REVIEW
General Concentric Black Rings
High Energy Physics - Theory · Submitted 16 Aug 2004 · 30 references
Pith's verdict: A clean, explicit generalization of supersymmetric black rings to arbitrary vector multiplets; the one real gap is a routine near-horizon expansion detail, not a threat to the result.
T0 · AI review
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Referee report
The paper
The paper claims
The paper constructs supersymmetric black ring solutions of five-dimensional supergravity coupled to any number of vector multiplets, and argues that concentric stacks of such rings have regular $S^1$ x $S^2$ horizons free of closed time-ll
The reading
What carries the argument
The load-bearing device is the Gibbons-Hawking reduction of the hyper-Kähler base: assuming the base admits a tri-holomorphic Killing vector $partial_5$ that also generates a symmetry of the full solution, the four-dimensional base becomes $H^{-1}(dx^5+chi)^2 + H delta_{ij}dx^i dx^j$ with $nabla times chi = nabla H$, and every field is determined by harmonic functions on $R^3$. Closure of the self-dual two-forms $Theta^I$ forces $H W^I$ harmonic, hence $W^I = H^{-1}K^I$; the Maxwell equations force $f^{-1}X^I = frac{1}{24}H^{-1}C^{IPQ}K_P K_Q + L^I$; and the self-duality of $domega$ fixes $omega_5$ and the vector part $hat{omega}$ up to a gradient. The symmetric-space identity $C_{IJK}C_{J'(LM}C_{PQ)K'}delta^{JJ'}delta^{KK'} = frac{4}{3}delta_{I(L}C_{MPQ)}$ is what makes $f$ and the combination $f^{-3}H^{-1}-(omega_5)^2$ explicitly computable in Eq. (34)-(35), and hence what makes the near-horizon expansion and the asymptotic charge read-off possible at all.
Extended reading notes
Core claim
The central claim is that a specific harmonic-function ansatz, Eq. (38), on a flat Gibbons-Hawking base gives asymptotically flat supersymmetric solutions of $D=5$ ungauged supergravity coupled to $n-1$ abelian vector multiplets, and that each pole of those harmonic functions is a regular Killing horizon of topology $S^1$ x $S^2$ carrying no closed time-like curves. The construction follows the classification of supersymmetric solutions: in the time-like case the metric takes the form $-f^2(dt+omega)^2 + f^{-1}ds^2(M_4)$ over a hyper-Kähler base, and the whole solution is fixed by $2n+2$ harmonic functions $H$, $K^I$, $L_I$, $M$ on $R^3$. Choosing $H = 1/r$ makes the base flat $R^4$, and the ansatz then places $N$ rings at arbitrary points. The authors expand around each pole, find a coordinate singularity, exhibit the coordinate change of Eqs. (49)-(50) that removes it, and show that the resulting horizon metric is $S^1$ x $S^2$ with $S^1$ radius $ell_i$ and $S^2$ radius $nu_i/2$, giving horizon area $2 pi^2 ell_i nu_i^2$. They also compute the conserved charges, verify the BPS bound is met, and show that for poles on the $z$-axis the moduli dependence of $ell_i$ can be rewritten in terms of $j_i = q^I_i lambda_I R_i^2$, which fixes that ring's contribution to $J_2 - J_1$.
Load-bearing premise
The regularity claim rests on the assertion that the metric determinant is analytic in the expansion parameter near each pole and that the leftover $1/epsilon_i$ divergence in $g_{epsilon_i epsilon_i}$ is removed by a constant $b_1$ whose value the paper declines to display, so the extension through $epsilon_i = 0$ is asserted rather than exhibited.
Editorial extensions
If this is right
- Any five-dimensional supergravity obtained by reducing M-theory on a Calabi-Yau threefold inherits these solutions, so concentric multi-charge ring configurations exist in that setting and can be uplifted to $D=11$ and, after reduction and T-duality, to type IIB on a five-torus.
- Because the horizon area depends on the asymptotic scalars only through $J_2 - J_1$, a single multi-charge ring has moduli-independent entropy, matching the expectation that entropy should not track adiabatic changes of the environment.
- Concentric rings can carry the same conserved charges as a rotating black hole while having smaller, equal, or larger entropy, so asymptotic charges alone do not label the microstates behind the entropy.
- Setting one pole at the origin reproduces the general rotating BPS black hole of the same theory at the centre of the rings, so black holes and rings appear as members of one family of solutions.
- In the three-charge STU model with the constraint of Eq. (62) imposed, the geometry with $ell_i = 0$ is a candidate regular three-charge supertube without a horizon, which would make the family include supertube-ring-black-hole superpositions.
Reading between the lines
- The moduli-independence result suggests a sharper statement than the paper proves: the entropy of any such ring should be a function of the quantised charges and of $J_2 - J_1$ alone, which is exactly the form a microscopic state count would need to reproduce.
- The paper leaves global causal structure open for non-collinear poles, and that is where a counterexample would most plausibly live; a numerical scan of the metric determinant and of $g_{tt}$ over the exterior for three or more non-collinear rings would be a cheap first test.
- Because the whole solution is fixed by harmonic functions on $R^3$, these configurations sit in the same mathematical family as multi-centre instanton and multi-black-hole solutions, so the standard bubble-equilibrium conditions for absence of conical singularities should have a direct analogue here.
- The fact that $ell_i$ can be traded for $j_i$ hints that the ring circumference is a collective coordinate conjugate to the angular-momentum splitting, which would make $J_2 - J_1$ the natural variable in which to write a ring's thermodynamic first law.
Original abstract
Supersymmetric black ring solutions of five dimensional supergravity coupled to an arbitrary number of vector multiplets are constructed. The solutions are asymptotically flat and describe configurations of concentric black rings which have regular horizons with topology $S^1 \times S^2$ and no closed time-like curves at the horizons.
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
Supersymmetric black ring solutions of five dimensional supergravity coupled to an arbitrary number of vector multiplets are constructed. The solutions are asymptotically flat and describe configurations of concentric black rings which have regular horizons with topology $S^1 \times S^2$ and no closed time-like curves at the horizons.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Supersymmetric black ring solutions of five dimensional supergravity coupled to an arbitrary number of vector multiplets are constructed. | contribution 0.99 | 0.93 | 0.59 | asserted 1.00 |
| s1 | The solutions are asymptotically flat and describe configurations of concentric black rings which have regular horizons with topology $S^1 \times S^2$ and no closed time-like curves at the horizons. | result 0.94 | 0.86 | 0.40 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v3
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- relations to other papers
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References
How later papers used it
90 later papers cite it
1 later paper by other authors is listed below: 1 compared its own results with it.
Compares with it 1 of 4 papers with this label
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Said its solution, built with ring source harmonics on four-dimensional Euclid space, is analogous to the case of this paper's supersymmetric black ring solution.
Dynamical Black Rings with a Positive Cosmological Constant (2009) · 0.89 · paraphrased: the licence does not allow quoting
How the literature uses it
79 arXiv papers cite this in 130 sentences
Papers citing it · 74 sentences without a confident label
Builds on it 2 papers
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We have thus generalized the results of [ref] to completely general, fluctuating electric charge densities.
An Infinite-Dimensional Family of Black-Hole Microstate Geometries · The general back-reacted solution with fluctuating electric charge densities · 1.00
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If one replaces the internal space $T^6=(T^2)^3$ by a Calabi-Yau 3-fold $X$, most of our formulas remain valid as long as we replace $C_IJK$ by the triple intersection numbers of $X$ [ref] .
Non-Abelian Supertubes · The harmonic solution · 0.83
Compares with it 3 papers
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In this way, eq.([ref]) is simply the ADM mass[ Even though [ref] , [ref] only refer to concentric rings, the above comparison is still meaningful because effects due to non-concentricity only start showing up for quantities involving the position vector $L⃗$, such as angular momentum, entropy, etc. ] of [ref] , [ref] .
More Rings to rule them all : Fragmentation, 4D/5D and Split-Spectral Flows · 5D multi-black ring charges and harmonic functions from fragmentation · 0.97
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The above three equations are similar to those in the Maxwell theory [ref] .
The Geometry of Dyonic Instantons in 5-dimensional Supergravity · General properties of supersymmetric solutions · 0.88
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Our solution has constructed by use of ring source harmonics on four-dimensional Euclid space and this is analogous to the case of super-symmetric black ring solution [ref] .
Dynamical Black Rings with a Positive Cosmological Constant · Summary and Discussion · 0.89
Cites it as background 42 papers
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In this section we first review the three-charge, 1/8-BPS, fractionally spectral flowed supertube solutions constructed and studied in [ref] , as well as their decomposition into two-center solutions of the multi-center formalism of [ref] .
Shockwaves in black hole microstate geometries · Shockwaves in fractionally spectral flowed supertubes · 0.95
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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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From the supergravity analysis [ref] , it is known that a three-charge system [ref] can contain a supersymmetric black ring that has three dipole charges, corresponding to D5($1234θ$), D1($θ$), and KKM($1234θ;z$), where $θ$ is the direction of the ring.
Double, Double Supertube Bubble · First supertube transition · 0.90
Discussion
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