background. The Einstein Equation supplies the equilibrium and perturbation equations; Killing Fields and Isometry Groups supplies stationary and axisymmetric conserved quantities.

Helpful background. Flat-Background Hilbert Stress-Energy from Metric Variation supplies material variations; Relativistic Matter and Compact Objects supplies the EOS setting; Computational Relativity supplies eigenvalue and convergence checks.

Radial modes of a spherical barotrope

Assume a regular, static, spherically symmetric perfect-fluid star with a sufficiently differentiable equilibrium EOS and a specified adiabatic response. Equating the perturbative derivative with the slope of the barotropic equilibrium sequence is an extra assumption, needed for the usual identification of a mass extremum with a zero mode. Let $\xi(r)e^{-i\omega t}$ be a Lagrangian radial displacement that is regular at the center and makes the Lagrangian pressure perturbation vanish at the surface. The linearized Einstein–Euler equations can be cast in self-adjoint form,

$$\frac{d}{dr}\left(\Pi\frac{d\zeta}{dr}\right) +\left(Q+\omega^2W\right)\zeta=0,$$

where $\zeta=r^2e^{-\Phi}\xi$. With $e^{2\Lambda}=(1-2Gm/r)^{-1}$, the coefficients are

$$\begin{aligned} \Pi&=\frac{\Gamma_1p}{r^2}e^{\Lambda+3\Phi},\\ W&=\frac{\epsilon+p}{r^2}e^{3\Lambda+\Phi},\\ Q&=\frac{e^{\Lambda+3\Phi}}{r^2} \left[\frac{(p')^2}{\epsilon+p} -\frac{4p'}{r}-8\pi G e^{2\Lambda}p(\epsilon+p)\right]. \end{aligned}$$

Here $\Gamma_1=(\epsilon+p)c_s^2/p$ uses the adiabatic derivative appropriate to the disturbance. These expressions are Kokkotas and Ruoff 2001, §2.1, Eqs. (13)–(17), with their $\rho,\nu,\lambda$ replaced by $\epsilon,\Phi,\Lambda$, $G$ restored, and the time sign reversed. Only $\omega^2$ enters the conservative radial problem, so that reversal leaves the eigenvalue equation unchanged.

The center condition is $\zeta=O(r^3)$, not an arbitrary finite value. At the free surface,

$$\Delta p=-\frac{\Gamma_1p\,e^\Phi}{r^2}\zeta'\longrightarrow0.$$

For smooth material with $p\gt0$, $\epsilon+p\gt0$, and $c_s^2\gt0$, $\Pi\gt0$ and $W\gt0$ in $0\lt r\lt R$; they may have singular or vanishing endpoint limits. With these boundary conditions, the eigenvalues are real and ordered. The star is radially dynamically stable exactly when the lowest eigenvalue satisfies $\omega_0^2\gt0$; $\omega_0^2\lt0$ gives exponential growth under the site’s $e^{-i\omega t}$ convention.

Multiplying by $\zeta$, integrating, and using the endpoint conditions gives the variational quotient

$$\omega^2 =\frac{\int_0^R[\Pi(\zeta')^2-Q\zeta^2]\,dr} {\int_0^R W\zeta^2\,dr}.$$

Its minimum over admissible trial displacements is $\omega_0^2$. One negative trial value proves radial instability; one positive trial value does not prove stability, because another displacement may lower the quotient. The underlying variational treatment originates in Chandrasekhar 1964, §VI, Eqs. (59)–(62), p. 424. At a simple equilibrium extremum a static perturbation can give a zero mode, but identifying it as the fundamental mode requires following the mode count from a known stable part of the sequence.

A cold beta-equilibrated background need not have the equilibrium derivative during a fast pulsation: weak reactions may leave composition frozen. Then $\Gamma_1$ differs from the equilibrium slope and the mass maximum need not coincide with dynamical onset. This limitation is emphasized in Kokkotas and Ruoff 2001, §1. Sharp phase interfaces additionally need displacement and traction conditions appropriate to rapid or slow phase conversion; the smooth operator alone does not supply them.

The constrained turning-point statement

Now assume an asymptotically flat family of stationary, axisymmetric, uniformly rotating, isentropic barotropic perfect-fluid equilibria. The stars are regular, have a well-defined surface, and are labeled smoothly by one parameter $\lambda$. Let $M$, baryon number $N$, and angular momentum $J$ be their conserved global quantities. Perturbations are axisymmetric and respect the conserved quantities relevant to the chosen sequence.

The useful rigorous conclusion is a sufficient condition for secular instability on one side, not a proof that the other side is stable. For a fixed-$J$ sequence, write primes for $d/d\lambda$ and assume

$$M'(\lambda_0)=0,\qquad \mu_\infty(\lambda_0)\gt0,\qquad \left.(\mu_\infty' M')'\right|_{\lambda_0}\ne0.$$

Then nearby configurations on the side with $\mu_\infty'M'\gt0$ are axisymmetrically secularly unstable. This is the fixed-angular-momentum theorem in Friedman, Ipser, and Sorkin 1988, §II, theorem and proof, p. 723. The adjacent lemma gives the analogous fixed-$N$ statement with $\Omega'J'\gt0$, provided its corresponding nondegeneracy condition holds. At nonzero $\Omega$, extrema of $J$ and $M$ on such a fixed-$N$ sequence coincide.

An independently known stable branch and exclusion of other earlier instabilities are needed before calling a turning point the onset of stability loss. The criterion does not locate nonaxisymmetric instabilities, prove nonlinear collapse, or replace a dynamical mode calculation.

Why the proof works

The proof architecture begins with the first law for neighboring uniformly rotating equilibria,

$$\delta M=\Omega\,\delta J+\mu_\infty\,\delta N,$$

where $\Omega$ is the uniform angular velocity and $\mu_\infty$ is a redshifted chemical potential. Along a fixed-$J$ sequence, $dM/d\lambda=0$ therefore implies $dN/d\lambda=0$ when $\mu_\infty\ne0$. The tangent to the equilibrium curve at that point is consequently a first-order perturbation preserving both charges. It is a neutral direction of the constrained energy functional.

Near the turning point, a constrained second-variation argument constructs a negative-energy direction on the side selected by the derivative test. Dissipative angular-momentum redistribution can explore that direction while preserving total $J$ and $N$, which is why the result concerns secular stability. The theorem itself does not assume completeness of a normal-mode expansion. Establishing the precise dynamical boundary instead requires the perturbation problem with its stronger, local conservation constraints.

Scope, caveats, and counterexamples

Uniform rotation is consequential. A differentially rotating star carries an angular-momentum distribution, not only a total $J$; a perturbation conserving $J$ may rearrange that distribution and explore directions absent from the one-parameter proof. Likewise, nonbarotropic matter has entropy and composition degrees of freedom, multifluids have separately conserved currents, and magnetized stars carry flux and topology constraints. A mass extremum on such a projected curve need not satisfy the theorem’s hypotheses.

A two-parameter surface gives a simple logical counterexample to the naive rule. A contour can have an extremum of $M$ with respect to one plotting coordinate while its tangent in another admissible direction still decreases the constrained energy. The plotted maximum is then not sufficient to infer a mode crossing. Degenerate turning points, branch intersections, and EOS phase discontinuities also require separate regularity and mode checks.

Check a fixed-angular-momentum turning point

Along a smooth sequence with $dJ/d\lambda=0$, suppose $dM/d\lambda=0$ and $\mu_\infty\gt0$. Show what the first law implies, and state one conclusion that still does not follow.

Solution

Differentiating the first law along the sequence gives

$$\frac{dM}{d\lambda} =\Omega\frac{dJ}{d\lambda} +\mu_\infty\frac{dN}{d\lambda}.$$

The first two derivatives vanish, so positivity of $\mu_\infty$ implies $dN/d\lambda=0$. The sequence tangent therefore preserves $J$ and $N$ to first order and supplies the neutral direction used by the turning-point argument. This alone does not prove a dynamical instability: the regularity, barotropy, uniform-rotation, and nondegeneracy hypotheses must hold, and the theorem gives a sufficient condition for axisymmetric secular instability. A mode count is additional information needed to identify which dynamical mode, if any, changes stability.

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. Open PDF.
• Kokkotas, Kostas D., and Johannes Ruoff. “Radial Oscillations of Relativistic Stars.” Astronomy and Astrophysics 366 (2001): 565–572. DOI. Open PDF.