Abstract

A stationary and axisymmetric (in fact circular) metric is reviewed which describes the first-order perturbation of a Schwarzschild black-hole space-time due to a rotating finite thin disc encircling the hole symmetrically. The key Green functions of the problem (corresponding to an infinitesimally thin ring)—the one for the gravitational potential and the one for the dragging angular velocity—were already derived, in terms of infinite series, by Will in 1974, but we have now put them into closed forms using elliptic integrals. Such forms are more practical for numerical evaluation and for integration in problems involving extended sources. This last point mostly remains difficult, but we illustrate that it may be workable by using the simple case of a finite thin disc with constant Newtonian surface density.

Highlights

  • A stationary and axisymmetric metric is reviewed which describes the first-order perturbation of a Schwarzschild black-hole space-time due to a rotating finite thin disc encircling the hole symmetrically

  • It may be problematic to assume isolation, namely to approximate the gravitational field by that solely generated by the black hole, because even a very low-mass additional source can be crucial for certain features of the field

  • The Green functions, written there in terms of infinite series, we put into closed forms using elliptic integrals.3. Such forms are better for numerical study, but mainly they are more convenient when trying to solve problems involving extended sources. We demonstrated that such an integration can in simple cases really be performed, on the example of a linear perturbation of the Schwarzschild black hole due to a constant-density finite thin circular disc extending between two concentric radii

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Summary

Static and Stationary Sources around Black Holes

Many highly active sources in the Universe appear to be driven by black holes strongly interacting with the surrounding matter and fields. Equations (6) and (7) simplify considerably in the static case, yet they still remain non-linear (in gradient of ν) and their line integration can usually be only done numerically. We report some recent results on perturbative treatment of circular (stationary, axisymmetric and orthogonally transitive) problem, starting from the Schwarzschild space-time as the seed. Such forms are better for numerical study, but mainly they are more convenient when trying to solve problems involving extended sources (when the Green functions have to be integrated out, together with the respective density, over the source volume) We demonstrated that such an integration can in simple cases really be performed, on the example of a linear perturbation of the Schwarzschild black hole due to a constant-density finite thin circular disc extending between two concentric radii.

Metric for the Rotating-Disc Perturbation of Schwarzschild
Mass and Angular Momentum
Horizon Geometry
Static Limit and Singularity
Interpretation of the Disc
Illustrations
Validity of the Linear Approximation
Outlook
Full Text
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