Stellar Oscillations and Instabilities

A compact star can be in hydrostatic equilibrium and still reveal its interior through small motions—or lose stability through one of them. This chapter builds the linear eigenvalue problem from radial pulsations to nonradial fluid, spacetime, and interface modes, then explains how rotation and dissipation can turn a mode into an instability. Its final pages separate robust mode physics from the microphysical and observational assumptions needed for asteroseismology.

Helpful background. Stress–Energy and Conservation supplies conserved perturbative energies; the Einstein equation supplies the equilibrium spacetime; Relativistic Matter and Compact Objects orients stellar structure; Black-Hole Perturbation Theory supplies quasinormal-mode methods; Gravitational Radiation and Waveforms supplies outgoing fluxes; and Computational Relativity supplies numerical checks.

Normal modes as stability probes

Linearize the fluid and Einstein equations about an equilibrium star and seek a displacement with time dependence $e^{-i\omega t}$. In a conservative, nonrotating problem the result can often be organized schematically as

$${\cal L}\boldsymbol{\xi}=-\omega^2{\cal W}\boldsymbol{\xi},$$

where ${\cal W}$ is a positive kinetic-energy weight and the boundary conditions impose regularity at the center, vanishing Lagrangian pressure at a free surface, and suitable metric matching. A positive $\omega^2$ gives an oscillation; a negative $\omega^2$ gives an exponentially growing radial or fluid displacement. Chandrasekhar’s relativistic radial analysis relates this spectral statement to a variational principle in Chandrasekhar 1964, §IV, Eqs. (59)–(62), p. 424.

Nonradial perturbations add angular structure and radiation. Fluid f-, p-, and g-modes probe different restoring forces, while w-modes are predominantly spacetime oscillations and interface modes concentrate motion near a material transition. Outgoing-wave boundary conditions make $\omega$ complex. With the chapter convention $e^{-i\omega t}$, a freely damped mode has $\operatorname{Im}\omega<0$ and a growing one has $\operatorname{Im}\omega>0$. Kokkotas and Schmidt survey the classification and boundary-value structure in Kokkotas and Schmidt 1999, §§4.1–4.2.

Route through the chapter

Read the six pages in order because each changes one ingredient of the same spectral problem.

1. Radial Pulsations reduces spherical perturbations to a one-dimensional Sturm–Liouville problem and identifies the zero-frequency stability boundary.
2. Nonradial f-, p-, and g-Modes introduces angular harmonics and distinguishes restoring forces, nodes, and asymptotic frequency orderings.
3. Spacetime and Interface Modes explains radiative w-modes and modes localized by density, elasticity, or phase interfaces.
4. The Chandrasekhar–Friedman–Schutz Instability derives the frame-reversal sign that permits radiation to amplify a rotating mode.
5. r-Modes, Viscosity, and Saturation turns that sign into a temperature–spin competition among radiation, viscosity, superfluidity, crustal coupling, and nonlinear transfer.
6. Compact-Star Asteroseismology connects a forward spectrum to calibrated data, selection, likelihoods, priors, and systematic uncertainty.

The first four pages establish analytic structure. The last two are deliberately more conditional: a linear instability criterion does not determine a saturation amplitude, and an observed spectral peak does not identify a mode without an instrument and source model.

Rotation changes the energy question

For an azimuthal dependence $e^{im\phi}$ in a uniformly rotating star,

$$\omega_{\rm rot}=\omega-m\Omega.$$

A pattern can therefore be retrograde relative to the fluid but prograde at infinity. Friedman and Schutz show how canonical energy and angular momentum make this distinction invariant and how outgoing radiation drives a negative-canonical-energy mode in Friedman and Schutz 1978, §IV, Eqs. (60)–(62), pp. 291–293. The sign condition is robust, but the growth time is not: viscosity, magnetic fields, crustal elasticity, and superfluid components modify the eigenfunction and the dissipative budget.

This is the chapter’s most important separation of questions. Existence asks whether a mode solves the linear boundary-value problem. Spectral stability asks for the sign of its imaginary frequency or canonical energy. Secular growth asks whether driving beats damping. Nonlinear survival asks where energy goes after the linear regime. Observability finally asks whether excitation, distance, orientation, detector response, and selection allow the signal to be distinguished from noise.

Assumptions and independent checks

Unless a page states otherwise, the background is an isolated equilibrium star in general relativity with explicit $G$ and $c=\hbar=k_B=1$. Frequencies have dimension $L^{-1}$, so $GM\omega$ is dimensionless when $M$ is a physical mass. The global metric signature is $(-,+,+,+)$. A normalization of the displacement can change a mode amplitude but cannot change an eigenfrequency; physical energies and detector strains must state their normalization.

Several checks recur. Regular solutions must have the correct center power law. Surface traction and metric matching must be satisfied rather than imposed after the eigenvalue search. In the Newtonian or weak-field limit, the relativistic equations should recover the appropriate classical pulsation problem. A conservative calculation should give real $\omega^2$ before outgoing radiation or dissipation is added. For rotating patterns, transforming $\phi'=\phi-\Omega t$ independently verifies the frame sign. For numerical spectra, eigenvalues should converge under resolution and domain changes, while spurious roots often drift.

Review the chapter

Consider a mode with $m=2$, inertial frequency $\omega=1.5\Omega$, and a positive viscous damping rate $1/\tau_{\rm visc}=0.02\ {\rm s}^{-1}$. Radiation contributes $1/\tau_{\rm GR}=-0.03\ {\rm s}^{-1}$. Is the pattern in the CFS frame-reversal regime, and does its amplitude grow in this two-channel model?

Solution

The rotating-frame frequency is

$$\omega_{\rm rot}=\omega-m\Omega =1.5\Omega-2\Omega=-0.5\Omega .$$

Thus $\omega>0$ but $\omega_{\rm rot}<0$: the pattern is prograde at infinity and retrograde in the fluid, so it has the CFS sign. The signed inverse timescale is

$$\frac{1}{\tau}=\frac{1}{\tau_{\rm GR}} +\frac{1}{\tau_{\rm visc}} =-0.03+0.02=-0.01\ {\rm s}^{-1}.$$

With $d\ln\alpha/dt=-1/\tau$, the amplitude grows at $0.01\ {\rm s}^{-1}$, an e-folding time of $100\ {\rm s}$. The sign and the timescale answer different questions: frame reversal permits radiation driving, while the rate balance decides whether that driving wins here.

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., and Bernard F. Schutz. “Secular Instability of Rotating Newtonian Stars.” Astrophysical Journal 222 (1978): 281–296. DOI.
• Kokkotas, Kostas D., and Bernd G. Schmidt. “Quasi-Normal Modes of Stars and Black Holes.” Living Reviews in Relativity 2 (1999): 2. DOI.