Spiral Density Waves and Torque Balance in the Kerr Geometry

TL;DR

This work delivers the first fully relativistic fluid calculation for an extreme mass-ratio inspiral embedded in a Kerr disc by integrating self-force methods, BH perturbation theory, and a master enthalpy formalism. The authors derive a relativistic master equation for fluid perturbations $\mathfrak{h}^{(1,0)}$ driven by metric perturbations from a small companion, and solve it using height-averaged projections (PMLA) to obtain spiral-density waves and detailed torque densities. They establish a relativistic torque-balance framework by linking locally computed matter torques $\partial T^{\phi}_{\mathrm{Mat}}$ with advected angular momentum flux $\partial_r\dot{J}^r_{\mathrm{Adv}}$, showing strong agreement and revealing relativistic Lindblad resonances and spin-dependent torques. This framework enables direct connections between disc perturbations and orbital evolution in strong gravity, offering a path toward torque-balance relations in relativistic discs and guiding future extensions to higher modes, static second-order effects, MHD, and dissipative physics.

Abstract

Extreme mass-ratio inspirals (EMRIs) in relativistic accretion discs are a key science target for the upcoming LISA mission. Existing models of disc-EMRI interactions typically rely on crude dynamical friction or Newtonian planetary migration prescriptions, which fail to capture the relativistic fluid response induced by the binary potential. In this work we address this gap by providing the relativistic calculation. We apply standard methods from self-force theory, black hole perturbation theory, and relativistic stellar perturbation theory to perform the full fluid calculation of the relativistic analogue of planetary migration for the first time. We calculate the response of a fluid in the perturbing potential of an EMRI consistently incorporating pressure effects. Using a master enthalpy-like variable and linearised fluid theory, we reconstruct the fluid perturbations and relativistic spiral arm structure for a range of spin values in the Kerr geometry. We conclude by deriving a relativistic torque-balance equation that enables computation and comparison of local torques with advected angular momentum through the disc. This opens a promising route towards establishing torque-balance relations between integrated disc torques arising from fluid perturbations and the forces acting on EMRIs embedded in matter.

Figures (9)

• Figure 1: Equatorial slices of the metric perturbation for a single $m$-mode, constructed using varying numbers of input $l$-modes. The lightest line corresponds to $l_{\rm max} = 2$, with successively darker lines representing $l_{\rm max} = 4, 6, \dots$ up to $l_{\rm max} = 28$. A characteristic logarithmic divergence is overlaid to illustrate the expected scaling of the source up to the softening length.
• Figure 2: Plot exemplifying the dominating structure of the potential piece of the enthalpy wave equation for the fiducial parameter set, $r_p = 15$, $h = 0.1$, and $a = 0.6$, for a range of $m$-modes. Key to this plot is the identification of wave regions, i.e., when the potential is positive, as opposed to excitation regions where the potential is negative. Naturally, for higher $m$-modes, the excitation region contracts towards the location of the secondary. Here the lightest lines, with divergences further from the particle location, correspond to the $m=1$ potential while subsequent darker lines, which diverge closer to the secondary, represent the potential of the $m=2$, $m=3 \cdots,m=15$ potential functions. The differential in colour schemes is simply to make clear the positions at which each equation transitions from a wavelike to an exponentially growing character.
• Figure 3: Illustration of the excitation regions arising between Lindblad resonances for a range of black hole spins at $r_p=15M$, and for a reference Newtonian analogue. The outermost region, shown as the lightest (non-white) colour on each bar, corresponds to the area between the $m=1$ inner and outer Lindblad resonances. Moving inward from the edges toward the secondary location, each successive darkening of the colour indicates crossing an additional resonant point. This colour progression provides a visual representation of how the resonance structure and excitation regions evolve with black hole spin compared to the Newtonian case.
• Figure 4: The independent $m$-mode solutions to the homogeneous enthalpy equations using the same lighter-to-darker colour scheme employed in previous figures to indicate increasing $m$-mode contributions. This example corresponds to the fiducial parameter set $r_p = 15$, $h = 0.1$, and $a = 0.6$. Solutions satisfying outgoing boundary conditions at the outer boundary are shown in orange, while those satisfying ingoing boundary conditions at the inner boundary are shown in blue. The ingoing solutions exhibit smooth behaviour within the domain interior to the particle (opaque blue) and rapidly oscillatory behaviour in the exterior domain (translucent blue). The use of differing opacities allows for easier identification of the uniform solution behaviour across the respective domains.
• Figure 5: Plot exemplifying the sourcing function, Eq. (\ref{['eq:MasterEquationsSource']}), of the master enthalpy equation, with contributions from a high-accuracy interpolant of the effective forcing function, (\ref{['eq:Effective force']}), contracted with the inversion matrix $\mathcal{Q}$ in the PMLA. The example is given for the fiducial parameter set $r_p = 15$, $h = 0.1$, and $a = 0.6$ for a range of $m$-modes. Notably, this plot explicitly shows the Lindblad resonant locations.