Ineffective higher derivative black hole hair

Inspired by the possibility that the Schwarzschild black hole may not be the unique spherically symmetric vacuum solution to generalizations of general relativity, we consider black holes in pure fourth order higher derivative gravity treated as an effective theory. Such solutions may be of interest in addressing the issue of higher derivative hair or during the later stages of black hole evaporation. Non-Schwarzschild solutions have been studied but we have put earlier results on a firmer footing by finding a systematic asymptotic expansion for the black holes and matching them with known numerical solutions obtained by integrating out from the near-horizon region. These asymptotic expansions can be cast in the form of trans-series expansions which we conjecture will be a generic feature of non-Schwarzschild higher derivative black holes. Excitingly we find a new branch of solutions with lower free energy than the Schwarzschild solution, but as found in earlier work, solutions only seem to exist for black holes with large curvatures, meaning that one should not generically neglect even higher derivative corrections. This suggests that one effectively recovers the nonhair theorems in this context.

Figure 1

Log plot of $|A_{(n,0)}|$ for $n=1$, 2, 3. In the expansion, (3.19), $A_{(n,0)}$ is multiplied by a factor $M^0(\lambda /L_{\alpha }{)}^n$; i.e. $|A_{(n,0)}|$ are the pieces of the metric which depend only on $\lambda$ and not $M$. The solid line is $|A_{(1,0)}|$, the dashed line is $|A_{(2,0)}|$ and the dotted line is $|A_{(3,0)}|$.

Figure 2

Log plot of the first order terms $|A_{(1,i)}|$ for $i=0$, 1. In the expansion, (3.19), $A_{(1,i)}$ is multiplied by a factor $M^i{\lambda }^{(1-i)}$. The solid line is $|A_{(1,1)}|$ and the dotted line is $|A_{(1,0)}|$. For the Schwarzschild black hole, one only has the term $A_{(1,1)}$ at this order.

Figure 3

Log plot of the second order terms $|A_{(2,i)}|$ for $i=0$, 1, 2. In the expansion, (3.19), $A_{(2,i)}$ is multiplied by a factor $M^i{\lambda }^{(2-i)}$. The solid line is $|A_{(2,2)}|$, the dashed line is $|A_{(2,1)}|$ and the dotted line is $|A_{(2,0)}|$. For the Schwarzschild black hole, one only has the term, $A_{(2,2)}=\frac{4}{{\rho }^2}$, at this order. The funny blip in $|A_{(2,1)}|$ corresponds to a sign change in $A_{(2,1)}$.

Figure 4

Log plot of the third order terms $|A_{(3,i)}|$ for $i=0$, 1, 2, 3. In the expansion, (3.19), $A_{(3,i)}$ is multiplied by a factor $M^i{\lambda }^{(3-i)}$. The solid line is $|A_{(3,3)}|$, the dashed line is $|A_{(3,2)}|$, the dashed-dotted line is $|A_{(3,1)}|$ and the dotted line is $|A_{(3,0)}|$. For the Schwarzschild black hole, one only has the term $A_{(3,3)}=\frac{8}{{\rho }^3}$ at this order. The funny blip in $|A_{(3,1)}|$ corresponds to a sign change in $A_{(3,1)}$.

Figure 5

Numerical plot of $f(r)$ (with ${\rho }_0=0.9$) showing the procedure for tuning $\delta$ and extending how far one can integrate before the solution diverges. The dotted plots correspond to $0.06584\le \delta \le 0.6616$. The dashed plots show the next step with $0.066032\le \delta \le 0.66096$ and the solid plots show a further step with $0.0660704\le \delta \le 0.0660832$.

Figure 6

Plots of the near-horizon expansion (2.7) for $f(\rho )$ compared with results of the numerical integration with ${\rho }_0=1.02$ and $\delta =0.4266\dots$. The numerical plot is shown as a solid line and the dashed lines are plots of the near-horizon expansion up to orders 9, 12, 15 and 18 as labeled. The $\mathcal{O}(18)$ expansion was used to define boundary conditions for the numerical integration which started at $\rho =1.12$. The point at which we started the numerical integration is denoted by a dashed grey vertical line in the plot.

Figure 7

A repeat of Fig. 6 with the fit of the $\mathcal{O}(2)$ (blue line with larger dashes) and $\mathcal{O}(3)$ (red dotted-dashed line) asymptotic expansions added.

Figure 8

Plot of the numerical integration of $f(\rho )$ against fits of the second and third order asymptotic expansion. The numerical integration is denoted by a solid back line, the third order fit is a mustard dashed line and the second order fit is a blue dotted line. A vertical dotted line denotes where we started the numerical integration and the dashed-dotted grey line is the near-horizon expansion. The horizon radius was ${\rho }_0=0.64$ and we found $\delta =-0.552155\dots$. The fitted values for $\lambda$ at second and third order were ${\lambda }_{\mathcal{O}(2)}=1.48\dots$ and ${\lambda }_{\mathcal{O}(3)}=0.42\dots$, respectively. The asymptotic expansions were fitted to the numerical results over the range ${\rho }_{*}=4{\rho }_0$ to ${\rho }_f=40{\rho }_0$.

Figure 9

Log-log plots of $|\mathrm{\Delta }f|/f$ against $\rho$ for the second (blue dashed line) and third order (red dotted-dashed line) asymptotic expansions [see (4.1)–(4.2) for the definition of $|\mathrm{\Delta }f|/f$]. The fits for $\lambda$ where, ${\lambda }_{\mathcal{O}(2)}=0.79\dots$ and ${\lambda }_{\mathcal{O}(3)}=0.71\dots$, over the range ${\rho }_{*}=2.5{\rho }_0$ to ${\rho }_f=40{\rho }_0$.

Figure 10

Log-log plots of $|\mathrm{\Delta }f|/f$ against $\rho$ for the third order asymptotic expansion fitted over the range ${\rho }_{*}$ to ${\rho }_f=40{\rho }_0$, for various values of ${\rho }_{*}$. The fitted values for $\lambda$ where, ${\lambda }_{{\rho }_{*}=2.5}=0.79\dots$ (red dotted-dashed line), ${\lambda }_{{\rho }_{*}=5.0}=0.70\dots$ (pink dashed line) and ${\lambda }_{{\rho }_{*}=7.5}=0.45\dots$ (orange dotted line).

Figure 11

Plot of temperature $T$ (in units of $1/L_{\alpha }$) against horizon radius ${\rho }_0$ for the Schwarzschild and non-Schwarzschild black holes. The dashed line denotes the Schwarzschild solutions whereas the red and blue solid lines denote the hot and cold non-Schwarzschild branches respectively.

Figure 12

Plot of mass $M$ (in units of $L_{\alpha }$) against horizon radius ${\rho }_0$ for the Schwarzschild and non-Schwarzschild black holes. The dashed line denotes the Schwarzschild solutions whereas the red and blue solid lines denote the hot and cold non-Schwarzschild branches respectively.

Figure 13

Plot of Wald entropy $S$ (in units of $L_{\alpha }^2$) against temperature $T$ (in units of $1/L_{\alpha }$), for the Schwarzschild and non-Schwarzschild black holes. The dashed line denotes the Schwarzschild solutions whereas the red and blue solid lines denote the hot and cold non-Schwarzschild branches respectively.

Figure 14

Plot of free energy, $F=M-TS$, against temperature $T$ (in units $L_{\alpha }=1$), for the Schwarzschild and non-Schwarzschild black holes. The dashed line denotes the Schwarzschild solutions whereas the red and blue solid lines denote the hot and cold non-Schwarzschild branches respectively.