REVIEW
Rotating Charged Black Hole Solution in Heterotic String Theory
High Energy Physics - Theory · Submitted 15 Apr 1992 · 9 references
Pith's verdict: A clean, correct-looking solution-generating paper: the first exact rotating charged black hole in 4D heterotic string theory, with g=2 and a finite-area extremal limit, resting on one unpublished transformation theorem.
T0 · AI review
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Referee report
The paper
The paper claims
Sen constructs an exact rotating, charged black hole solution in four-dimensional heterotic string theory by twisting the Kerr solution, and shows its extremal limit keeps a horizon whose area depends only on angular momentum.
The reading
What carries the argument
The engine is the twisting transformation of eqs. (5)–(9). One assembles a $9×9$ matrix $M$ from the metric $G$, the antisymmetric tensor $B$ and the gauge field $A$ through $K = -B - G - (1/4)A A^T$ and the fixed matrix $η = diag(1,1,1,-1)$, then acts with a boost $Ω$ that is the identity on seven entries and a hyperbolic rotation of angle $α$ on the last two: $M' = Ω M Ω^T$, together with the dilaton shift $Φ' - ln det G' = Φ - ln det G$. Because $Ω$ is a symmetry of the equations of motion, the transformed fields solve the same equations as the seed. Feeding in the Kerr solution with $Φ = 0$, $B = 0$, $A = 0$ produces the charged, spinning solution, with the boost parameter $α$ playing the role of the charge and $a$ the role of the spin. The same transformation also generates the antisymmetric tensor automatically, which the paper attributes to the coupling of $H$ to the gauge Chern-Simons three-form: a rotating charged hole carries a magnetic dipole moment, and that sources $B_{tφ}$.
Extended reading notes
Core claim
The central claim is that the field configuration of eqs. (11)–(15) — metric, dilaton, $U(1)$ gauge field and antisymmetric tensor — solves the classical equations of motion of the truncated four-dimensional heterotic action of eq. (1), and describes a black hole carrying mass $M$, charge $Q$, angular momentum $J$ and magnetic dipole moment $μ$, with $M = (m/2)(1+cosh α)$, $Q = (m/√2) sinh α$, $J = (ma/2)(1+cosh α)$ and $μ = (ma/√2) sinh α$, so that the gyromagnetic ratio $g ≡ 2μM/(QJ)$ equals exactly 2. The horizon sits at $ρ = m ± √(m^2-a^2)$, written in physical variables as $ρ^{±}_H = M - Q^2/(2M) ± √((M - Q^2/(2M))^2 - J^2/M^2)$, and the outer horizon has area $A = 8πM(M - Q^2/(2M) + √((M - Q^2/(2M))^2 - J^2/M^2))$. A horizon exists only when $|J| ≤ M^2 - Q^2/2$, so the extremal limit is $|J| → M^2 - Q^2/(2M)$; there the area tends to $8π|J|$, the angular velocity tends to $sign(J)/(2M)$, and the surface gravity tends to zero for nonzero $J$, while for $J = 0$ one recovers $κ = 1/(4M)$ as in the earlier charged black hole solutions.
Load-bearing premise
The whole construction rests on the claim, taken from a cited companion paper and not re-derived here, that the boost transformation of eqs. (5)–(9) maps any time-independent solution of the action to another solution of the same action, including when the antisymmetric tensor carries the gauge Chern-Simons term.
Editorial extensions
If this is right
- Every rotating charged black hole in this theory has gyromagnetic ratio exactly 2, independent of the values of $M$, $Q$ and $J$, because the ratio is fixed by the transformation rather than by the details of the solution.
- In the extremal limit with nonzero $J$ the horizon retains finite area $8π|J|$ and the Hawking temperature goes to zero, so the extremal state behaves like an extremal rotating hole rather than an extremal charged one.
- The extremal horizon area depends only on the angular momentum, not on the charge, matching an earlier result obtained in a different model.
- Spinning holes with $|J| > M^2 - Q^2/2$ have no horizon at all, giving a concrete bound on angular momentum at fixed mass and charge.
- Switching off the spin recovers the previously known charged black hole solutions, including the surface gravity $κ = 1/(4M)$ at extremality.
Reading between the lines
- Because the transformation is a symmetry of the action rather than an integration of it, the same recipe should produce $g = 2$ for any seed solution it is applied to; checking this on a different seed would test whether the result is a property of the transformation or of Kerr specifically.
- The agreement of the extremal area $8π|J|$ with an independent calculation in a different model suggests the area law may be generic to extremal rotating holes rather than an artifact of the dilaton and antisymmetric tensor, which is a comparison worth pushing further.
- One could isolate how much of the solution is genuinely stringy by dropping the Chern-Simons term from $H$ and repeating the twist; if the metric and dilaton survive unchanged while only $B_{tφ}$ shifts, the string-specific content is confined to the antisymmetric tensor sector.
- The paper does not compute the entropy or check thermodynamic stability of the extremal configuration, so a natural next step is to compare its horizon area and temperature against the Kerr-Newman hole of ordinary Einstein-Maxwell theory at the same $M$, $Q$ and $J$.
Original abstract
We construct a solution of the classical equations of motion arising in the low energy effective field theory for heterotic string theory. This solution describes a black hole in four dimensions carrying mass $M$, charge $Q$ and angular momentum $J$. The extremal limit of the solution is discussed.
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
We construct a solution of the classical equations of motion arising in the low energy effective field theory for heterotic string theory. This solution describes a black hole in four dimensions carrying mass $M$, charge $Q$ and angular momentum $J$. The extremal limit of the solution is discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We construct a solution of the classical equations of motion arising in the low energy effective field theory for heterotic string theory. | contribution 1.00 | 0.95 | 0.29 | asserted 1.00 |
| s1 | This solution describes a black hole in four dimensions carrying mass $M$, charge $Q$ and angular momentum $J$. | contribution 0.54 | 0.79 | 0.38 | asserted 1.00 |
| s2 | The extremal limit of the solution is discussed. | contribution 0.92 | 0.77 | 0.11 | asserted 0.61 |
What was examined, and what was not
- arXiv listing abstract v1
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References
How later papers used it
121 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 own solution is formally identical to this paper's charged rotating black hole solution of normal EMDA, differing only in the sign of the dilatonic charge.
Phantom Black Holes and Sigma Models (2011) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
85 arXiv papers cite this in 133 sentences
Papers citing it · 62 sentences without a confident label
Uses its data or values 1 paper
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In Boyer–Lindquist coordinates the string frame Kerr-Sen black hole solution reads [ref] ds^2 = e^Φ{-Δ/ρ_b^2(dt-asin^2θdφ)^2+sin^2θ/ρ_b^2[a dt-(r^2+2br+a^2)dφ]^2+ρ_b^2/Δdr^2+ρ_b^2 dθ^2} , H = -2ba/ρ_b^4 dt∧dφ∧[(r^2-a^2cos^2θ)sin^2θdr-rΔsin2θdθ] , A = -Qr/ρ_b^2(dt-asin^2θdφ) , Φ = 2ln(ρ/ρ_b) , where ρ^2=r^2+a^2cos^2θ , ρ_b^2=ρ^2+2br , Δ=r^2-2(M-b)r+a^2 .
Generalized hidden symmetries and the Kerr-Sen black hole · Metric and fields · 0.83
Compares with it 3 papers
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where, $Δ= r(r+r_2) -2Mr + a^2$ and $Σ̃=r(r+r_2) + a^2 cos^2 θ$ Comparing with the solution obtained from [ref] , we have the electromagnetic potential as, [equation] The dilaton field $Φ$ is [equation] where the constant multiplicative term $Φ_0$ is defined as the asymptotic value of the dilaton.
Penrose process in charged axion-dilaton coupled black hole · Axion-dilaton coupled gravity · 0.81
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It is formally identical to the charged rotating black hole solution of normal EMDA [ref] , the only difference being that the dilatonic charge $D$ is negative in the case of EMDA ($Q^2 = -2MD$) and positive in the present case of E$M$DA ($Q^2 = 2MD$).
Phantom Black Holes and Sigma Models · Stationary E$M$DA theory · 1.00
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In fact the rotation is a close cousin of the Sen transformation, that takes the Kerr black hole to the Kerr-Sen black hole [ref] .
Non-Kahler heterotic rotations · Heterotic solution-generating method · 0.99
Cites it as background 51 papers
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In a low-energy limit of heterotic string theory, a rotating black hole solution has been found in Ref. [ref] , known as the Kerr-Sen (KS) black hole.
Observational signature of a near-extremal Kerr-Sen black hole in the heterotic string theory · Introduction · 0.87
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The line element of the Kerr-Sen black hole is [ref] [equation] where [equation] The inner and outer horizons of the black hole are given as [equation] When $|a| = |M-c|$, the Kerr-Sen black hole becomes extreme.
Escape probability of particle from Kerr-Sen black hole · Four-momentum of the particle in the Kerr-Sen spacetime · 0.80
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The EMDA theory generalizes the Einstein-Maxwell theory by introducing a coupling between the Maxwell electromagnetic field tensor $F_μν$, the scalar dilaton field $ξ$ and the pseudo scalar axion field $ϕ$, and it can be constructed from the following four dimensional effective action [ref] [equation] where $R$ is the Ricci scalar and $g$ is the metric tensor determinant.
The Exact Relativistic Scalar Quasibound States of The Dyonic Kerr-Sen Black Hole: Quantized Energy, and Hawking Radiation · Introduction · 0.83
Discussion
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