Hamiltonian evolution and quantization for extremal black holes

We present and contrast two distinct ways of including extremal black holes in a Lorentzian Hamiltonian quantization of spherically symmetric Einstein-Maxwell theory. First, we formulate the classical Hamiltonian dynamics with boundary conditions appropriate for extremal black holes only. The Hamiltonian contains no surface term at the internal infinity, for reasons related to the vanishing of the extremal hole surface gravity, and quantization yields a vanishing black hole entropy. Second, we give a Hamiltonian quantization that incorporates extremal black holes as a limiting case of nonextremal ones, and examine the classical limit in terms of wave packets. The spreading of the packets, even the ones centered about extremal black holes, is consistent with continuity of the entropy in the extremal limit, and thus with the Bekenstein-Hawking entropy even for the extremal holes. The discussion takes place throughout within Lorentz-signature spacetimes.     

Area spectra of near extremal black holes

Motivated by Maggiore’s new interpretation of quasinormal modes, we investigate area spectra of a near extremal Schwarzschild–de Sitter black hole and a higher-dimensional near extremal Reissner–Nordstrom–de Sitter black hole. The result shows that the area spectra are equally spaced and irrelevant to the parameters of the black holes.     

Entropy of an extremal regular black hole

We introduce a magnetically charged extremal regular black hole in the coupled system of Einstein gravity and nonlinear electrodynamics. Its near horizon geometry is given by AdS₂×S²${\mathit{AdS}}_2\times S^2$. It turns out that the entropy function approach does not automatically lead to a correct entropy of the Bekenstein–Hawking entropy. This contrasts to the case of the extremal Reissner–Norström black hole in the Einstein–Maxwell theory. We conclude that the entropy function approach does not work for a magnetically charged extremal regular black hole without singularity, because of the nonlinearity of the entropy function.     

Critical behavior of extremal Kerr-Newman black holes

The previously suggested existence of second-order phase transitions in a series of Kerr-Newman holes is re-examined in the framework of equilibrium black-hole thermodynamics, to distinguish a true transition from another confusing phenomenon. By adopting a physical interpretation unique to the black-hole thermodynamics, various critical exponents are calculated for one side of the transition which is shown actually very likely to occur at the extremal limit.     

Weak Gravity Conjecture and extremal black holes

Motivated by the desire to improve our understanding of the Weak Gravity Conjecture, we compute the one-loop correction of charged particles to the geometry and entropy of extremal black holes in 4D. We use the entropy function formalism to extend previous analysis that dealt with neutral particles, and obtain the corrections to the horizon entropy for different regimes of black hole masses. These corrections are small in general. They are furthermore reduced when supersymmetry is present, and disappear in N = 4 supergravity. We provide some speculative arguments that in a theory with only sub-extremal particles, classical Reissner-Nordstrom black holes actually possess an infinite microcanonical entropy, though only a finite amount is visible to an external observer, as shown by the horizon entropy computation.     

Stationary scalar configurations around extremal charged black holes

We consider the minimally coupled Klein-Gordon equation for a charged, massive scalar field in the non-extremal Reissner-Nordström background. Performing a frequency domain analysis, using a continued fraction method, we compute the frequencies $\omega $ for quasi-bound states. We observe that, as the extremal limit for both the background and the field is approached, the real part of the quasi-bound states frequencies $\mathcal{R }(\omega )$ tends to the mass of the field and the imaginary part $\mathcal{I }(\omega )$ tends to zero, for any angular momentum quantum number $\ell $ . The limiting frequencies in this double extremal limit are shown to correspond to a distribution of extremal scalar particles, at stationary positions, in no-force equilibrium configurations with the background. Thus, generically, these stationary scalar configurations are regular at the event horizon. If, on the other hand, the distribution contains scalar particles at the horizon, the configuration becomes irregular therein, in agreement with no hair theorems for the corresponding Einstein-Maxwell-scalar field system.     

The entropy function for the extremal Kerr-(anti-)de Sitter black holes

Based on Sen's entropy function formalism, we consider the Bekenstein-Hawking entropy of the extremal Kerr-(anti-)de Sitter black holes in 4-dimensions. Unlike the extremal Kerr black hole case with flat asymptotic geometry, where the Bekenstein-Hawking entropy S is proportional to the angular momentum J, we get a quartic algebraic relation between S and J by using the known solution to the Einstein equation. We recover the same relation in the entropy function formalism. Instead of full geometry, we write down an ansatz for the near horizon geometry only. The exact form of the unknown functions and parameters in the ansatz are obtained by solving the differential equations which extremize the entropy function. The results agree with the nontrivial relation between S and J.We also study the Gauss-Bonnet correction to the entropy exploiting the entropy function formalism. We show that the term, though being topological thus does not affect the solution, contributes a constant addition to the entropy because the term shifts the Hamiltonian by that amount.     

Absorption of scalars by extremal black holes in string theory

We show that the low frequency absorption cross section of minimally coupled test massless scalar fields by extremal spherically symmetric black holes in d dimensions is equal to the horizon area, even in the presence of string-theoretical \(\alpha '\) corrections. Classically one has the relation \(\sigma = 4 GS\) between that absorption cross section and the black hole entropy. By comparing in each case the values of the horizon area and Wald’s entropy, we discuss the validity of such relation in the presence of higher derivative corrections for extremal black holes in many different contexts: in the presence of electric and magnetic charges; for nonsupersymmetric and supersymmetric black holes; in \(d=4\) and \(d=5\) dimensions. The examples we consider seem to indicate that this relation is not verified in the presence of \(\alpha '\) corrections in general, although being valid in some specific cases (electrically charged maximally supersymmetric black holes in \(d=5\)). We argue that the relation \(\sigma = 4 GS\) should in general be valid for the absorption cross section of scalar fields which, although being independent from the black hole solution, have their origin from string theory, and therefore are not minimally coupled.     

Quasi-normal modes of extremal black holes from hidden conformal symmetry

In this Letter, we construct the quasi-normal modes of three-dimensional extremal black holes in an algebraic way. We show that the infinite towers of the quasi-normal modes of scalar, vector and tensor could be constructed as the descendents of the highest weight modes. Our investigation shows that the hidden conformal symmetry suggested in Chen et al. (2010) [5] is an intrinsic property of the extremal black hole. Moreover, we notice that we need to fix the freedom in defining the local vector fields and find the right hidden conformal symmetry to obtain the physical quasi-normal modes.     

Topological aspect of Chern--Simons p-branes

By generalizing the topological current of Abelian Chern--Simons (CS) vortices, we present a topological tensor current of CS p-branes based on the \phi -mapping topological current theory. It is revealed that CS p-branes are located at the isolated zeros of the vector field \phi(x), and the topological structure of CS p-branes is characterized by the winding number of the \phi-mappings. Furthermore, the Nambu--Goto action and the equation of motion for multi CS p-branes are obtained.     

First order flows for N=2 extremal black holes and duality invariants

We derive explicitly the superpotential W for the non-BPS branch of $N=2$ extremal black holes in terms of duality invariants of special geometry. Although this is done for a one-modulus case (the $t^3$ model), the example gives $Z\ne 0$ black holes and captures the basic distinction from previous attempts on the quadratic series (vanishing C tensor) and from the other $Z=0$ cases. The superpotential W turns out to be a non-polynomial expression (containing radicals) of the basic duality invariant quantities. These are the same which enter in the quartic invariant $I_4$ for $N=2$ theories based on symmetric spaces. Using the flow equations generated by W, we also provide the analytic general solution for the warp factor and for the scalar field supporting the non-BPS black holes.     

Self-duality in the theory of the bosonic p-branes

It is shown that the Nambu-Goto lagrangian for the p-branes embedded in (p+1)-dimensional euclidean spacetime has a topological character and the dynamics is determined completely by the constraints. Self-duality equations are proposed whose solutions satisfy the constraints equations and minimize the action.