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New frontiers in Numerical Relativity
General Relativity and Quantum Cosmology · Submitted 8 Jan 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this thesis, we take this field to new frontiers by exploring its extensions to higher dimensions, non-asymptotically flat spacetimes and Einstein-Maxwell theory.
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The abstract, sentence by sentence
The first attempts at solving a binary black hole spacetime date back to the 1960s, with the pioneering works of Hahn and Lindquist. In spite of all the computational advances and enormous efforts by several groups, the first stable, long-term evolution of the orbit and merger of two black holes was only accomplished over 40 years later, in 2005. Since then, the field of Numerical Relativity has matured, and been extensively used to explore and uncover a plethora of physical phenomena in various scenarios. In this thesis, we take this field to new frontiers by exploring its extensions to higher dimensions, non-asymptotically flat spacetimes and Einstein-Maxwell theory. We start by reviewing the usual formalism and tools, including the "3+1" decomposition, initial data construction, the BSSN evolution scheme and standard wave extraction procedures. We then present a dimensional reduction procedure that allows one to use existing numerical codes (with minor adaptations) to evolve higher-dimensional systems with enough symmetry, and show corresponding results obtained for five-dimensional head-on collisions of black holes. Finally, we show evolutions of black holes in non-asymptotically flat spacetimes, and in Einstein-Maxwell theory.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The first attempts at solving a binary black hole spacetime date back to the 1960s, with the pioneering works of Hahn and Lindquist. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s1 | In spite of all the computational advances and enormous efforts by several groups, the first stable, long-term evolution of the orbit and merger of two black holes was only accomplished over 40 years later, in 2005. | background 0.97 | 0.04 | 0.03 | asserted 1.00 |
| s2 | Since then, the field of Numerical Relativity has matured, and been extensively used to explore and uncover a plethora of physical phenomena in various scenarios. | background 1.00 | 0.06 | 0.04 | asserted 1.00 |
| s3 | In this thesis, we take this field to new frontiers by exploring its extensions to higher dimensions, non-asymptotically flat spacetimes and Einstein-Maxwell theory. | contribution 1.00 | 0.56 | 0.27 | asserted 0.97 |
| s4 | We start by reviewing the usual formalism and tools, including the "3+1" decomposition, initial data construction, the BSSN evolution scheme and standard wave extraction procedures. | method 0.70 | 0.39 | 0.02 | asserted 0.96 |
| s5 | We then present a dimensional reduction procedure that allows one to use existing numerical codes (with minor adaptations) to evolve higher-dimensional systems with enough symmetry, and show corresponding results obtained for five-dimensional head-on collisions of black holes. | contribution 0.55 | 0.94 | 0.12 | asserted 1.00 |
| s6 | Finally, we show evolutions of black holes in non-asymptotically flat spacetimes, and in Einstein-Maxwell theory. | contribution 0.78 | 0.92 | 0.15 | asserted 1.00 |
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The paper
Original abstract
The first attempts at solving a binary black hole spacetime date back to the 1960s, with the pioneering works of Hahn and Lindquist. In spite of all the computational advances and enormous efforts by several groups, the first stable, long-term evolution of the orbit and merger of two black holes was only accomplished over 40 years later, in 2005. Since then, the field of Numerical Relativity has matured, and been extensively used to explore and uncover a plethora of physical phenomena in various scenarios. In this thesis, we take this field to new frontiers by exploring its extensions to higher dimensions, non-asymptotically flat spacetimes and Einstein-Maxwell theory. We start by reviewing the usual formalism and tools, including the "3+1" decomposition, initial data construction, the BSSN evolution scheme and standard wave extraction procedures. We then present a dimensional reduction procedure that allows one to use existing numerical codes (with minor adaptations) to evolve higher-dimensional systems with enough symmetry, and show corresponding results obtained for five-dimensional head-on collisions of black holes. Finally, we show evolutions of black holes in non-asymptotically flat spacetimes, and in Einstein-Maxwell theory.
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References
How the literature uses it
4 arXiv papers cite this in 4 sentences
Papers citing it · 2 sentences without a confident label
Cites it as background 2 papers
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This can be achieved in practice by either (i) using a specific form of the line element that directly imposes the symmetry in question (see e.g. [ref] ), (ii) starting with a generic line element and applying dimensional reduction through isometry (see e.g. [ref] ) or (iii) implementing the symmetry through a so-called Cartoon method [ref] .
Dimensional reduction in numerical relativity: Modified cartoon formalism and regularization · Introduction · 0.82
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Numerical relativity for higher-dimensional space-times has seen an increasing amount of development in recent years [ref] , but so far black hole collisions have only been successfully computed at low energies [ref] .
Radiation from a $D$-dimensional collision of gravitational shock waves · Collision of gravitational shock waves in $D$ dimensions · 0.83
What it cites · 155 sentences without a confident label
Contests 2 papers
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We can compute directly $Φ_,t$ with [ref] [Note that there is a factor $r$ missing in equation (3.15) of reference [ref] .] [equation] where $k^2=l(l+D-3)$.
Numerical relativity for D dimensional space-times: head-on collisions of black holes and gravitational wave extraction · Extracting gravitational waves · contest 0.78
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We can compute directly $Φ_,t$ with [ref] [Note that there is a factor $r$ missing in equation (3.15) of reference [ref] .] [equation] where $k^2=l(l+D-3)$.
A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions · Extracting gravitational waves · contest 0.78
Builds on 9 papers
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Building on the results outlined in chapter [ref] (based on [ref] ), we now present a generalisation of the spectral solver in [ref] that generates initial data for black hole binaries in $D≥5$ dimensions with non-vanishing initial boost [ref] .
Higher-dimensional puncture initial data · Higher-dimensional initial data · 0.98
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Building on the results outlined in chapter [ref] (based on [ref] ), we now present a generalisation of the spectral solver in [ref] that generates initial data for black hole binaries in $D≥5$ dimensions with non-vanishing initial boost [ref] .
A single-domain spectral method for black hole puncture data · Higher-dimensional initial data · 0.98
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Building on the results outlined in chapter [ref] (based on [ref] ), we now present a generalisation of the spectral solver in [ref] that generates initial data for black hole binaries in $D≥5$ dimensions with non-vanishing initial boost [ref] .
Close-limit analysis for head-on collision of two black holes in higher dimensions: Brill-Lindquist initial data · Higher-dimensional initial data · 0.98
Uses the method or tool of 16 papers
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We now study the numerical results as obtained for $D=4,5,6,7$ with these adaptations of the spectral solver of [ref] .
A single-domain spectral method for black hole puncture data · Results · 0.91
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We will adopt the approach outlined in [ref] to evolve the electro-vacuum Einstein-Maxwell equations which incorporates suitably added additional fields to ensure the evolution will preserve the constraints.
Multi-dimensional Numerical Scheme for Resistive Relativistic MHD · Evolution equations · 1.00
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We will adopt the approach outlined in [ref] to evolve the electro-vacuum Einstein-Maxwell equations which incorporates suitably added additional fields to ensure the evolution will preserve the constraints.
Understanding possible electromagnetic counterparts to loud gravitational wave events: Binary black hole effects on electromagnetic fields · Evolution equations · 1.00
Uses data or values from 6 papers
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In [ref] it was shown that this number can be as large as 35% in $D=4$.
Cross section, final spin and zoom-whirl behavior in high-energy black hole collisions · Discussion · 0.87
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In [ref] it was shown that this number can be as large as 35% in $D=4$.
High-velocity collision of two black holes · Discussion · 0.87
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Indeed, in the four dimensional case, it is known that ultra-relativistic head-on collisions of equal mass non-rotating black holes release up to 14% of the initial centre of mass energy into gravitational radiation [ref] .
The high-energy collision of two black holes · Discussion · 0.82
Compares with 9 papers
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in agreement to within a factor of $2$ with numerical simulations (see [ref] and table [ref] below; the agreement could be improved by assuming $α_b∼1.3$).
Numerical relativity for D dimensional space-times: head-on collisions of black holes and gravitational wave extraction · Analytic predictions · 0.99
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This value is in excellent agreement with perturbative calculations, which predict a lowest quasinormal frequency $r_S ω=0.9477- i 0.2561$ for $l=2$ [ref] .
Quasinormal modes of black holes and black branes · Waveforms · 0.95
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This value is in excellent agreement with perturbative calculations, which predict a lowest quasinormal frequency $r_S ω=0.9477- i 0.2561$ for $l=2$ [ref] .
Close-limit analysis for head-on collision of two black holes in higher dimensions: Brill-Lindquist initial data · Waveforms · 0.95
Cites as support 2 papers
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It has actually been shown [ref] that there is no foliation of the Kerr spacetime that is axisymmetric, conformally flat, and reduces smoothly to the Schwarzschild solution in the non-rotating limit.[For the four-dimensional Kerr solution, but there is no reason to believe that the higher-dimensional case is any different.] This means that our Bowen-York solution with $J^ab ≠0$ does represent a rotating black hole, but not a stationary one.
Nonexistence of conformally flat slices of the Kerr spacetime · General case · 0.99
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This conjecture (or rather, the classical variant thereof) has recently gained more support with the work by Choptuik and Pretorius [ref] , where it was shown that collisions of boson stars do form black holes, for sufficiently high boost parameter.
Ultra Relativistic Particle Collisions · Motivation and historical background · 0.99
Cites as background 42 papers
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Numerical relativity in higher dimensions has only recently started being explored, with pioneering works including those in [ref] .
Higher-dimensional numerical relativity: Formulation and code tests · Horizon finding · 0.94
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Numerical relativity in higher dimensions has only recently started being explored, with pioneering works including those in [ref] .
Towards the Final Fate of an Unstable Black String · Horizon finding · 0.94
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Numerical relativity in higher dimensions has only recently started being explored, with pioneering works including those in [ref] .
Numerical relativity for D dimensional space-times: head-on collisions of black holes and gravitational wave extraction · Horizon finding · 0.94
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