Spacetime and Interface Modes

A compact star can ring without behaving like a single compressible fluid. Curvature-led w-modes store most of their dynamical amplitude in the spacetime; crustal, superfluid, and discontinuity modes live on additional material degrees of freedom or interfaces. The useful distinction is operational: identify the restoring term, impose the correct junction conditions, and test the limit in which that term disappears.

Helpful background. Stress–Energy and Conservation supplies material traction and currents; the Einstein equation supplies curvature dynamics; Relativistic Matter and Compact Objects orients the stellar models; Black-Hole Perturbation Theory supplies outgoing-mode methods; Gravitational Radiation and Waveforms supplies radiative observables; and Computational Relativity supplies spectral and evolution checks.

Curvature-led w-modes

For polar or axial perturbations of a spherical relativistic star, metric variables satisfy wave equations coupled to matter in the interior and to a vacuum curvature potential outside. A w-mode is identified by weak bulk-fluid motion compared with its spacetime perturbation and by an outgoing exterior solution. Its frequency and damping depend strongly on compactness, but it is not a black-hole quasinormal mode: regular-center data replace the ingoing horizon condition.

With time dependence $e^{-i\omega t}$, a freely damped mode has $\operatorname{Im}\omega<0$. The curvature-wave crossing time gives the scale $\operatorname{Re}\omega\sim R^{-1}$, whereas leakage to infinity sets the damping. Kokkotas and Schmidt describe stellar w-mode families and their fluid–spacetime separation in Kokkotas and Schmidt 1999, §§4.1–4.2.2. A Cowling calculation, which freezes the metric perturbation, cannot recover this family; its disappearance is a defining check rather than a numerical failure.

There are two unrelated uses of “interface mode” in this literature. The $w_{\rm II}$ family discussed in that review consists of strongly damped spacetime modes associated with scattering at the stellar surface. A material discontinuity mode, discussed below, is restored by buoyancy at a density jump. The shared name does not imply a common restoring force or boundary condition.

Elastic, superfluid, and interface restoring forces

A solid crust adds a shear stress. In a local isotropic approximation, a torsional displacement has wave speed

$$v_s^2=\frac{\mu}{\varepsilon+p},$$

where $\mu$ is the shear modulus. Frequencies scale roughly as $v_s/R$ and depend on crust thickness and traction conditions. The perfect-fluid limit $\mu\to0$ must remove pure shear modes.

A superfluid mixture has at least two independently conserved currents over times shorter than reaction and mutual-friction timescales. Co-moving motion resembles an ordinary fluid mode; counter-moving motion samples entrainment and chemical imbalance. Calling both “fluid modes” without stating which currents are locked hides the essential material input.

At a density or phase discontinuity, integrating the perturbation equations across the interface supplies junction conditions. The induced metric and appropriate normal traction remain continuous unless a surface layer is explicitly present, while the Lagrangian pressure condition follows the material interface. For a slowly converting phase boundary that is carried with the fluid, radial displacement and Lagrangian normal traction match on both sides. A rapidly converting boundary can move relative to fluid elements, so its displacement condition differs; the slow and rapid matching conditions are derived in Tonetto and Lugones 2020, §III, Eqs. (32)–(42). A density jump alone does not determine an interface spectrum until that conversion assumption is specified. A material discontinuity mode can localize at the jump; removing the jump removes that buoyancy source or lets the branch merge with another family.

Coupling and mixed character

Realistic stars couple these sectors. A magnetic field couples crustal shear to an Alfvén continuum; entrainment mixes lattice and superfluid motion; and a sharp phase transition can exchange character with a nearby g-mode. An avoided crossing is therefore not evidence for a new degree of freedom by itself. Track the mode using displacement, traction, current counterflow, metric energy, and overlap with neighboring models.

The decomposition also depends on approximation and normalization. “Mostly spacetime” can mean a small coordinate displacement, a small canonical fluid energy, or a small density perturbation; those statements are not equivalent until gauge and norm are declared. A robust comparison fixes the same surface and outgoing boundary conditions and normalizes modes by a common energy or wave amplitude.

Dimensions expose mismatches. In $c=1$ units, $\mu/(\varepsilon+p)$ is dimensionless and every mode frequency has dimension $L^{-1}$. Restoring $G$ in a compactness comparison requires $C=GM/R$, not $M/R$ when $M$ is physical mass.

Boundary and limiting checks

At every material interface, integrate the linearized conservation law through a thin Gaussian pillbox. A finite jump in traction without a registered surface stress is inconsistent. At the stellar surface, impose the correct Lagrangian traction condition and match the metric to the vacuum exterior. At the center, retain only regular harmonic behavior; at infinity, select an outgoing wave.

Then take discriminating limits. Setting $\mu\to0$ removes crustal shear, locking all material currents removes counter-moving superfluid branches, smoothing a density jump removes its interface mode, and freezing the metric removes w-modes. A mode that survives its only restoring force has been misidentified or is numerically contaminated.

Check the disappearing force

An axial mode remains at the same nonzero frequency as the crustal shear modulus is reduced by four orders of magnitude, and the calculation contains neither magnetic fields nor rotation. What should be suspected?

Solution

A pure torsional crust mode should scale with $v_s\propto\sqrt{\mu/(\varepsilon+p)}$ and move toward zero as $\mu\to0$. Persistence at fixed frequency indicates a boundary-condition error, an unintended restoring force, or a different family such as a curvature-led axial w-mode. Inspect the metric amplitude, verify traction continuity, and repeat the explicit $\mu=0$ limit.

References

• Kokkotas, Kostas D., and Bernd G. Schmidt. “Quasi-Normal Modes of Stars and Black Holes.” Living Reviews in Relativity 2 (1999): 2. DOI.
• Tonetto, Lucas, and Germán Lugones. “Discontinuity Gravity Modes in Hybrid Stars: Assessing the Role of Rapid and Slow Phase Conversions.” Physical Review D 101 (2020): 123029. DOI; Open PDF.