Dynamical black holes in $2\mathbf{+}1$ dimensions

We investigate the global structure of a recently discovered simple exact, nonstationary solution of topologically massive and new massive gravity with the asymptotic charges of an undeformed Bañados-Teitelboim-Zanelli black hole. We establish the existence of a timelike singularity in the causal structure of the spacetime even in the absence of angular momentum. The dynamical trapping and event horizons are determined, and we investigate the evolution of the outer horizon showing that it may increase or decrease with time, depending on the value of the mass parameter. Finally, we test two proposals for dynamical entropy on this solution, one of them depending on the Kodama vector. In addition we show that the Kodama vector leads to the correct entropy for all stationary black holes in $2+1$ dimensions.

Figure 1

Diagrams for $\mu =1/2$ on the left and $\mu =-3/2$ on the right. Event horizons are depicted as solid (black) lines, trapping horizons as dashed (black) lines and the singularity is the lowest (red) line. The projections of several null geodesics of maximal slope in the $z\mathrm{\text{−}}R$ plane are drawn as thin (blue) lines.

Figure 2

Radii $r=\sqrt{g_{yy}}$ of the outer event horizon as a function of $z$ for $\mu =1/2$ (solid) and $\mu =-3/2$ (dashed). Smaller values of $z$ correspond to later times.

Figure 3

Dynamic surface gravity $\kappa (z)$ as described in (21) for two representative values of $\mu$.

Figure 4

$\mathcal{S}(z)$ evaluated on the outer trapping horizon following Hayward’s approach for different values of $\mu$, see (16). The dynamic entropy $\mathcal{S}(z)$ is shown as a solid (red) line, the constant entropy value $\overline{\mathcal{S}}$ of the background metric ${\overline{g}}_{\mu \nu }$ for the respective value of $m^2={\mu }^2-\frac{1}{2}$ is shown as a dashed (blue) line. $\mathcal{S}(z)$ and $\overline{\mathcal{S}}$ are identical for $\mu =0$.

Figure 5

$\mathcal{S}(z)$ in the framework of TMG evaluated on the outer event horizon following the Iyer-Wald approach for different values of $\mu$. The dynamic entropy $\mathcal{S}(z)$ is shown as a solid (red) line, the contribution from the Einstein-Hilbert term of the action (proportional to the horizon circumference) is shown as a dashed (blue) line, the contribution from the Chern-Simons term (b4) is shown as a dot-dashed (purple) line.