# Near-\(AdS_2\) perturbations and the connection with near-extreme Reissner–Nordstrom

## Abstract

The geometry very near the horizon of a near-extreme Reissner–Nordstrom black hole is described by the direct product of a near-\(AdS_2\) spacetime with a two-sphere. While near-\(AdS_2\) is locally diffeomorphic to \(AdS_2\) the two connect differently with the asymptotically flat part of the geometry of (near-)extreme Reissner–Nordstrom. In previous work, we solved analytically the coupled gravitational and electromagnetic perturbation equations of \(AdS_2\times S^2\) and the associated connection problem with extreme Reissner–Nordstrom. In this paper, we give the solution for perturbations of near-\(AdS_2\times S^2\) and make the connection with near-extreme Reissner–Nordstrom. Our results here may also be thought of as computing the classical scattering matrix for gravitational and electromagnetic waves which probe the region very near the horizon of a highly charged spherically symmetric black hole.

## 1 Introduction

The direct products of a two-sphere with \(AdS_2\) or near-\(AdS_2\) are exact solutions of four-dimensional Einstein–Maxwell theory without a cosmological constant. In [1] we analytically solved the coupled gravitational and electromagnetic perturbation equations of \(AdS_2\times S^2\) in this theory. While \(AdS_2\) and near-\(AdS_2\) are locally diffeomorphic, the solutions to the corresponding perturbation equations give rise to distinct answers for the so-called connection problem. This is the problem of extending anti-de Sitter solutions away from the near-horizon region of (near-)extreme black holes and connecting them with solutions in the far asymptotically flat region. In [1] we solved the connection problem for the gravitational and electromagnetic perturbations of \(AdS_2\times S^2\). Namely, we produced the \(AdS_2\times S^2\) perturbation equations as an appropriate near-horizon approximation of the corresponding equations for the extreme Reissner–Nordstrom black hole and then, using matched asymptotic expansions, we analytically extended the \(AdS_2\times S^2\) solutions away from the near-horizon region connecting them with solutions in the far asymptotically flat region. In this paper, we solve the connection problem for perturbations of near-\(AdS_2\), making the connection with near-extreme Reissner–Nordstrom.

From the high-energy theory viewpoint, the motivation for studying the connection problem stems from a desire to pave the way for transferring results from (near-)\(AdS_2\) holography to the realm of four-dimensional classical and quantum gravity in asymptotically flat spacetimes. This viewpoint is explained in more detail in [1], where one may also find a more extensive list of references. On the other hand, it is worthwhile to emphasize that the results contained here – and in [1] – are of interest from a purely classical gravitational physics viewpoint as well. In particular, here we present a new take on the venerable problem of Reissner–Nordstrom perturbations whereby the perturbation equations are reduced to a single fourth-order radial differential equation on the metric perturbation that measures the deviation in the size of the background two-sphere. Moreover, for a background that is nearly extremal and at low energies we solve this equation analytically using the method of matched asymptotic expansions.

In Sect. 2 we set notation, briefly reviewing the \(AdS_2\times S^2\) and near-\(AdS_2\times S^2\) throat geometries of near-extreme Reissner–Nordstrom (NERN), as well as the Regge-Wheeler-Zerilli gauge for the pertubation equations. Section 3 offers a reduction of the perturbation equations for a general Reissner–Nordstrom black hole to a single fourth-order radial differential equation. This is an alternative to the well known reductions to two second-order equations. In Sect. 4 we obtain the exact analytic answer for near-\(AdS_2\times S^2\) perturbations and solve the connection problem using matched asymptotic expansions in NERN. Section 5 specializes the connection formulas to a basis of purely ingoing near the horizon and purely outgoing near infinity solutions.

## 2 Near-Extreme Reissner–Nordstrom, \(AdS_2\times S^2\), and \(NAdS_2\times S^2\)

*M*and charge

*Q*is given by

^{1}:

*NAdS*” stands for “nearly anti-de Sitter.” Locally, \(NAdS_2\) and \(AdS_2\) are diffeomorphic and on a Penrose diagram of the throat one finds that (5) covers a Rindler patch while (6) covers a Poincare patch.

^{2}From now on we set \(M=1\).

^{3}

## 3 Reduction of the perturbation equations

*K*,

Note that the above reduction of the perturbation equations does not assume near-extremality and is valid for the general Reissner–Nordstrom black hole. It is an alternative to the well known reductions to two second-order equations and contains the same information.

For readers’ convenience a *Mathematica* notebook with Eqs. (10–14) and the coefficients \(a_i(r)\) from Appendix A is included with this paper.

## 4 \(NAdS_2\times S^2\) answers and the connection problem solved

*H*,

### 4.1 Near region

^{4}

### 4.2 Static and Far regions

### 4.3 Overlap regions, matching, and the solution to the connection problem

*i.e.*the following linear relation between the \(C_i^f\)’s and \(C_i^n\)’s:

The connection formulas (25) may also be thought of as computing the classical scattering matrix for gravitational and electromagnetic waves which probe the near-horizon \(NAdS_2\times S^2\) region of a NERN black hole. Notice that while the Far solution (22) is identical for the near-extreme black hole in this paper and the exactly extreme one in [1], the connection formulas with \(NAdS_2\times S^2\) solutions here lead to the markedly distinct results of this paper compared to the results obtained by making the connection with \(AdS_2\times S^2\) solutions in [1].

## 5 A useful basis of solutions

^{5}This basis may be defined as follows:

## 6 Conclusion

In this paper we have analytically solved the coupled gravitational and electromagnetic perturbation equations for low energy modes propagating in the spacetime of a near-extreme Reissner–Nordstrom black hole. The computations herein generalize the ones obtained for extreme Reissner–Nordstrom in [1] and as such they pave the way for transferring results from studies of near-\(AdS_2\) holography to the realm of four-dimensional classical and quantum gravity in asymptotically flat spacetimes. It is worth emphasizing that while the Near solution in this paper is diffeomorphic to the Near solution in [1], and the Static and Far solutions are identical, we have seen that the full Reissner–Nordstrom solution is markedly distinct. Holographically, the reason is that the mapping of Near solutions from Poincare to Rindler AdS is nontrivial: it is dual to a field theory mapping from zero to finite temperature, which when glued onto the Static and Far solutions gets propagated to null infinity.

From a purely classical gravitational physics viewpoint, this paper also provides a fresh look into the perturbation problem of generic Reissner–Nordstrom black holes. To date, this problem has been studied in terms of the well known reductions to two second-order radial equations first derived in the seventies (see e.g. the monograph [8]). In this paper, we have found a new reduction of the perturbation equations to a single fourth-order equation (Sect. 3). Moreover, in the case of a black hole near extremality, we have found an analytic solution for low energy modes (Sect. 4). Analytic solutions of Einstein’s equations are among the rarest gems and they may inform even numerical studies, particularly near extremality where the numerics begin to struggle.

The results contained in this paper – and in [1] – may prove useful in analytic studies of a wide range of applications that hinge on the perturbative solution of Einstein’s equations around extreme and near-extreme Reissner–Nordstrom black holes. This includes computations of radiation fluxes and absorption cross-sections [9, 10], quasinormal modes [11, 12, 13], late-time tails [14, 15], backreaction and self-force effects [16, 17], supergravity scattering amplitudes relations [18], and more.

## Footnotes

- 1.
\(G=c=1\)

- 2.
- 3.
The technical reason for this are the special facts that \(\partial _\theta Y_{0,0}=0\) and \(\partial ^2_\theta Y_{1,0}=-Y_{1,0}\) which imply that certain components of the linearized Einstein–Maxwell equations vanish identically for the \(l=0,1\) modes.

- 4.Recall that for \(m=0,1,2,\ldots \) and \(c\ne 0,-1,-2,\ldots \) we have the following polynomial:with the (rising) Pochhammer symbol defined as \((a)_k=a(a+1)(a+2)\cdots (a+k-1)\).$$\begin{aligned} _2F_1(-m,b;c;z)=\sum _{k=0}^{m}(-1)^k\left( {\begin{array}{c}m\\ k\end{array}}\right) \frac{(b)_k}{(c)_k}z^k \end{aligned}$$
- 5.
The caveats explained in [1] regarding the boundary conditions obeyed by the analogous basis in the case of extreme Reissner–Nordstrom apply here as well.

## Notes

### Acknowledgements

This work is supported by NSF Grant PHY-1504541

## Supplementary material

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