Explicit black hole solutions in higher-derivative gravity

We present, in an explicit form, the metric for all spherically symmetric Schwarzschild-Bach black holes in Einstein-Weyl theory. In addition to the black hole mass, this complete family of spacetimes involves a parameter that encodes the value of the Bach tensor on the horizon. When this additional “non-Schwarzschild parameter” is set to zero, the Bach tensor vanishes everywhere, and the “Schwa-Bach” solution reduces to the standard Schwarzschild metric of general relativity. Compared with previous studies, which were mainly based on numerical integration of a complicated form of field equations, the new form of the metric enables us to easily investigate geometrical and physical properties of these black holes, such as specific tidal effects on test particles, caused by the presence of the Bach tensor, as well as fundamental thermodynamical quantities.

Figure 1

The functions $\mathcal{H}(r)$ and $\mathrm{\Omega }(r)$ for the Schwa-Bach black hole in the form (2.2). The first 20 terms in (5.3) for $\mathcal{H}$ agree with a numerical solution with precision ${10}^{-4}$, and the first 40 terms in (5.2) for $\mathrm{\Omega }$ agree with precision ${10}^{-5}$ on $[-1,-0.5]$. The horizon is at $r_h=-1$, and $k=0.5$, $b=0.3633018769168$, which are the same values as in Ref. [6].

Figure 2

To demonstrate the rapid convergence in the near-horizon region, we plot the function $h(\overline{r})$ of the metric (2.1) expressed using (2.4). The first 20 (red), 40 (orange), 60 (green), 80 (blue), and 100 (violet) terms in the series are compared with the numerical solution of Ref. [6] (black). The horizon is located at ${\overline{r}}_h=1$ (that is $r_h=-1$). The scaling freedom with ${\sigma }^2\approx 2.18$ has been used to obtain $h\to 1$ asymptotically.

Figure 3

The Bach tensor components (3.3), entering (6.1) and (6.2), as functions of $\overline{r}$. For the special value of $b$ as in Fig. 1, they approach zero for large $\overline{r}$.