Leonard E Parker

Center for Gravitation, Cosmology & Astrophysics

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Leonard E Parker Center for Gravitation, Cosmology and Astrophysics

Calculating the Gravitational Self-Force

Gravitational Self-force in a Radiation Gauge Tobias S. Keidl, Abhay G. Shah, John L. Friedman, Dong-Hoon Kim, and Larry R. Price (Paper)
Self force picture

When a stellar object comes too close to a black hole, it is slowly captured and spirals inward until the object and black hole merge. Measurements of gravitational waves from these mergers will teach us about the geometry of spacetime near black holes. Credit:NASA/GSFC

The merger of a stellar mass compact object with a supermassive black hole is known as an extreme mass-ratio inspiral (EMRI). Future gravitational wave detectors (such as LISA, the Laser Interferometer Space Antenna) could potentially measure hundreds of EMRI events. These events have a wide range of astrophysical and fundamental implications include determination of the Hubble constant, as well as measurements related to the geometry of spacetime near a galactic black hole. Such studies depend on good parameter estimation, for which it is essential to have accurate gravitational waveforms available. These require an accurate calculation of the gravitational self-force experienced by the particle, which is the subject of this work.

Since the mass ratio m/M of stellar to galactic black hole is very small, the system can be described by a perturbative expansion in the parameter m/M. In the limit when m -> 0, the stellar object, now a test-mass, follows a geodesic of the background. To first order in m/M, the object experiences a self-force which accelerates it away from the geodesic. The gravitational self-force contains both dissipative and conservative terms. The dissipative part is responsible for the loss of energy and angular momentum that are carried away by the gravitational wave to the horizon and infinity. The dissipative part is calculated from the half-retarded minus the half-advanced fields. Due to this subtraction, the singularities present in these fields at the position of the particle cancel each other. The resultant field is smooth along the particle's trajectory and relatively easy to calculate. The conservative part of the self-force is responsible for the change in phase of the orbit and, hence, the phase of the emitted gravitational wave. It is associated with the half-retarded plus half-advanced fields and hence singular at the worldline of the particle.

Although the Lorenz gauge is particularly useful for sorting out formal issues, it does not lead to separable, decoupled equations in the Kerr spacetime (which describes spacetime near a rotating black hole). Therefore, the authors chose to work in a radiation gauge, which allowed them to use a single separable differential equation (the Teukolsky equation) instead of ten partial differential equations (which would happen, for example, in the Lorenz gauge).

The methods discussed in this paper have been used to find the self-force on a particle in circular orbit in a Schwarzschild spacetime. The retarded part of the field was easily computed in the radiation gauge, however the gauge makes it difficult to analytically compute the singular field. To avoid this challenge, the singular field was determined by a numerical matching procedure. A companion paper is being prepared that shows that the numerical matching is appropriate and accurate. Work is now underway to calculate the self-force for a Kerr background.

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