REVIEW
A supersymmetric black ring
High Energy Physics - Theory · Submitted 9 Jul 2004 · 13 references
Pith's verdict: A genuine new exact solution — the first asymptotically flat BPS black hole with S1×S2 horizon — with the checks that matter actually present in the paper.
Why Pith says this
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Referee report
The paper
The paper claims
Five-dimensional supergravity admits a supersymmetric black hole whose horizon is shaped like a ring — a circle times a sphere — rather than a sphere, and the authors write the solution down explicitly.
The reading
What carries the argument
The load-bearing device is the general form of any supersymmetric solution of five-dimensional minimal supergravity in a region where the supersymmetric Killing vector $V=partial/partial t$ is timelike: $ds^2 = -f^2(dt+omega)^2 + f^{-1} ds^2(M_4)$, with $M_4$ hyper-Kähler, subject to $dG^+ = 0$ and $Delta f^{-1} = (4/9)(G^+)^2$ where $G^+ = frac{1}{2} f(domega + star domega)$. Taking $M_4$ to be flat $R^4$ in the ring coordinates of Eq. (3), the authors supply $f^{-1}$ and the two components of $omega$ and check the pair of equations, with the Maxwell field then fixed by $f$ and $omega$. The second device is the near-horizon analysis: a coordinate change $dt = dv - B(r)dr$, $dphi = dphi' - C(r)dr$, $dpsi = dpsi' - C(r)dr$ with $B$ and $C$ chosen order by order in $r$ removes the divergences and leaves a metric analytic at $r=0$, which is what upgrades a coordinate pathology at $y=-infty$ into a genuine Killing horizon. The third is the closed-timelike-curve check: positivity of $g_{psi psi}$ as $y to -infty$, together with positivity of the determinant of the $(phi,psi)$ block, is shown to hold throughout the exterior when $R < (Q-q^2)/(2q)$.
Extended reading notes
Core claim
The central claim is that the fields in Eqs. (4)–(6) — a scalar $f$, a one-form $omega$ and a gauge potential $A$ built on flat $R^4$ written in ring coordinates — constitute a supersymmetric, asymptotically flat solution of five-dimensional minimal supergravity whose event horizon has topology $S^1 times S^2$. This is stated as the first example of a supersymmetric, asymptotically flat black hole of non-spherical topology, and it sits exactly in case (ii) of the earlier classification of allowed supersymmetric horizon geometries. The authors compute the charges: $M = 3pi Q/(4G)$, $J_phi = pi q(3Q-q^2)/(8G)$ and $J_psi = pi q(6R^2+3Q-q^2)/(8G)$, so the BPS inequality is saturated while the two angular momenta remain independent. They show the horizon is regular by exhibiting a coordinate transformation with explicitly chosen constants that makes the metric analytic at $r=0$, with $V=partial/partial v$ null and normal there; the spatial cross-section is $S^1 times S^2$ with circle radius $L$ and sphere radius $q/2$, and the area is $A = 2pi^2 L q^2$. The authors acknowledge that if $L=0$ the same construction produces a null orbifold singularity rather than a horizon, so regularity of the ring requires $L>0$, which the no-closed-timelike-curves bound $R < (Q-q^2)/(2q)$ guarantees. Setting $R=0$ collapses the ring to the BMPV black hole, which is why the ring's charges can approach but never equal those of the spherical solution.
Load-bearing premise
The load-bearing premise is that the coordinate change with the quoted finite-part constants really does make the metric analytic at $r=0$, so that the surface is a regular Killing horizon rather than a null orbifold singularity; the paper asserts this rather than deriving the constants, and it notes that when $L=0$ the same construction gives a singularity instead of a horizon.
Editorial extensions
If this is right
- Supersymmetric black holes in five dimensions are not unique to spherical topology: for fixed electric charge there is a two-parameter family of BPS black rings spanning a region of the $J_phi$–$J_psi$ plane, bounded by $L to 0$, by $q to 0$, and by $R to 0$.
- The near-horizon geometry is locally $AdS_3 times S^2$ with $AdS_3$ radius $q$ and sphere radius $q/2$, matching the expectation that supersymmetric horizons of this topology factorize this way.
- The entropy is discontinuous across $R=0$: the ring entropy approaches a limit that is strictly below the BMPV entropy at the same charges, in analogy with coincident sources in multi-centre extremal Reissner–Nordström solutions.
- The infinite-radius limit $R to infty$ with fixed charge density reproduces the previously conjectured black-string solution of Bena, giving that solution a finite-radius parent with two independent angular momenta.
- Because the solution is supersymmetric and has a regular finite-area horizon, its entropy is a candidate target for a statistical-mechanical counting, which the authors leave for future work.
Reading between the lines
- If the ring is stable as the authors expect from supersymmetry, then the space of BPS states at fixed charge contains at least two disconnected horizon topologies, so any entropy formula that depends only on conserved charges cannot distinguish them — a sharp test for microscopic counting.
- The discontinuity of entropy at $R to 0$ suggests a first-order-like transition between ring and sphere phases at fixed charge; it would be worth checking whether a non-extremal five-parameter family interpolates between them and where the ring actually dominates.
- The quantization remark — that $q$ counts Kaluza–Klein monopoles wrapping the ring circle and should be quantized in units of the compactification radius — implies the allowed ring charges form a discrete lattice inside the continuous $J_phi$–$J_psi$ region, which is checkable once the six-dimensional uplift is fixed.
- Since the two angular momenta are unconstrained by supersymmetry, the same construction should extend to a three-charge, three-dipole family whose conserved charges no longer determine the solution; the equal-charge case presented here would then be a measure-zero slice with unusually simple uniqueness properties.
Original abstract
A new supersymmetric black hole solution of five-dimensional supergravity is presented. It has an event horizon of topology S1xS2. This is the first example of a supersymmetric, asymptotically flat black hole of non-spherical topology. The solution is uniquely specified by its electric charge and two independent angular momenta. These conserved charges can be arbitrarily close, but not exactly equal, to those of a supersymmetric black hole of spherical topology.
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
A new supersymmetric black hole solution of five-dimensional supergravity is presented. It has an event horizon of topology S1xS2. This is the first example of a supersymmetric, asymptotically flat black hole of non-spherical topology. The solution is uniquely specified by its electric charge and two independent angular momenta. These conserved charges can be arbitrarily close, but not exactly equal, to those of a supersymmetric black hole of spherical topology.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A new supersymmetric black hole solution of five-dimensional supergravity is presented. | contribution 0.98 | 0.93 | 0.58 | asserted 1.00 |
| s1 | It has an event horizon of topology S1xS2. | result 0.78 | 0.87 | 0.15 | asserted 1.00 |
| s2 | This is the first example of a supersymmetric, asymptotically flat black hole of non-spherical topology. | contribution 0.46 | 0.85 | 0.60 | asserted 1.00 |
| s3 | The solution is uniquely specified by its electric charge and two independent angular momenta. | result 0.96 | 0.84 | 0.13 | asserted 1.00 |
| s4 | These conserved charges can be arbitrarily close, but not exactly equal, to those of a supersymmetric black hole of spherical topology. | result 0.88 | 0.81 | 0.11 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
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- P abstract
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Figures
Figures reproduced from arXiv:hep-th/0407065 by the authors.
References
How later papers used it
110 later papers cite it
2 later papers by other authors are listed below: 1 used its method or tool and 1 compared its own results with it.
Uses its method or tool 1 of 3 papers with this label
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Used the canonical coordinates adapted to the symmetries of the black ring in this paper's flat metric of the ℝ4 base.
Supertubes in Bubbling Backgrounds: Born-Infeld Meets Supergravity (2008) · 0.80 · paraphrased: the licence does not allow quoting
Compares with it 1 of 16 papers with this label
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Showed the allowed regions in the (jψ,jφ)-plane for the σ→1 limit of its own solution and this paper's supersymmetric black ring solution.
New black ring with all independent conserved charges in five-dimensional minimal supergravity (2024) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
101 arXiv papers cite this in 181 sentences
Papers citing it · 110 sentences without a confident label
Builds on it 1 paper
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We focus on the case $n=3$, which has one more charge and dipole with the respect to the original black ring of the STU-model [ref] .
Adding new hair to the 3-charge black ring · Introduction · 0.85
Uses its method or tool 3 papers
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One can then express $Z_fake$ in the $x$-$y$ coordinate system used to find the black ring in $^4$ [ref] , solve the corresponding equations[Equations (46) and (47) in [ref] .] for $k_1$ and $k_2$, and express the $^4$ solution as a solution of the almost BPS equations to read off $μ_5 V$ and $ω⃗_5$.
Non-BPS Black Rings and Black Holes in Taub-NUT · Solving the equations · 0.92
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Now we can find the near horizon geometry with explicit $AdS_2$ symmetry as in [ref] , by re-defining
Charges from Attractors · Non-equal Angular Momenta: Supersymmetric Black Holes · 0.90
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We use the canonical coordinates that are adapted to the symmetries of the black ring in the flat metric of the $ℝ^4$ base [ref] :
Supertubes in Bubbling Backgrounds: Born-Infeld Meets Supergravity · The black-ring solution · 0.80
Uses its data or values 1 paper
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The Bekenstein-Hawking entropy of the black ring is given by [ref] [equation] which is real and positive if we demand $R<(Q-q^2)/(2q)$.
Black hole solutions in string theory · Timelike family · 0.81
Compares with it 16 papers
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Thus, using the algebra of conserved charges, we computer the conserved charges of supersymmetric black rings in minimal $𝒩=1$ supergravity and obtain the result consistent with [ref] .
Supersymmetric black rings and non-linear sigma models · Conserved charges · 0.99
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These agree precisely with the corresponding asymptotic angular momenta of [ref] .
Charges from Attractors · Non-equal Angular Momenta: Supersymmetric Black Holes · 0.93
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This black ring solution, derived through the Ehlers-Harrison transformations, encompasses both well-known and unfamiliar physically significant solutions: In the neutral limit, the solution coincides with the vacuum doubly spinning black ring solution, referred to as the Pomeransky-Sen'kov solution [ref] ; in the BPS limit, it yields the restrictive supersymmetric black ring [ref] , in the sense that the electric charge (mass) and the dipole charge are not independent; in an extremal limit, distinct from the BPS limit, it becomes the charged doubly spinning black ring with two nonzero angular velocities, including the extremal Pomeransky-Sen'kov black ring.
New black ring with all independent conserved charges in five-dimensional minimal supergravity · Summary · 0.89
Cites it as support 2 papers
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The motivation behind studying pressureless black strings comes from the fact that all known smooth black rings with charges [ref] and dipoles [ref] become pressureless strings in the infinite radius limit.
Extremal solutions of the S3 model and nilpotent orbits of G2(2) · Introduction and summary · 0.97
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Indeed, this is what happens for supersymmetric black rings, which have near-horizon geometry locally isometric to $AdS_3 ×S^2$ [ref] .
Near-horizon symmetries of extremal black holes · Static black holes · 0.96
Cites it as background 39 papers
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On the one hand, the macroscopic limit of this cosmic string becomes the EEMR/BW/GG black ring [ref] ; on the other hand, the low energy limit of the M5 brane worldvolume theory is the Maldacena-Strominger-Witten (MSW) CFT [ref] .
Blackhole/blackring transition · Black ring microstates with unit M5 charge · 0.87
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In more than four dimensions supersymmetric black holes and the possible non-extremal rotating black hole saddles exhibit much richer dynamics [ref] .
The gravitational index of a small black ring · Introduction · 0.93
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Remarkably, the discovery of a supersymmetric black ring solution of five-dimensional minimal supergravity was introduced in [ref] .
Supersymmetric black rings and non-linear sigma models · Introduction · 0.93
Discussion
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