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Topology change in general relativity and the blackhole blackstring transition
, 2005
"... In the presence of compact dimensions massive solutions of General Relativity may take one of several forms including the blackhole and the blackstring, the simplest relevant background being R 3+1 × S 1. It is shown how Morse theory places constraints on the qualitative features of the phase dia ..."
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Cited by 56 (14 self)
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In the presence of compact dimensions massive solutions of General Relativity may take one of several forms including the blackhole and the blackstring, the simplest relevant background being R 3+1 × S 1. It is shown how Morse theory places constraints on the qualitative features of the phase diagram, and a minimalistic diagram is suggested which describes a first order transition whose only stable phases are the uniform string and the blackhole. The diagram calls for a topology changing “merger ” transition in which the blackhole evolves continuously into an unstable blackstring phase. As evidence a local model for the transition is presented in which the cone over S 2 × S 2 plays a central role. Horizon cusps do not appear as precursors to black hole merger. A generalization to higher dimensions finds that whereas the cone has a tachyon function for d = 5, its stability depends interestingly on the dimension it is unstable for d < 10, and stable for d> 10.
Caged black holes: black holes in compactified spacetimes II – 5d numerical implementation
"... Abstract: In backgrounds with compact dimensions there may exist several phases for black objects including the blackhole and the blackstring, and the phase transition between them raises puzzles and touches fundamental issues such as topology change, uniqueness and cosmic censorship. No analytic ..."
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Cited by 30 (11 self)
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Abstract: In backgrounds with compact dimensions there may exist several phases for black objects including the blackhole and the blackstring, and the phase transition between them raises puzzles and touches fundamental issues such as topology change, uniqueness and cosmic censorship. No analytic solution is known for the black hole, and moreover, one can expect approximate solutions only for very small black holes, while the phase transition physics happens when the black hole is large. Hence we turn to numerical solutions. Here some theoretical background to the numerical analysis is given, while the results will appear in a forthcoming paper. Goals for a numerical analysis are set, and the a scalar charge is defined and used as an improved order parameter which puts both the black hole and the black string at finite values. Predictions for small black holes are presented. The integrated first law (Smarr’s formula) is derived and will be used to estimate the “overall numerical error”. Expressions for physical quantities in terms of the numerical ones are supplied. Techniques include “method of equivalent charges”, dimensional reduction, analytic perturbation for small black holes, and free energy. Contents
Axisymmetric Numerical Relativity
, 2005
"... Chapters 2, 3 and 6 contain work done in collaboration with my supervisor and published in a joint paper [119]. The dynamical shift conditions in chapter 6 are a later addition by myself. The remaining chapters are my own work. All computer programmes were written by myself unless otherwise stated. ..."
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Cited by 18 (8 self)
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Chapters 2, 3 and 6 contain work done in collaboration with my supervisor and published in a joint paper [119]. The dynamical shift conditions in chapter 6 are a later addition by myself. The remaining chapters are my own work. All computer programmes were written by myself unless otherwise stated. c○Oliver Rinne, 2005 This thesis is concerned with formulations of the Einstein equations in axisymmetric spacetimes which are suitable for numerical evolutions. The common basis for our formulations is provided by the (2+1)+1 formalism. General matter sources and rotational degrees of freedom are included. A first evolution system adopts elliptic gauge conditions arising from maximal slicing and conformal flatness. The numerical implementation is based on the finitedifference approach, using a Multigrid algorithm for the elliptic equations and the method of lines for the hyperbolic evolution equations.
Investigations in Numerical Relativity
, 1993
"... Numerical relativity has come a long way in the last three decades and is now reaching a state of maturity. We are gaining a deeper understanding of the fundamental theoretical issues related to the field, from the well posedness of the Cauchy problem, to better gauge conditions, improved boundary t ..."
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Cited by 10 (0 self)
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Numerical relativity has come a long way in the last three decades and is now reaching a state of maturity. We are gaining a deeper understanding of the fundamental theoretical issues related to the field, from the well posedness of the Cauchy problem, to better gauge conditions, improved boundary treatment, and more realistic initial data. There has also been important work both in numerical methods and software engineering. All these developments have come together to allow the construction of several advanced fully threedimensional codes capable of dealing with both matter and black holes. In this manuscript I make a brief review the current status of the field. 1.
Adaptive mesh refinement for characteristic codes
 J. Comput. Phys
, 2004
"... The use of adaptive mesh refinement (AMR) techniques is crucial for accurate and efficient simulation of higher dimensional spacetimes. In this work we develop an adaptive algorithm tailored to the integration of finite difference discretizations of wavelike equations using characteristic coordinat ..."
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Cited by 8 (1 self)
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The use of adaptive mesh refinement (AMR) techniques is crucial for accurate and efficient simulation of higher dimensional spacetimes. In this work we develop an adaptive algorithm tailored to the integration of finite difference discretizations of wavelike equations using characteristic coordinates. We demonstrate the algorithm by constructing a code implementing the EinsteinKleinGordon system of equations in spherical symmetry. We discuss how the algorithm can trivially be generalized to higher dimensional systems, and suggest a method that can be used to parallelize a characteristic code. I.
Transient Pulses from Exploding Primordial Black Holes as a Signature of an Extra Dimension
, 801
"... An evaporating black hole in the presence of an extra spatial dimension would undergo an explosive phase of evaporation. We show that such an event, involving a primordial black hole, can produce a detectable electromagnetic pulse, signaling the existence of an extra dimension of size L ∼ 10 −18 − 1 ..."
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Cited by 5 (5 self)
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An evaporating black hole in the presence of an extra spatial dimension would undergo an explosive phase of evaporation. We show that such an event, involving a primordial black hole, can produce a detectable electromagnetic pulse, signaling the existence of an extra dimension of size L ∼ 10 −18 − 10 −20 m. We derive a generic relationship between the Lorentz factor of a pulseproducing “fireball ” and the TeV energy scale. For a toroidally compactified extra dimension, transient radiopulse searches probe the electroweak energy scale (∼0.1 TeV), enabling comparison with the Large Hadron Collider. The enormous challenges of detecting quantum gravitational effects, and exploring electroweakscale physics, make this a particularly attractive possibility.
Generalized harmonic formulation in spherical symmetry
, 908
"... Abstract: In this pedagogically structured article, we describe a generalized harmonic formulation of the Einstein equations in spherical symmetry which is regular at the origin. The generalized harmonic approach has attracted significant attention in numerical relativity over the past few years, es ..."
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Cited by 3 (1 self)
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Abstract: In this pedagogically structured article, we describe a generalized harmonic formulation of the Einstein equations in spherical symmetry which is regular at the origin. The generalized harmonic approach has attracted significant attention in numerical relativity over the past few years, especially as applied to the problem of binary inspiral and merger. A key issue when using the technique is the choice of the gauge source functions, and recent work has provided several prescriptions for gauge drivers designed to evolve these functions in a controlled way. We numerically investigate the parameter spaces of some of these drivers in the context of fully nonlinear collapse of a real, massless scalar field, and determine nearly optimal parameter settings for specific situations. Surprisingly, we find that many of the drivers that perform well in 3+1 calculations that use Cartesian coordinates, are considerably less effective in spherical symmetry, where some of them are, in fact, unstable
Generating initial data in general relativity using adaptive finite element methods
, 2008
"... The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite elementtype numerical methods for the resulting coupled nonlinear elliptic system. We derive weak formulations of the coupl ..."
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Cited by 3 (0 self)
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The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite elementtype numerical methods for the resulting coupled nonlinear elliptic system. We derive weak formulations of the coupled constraints, and review some new developments in the solution theory for the constraints in the cases of constant mean extrinsic curvature (CMC) data, nearCMC data, and arbitrarily prescribed mean extrinsic curvature data. We then outline some recent results on a priori and a posteriori error estimates for a broad class of Galerkintype approximation methods for this system which includes techniques such as finite element, wavelet, and spectral methods. We then use these estimates to construct an adaptive finite element method (AFEM) for solving this system numerically, and outline some new convergence and optimality results. We then describe in some detail an implementation of the methods using the FETK software package, which is an adaptive multilevel finite element code designed to solve nonlinear elliptic and parabolic systems on Riemannian
Black Hole Interaction Energy
, 2008
"... The interaction energy between two black holes at large separation distance is calculated. The first term in the expansion corresponds to the Newtonian interaction between the masses. The second term corresponds to the spinspin interaction. The calculation is based on the interaction energy defined ..."
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Cited by 2 (0 self)
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The interaction energy between two black holes at large separation distance is calculated. The first term in the expansion corresponds to the Newtonian interaction between the masses. The second term corresponds to the spinspin interaction. The calculation is based on the interaction energy defined on the two black holes initial data. No test particle approximation is used. The relation between this formula and cosmic censorship is discussed. 1