Multi-black holes from nilpotent Lie algebra orbits

For N 3 2{\mathcal{N}\ge 2} supergravities, BPS black hole solutions preserving four supersymmetries can be superposed linearly, leading to well defined solutions containing an arbitrary number of such BPS black holes at arbitrary positions. Being stationary, these solutions can be understood via associated non-linear sigma models over pseudo-Riemannian spaces coupled to Euclidean gravity in three spatial dimensions. As the main result of this paper, we show that whenever this pseudo-Riemannian space is an irreducible symmetric space \mathfrakG/\mathfrakH^*{\mathfrak{G}/\mathfrak{H}^*}, the most general solutions of this type can be entirely characterised and derived from the nilpotent orbits of the associated Lie algebra \mathfrakg{\mathfrak{g}}. This technique also permits the explicit computation of non-supersymmetric extremal solutions which cannot be obtained by truncation to N=2{\mathcal{N}=2} supergravity theories. For maximal supergravity, we not only recover the known BPS solutions depending on 32 independent harmonic functions, but in addition find a set of non-BPS solutions depending on 29 harmonic functions. While the BPS solutions can be understood within the appropriate N=2{\mathcal{N}=2} truncation of N=8{\mathcal{N}=8} supergravity, the general non-BPS solutions require the whole field content of the theory.     

Schwarzschild black hole with global monopole charge

We derive the metric for a Schwarzschild black hole with global monopole charge by relaxing asymptotic flatness of the Schwarzschild field. We then study the effect of global monopole charge on particle orbits and the Hawking radiation. It turns out that existence, boundedness and stability of circular orbits scale up by (1−8πη ²)⁻¹, and the perihelion shift and the light bending by (1−8πη ²)^−3/2, while the Hawking temperature scales down by (1−8πη ²)² the Schwarzschild values. Hereη is the global charge.     

Interacting non-BPS black holes

We explain how to exploit systematically the structure of nilpotent orbits to obtain a solvable system of equations describing extremal solutions of (super-)gravity theories, i.e. systems that can be solved in a linear way. We present the procedure in the case of the STU model, where we show that all extremal solutions with a flat three-dimensional base are fully described with the help of three different nilpotent orbits: the BPS, the almost-BPS and the composite non-BPS. The latter describes a new class of solutions for which the orientation of half of the constituent branes have been inverted with respect to the BPS one, such that all the centres are intrinsically non-BPS, and interact with each others. We finally recover explicitly the ensemble of the almost-BPS solutions in our formalism and present an explicit two-centre solution of the new class.     

How do black holes predict the sign of the Fourier coefficients of siegel modular forms?

Single centered supersymmetric black holes in four dimensions have spherically symmetric horizon and hence carry zero angular momentum. This leads to a specific sign of the helicity trace index associated with these black holes. Since the latter are given by the Fourier expansion coefficients of appropriate meromorphic modular forms of Sp(2,\mathbbZ){Sp(2,{\mathbb{Z}})} or its subgroup, we are led to a specific prediction for the signs of a subset of these Fourier coefficients which represent contributions from single centered black holes only. We explicitly test these predictions for the modular forms which compute the index of quarter BPS black holes in heterotic string theory on T ⁶, as well as in \mathbbZ_N{{\mathbb{Z}}_N} CHL models for N = 2, 3, 5, 7.     

M‐theory solutions invariant under D(2,1; γ) ⊕ D(2,1;γ)

We simplify and extend the construction of half‐BPS solutions to 11‐dimensional supergravity, with isometry superalgebra D(2,1;γ) ⊕ D(2,1;γ). Their space‐time has the form AdS₃× S³× S³ warped over a Riemann surface Σ. It describes near‐horizon geometries of M2 branes ending on, or intersecting with, M5 branes along a common string. The general solution to the BPS equations is specified by a reduced set of data (γ, h, G), where γ is the real parameter of the isometry superalgebra, and h and G are functions on Σ whose differential equations and regularity conditions depend only on the sign of γ. The magnitude of γ enters only through the map of h,G onto the supergravity fields, thereby promoting all solutions into families parametrized by |γ|. By analyzing the regularity conditions for the supergravity fields, we prove two general theorems: (i) that the only solution with a 2‐dimensional CFT dual is AdS₃× S³× S³× ℝ², modulo discrete identifications of the flat ℝ², and (ii) that solutions with γ < 0 cannot have more than one asymptotic higher‐dimensional AdS region. We classify the allowed singularities of h and G near the boundary of Σ, and identify four local solutions: asymptotic AdS₄/Z₂ or AdS₇^′ regions; highly‐curved M5‐branes; and a coordinate singularity called the “cap”. By putting these “Lego” pieces together we recover all known global regular solutions with the above symmetry, including the self‐dual strings on M5 for γ <0, and the Janus solution for γ > 0, but now promoted to families parametrized by |γ|. We also construct exactly new regular solutions which are asymptotic to AdS₄/Z₂ for γ < 0, and conjecture that they are a different superconformal limit of the self‐dual string. Finally, we construct exactly γ > 0 solutions with highly curved M5‐brane regions, which are the formal continuation of the self‐dual string solutions across the decompactification point at γ = 0.     

Entropy function approach to charged BTZ black hole

We find solution to the metric function f(r) = 0 of charged BTZ black hole making use of the Lambert function. The condition of extremal charged BTZ black hole is determined by a non-linear relation of M _e (Q) = Q ²(1 − ln Q ²). Then, we study the entropy of extremal charged BTZ black hole using the entropy function approach. It is shown that this formalism works with a proper normalization of charge Q for charged BTZ black hole because AdS₂ × S¹ represents near-horizon geometry of the extremal charged BTZ black hole. Finally, we introduce the Wald’s Noether formalism to reproduce the entropy of the extremal charged BTZ black hole without normalization when using the dilaton gravity approach.     

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Abstract

We classify the Noether point symmetries of the generalized Lane-Emden equation y″+ ny′/x+ f(y)?=?0 with respect to the standard Lagrangian L = xⁿy′²/2 — xⁿ ∫f(y)dy for various functions f(y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f(y). One of them is f(y) = αy^r , where r ≠ 0,1. The case r?=?5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r?= 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.     

Charged balanced black rings in five dimensions

We present balanced black ring solutions of pure Einstein–Maxwell theory in five dimensions. The solutions are asymptotically flat, and their tension and gravitational self-attraction are balanced by the repulsion due to rotation and electrical charge. Hence the solutions are free of conical singularities and possess a regular horizon which exhibits the ring topology S¹×S²$S^1\times S^2$. We discuss the global charges and the horizon properties of the solutions and show that they satisfy a Smarr relation. We construct these black rings numerically, restricting to the case of black rings with a rotation in the direction of the S¹$S^1$ and large black rings. We compare these to the blackfold results.     

Nilpotent orbits in real symmetric pairs and stationary black holes

In the study of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determine the nilpotent orbits of a real symmetric pair. We apply our methods to an explicit example, and thereby classify the nilpotent orbits of acting on the fourth tensor power of the natural 2‐dimensional ‐module. This makes it possible to classify all stationary solutions of the so‐called STU‐supergravity model.     

Abelian and non‐abelian D‐brane effective actions

In this Ph.D. thesis, accepted at the Vrije Universiteit Brussel, we review and elaborate on a method to find the D‐brane effective action, based on BPS equations. Firstly, both for the Yang‐Mills action and the Born‐Infeld action it is shown that these configurations are indeed BPS, i.e. solutions to these equations saturate a Bogomolny bound and leave some supersymmetry unbroken. Next, we use the BPS equations as a tool to construct the D‐brane effective action and require that (a deformation of) these equations should still imply the equations of motion in more general cases. In the abelian case we managed to calculate all order in α′ four‐derivative corrections to the effective action and the BPS equations while in the non‐abelian case we obtained the effective action up to order α′⁴. Furthermore, we discuss a check based on the spectrum of strings stretching between intersecting branes. Finally, this Ph.D. thesis also discusses the construction of a boundary superspace which would be the first step to use the method of Weyl invariance in N = 2 superspace in order to again construct the D‐brane effective action. A more detailed summary of each section can be found in the introduction.     

Uniqueness and Stability of Riemann Solutions¶with Large Oscillation in Gas Dynamics

We prove the uniqueness of Riemann solutions in the class of entropy solutions in with arbitrarily large oscillation for the 3 × 3 system of Euler equations in gas dynamics. The proof for solutions with large oscillation is based on a detailed analysis of the global behavior of shock curves in the phase space and the singularity of centered rarefaction waves near the center in the physical plane. The uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large perturbation of the Riemann initial data, as long as the corresponding solutions are in L ^∞ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is needed. The uniqueness result for Riemann solutions can easily be extended to entropy solutions U(x,t), piecewise Lipschitz in x, for any t > 0, with arbitrarily large oscillation. Received: 23 April 2001 / Accepted: 20 September 2001     

Decomposition of Quasi-regular Representations Induced from Discrete Subgroups of Nilpotent Lie Groups

We describe the direct integral decomposition of a quasi regular representation of a connected and simply connected nilpotent Lie group G, which is induced from a discrete subgroup Γ. The solution is given explicitly in terms of orbital parameters. That is, the spectrum, multiplicity and spectral measure that constitute the decomposition are described completely in terms of natural objects associated to the co-adjoint orbits of G. We conclude with a study of the multiplicity function in certain cases.      

The extremal black holes of {\mathcal{N} = 4} supergravity from {\mathfrak{so}(8, 2 + n)} nilpotent orbits

We consider the stationary solutions of ${\mathcal{N} = 4}$ supergravity coupled to n vector multiplets that define linear superpositions of non-interacting extremal black holes. The most general solutions of this type are derived from the graded decompositions of ${\mathfrak{so}(8, 2 + n)}$ associated to its nilpotent orbits. We illustrate the formalism by giving explicitly asymptotically Minkowski non-BPS solutions of the most exotic class depending on 6 + n harmonic functions.     

Static spherically symmetric solutions of the Einstein-Yang-Mills equations

We study the global behaviour of static, spherically symmetric solutions of the Einstein-Yang-Mills equations with gauge groupSU(2). Our analysis results in three disjoint classes of solutions with a regular origin or a horizon. The 3-spaces (t=const.) of the first, generic class are compact and singular. The second class consists of an infinite family of globally regular, resp. black hole solutions. The third type is an oscillating solution, which although regular is not asymptotically flat.This article was processed by the author using the Springer-Verlag TEX CoMaPhy macro package 1991.     

Multipole analysis of kicks in collision of spinning binary black holes

Thorne and Kidder give expressions which allow for analytical estimates of the “kick”, i.e. the recoil, produced from asymmetrical gravitational radiation during the interaction of black holes, or in fact any gravitating compact bodies. (The Thorne-Kidder formula uses momentum flux calculations based on the linearized General Relativity of gravitational radiation). We specifically treat kicks arising in the binary interaction of equal mass black holes, when at least one of the black holes has significant spin, a. Such configurations can produce very large kicks in computational simulations. We consider both fly-by and quasicircular orbits. For fly-by orbits we find substantial kicks from those Thorne-Kidder terms which are linear in a. For the quasi-circular case, we consider in addition the nonlinear contribution (O(a ²)) to the kicks, and provide a dynamical explanation for such terms discovered and displayed by [2]. However, in the cases of maximal kick velocities, the dependence on spin is largely linear (reproduced in numerical results [6]).     

Black holes and large N species solution to the hierarchy problem

We provide the perturbative and non‐perturbative arguments showing that theories with large number of species of the quantum fields, imply an inevitable hierarchy between the masses of the species and the Planck scale, shedding a different light on the hierarchy problem. In particular, using the black hole physics, we prove that any consistent theory that includes N Z₂‐conserved species of the quantum fields of mass Λ, must have a value of the Planck mass, which in large N limit is given by M_P² \gsim N Λ². An useful byproduct of this proof is that any exactly conserved quantum charge, not associated with a long‐range classical field, must be defined maximum modulo N, with N \gsim (M_P/m)², where m is the mass of the unit charge. For example, a continuous global U(1) ‘baryon number’ symmetry, must be explicitly broken by gravity, at least down to a Z_N subgroup, with N \lsim (M_P/m_b)², where m_b is the baryon mass. The same constraint applies to any discrete gauge symmetry, as well as to other quantum‐mechanically‐detectable black hole charges that are associated with the massive quantum hair of the black hole. We show that the gravitationally‐coupled N‐species sector that solves the gauge hirearchy problem, should be probed by LHC.     

Non-supersymmetric extremal RN-AdS black holes in $$ \mathcal{N} = 2 $$ gauged supergravity

We investigate extremal Reissner-Nordström-AdS black holes in fourdimensional \( \mathcal{N} = 2 \) abelian gauged supergravity. We find a new attractor equation which is not reduced to the one in the asymptotically flat spacetime. We also argue a formula which is available even in the presence of the scalar potential. We apply them to the T³-model and the STU-model in generic black hole charge distributions. In addition, focusing on the so-called T³-model with a single neutral vector multiplet, we obtain non-supersymmetric extremal Reissner-Nordström-AdS black hole solutions with regular event horizons in the D0-D4 and the D2-D6 black hole charge configurations. The negative cosmological constant emerges even without the Fayet-Iliopoulos parameters.     

Phase-plane analysis of test particle orbits in regular black holes

We investigate phase-plane analysis of general relativistic orbits in a gravitational field of the Reissner–Nordstr?m-type regular black hole spacetime. We employ phase-plane analysis to obtain different phase-plane diagrams of the test particle orbits by varying charge q and dimensionless parameter β, where β contains angular momentum of the test particle. We compute numerical values of radii for the innermost stable orbits and corresponding values of energy required to place the test particle in orbits. Later on, we employ similar analysis on an Ayón–Beato–García(ABG) regular black hole and a comparison regarding key results is also included.