New self-force method via elliptic partial differential equations for Kerr inspiral models

Thomas Osburn

Associate professor of physics Thomas Osburn (Image provided)


Author (Has Faculty Page)

Additional Authors and Editors

Nami Nishimura '21


Journal/Publication and Year

American Physical Society (2022)


The researchers developed a new computational method to predict the behavior of astrophysics systems involving a compact celestial body orbiting a supermassive black hole.


The reseachers present a new method designed to avoid numerical challenges that have impeded calculation of the Lorenz gauge self-force acting on a compact object inspiraling into a Kerr black hole. This type of calculation is valuable in creating waveform templates for extreme mass-ratio inspirals, which are an important source of gravitational waves for the upcoming Laser Interferometer Space Antenna mission. Prior hyperbolic partial differential equation (PDE) formulations encountered numerical instabilities involving unchecked growth in time; our new method is based on elliptic PDEs, which do not exhibit instabilities of that kind. For proof of concept, we calculate the self-force acting on a scalar charge in a circular orbit around a Kerr black hole. The researchers anticipate this method will subsequently facilitate calculation of first-order Lorenz gauge Kerr metric perturbations and self-force, which could serve as a foundation for second-order Kerr self-force investigations.

Primary research questions:

1. What pattern will the gravitational wave signal (that can be detected by astronomical equipment) follow? 
2. How will the motion of the smaller body be influenced by radiation reaction forces? 
3. How can knowledge of first order gravitational perturbations be used as a foundation to calculate higher order gravitational perturbations?

What the research builds on:

Prior work used computational models to describe the dynamics of the system by evolving forward in time. This time-domain approach encountered numerical instabilities where the simulated gravitational perturbations artificially grew without bound as time increased.

What the research adds to the field:

This work enters the frequency domain, which leads to a different type of differential equation. The researchers were the first use that type of differential equation to solve the mathematical equations governing gravitational fields for this type of system. Having a viable strategy to describe the gravitational perturbations induced by these astronomical bodies will lead to more accurate theoretical predictions for the gravitational wave signals pattern.

Novel methodology:

The main theme is introducing a new method that leads to a different type of differential equation governing the system's behavior. Prior methods involved computational simulations in the time domain, but they encountered time instabilities. This work proposed and applied a method that entered the frequency domain. This frequency domain approach led to a different type of differential equation that avoided the time instability.

Implications for future research:

Future research can use this method to improve the accuracy of predicted gravitational wave signals emitted when a celestial body orbits a supermassive black hole.

Policy implications:

These research findings could affect NASA's ability to support an upcoming space mission, the Laser Interferometer Space Antenna, which could affect NASA funding (very indirectly).


This work was funded by the National Science Foundation (NSF) grant no. 2309020



Thomas Osburn and Nami Nishimura. "New self-force method via elliptic partial differential equations for Kerr inspiral models." Physical Review D 106, 044056.