Non-extremal Black Holes Research Articles

M-branes, anti-M-branes and non-extremal black holes

An M-brane and anti-M-brane scheme is proposed to study non-extremal 4D and 5D black holes. The improved non-extremal intersecting M-brane solutions proposed here involve two sets of harmonic functions. The constraints among the pressures are found, and new features in the M-brane and anti-M-brane picture are demonstrated, which resolve the discrepancy in the number of free parameters in the D-brane picture. In terms of the “numbers” of M-branes and anti-M-branes, the prefactors of the entropies are found to be model independent, and the Bekenstein-Hawking entropy assumes the duality invariant form which is consistent with the microscopic explanation of the black hole entropy.

The classical stability of charged (1 + 1)-dimensional black holes

We discuss the classical stability of charged, two-dimensional, black holes. Here is the dilaton - Maxwell coupling parameter. We introduce a tachyon field, which arises from string theories, as a test field. One finds barrier - well type potentials surrounding the extremal black holes, which induce the classical instability. For the non-extremal black holes, one finds the barrier - well type potentials between the inner (Cauchy) horizon and outer (event) horizon, while potential barriers appear outside the event horizon. This implies that the inner horizons are unstable and the outer horizons are stable. However, this means that the non-extremal black holes themselves are stable because the stability of a black hole depends only on the outer horizon. Finally we mention that the instability of extremal black holes may originate from the unstable inner horizon of non-extremal black holes.

Black p-branes and their vertical dimensional reduction

We construct multi-center solutions for charged, dilatonic, non-extremal black holes in D = 4. When an infinite array of such non-extremal black holes are aligned periodically along an axis, the configuration becomes independent of this coordinate, which can therefore be used for Kaluza-Klein compactification. This generalises the vertical dimensional reduction procedure to include non-extremal black holes. We then extend the construction to multi-center non-extremal ( D − 4)-branes in D dimensions, and discuss their vertical dimensional reduction.

The string spectrum on the horizon of a non-extremal black hole

We investigate the conformal string σ-model corresponding to a general five-dimensional non-extremal black hole solution. In the horizon region the theory reduces to an exactly solvable conformal field theory. We determine the modular invariant spectrum of physical string states, which expresses the Rindler momentum operator in terms of three charges and string oscillators. For black holes with winding and Kaluza-Klein charges, we find that states made with only right-moving excitations have ADM mass equal to the black hole ADM mass, and thus they can be used as sources of the gravitational field. A discussion on statistical entropy is included.

Brane-antibrane democracy.

We suggest a duality invariant formula for the entropy and temperature of non-extreme black holes in supersymmetric string theory. The entropy is given in terms of the duality invariant parameter of the deviation from extremality and 56 SU(8) covariant central charges. It interpolates between the entropies of Schwarzschild solution and extremal solutions with various amount of unbroken supersymmetries and therefore serves for classification of black holes in supersymmetric string theories. We introduce the second auxiliary 56 via E(7) symmetric constraint. The symmetric and antisymmetric combinations of these two multiplets are related via moduli to the corresponding two fundamental representations of E(7): brane and anti-brane "numbers." Using the CPT as well as C symmetry of the entropy formula and duality one can explain the mysterious simplicity of the non-extreme black hole area formula in terms of branes and anti-branes.

Non-extreme black holes from non-extreme intersecting M-branes

We present non-extreme generalisations of intersecting p-brane solutions of eleven-dimensional supergravity which upon toroidal compactification reduce to non-extreme static black holes in dimensions D = 4, D = 5 and 6 ⩽ D ⩽ 9, parameterised by four, three and two charges, respectively. The D = 4 black holes are obtained either from a non-extreme configuration of three intersecting five-branes with a boost along the common string or from a non-extreme intersecting system of two two-branes and two five-branes. The D = 5 black holes arise from three intersecting two-branes or from a system of an intersecting two-brane and five-brane with a boost along the common string. The five-brane and two-brane with a boost along one direction reduce to black holes in D = 6 and D = 9, respectively, while a D = 7 black hole can be interpreted in terms of a non-extreme configuration of two intersecting two-branes. We discuss the expressions for the corresponding masses and entropies.

Nonextremal black hole microstates and U-duality

A six-parameter family of five-dimensional black hole solutions is constructed which are labeled by their mass, two asymptotic scalar fields and three charges. It is shown that the Bekenstein-Hawking entropy is exactly matched, arbitrarily far from extremality, by a simple but mysterious duality-invariant extension of previously derived formulae for the number of D-brane states in string theory.

Gravitational lensing by p-branes

The scattering of R-R gauge bosons off of Dirichlet p-branes is computed to leading order in the string coupling. The results are qualitatively similar to those found in the scattering of massless NS-NS bosons: all p-branes with p ≥ 0 exhibit stringy properties, in particular the Regge behavior. Both the R-R and NS-NS scattering amplitudes agree in the limit of small momentum transfer with scattering off the extremal R-R charged p-brane solutions found in the low-energy supergravities. We interpret this as evidence that Dirichlet-branes are an exact worldsheet description of the extremal p-branes. The −1-brane (D-instanton) is a special object which, unlike all other Dirichlet-branes, exhibits point-like behavior. We show that the field theoretic scattering off of the R-R charged instanton solution to type IIB supergravity exactly reproduces the full stringy calculation. As an aside, we include a discussion of the entropy of nonextremal black holes in ten dimensions, produced by exciting the 0-brane.

The gravitational Hamiltonian, action, entropy and surface terms

We give a derivation of the gravitational Hamiltonian starting from the Einstein - Hilbert action, keeping track of all surface terms. This derivation can be applied to any spacetime that asymptotically approaches a static background solution. The surface term that arises in the Hamiltonian can be taken as the definition of the `total energy', even for spacetimes that are not asymptotically flat. (In the asymptotically flat case, it agrees with the usual ADM energy.) We also discuss the relation between the Euclidean action and the Hamiltonian when there are horizons of infinite area (e.g. acceleration horizons) as well as the usual finite area black hole horizons. Acceleration horizons seem to be more analogous to extreme than nonextreme black holes, since we find evidence that their horizon area is not related to the total entropy.

Black hole geometry around an elementary BPS string state

We probe the geometry around an elementary BPS (EBPS) state in heterotic string theory compactified on a six-torus by scattering a massless scalar off it and comparing with the corresponding experiment in which the EBPS state is replaced by a classical extremal black hole background satisfying the BPS condition. We find that the low energy limit of the scattering amplitudes precisely agree if one takes the limit m bh ⪢ m p . In the classical experiment, beyond a certain frequency of the incident wave, part of the wave is found to be absorbed by the black hole, whereas in case of the string scattering there is a critical frequency (inelastic threshold) of the probe beyond which the EBPS state gets excited to a higher mass non-BPS elementary state. The classical absorption threshold matches exactly with the inelastic threshold in the limit of maximum degeneracy of the EBPS state of a given mass. In that limit we can therefore identify absorption by the black hole as excitation of the elementary string state to the next vibrational state of the string and consequently also identify the non-BPS string state as a non-extremal black hole.

Pair creation rate for U(1)2 black holes.

I consider a truncation of low-energy string theory which contains two U(1) gauge fields. After making some general comments on the theory, I describe a previously obtained instanton for the pair creation of black holes when both gauge fields are nonzero, and obtain the pair creation rate by calculating its action. This calculation agrees qualitatively with the earlier calculation of the pair creation rate for black holes in Einstein-Maxwell theory. That is, the pair creation is strongly suppressed in realizable circumstances, and it reduces to the Schwinger result in the point-particle limit. The pair creation of nonextreme black holes is enhanced over that of extreme black holes by ${\mathit{e}}_{\mathrm{BH}}^{\mathit{s}\mathit{c}\mathit{r}\mathit{A}}$/4. \textcopyright{} 1995 The American Physical Society.

All the Four-Dimensional Static, Spherically Symmetric Solutions of Abelian Kaluza-Klein Theory

We present the explicit form for all the four dimensional, static, spherically symmetric solutions in $(4+n)$-d Abelian Kaluza-Klein theory by performing a subset of $SO(2,n)$ transformations corresponding to four $SO(1,1)$ boosts on the Schwarzschild solution, supplemented by $SO(n)/SO(n-2)$ transformations. The solutions are parameterized by the mass $M$, Taub-Nut charge $a$, $n$ electric $\vec{\cal Q}$ and $n$ magnetic $\vec{\cal P}$ charges. Non-extreme black holes (with zero Taub-NUT charge) have either the Reissner-Nordstr\" om or Schwarzschild global space-time. Supersymmetric extreme black holes have a null or naked singularity, while non-supersymmetric extreme ones have a global space-time of extreme Reissner-Nordstr\" om black holes.

Erratum: Action and entropy of extreme and nonextreme black holes

The Hamiltonian actions for extreme and non-extreme black holes are compared and contrasted and a simple derivation of the lack of entropy of extreme black holes is given. In the non-extreme case the wave function of the black hole depends on horizon degrees of freedom which give rise to the entropy. Those additional degrees of freedom are absent in the extreme case.

Thermal divergences on the event horizons of two-dimensional black holes.

The expectation value of the stress-energy tensor $\langleT_{\mu\nu}\rangle$ of a free conformally invariant scalar field is computed in a general static two-dimensional black hole spacetime when the field is in either a zero temperature vacuum state or a thermal state at a nonzero temperature. It is found that for every static two-dimensional black hole the stress-energy diverges strongly on the event horizon unless the field is in a state at the natural black hole temperature which is defined by the surface gravity of the event horizon. This implies that both extreme and nonextreme two-dimensional black holes can only be in equilibrium with radiation at the natural black hole temperature.

Two-dimensional dilaton gravity coupled to an Abelian gauge field.

The most general two-dimensional dilaton gravity theory coupled to an Abelian gauge field is considered. It is shown that, up to spacetime diffeomorphisms and $U(1)$ gauge transformations, the field equations admit a two-parameter family of distinct, static solutions. For theories with black hole solutions, coordinate invariant expressions are found for the energy, charge, surface gravity, Hawking temperature and entropy of the black holes. The Hawking temperature is proportional to the surface gravity as expected, and both vanish in the case of extremal black holes in the generic theory. A Hamiltonian analysis of the general theory is performed, and a complete set of (global) Dirac physical observables is obtained. The theory is then quantized using the Dirac method in the WKB approximation. A connection between the black hole entropy and the imaginary part of the WKB phase of the Dirac quantum wave functional is found for arbitrary values of the mass and $U(1)$ charge. The imaginary part of the phase vanishes for extremal black holes and for eternal, non-extremal Reissner-Nordstrom black holes.

Cosmological production of charged black hole pairs.

We investigate the pair creation of charged black holes in a backgrond with a positive cosmological constant. We consider C metrics with a cosmological constant, and show that the conical singularities in the metric only disappear when it reduces to the Reissner--Nordstr\"om--de Sitter metric. We construct an instanton describing the pair production of extreme black holes and an instanton describing the pair production of nonextreme black holes from the Reissner--Nordstr\"om--de Sitter metric, and calculate their actions. There are a number of striking similarities between these instantons and the Ernst instantons, which describe pair production in a background electromagnetic field. We also observe that the type I instanton in the ordinary C metric with zero cosmological constant is actually the Reissner-Nordstr\"om solution.

Action and entropy of extreme and nonextreme black holes.

The Hamiltonian actions for extreme and nonextreme black holes are compared and contrasted and a simple derivation of the lack of entropy of extreme black holes is given. In the nonextreme case the wave function of the black hole depends on horizon degrees of freedom which give rise to the entropy. Those additional degrees of freedom are absent in the extreme case.

Entropy, area, and black hole pairs.

We clarify the relation between gravitational entropy and the area of horizons. We first show that the entropy of an extreme Reissner-Nordstr\"om black hole is zero, despite the fact that its horizon has nonzero area. Next, we consider the pair creation of extremal and nonextremal black holes. It is shown that the action which governs the rate of this pair creation is directly related to the area of the acceleration horizon and (in the nonextremal case) the area of the black hole event horizon. This provides a simple explanation of the result that the rate of pair creation of nonextreme black holes is enhanced by precisely the black hole entropy. Finally, we discuss black hole annihilation, and argue that Planck scale remnants are not sufficient to preserve unitarity in quantum gravity.

4-DIMENSIONAL AND 2-DIMENSIONAL EVAPORATING DILATONIC BLACK HOLES

The picture of S-wave scattering from a 4-D extremal dilatonic black hole is examined. Classically, a small matter shock wave will form a non-extremal black hole. In the “throat region” the r-t geometry is exactly that of a collapsing 2-D black hole. The 4-D Hawking radiation (in this classical background) gives the 2-D Hawking radiation exactly in the throat region. The 4-D geometry outside the throat region is almost the extremal one, and the deviations can be calculated using a linear approximation. Inclusion of the back-reaction changes this picture: the linear approximation is valid only at the beginning of the evaporating process. We give (explicitly) that linear 4-D solution. The linear approximation breaks down even before an apparent horizon is formed, which suggests that the 4-D semiclassical solution may be quite different from the 2-D one.

Nonextreme and ultraextreme domain walls and their global space-times.

Nonextreme walls (bubbles with two insides) and ultraextreme walls (bubbles of false vacuum decay) are discussed. Their respective energy densities are higher and lower than that of the corresponding extreme (supersymmetric), planar domain wall. These singularity-free space-times exhibit nontrivial causal structure analogous to certain nonextreme black holes. We focus on anti-de Sitter-Minkowski walls and comment on Minkowski-Minkowski walls with trivial extreme limit, as well as walls adjacent to de Sitter space-times with no extreme limit.