Linear perturbation theory of black holes

1.3 Linear perturbation theory of black holes

In the black hole perturbation approach, we deal with gravitational waves from a particle of mass $μ$ orbiting a black hole of mass $M$ , assuming $μ ≪ M$ . The perturbation of a black hole spacetime is evaluated to linear order in $μ∕M$ . The equations are essentially in the form of Equation (2

) with $ημν$ replaced by the background black hole metric $gBμHν$ and the higher order terms $Λ(h )μν$ neglected. Thus, apart from the assumption $μ ≪ M$ , the black hole perturbation approach is not restricted to slow-motion sources, nor to small deviations from the Minkowski spacetime, and the Green function used to integrate the Einstein equations contains the whole curved spacetime effect of the background geometry.

The black hole perturbation theory was originally developed as a metric perturbation theory. For non-rotating (Schwarzschild) black holes, a single master equation for the metric perturbation was derived by Regge and Wheeler [46 ] for the so-called odd parity part, and later by Zerilli [67 ] for the even parity part. These equations have the nice property that they reduce to the standard Klein–Gordon wave equation in the flat-space limit. However, no such equation has been found in the case of a Kerr black hole so far.

Then, based on the Newman–Penrose null-tetrad formalism, in which the tetrad components of the curvature tensor are the fundamental variables, a master equation for the curvature perturbation was first developed by Bardeen and Press [4 ] for a Schwarzschild black hole without source ( $μν T = 0$ ), and by Teukolsky [61 ] for a Kerr black hole with source ( $T μν ⁄= 0$ ). The master equation is called the Teukolsky equation, and it is a wave equation for a null-tetrad component of the Weyl tensor $ψ0$ or $ψ4$ . In the source-free case, Chrzanowski [12 ] and Wald [65 ] developed a method to construct the metric perturbation from the curvature perturbation.

The Teukolsky equation has, however, a rather complicated structure as a wave equation. Even in the flat-space limit, it does not reduce to the standard Klein–Gordon form. Later, Chandrasekhar showed that the Teukolsky equation can be transformed to the form of the Regge–Wheeler or Zerilli equation for the source-free Schwarzschild case [10 ]. A generalization of this to the Kerr case with source was done by Sasaki and Nakamura [49 , 50 ]. They gave a transformation that brings the Teukolsky equation to a Regge–Wheeler type equation that reduces to the Regge–Wheeler equation in the Schwarzschild limit. It may be noted that the Sasaki–Nakamura equation contains an imaginary part, suggesting that either it is unrelated to a (yet-to-be-found) master equation for the metric perturbation for the Kerr geometry or implying the non-existence of such a master equation.

As mentioned above, an important difference between the black-hole perturbation approach and the conventional post-Newtonian approach appears in the structure of the Green function used to integrate the wave equations. In the black-hole perturbation approach, the Green function takes account of the curved spacetime effect on the wave propagation, which implies complexity of its structure in contrast to the flat-space Green function. Thus, since the system is linear in the black-hole perturbation approach, the most non-trivial task is the construction of the Green function.

There are many papers that deal with a numerical evaluation of the Green function and calculations of gravitational waves induced by a particle. See Breuer [9 ], Chandrasekhar [11 ], and Nakamura, Oohara, and Kojima [35 ] for reviews and for references on earlier papers.

Here, we are interested in an analytical evaluation of the Green function. One way is to adopt the post-Minkowski expansion assuming $2 GM ∕c ≪ r$ . Note that, for bound orbits, the condition $GM ∕c2 ≪ r$ is equivalent to the condition for the post-Newtonian expansion, $v2∕c2 ≪ 1$ . If we can calculate the Green function to a sufficiently high order in this expansion, we may be able to obtain a rather accurate approximation of it that can be applicable to a relativistic orbit fairly close to the horizon, possibly to a radius as small as the inner-most stable circular orbit (ISCO), which is given by $2 rISCO = 6GM ∕c$ in the case of a Schwarzschild black hole.

It turns out that this is indeed possible. Though there arise some complications as one goes to higher PN orders, they are relatively easy to handle as compared to situations one encounters in the conventional post-Newtonian approaches. Thus, very interesting relativistic effects such as tails of gravitational waves can be investigated easily. Further, we can also easily investigate convergence properties of the post-Newtonian expansion by comparing a numerically calculated exact result with the corresponding analytic but approximate result. In this sense, the analytic black-hole perturbation approach can provide an important test of the post-Newtonian expansion.