Geometry and dynamics of a tidally deformed black hole

The metric of a nonrotating black hole deformed by a tidal interaction is calculated and expressed as an expansion in the strength of the tidal coupling. The expansion parameter is the inverse length scale ${\mathcal{R}}^{-1}$, where $\mathcal{R}$ is the radius of curvature of the external spacetime in which the black hole moves. The expansion begins at order ${\mathcal{R}}^{-2}$, and it is carried out through order ${\mathcal{R}}^{-4}$. The metric is parametrized by a number of tidal multipole moments, which specify the black hole’s tidal environment. The tidal moments are freely-specifiable functions of time that are related to the Weyl tensor of the external spacetime. At order ${\mathcal{R}}^{-2}$ the metric involves the tidal quadrupole moments ${\mathcal{E}}_{ab}$ and ${\mathcal{B}}_{ab}$. At order ${\mathcal{R}}^{-3}$ it involves the time derivative of the quadrupole moments and the tidal octupole moments ${\mathcal{E}}_{abc}$ and ${\mathcal{B}}_{abc}$. At order ${\mathcal{R}}^{-4}$ the metric involves the second time derivative of the quadrupole moments, the first time derivative of the octupole moments, the tidal hexadecapole moments ${\mathcal{E}}_{abcd}$ and ${\mathcal{B}}_{abcd}$, and bilinear combinations of the quadrupole moments. The metric is presented in a light-cone coordinate system that possesses a clear geometrical meaning: The advanced-time coordinate $v$ is constant on past light cones that converge toward the black hole; the angles $\theta$ and $\varphi$ are constant on the null generators of each light cone; and the radial coordinate $r$ is an affine parameter on each generator, which decreases as the light cones converge toward the black hole. The coordinates are well-behaved on the black-hole horizon, and they are adjusted so that the coordinate description of the horizon is the same as in the Schwarzschild geometry: $r=2M+O({\mathcal{R}}^{-5})$. At the order of accuracy maintained in this work, the horizon is a stationary null hypersurface foliated by apparent horizons; it is an isolated horizon in the sense of Ashtekar and Krishnan. As an application of our results we examine the induced geometry and dynamics of the horizon, and calculate the rate at which the black-hole surface area increases as a result of the tidal interaction.

Figure 1

Light-cone coordinates centered on a world line $\gamma$. The figure shows a light cone $v=\mathrm{\text{constant}}$ that intersects the world line at the point $z(v)$. It shows also one of the light cone’s generators, along which ${\theta }^A$ is constant; the affine parameter $r$ decreases to zero as the generator approaches the world line.

Figure 2

Light-cone coordinates centered on a black hole. The figure shows the world tube traced by the black-hole horizon. It shows also the light cone $v=\mathrm{\text{constant}}$ and one of its generators. As in Fig. 1, the angles ${\theta }^A$ are constant and the affine parameter $r$ decreases on the generator.