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Nonrotating black hole in a post-Newtonian tidal environment

Stephanne Taylor, Eric Poisson

Abstract

We examine the motion and tidal dynamics of a nonrotating black hole placed within a post-Newtonian external spacetime. The tidal perturbation created by the external environment is treated as a small perturbation. At a large distance from the black hole, the gravitational field of the external distribution of matter is assumed to be sufficiently weak to be adequately described by the (first) post-Newtonian approximation to general relativity. There, the black hole is treated as a monopole contribution to the total gravitational field. There exists an overlap in the domains of validity of each description, and the black-hole and post-Newtonian metrics are matched in the overlap. The matching procedure produces the equations of motion for the black hole and the gravito-electric and gravito-magnetic tidal fields acting on the black hole. We first calculate the equations of motion and tidal fields by making no assumptions regarding the nature of the post-Newtonian environment; this could contain a continuous distribution of matter or any number of condensed bodies. We next specialize our discussion to a situation in which the black hole is a member of a post-Newtonian two-body system. As an application of our results, we examine the geometry of the deformed event horizon and calculate the tidal heating of the black hole, the rate at which it acquires mass as a result of its tidal interaction with the companion body.

Nonrotating black hole in a post-Newtonian tidal environment

Abstract

We examine the motion and tidal dynamics of a nonrotating black hole placed within a post-Newtonian external spacetime. The tidal perturbation created by the external environment is treated as a small perturbation. At a large distance from the black hole, the gravitational field of the external distribution of matter is assumed to be sufficiently weak to be adequately described by the (first) post-Newtonian approximation to general relativity. There, the black hole is treated as a monopole contribution to the total gravitational field. There exists an overlap in the domains of validity of each description, and the black-hole and post-Newtonian metrics are matched in the overlap. The matching procedure produces the equations of motion for the black hole and the gravito-electric and gravito-magnetic tidal fields acting on the black hole. We first calculate the equations of motion and tidal fields by making no assumptions regarding the nature of the post-Newtonian environment; this could contain a continuous distribution of matter or any number of condensed bodies. We next specialize our discussion to a situation in which the black hole is a member of a post-Newtonian two-body system. As an application of our results, we examine the geometry of the deformed event horizon and calculate the tidal heating of the black hole, the rate at which it acquires mass as a result of its tidal interaction with the companion body.

Paper Structure

This paper contains 30 sections, 143 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. This work and its context
  3. Our results
  4. Relevance of this work
  5. Organization of this paper
  6. Tidal scales and tidal moments
  7. Tidally deformed black hole
  8. Metric of a deformed black hole
  9. Derivation
  10. Post-Newtonian metric
  11. Transformation from barycenter frame to black-hole frame
  12. Coordinate transformation
  13. Post-Newtonian potentials in the black-hole frame
  14. Decomposition into irreducible pieces
  15. Matching the black-hole and post-Newtonian metrics
  16. ...and 15 more sections

Figures (1)

  • Figure 1: The post-Newtonian domain $\scr D$, the black-hole neighborhood $\scr B$, and the overlap region $\scr O$. The post-Newtonian domain is depicted in blue (light grey), and it includes the green (dark grey) annulus that surrounds the black hole. The black-hole neighborhood is drawn as a yellow (white) disk around the black hole, and it also includes the green (dark grey) annulus. The black hole is represented as a black disk. The overlap region is the union of ${\scr B}$ and ${\scr D}$ that is shown in green (dark grey), the union of the blue and yellow colors.