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A consequence of the gravitational self-force for circular orbits of the Schwarzschild geometry

Steven Detweiler

TL;DR

This paper computes the gravitational self-force effects on a small mass in circular orbits around a Schwarzschild black hole by solving for the regularized metric perturbation h^R_{ab} and evaluating gauge-invariant quantities. It shows that the split of the time component of the four-velocity, u^t = {}_0u^t + {}_1u^t, has a gauge-invariant O(μ) piece {}_1u^t that encodes self-force effects and agrees with post-Newtonian expansions up to 3PN, with additional higher-order coefficients inferred numerically. The analysis relies on a frequency-domain Regge-Wheeler framework, mode-sum regularization, and careful consideration of gauge independence, establishing a robust method to connect local self-force dynamics to observable redshift-like quantities. The results advance the modeling of extreme mass ratio inspirals by providing gauge-invariant, PN-consistent characterizations of self-force effects on orbital dynamics and offering a pathway toward more accurate gravitational-wave templates. The work also discusses limitations (noncircular orbits, Kerr spacetime) and suggests future directions, including second-order perturbations and extensions to more general orbital configurations.

Abstract

A small mass μin orbit about a much more massive black hole M moves along a world line that deviates from a geodesic of the black hole geometry by O(μ/M). This deviation is said to be caused by the gravitational self-force of the metric perturbation h_{ab} from μ. For circular orbits about a non-rotating black hole we numerically calculate the O(μ/M) effects upon the orbital frequency and upon the rate of passage of proper time on the worldline. These two effects are independent of the choice of gauge for h_{ab} and are observable in principle. For distant orbits, our numerical results agree with a post-Newtonian analysis including terms of order (v/c)^6.

A consequence of the gravitational self-force for circular orbits of the Schwarzschild geometry

TL;DR

This paper computes the gravitational self-force effects on a small mass in circular orbits around a Schwarzschild black hole by solving for the regularized metric perturbation h^R_{ab} and evaluating gauge-invariant quantities. It shows that the split of the time component of the four-velocity, u^t = {}_0u^t + {}_1u^t, has a gauge-invariant O(μ) piece {}_1u^t that encodes self-force effects and agrees with post-Newtonian expansions up to 3PN, with additional higher-order coefficients inferred numerically. The analysis relies on a frequency-domain Regge-Wheeler framework, mode-sum regularization, and careful consideration of gauge independence, establishing a robust method to connect local self-force dynamics to observable redshift-like quantities. The results advance the modeling of extreme mass ratio inspirals by providing gauge-invariant, PN-consistent characterizations of self-force effects on orbital dynamics and offering a pathway toward more accurate gravitational-wave templates. The work also discusses limitations (noncircular orbits, Kerr spacetime) and suggests future directions, including second-order perturbations and extensions to more general orbital configurations.

Abstract

A small mass μin orbit about a much more massive black hole M moves along a world line that deviates from a geodesic of the black hole geometry by O(μ/M). This deviation is said to be caused by the gravitational self-force of the metric perturbation h_{ab} from μ. For circular orbits about a non-rotating black hole we numerically calculate the O(μ/M) effects upon the orbital frequency and upon the rate of passage of proper time on the worldline. These two effects are independent of the choice of gauge for h_{ab} and are observable in principle. For distant orbits, our numerical results agree with a post-Newtonian analysis including terms of order (v/c)^6.

Paper Structure

This paper contains 23 sections, 111 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and summary
  2. Gravitational waveforms, perturbation analysis and the self-force
  3. Our limitations
  4. Our results
  5. Outline
  6. Geodesics of the perturbed Schwarzschild geometry
  7. Geodesic equation
  8. The radial component
  9. Quasi-circular orbits
  10. Physical interpretations of $u^t$
  11. $u^t$ as a constant of motion
  12. $u^t$ as a redshift measurement
  13. Gauge independence of our results
  14. Determination and regularization of $h_{ab}$
  15. The retarded field
  16. ...and 8 more sections

Figures (2)

  • Figure 1: The quantity ${}_1u^t$, which is the gauge independent $O(\mu)$ part of $u^t$ effected by the self-force, is given as a function of $R_{\Omega}$ for circular orbits in the Schwarzschild geometry. Also shown are ${}_1u^t$ as calculated with 1PN and 2PN analyses of Appendix \ref{['postnewtonian']} based upon Refs. BlanchetFayePonsot98 and BlanchetLR06.
  • Figure 2: The same as Fig. \ref{['20-150uT']}, but including the 3PN analysis.