Toward relativistic inspirals into black holes surrounded by matter

TL;DR

This paper develops a fully relativistic framework for extreme mass ratio inspirals (EMRIs) in gravitating environments by introducing a double perturbation expansion in the small object mass ratio $\varepsilon$ and an environmental parameter $\zeta$. It derives a modified Teukolsky equation at order $\mathcal{O}(\varepsilon\zeta)$ and applies it to a concrete pole-dipole ring model, producing a piecewise type-D spacetime separated by a matter shell. The solution splits into a smooth part solvable with standard Teukolsky methods and a shell-local singular part that requires metric reconstruction and matter perturbations; the ring’s oscillations contribute a dynamical shell source, and mode mixing arises from the ring’s angular momentum. These results establish a complete theoretical foundation for computing EMRI waveforms in axisymmetric environments and set the stage for future waveform construction and environmental constraint analyses with LISA.

Abstract

Extreme mass ratio inspirals, compact objects spiraling into massive black holes, represent key sources for future space-based gravitational-wave detectors such as LISA. The inspirals will occur within rich astrophysical environments containing gravitating matter. Motivated by this, we develop a fully relativistic framework for inspirals under the gravitational influence of matter environments. Our approach employs a two-parameter perturbation expansion in the mass ratio and an environmental parameter. This yields a modified Teukolsky equation capturing the leading cross-order. We then implement a simple pole-dipole approximation of an axisymmetric environment through a thin matter shell and restrict to non-rotating black holes. As a result, we obtain a piecewise type D spacetime. This enables the use of Teukolsky-based methods while accounting for junction physics. The presence of the matter shell leads to effectively non-separable boundary conditions for the Teukolsky scalar and introduces mode mixing between adjacent multipoles. Additionally, the shell oscillates under the wave perturbation of the inspiral, contributing to the overall flux. The framework provides novel insights into the global dynamics of gravitational radiation in tidal environments. Furthermore, it represents a complete theoretical foundation for a future computation of inspirals and waveforms in our environmental model.

Figures (6)

• Figure 1: A flowchart describing necessary steps to construct the source of the modified Teukolsky equation \ref{['Teukeq11source']} .
• Figure 2: An illustration of the physical setup investigated in this paper. The ring-like matter cloud (blue) is located at a characteristic radius $r_{\rm r}$ that is assumed to be much larger than the radius of the orbit $r_{\rm p}$ (red) of the secondary inspiraling into the black hole. As a result, the ring is characterized only by leading tidal multipoles.
• Figure 3: This flowchart shows how the modified Teukolsky equation is reduced to two independent problems due to the symmetries of the background metric \ref{['background']}. The key insight is that our piecewise spacetime allows us to solve standard Teukolsky equations in each region, with all environmental complexity confined to matching conditions at the matter shell. One part of the problem involves a solution to two standard Teukolsky equation matched on the shell $r=r_{\rm r}$ while the bottom part of the flowchart describes a construction of a complicated source living on the matter shell which is essential for finding $\Psi_{4(r_{\rm r})}^{(1,1)}$. The prediction for the environmental contribution to the GW flux is obtained from the sum of the outputs of these two independent calculations.
• Figure 4: Deformation of the ring due to GWs $h^{(0,1)}_{\mu\nu}$ sourced by a particle on a circular inclined orbit. The waves in the figure depict the polarization $h^{(0,1)}_{+}=\Re(h^{(0,1)}_{mm})$ as evaluated on the equatorial plane. The ring oscillates both in the radial and polar directions, and this leads to the $\mathfrak{T}^{(1,1)}_{\rm dynamical}$ source term in the modified Teukolsky equations.
• Figure 5: The radial displacement $\delta r^{(0,1)}$ of the matter ring caused by the gravity of an inspiraling object of mass $m_{\rm part}$. The inspiraling secondary is on an inclined circular orbit of radius $r_{\rm p}=12M$ and an inclination of $60^\circ$ with respect to the plane of the ring; the unperturbed ring radius is $r_{\rm r}=20 M$. (The location of the ring is placed relatively close to the orbit to obtain more pronounced effects.) The radial displacement at $t=0$ is depicted on the left while the oscillation of element $\phi=0$ is shown in the right figure. The oscillations are anharmonic, and the structural deformation has a non-trivial structure.