Notes on sums over horizons

Black holes typify the wondrous curved spacetime structures predicted by Einstein’s theory of general relativity. They are asymptotic to maximally symmetric spacetimes such as Minkowski, (anti–)de Sitter [(A)dS], but with an event horizon in the middle, which is typically determined as the largest positive root of the inverse radial function. While the celebrated Schwarzschild black hole has only one such root, many other examples including the Kerr or Reissner-Nordström (RN) black holes may have two positive roots, with the smaller one corresponding to the inner horizon. In higher dimensions, and/or including the cosmological constant, complex horizons associated with the complex roots may arise. Conventionally, only the event horizon, associated with the famous Hawking radiation, has received the central attention, while the inner horizon is usually ignored and the complex horizons are treated as unphysical. Nevertheless, there are several motivations to consider all the horizons. (1) Each horizon formally satisfies the first law of black hole thermodynamics [1]. (2) The wave equation depends on the poles at all the horizons even if one focuses only on the outer region. (3) In the Euclidean approach to black hole thermodynamics, the metrics for rotating black holes are necessarily complex. (4) There are also tantalizing connections to the holography of left- and right-moving modes in conformal field theories [2]. (5) Recent progress in resolving the black hole information paradox requires the knowledge of the inside of the black hole [3]. (6) For asymptotically dS spacetimes, all the horizons can be real and physically important as cosmological and black hole horizons.

Following works on the product of the entropies over all the horizons [4, 5], the sum of the entropies was identified to obey a universal formula that depends only on the cosmological constant and the horizon topology [6]. This universal relation was tested for many examples and several works were dedicated to its computation [7, 8]. Other entropy relations were explored [9, 10]. More recently, a closely related quantity, the sum of the Euclidean actions, was found to satisfy the same universal relation. This was observed numerically by Hristov for the Kerr-Newman-AdS metric [11]. In the asymptotically flat spacetime, the sum of the Euclidean actions is a simple function of the mass and electric charge [12].

The Euclidean action is of independent interest from the entropy as it is a quantum mechanical quantity that is related to the thermodynamic free energy via a quantum statistical relation. It is also a useful order parameter for phase transitions. The simplicity of the answer suggests that the sum over horizons is a property of the asymptotically (A)dS spacetimes, and an analytic proof would be desirable. In this note, we present an elementary proof based on the residue theorem. The main results are summarized as follows:

For a large class of black-hole solutions in Einstein gravity with a cosmological constant $\Lambda$, we prove that the sum of the inverse temperatures $\beta_{h}$ over all the horizons vanishes. The sum of the Euclidean actions $I_{h}$ is equal to the negative of the sum of the entropies. We present a new method to compute the sum of the entropies using a generalized Smarr formula that simplifies previous computations. We find

Here $\mathcal{A}_{D}$ is the volume of the unit $D$ sphere and $L$, the scale of the (A)dS spacetime, is related to the cosmological constant as $\Lambda=-(D-1)(D-2)/2L^{2}$. Examples include the Schwarzschild, RN, and Kerr metrics in $D\geq 4$, the Kerr-Newman metric in 4D, charged, rotating black holes in $D=5,6,7$ supergravities, and and the Bañados-Teitelboim-Zanelli (BTZ) black hole in 3D.

In the asymptotically flat case, the sums are determined by the mass $M$ and electric charge $Q$ of the black hole as

We also discuss the higher-curvature corrections in Gauss-Bonnet gravity and obtain exact formulas for the sum of the Euclidean actions in all dimensions.

In the path-integral approach to quantum gravity, the partition function over fields that are periodic in the imaginary time is identified with the thermal partition function of the grand canonical ensemble [13]. The dominating contribution to the thermodynamic potential is given by the Euclidean action evaluated at the solutions of the classical field equations. Since the thermodynamic potential is defined at each horizon, the following relation is expected to hold in leading order:

Here $H$ is the enthalpy for the Gibbs free energy, $S$ is the entropy, $\mu_{i}$ are the chemical potentials such as the angular velocity $\Omega$ and electric potential $\Phi$, and $C_{i}$ are the conserved charges such as the angular momentum $J$ and the electric charge $Q$.

We denote the thermodynamic quantities at each horizon as $\beta_{h}\equiv\beta(r_{h})$, $\Omega_{h}\equiv\Omega(r_{h})$, etc. The goal is to bring the inverse temperature and the chemical potentials to the form

for some holomorphic functions $t,s_{i}$ that are usually polynomials and a discriminant polynomial $\Delta$. The horizons are located at the roots of $\Delta$. We consider the case where $\Delta$ contains only simple roots. The sum over all the horizons may be expressed as a contour integral encircling all the simple poles. The contour may be inverted to receive a contribution only from the pole at infinity. By the residue theorem applied to a ratio of two holomorphic functions $F,G$, which has only simple poles,

It follows that the residue at infinity determines the sum. If they vanish, as in spacetimes with a cosmological constant, then the sum of the Euclidean actions is reduced to the sum of the entropies.

We may prove that the inverse temperature takes this form for a large class of metrics, while the chemical potentials will be examined case by case in examples. Consider a stationary, axisymmetric spacetime whose metric is of the form

$h$ may differ from $f$ but we require $h=f$ on $\theta=0$. The horizons are defined by the roots of the metric radial function $f(r,\theta)=g_{rr}^{-1}$. We consider the case where $f(r,0)$ contains only simple roots and no branch point. This includes the Kerr-Newman solution as well as a large class of charged, rotating black holes with scalar hairs and/or in higher dimensions.

By the zeroth law of black hole thermodynamics, the temperature is constant on the horizon. We may evaluate it at $\theta=0$. Expanding the metric around $(r,\theta)=(r_{h},0)$ as $r=r_{h}+\rho^{2}$, the metric is conformal to $(\partial_{r}f(r,0))^{2}\rho^{2}d\tau^{2}+d\rho^{2}$. To eliminate the deficit angle, the time coordinate must be periodic with period $t\sim t+i\beta$. The deficit-angle argument shows that

Since the zeroes of $f(r,0)$ are encoded in $\Delta$, $\beta_{h}$ must be of the form (4).

We will show that the chemical potential terms $\beta_{h}\mu_{i,h}$ are also of this form. Their sums over the horizons vanish in all the examples with a cosmological constant. The sum of the Euclidean actions is then equal to the negative of the sum of the entropies. The calculation of the sum of the entropies requires additional technicalities, especially when rotation is turned on. We present a simpler method using a generalized Smarr formula [14]

For spacetimes with a cosmological constant, $P=-\Lambda/8\pi$ is interpreted as the pressure that is conjugate to the thermodynamic volume $V$ of the black hole. Note that the black hole mass is identified with the enthalpy of the system when $\Lambda$ is allowed to vary.

The thermodynamic volume coincides with the geometric volume for static black holes but receives corrections due to rotation. If the sum of all the other thermodynamic quantities vanishes, then the sum of the entropies may be evaluated from the thermodynamic volume as

which may then be converted into a contour integral and evaluated at the pole at infinity.

In this work, we have rigorously derived a universal result that applies to a large class of black-hole spacetimes. Individually on each “unphysical” horizon, the thermodynamic quantities such as temperature and entropy may be negative or even complex and can only be interpreted formally at present. The point is is that it can be physically relevant when all the horizons are taken in account.

As we have pointed out in the introduction, in the scalar wave equation, which is an important probe of a black hole, all the inner roots, real or complex, has effect on the equation even if we only examine the outer region. This and our results suggest that in order to understand the properties of black holes completely, it is necessary to complexify the metrics and take into account of all the special points in the complex space.

For asymptotically (A)dS spacetimes, the dependence of the sum of the Euclidean actions only on $\Lambda$ is suggestive of the property of the global AdS vacuum. Indeed, consider a pure (A)dS spacetime, with the metric

where $d\Omega_{D-2,k}$ is the volume element on a space with constant curvature and $k=1,0,-1$ corresponds to a sphere, Ricci flat, and hyperbolic topology, respectively. $L^{2}$ is positive for AdS and negative for dS. The generalized horizons are at $r_{\pm}=\pm\sqrt{-kL^{2}}$, with

Since the vacua have zero mass, we conclude that

Here $\mathcal{A}_{D,k}$ is the volume of the constant-curvature space. For black holes, the topologies change, but the sum of the Euclidean actions over all the horizons remains the same. This indicates that the outer region of a black hole is insufficient to address the spacetime properties and a new symmetry may emerge to extend the metric into the complex regions [27].