REVIEW
Topological Black Holes -- Outside Looking In
General Relativity and Quantum Cosmology · Submitted 15 Sep 1997 · 28 references
Pith's verdict: The arbitrary-dimension construction is the real contribution; the numerical headline is confounded at small γ by a surface-gravity mismatch the paper doesn't control for.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
The paper gives a general anti-de Sitter identification construction for black holes whose horizons are toroidal or higher-genus surfaces, and reports numerical evidence that conformally coupled scalar waves decay differently outside球形 and
The reading
What carries the argument
The construction starts from anti-de Sitter space as the hypersurface $-x_0^2+x_1^2+cdots+x_n^2-x_{n+1}^2=-ell^2$ in flat signature $(n-2,2)$; identifying points in an $m$-dimensional Minkowski subspace along spacelike Killing directions produces a compact negative-curvature surface $H^{m-1}/Gamma$ and the quotient metric (19), with $N(R)=(R^2-R_+^2)/ell^2$ and an event horizon at $R=R_+$. In $(3+1)$ dimensions the charged genus-$g$ metric has lapse $V(r)=r^2/ell^2-(1-delta_{g,1}-2delta_{g,0})-2m/r+q^2/r^2$, and the conserved mass and charge are $M=m(|g-1|+delta_{g,1})$ and $Q=q(|g-1|+delta_{g,1})$. The numerical machinery reduces the conformally coupled scalar equation $nabla^2Psi=xi RPsi$ with $xi=1/6$ to a $1+1$ wave equation $partial_t^2psi-partial_x^2psi+V(x)psi=0$ after $Psi=r^{-1}psi(t,r)Y_l(Omega_g)$, using the tortoise coordinate $x=int dr/N(r)$ and finite differences with $Delta x>Delta t$ and Dirichlet or Neumann conditions at the anti-de Sitter boundary $x=0$.
Extended reading notes
Core claim
The central claim is twofold. First, topological black holes are not ad hoc solutions: they follow from a general identification procedure in anti-de Sitter space, and the resulting metric (19) contains the known $(2+1)$ BTZ black hole, the $(3+1)$ toroidal and genus-$gge 2$ black holes, and the constant-curvature black hole as special cases. Second, in $(3+1)$ dimensions the late-time radiative falloff of a conformally coupled scalar field is not universal across horizon topology. For large $gamma=3sqrt{3}M/ell$, spherical Schwarzschild anti-de Sitter and higher-genus topological black holes behave similarly, with ringing that decays exponentially and only quantitative differences in frequency and rate. For small $gamma$, the spherical case shows ringing followed by a power-law tail and later resurgence, whereas the higher-genus case loses the ringing and settles into a smooth exponential decay; the toroidal case retains some ringing. The author reads this as evidence that topology can affect mass inflation, because a sufficiently rapid exponential falloff can cut off the blueshift at the Cauchy horizon.
Load-bearing premise
The scalar-wave falloff is a faithful proxy for the gravitational and electromagnetic perturbations that actually drive mass inflation, and the paper asserts qualitative agreement without demonstrating it.
Editorial extensions
If this is right
- The metric (19) gives a single construction that contains the BTZ black hole, the toroidal black hole, the higher-genus black hole, and the constant-curvature black hole as special choices of $m$ and $n$.
- In $(3+1)$ dimensions, genus-$g$ black holes can exist with negative mass and with arbitrarily large charge relative to mass, unlike the spherical Reissner-Nordstrom anti-de Sitter case.
- Topological black holes can form from the gravitational collapse of pressureless dust provided $Lambda>8pi Grho_0$, and negative-energy dust can collapse to negative-mass black holes if the density is not too large.
- The late-time falloff of a conformally coupled scalar field outside higher-genus topological black holes can be a smooth exponential at small $gamma$, so mass inflation may be cut off for some range of $gamma$ and $|Lambda|$.
- The numerical falloff results are largely insensitive to the choice of Dirichlet or Neumann boundary conditions at spatial infinity.
Reading between the lines
- If gravitational and electromagnetic perturbations follow the scalar trend, then two black holes with the same mass and charge but different horizon genus could have different Cauchy-horizon stability, making topology a control parameter for mass inflation.
- The toroidal case sitting between the spherical and higher-genus behavior suggests a possible monotonic trend with genus or Euler characteristic, which could be checked by computing falloff for $g=2,3,4$ at fixed $gamma$.
- The existence of negative-mass horizons suggests a thermodynamic regime where mass is not a positive energy cost, which could be probed through boundary stress-tensor calculations for genus-$g$ anti-de Sitter black holes.
- The finite anti-de Sitter boundary and the use of Dirichlet or Neumann conditions could be extended to radiative boundary conditions to test whether the topology-dependent exponential envelope survives in a more realistic scattering setup.
Original abstract
I describe the general mathematical construction and physical picture of topological black holes, which are black holes whose event horizons are surfaces of non-trivial topology. The construction is carried out in an arbitrary number of dimensions, and includes all known special cases which have appeared before in the literature. I describe the basic features of massive charged topological black holes in $(3+1)$ dimensions, from both an exterior and interior point of view. To investigate their interiors, it is necessary to understand the radiative falloff behaviour of a given massless field at late times in the background of a topological black hole. I describe the results of a numerical investigation of such behaviour for a conformally coupled scalar field. Significant differences emerge between spherical and higher genus topologies.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
I describe the general mathematical construction and physical picture of topological black holes, which are black holes whose event horizons are surfaces of non-trivial topology. The construction is carried out in an arbitrary number of dimensions, and includes all known special cases which have appeared before in the literature. I describe the basic features of massive charged topological black holes in $(3+1)$ dimensions, from both an exterior and interior point of view. To investigate their interiors, it is necessary to understand the radiative falloff behaviour of a given massless field at late times in the background of a topological black hole. I describe the results of a numerical investigation of such behaviour for a conformally coupled scalar field. Significant differences emerge between spherical and higher genus topologies.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | I describe the general mathematical construction and physical picture of topological black holes, which are black holes whose event horizons are surfaces of non-trivial topology. | contribution 0.89 | 0.75 | 0.08 | asserted 0.98 |
| s1 | The construction is carried out in an arbitrary number of dimensions, and includes all known special cases which have appeared before in the literature. | contribution 0.78 | 0.81 | 0.05 | asserted 1.00 |
| s2 | I describe the basic features of massive charged topological black holes in $(3+1)$ dimensions, from both an exterior and interior point of view. | contribution 0.98 | 0.77 | 0.05 | asserted 0.95 |
| s3 | To investigate their interiors, it is necessary to understand the radiative falloff behaviour of a given massless field at late times in the background of a topological black hole. | background 0.70 | 0.13 | 0.04 | asserted 1.00 |
| s4 | I describe the results of a numerical investigation of such behaviour for a conformally coupled scalar field. | contribution 0.72 | 0.93 | 0.08 | asserted 0.89 |
| s5 | Significant differences emerge between spherical and higher genus topologies. | result 0.99 | 0.84 | 0.38 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:gr-qc/9709039 by the authors.
References
How the literature uses it
41 arXiv papers cite this in 51 sentences
Papers citing it · 23 sentences without a confident label
Builds on it 1 paper
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For example, for topological AdS black holes ($k≠1$ in [ref]), $ω_d$ would be replaced by the corresponding volume of the constant curvature space $ω_d^(k)$ [ref] .]
Black hole chemistry: thermodynamics with Lambda · Reverse isoperimetric inequality · 0.85
Compares with it 1 paper
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Comparing them with topological black holes within Einstein gravity [ref] , we observe that compact sections at fixed $r$ and $t$ can also be obtained for Lifshitz topological black holes by suitable identifications in the spatial coordinates.
On the role of boundary conditions within Ho\v{r}ava-Lifshitz gravity · Hor̆ava-Lifshitz solutions · 1.00
Cites it as background 24 papers
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The metric of the $D$-dimensional MTBH reads [equation] where $L$ is related to the negative cosmological constant $Λ$ by [equation] and $Σ^2_D-2$ stands for the line element of a $(D-2)$-dimensional compact Einstein space of negative curvature [ref] .
On the Time Times Temperature Bound · Introduction · 0.83
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The MTBH is a static solution to Einstein's equations with negative cosmological constant, whose mass is equal to zero and its event horizon is a compact and orientable manifold of negative curvature [ref] – [ref] .
Entropy spectrum of the D-dimensional massless topological black hole · Introduction · 0.81
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They can be seen as higher-dimensional generalizations of a static BTZ black hole [ref] .
Ground and thermal states for the Klein-Gordon field on a massless hyperbolic black hole with applications to the anti-Hawking effect · Introduction · 0.88
Discussion
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