Slowly rotating black holes in dynamical Chern-Simons gravity: Deformation quadratic in the spin

We derive a stationary and axisymmetric black hole solution to quadratic order in the spin angular momentum. The previously found, linear-in-spin terms modify the odd-parity sector of the metric, while the new corrections appear in the even-parity sector. These corrections modify the quadrupole moment, as well as the (coordinate-dependent) location of the event horizon and the ergoregion. Although the linear-in-spin metric is of Petrov type D, the quadratic-order terms render it of type I. The metric does not possess a second-order Killing tensor or a Carter-like constant. The new metric does not possess closed timelike curves or spacetime regions that violate causality outside of the event horizon. The new, even-parity modifications to the Kerr metric decay less rapidly at spatial infinity than the leading order in spin, odd-parity ones, and thus, the former are more important when considering black holes that are rotating moderately fast. We calculate the modifications to the Hamiltonian, binding energy and Kepler’s third law. These modifications are crucial for the construction of gravitational wave templates for black hole binaries, which will enter at second post-Newtonian order, just like dissipative modifications found previously.

Figure 1

Curves of zero velocity, $V_{\mathrm{eff}}=0$ for the Kerr metric. The enclosed regions show the allowed orbit region $V_{\mathrm{eff}}\ge 0$ for $E=0.95$, $L_z=3M$ and $\chi =0.3$. The left panel corresponds to the inner region while the right panel corresponds to the outer region. The thick solid lines at $r/M=1.955$ correspond to the location of the event horizon when $\chi =0.3$.

Figure 2

Curves of zero velocity, $V_{\mathrm{eff}}=0$ for the new BH metric with $\zeta =0.1$. As in Fig. 1, the enclosed regions show the allowed orbit region $V_{\mathrm{eff}}\ge 0$ for the same parameters. The middle panel corresponds to the inner region while the right panel corresponds to the outer region. The left panel zooms into the region around $\theta \approx (3/4)\pi$. Similar structure appears at around $\theta \approx (1/4)\pi$. The shaded areas are the allowed regions where test-particle orbits exist. The thick solid lines at $r/M=1.955$ correspond to the location of the event horizon when $\chi =0.3$.