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Superradiant instabilities of rotating black branes and strings

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Vitor Cardoso, Shijun Yoshida

High Energy Physics - Theory · Submitted 22 Feb 2005 · 31 references

Pith's verdict: A clean organizing criterion plus a real negative result for higher dimensions; the only genuine soft spot is that the abstract's flat 'stable for d>4' is a bit stronger than the d=5 argument actually delivered.

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arXiv IDhep-th/0502206 (version 2)
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CitationShijun Yoshida, Vitor Cardoso. "Superradiant instabilities of rotating black branes and strings." arXiv:hep-th/0502206 (2005). https://pith.science/paper/UEYZVW2Z
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The paper

The paper claims

The paper argues that a rotating black brane $Kerr_4$ x $R^p$ is unstable because massless waves get trapped in a potential well and amplified by superradiance, while the same construction in $d>4$ dimensions is stable because no such well,

The reading

A black brane is a black hole with flat extra dimensions glued on. The authors ask whether such objects suffer a second instability besides the known Gregory-Laflamme one, and argue that two ingredients decide the answer: an ergo-region, which makes waves scatter back with more energy than they carried in (superradiance), and a well in the effective potential, which traps those waves near the hole so they are amplified again and again. The key observation is that momentum along the flat extra directions acts exactly like a mass for the field in the four-dimensional part, so a massless field on the brane behaves like a massive field around an ordinary Kerr black hole. In four dimensions a massive field always has a well far from the hole, so the brane is unstable; in higher dimensions the well is absent, so it is not. The claim matters because it would extend the reach of black-hole superradiance from a four-dimensional curiosity to a statement about which extended rotating objects can survive in nature.

What carries the argument

The load-bearing step is a separation of variables. Writing the massless scalar on $Kerr_d$ x $R^p$ as a mode carrying momentum $mu_i$ along each flat direction reduces its wave equation to that of a massive scalar of mass $mu^2 = sum mu_i^2$ in a $d$-dimensional Kerr geometry, so the extra dimensions supply an effective mass. The instability criterion is then read off the effective potential $V$: a well is guaranteed when its derivative is positive at large $r$, which happens only for $n=0$. The numerical work uses Leaver's continued-fraction method, matching an expansion about the horizon to one at infinity and solving the coupled radial and angular continued fractions for the complex frequency and separation constant. The appendix recasts $V$ as $V = alpha(omega - V_+)(omega - V_-)$ with $alpha > 0$, so the allowed and forbidden regions can be read directly off the two factor potentials and the existence of a bound state becomes a question about whether $V_+$ has a local minimum.

Extended reading notes

Core claim

The paper establishes that the black brane $Kerr_4$ x $R^p$ is unstable against a superradiant mechanism, and reports numerical instability timescales for it: for $l=m=1$, $a=0.497$ and $mu=0.9$ the largest imaginary part found is about $10^{-6}$ times the real part of the frequency, and the instability switches off below a critical rotation where the superradiant factor $Re[omega] - m a/(r_+^2+a^2)$ changes sign. It then argues the opposite for $Kerr_d$ x $R^p$ with $d>4$: because the large-$r$ derivative of the effective potential is dominated by a term of the opposite sign when $n = d-4 > 0$, no well forms, there are no bound states, and the object is stable even though superradiance is still present. The physical reading offered is that higher-dimensional black holes have no stable circular orbits, so a wavepacket that would be gravitationally bound in four dimensions escapes instead. A factorized-potential analysis in an appendix supports the stability conclusion, showing no unstable bound states for $n >= 5$ and, in the marginal five-dimensional case, that any bound states that do exist sit above $m Omega$ and are therefore stable.

Load-bearing premise

The sweeping 'all stable for $d>4$' conclusion leans on a factorized-potential analysis that fixes the angular separation constant to $A_{lm} = l(l+1)$, an approximation the paper itself says holds only as $a^2(omega^2 - mu^2)$ goes to zero, and on the acknowledged inability to resolve numerically the marginal five-dimensional case where a well might appear at very large effective mass.

Editorial extensions

If this is right

  • Any four-dimensional extended rotating black object, not just the specific brane metric studied here, should inherit the instability, since the well comes from the effective mass term rather than from the details of the geometry.
  • The instability should be far stronger for gravitational perturbations than for scalar ones, because gravitational modes superradiate at roughly 138 percent against about 2 percent for scalars, suggesting timescales two orders of magnitude shorter.
  • Because the instability shuts off at a critical rotation, the endpoint is plausibly a slowly rotating or static black brane plus outgoing radiation carrying away the excess angular momentum.
  • The mechanism should persist all the way to extremality, unlike Gregory-Laflamme, so it can take over in regimes where the older instability is absent.
  • A draining-bathtub acoustic geometry with a suitably chosen sound-speed profile should reproduce the effect on a laboratory bench.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the four-dimensional result carries over to gravitational modes at the strength the authors estimate, then astrophysical black holes with large extra dimensions would spin down on timescales short enough to leave an observable preference for low rotation, which is in principle a clean observational handle.
  • The five-dimensional case is the interesting boundary: the paper's own factorized analysis allows bound states there at large effective mass, and only argues they are stable because they lie above $m Omega$. A dedicated search at large $mu$ would decide whether the boundary dimension behaves like $d=4$ or like $d>4$.
  • The same effective-mass trick should apply to any field with momentum in a compact or flat direction, including fields on Randall-Sundrum-like backgrounds, which the authors flag as a place where Gregory-Laflamme is absent but this mechanism may not be.
  • Because the argument only needs an ergo-region and a confining well, it should transfer to any rotating absorbing body with a trapping potential, which widens the class of analogue experiments that could see it.
Original abstract

Black branes and strings are generally unstable against a certain sector of gravitational perturbations. This is known as the Gregory-Laflamme instability. It has been recently argued that there exists another general instability affecting many rotating extended black objects. This instability is in a sense universal, in that it is triggered by any massless field, and not just gravitational perturbations. Here we investigate this novel mechanism in detail. For this instability to work, two ingredients are necessary: (i) an ergo-region, which gives rise to superradiant amplification of waves, and (ii) ``bound'' states in the effective potential governing the evolution of the particular mode under study. We show that the black brane Kerr_4 x R^p is unstable against this mechanism, and we present numerical results for instability timescales for this case. On the other hand, and quite surprisingly, black branes of the form Kerr_d x R^p are all stable against this mechanism for d>4. This is quite an unexpected result, and it stems from the fact that there are no stable circular orbits in higher dimensional black hole spacetimes, or in a wave picture, that there are no bound states in the effective potential. We also show that it is quite easy to simulate this instability in the laboratory with acoustic black branes.

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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,484 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.34 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.71 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Black branes and strings are generally unstable against a certain sector of gravitational perturbations. This is known as the Gregory-Laflamme instability. It has been recently argued that there exists another general instability affecting many rotating extended black objects. This instability is in a sense universal, in that it is triggered by any massless field, and not just gravitational perturbations. Here we investigate this novel mechanism in detail. For this instability to work, two ingredients are necessary: (i) an ergo-region, which gives rise to superradiant amplification of waves, and (ii) ``bound'' states in the effective potential governing the evolution of the particular mode under study. We show that the black brane Kerr_4 x R^p is unstable against this mechanism, and we present numerical results for instability timescales for this case. On the other hand, and quite surprisingly, black branes of the form Kerr_d x R^p are all stable against this mechanism for d>4. This is quite an unexpected result, and it stems from the fact that there are no stable circular orbits in higher dimensional black hole spacetimes, or in a wave picture, that there are no bound states in the effective potential. We also show that it is quite easy to simulate this instability in the laboratory with acoustic black branes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Black branes and strings are generally unstable against a certain sector of gravitational perturbations. background 1.00 0.06 0.03 asserted 0.95
s1 This is known as the Gregory-Laflamme instability. background 1.00 0.03 0.02 asserted 1.00
s2 It has been recently argued that there exists another general instability affecting many rotating extended black objects. background 0.97 0.05 0.04 hedged possibility 0.59
s3 This instability is in a sense universal, in that it is triggered by any massless field, and not just gravitational perturbations. background 1.00 0.08 0.05 asserted 0.99
s4 Here we investigate this novel mechanism in detail. contribution 1.00 0.53 0.03 asserted 0.67
s5 For this instability to work, two ingredients are necessary: (i) an ergo-region, which gives rise to superradiant amplification of waves, and (ii) ``bound'' states in the effective potential governing the evolution of the particular mode under study. background 0.77 0.13 0.06 asserted 1.00
s6 We show that the black brane Kerr_4 x R^p is unstable against this mechanism, and we present numerical results for instability timescales for this case. result 0.94 0.96 0.28 asserted 1.00
s7 On the other hand, and quite surprisingly, black branes of the form Kerr_d x R^p are all stable against this mechanism for d>4. result 1.00 0.88 0.26 asserted 1.00
s8 This is quite an unexpected result, and it stems from the fact that there are no stable circular orbits in higher dimensional black hole spacetimes, or in a wave picture, that there are no bound states in the effective potential. interpretation 0.95 0.72 0.20 asserted 1.00
s9 We also show that it is quite easy to simulate this instability in the laboratory with acoustic black branes. result 0.49 0.80 0.04 asserted 0.96

What kind of paper the classifier reports this is: theoretical result 0.76 · evidence from simulation or computation 0.91.

What was examined, and what was not

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  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. A typical form for the effective potential, here shown for l = m = 1 modes. We have set M = 1, so the rotation parameter a varies between 0 (Schwarzschild limit) and 1/2 (extremal limit). Here we plot the effective potential for the near extreme situation, a ∼ 0.5 and for µ = 0.7 and ω = 0.6878. 2.2.2 The potential well The effective potential in four dimensions, d = 4, as given by (2.10) is plotted in figure 1, for a ∼ 0.5 (in units of M), µ = 0.7 and ω = 0.6878. As can be seen from
Figure 2. Details for the instability in the four dimensional case. We have set M = 1, so the rotation parameter a varies between 0 (Schwarzschild limit) and 1/2 (extremal limit). In the left panel we plot the imaginary part of the unstable modes as a function of a for nine values of µ, and for l = m = 1. The same quantities are plotted in the right panel, but now we focus on the maximum instability region, for values of µ near unity. The instability is stronger for larger values of a and for µ ∼ 1. In fact, our results suggest that the imaginary part of the unstable modes attains its maximum…
Figure 3. More details of the instability in the four dimensional case. Here we plot the superradiant factor Re[ω] − ma a2+r 2 + as a function of a for l = m = 1 and for several values of µ. Again, we have set M = 1 and so the superradiant factor is, in four dimensions, Re[ω] − ma r+ . the instability disappears for a below a certain critical point, which is given by setting the superradiant factor to zero. On the right panel we make the same plot but on a logarithmic scale, for better visualization. Notice that the superradiant factor is negative, as it should for the instability to be tri
Figure 5. The curve of dV+/dr(r −1 , µ) = 0 for the parameter n = 0, l = m = 1, j = 0, and a = 0.4999. Now, let us move on to higher dimensions, taking n = 1 to be specific. Consider the case of l = m = 1 and j = 0 as an example. We are interested in finding out whether or not there can be bound states in the five dimensional case. In order to see the distribution of extremes of V+, which give us a criterion for nonexistence of the bound states, a curve of dV+/dr(r, µ) = 0 on the r −1 – µ plan is sketched in
Figure 6. The curve of dV+/dr(r −1 , µ) = 0 for the parameter n = 1, l = m = 1, j = 0, and a = 0.9. 0 0.5 1 1.5 2 r -1 -1.5 -1 -0.5 0 0.5 1 1.5 V V+ V￾0 0.2 0.4 0.6 0.8 1 r -1 1.74 1.75 V 1.745 +
Figure 7. The factorized potentials V± as a function of r −1 . The potentials are characterized by n = 1, l = m = 1, j = 0, µ = 1.75, and a = 0.9 (left panel). The right panel is the same as the left one, but, a close-up of the potential V+ near extremes. not found any unstable modes. For general n > 5 the trend is the same, there are no unstable bound states. References [1] D. Marolf and B. C. Palmer, Phys. Rev. D 70, 084045 (2004). [2] V. Cardoso and J. P. S. Lemos, hep-th/0412078. [3] T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957). [4] Ya. B. Zel’dovich, Pis’ma Zh. Eksp. Teor. Fi

Showing 6 of the paper's 7 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-th/0502206 by the authors.

References
How the literature uses it

81 arXiv papers cite this in 141 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 80 sentences without a confident label

Contests it 1 paper
  • These results clearly show that the four orders of magnitude discrepancy between the results of [ref] and [ref] are unphysical.

    The extremal black hole bomb · Point matching · contest 0.81

Uses its method or tool 10 papers
  • To judge how good the corrected solution compared to the numerical result, we follow the continued fraction method, which is firstly proposed by Leaver [ref] and developed in Refs. [ref] .

    Improved Analytic Solution of Black Hole Superradiance · Review of LO Calculation · 0.98

  • Quasi-bound state solutions with radial function $R^b_n_iℓ_i$ and their corresponding complex eigenfrequencies $ω:=ω_n_iℓ_i$ are found by imposing exponential decay at spatial infinity and purely ingoing waves at the event horizon and using the continued-fraction method developed in Refs. [ref] .

    Resonances as signatures of scalar clouds in eccentric extreme-mass-ratio inspirals · Introduction · 1.00

  • See [ref] for an initial study incorporating Leaver's continued fraction method [ref] for numerical calculations and Dolan's work [ref] for an extensive study of the expanded parameter space, providing numerical solutions using a three-term recurrence relation and the continued fraction method.

    Black Hole Spin Constraints on the Mass Spectrum and Number of Axion-like Fields · Numerical Solutions for Bound States · 0.87

Uses its data or values 1 paper
  • However, even within the linear regime, studying the onset of the scalar-field instability in the time domain is a difficult challenge, because (as anticipated from frequency-domain analyses [ref] ) the growth rate of the instability is rather slow, with a minimum e-folding time of $∼5.8 ×10^6 M$ (see Sec. [ref] and Fig. [ref]).

    Superradiant instabilities of rotating black holes in the time domain · Introduction · 0.80

Compares with it 5 papers
  • The results in [ref] exhibit, however, a huge discrepancy with respect to the numerical analysis in [ref] , with a maximum growth rate of $ω_I M≃1.7×10^-3$, four orders of magnitude larger than the one mentioned above.

    The extremal black hole bomb · Introduction · 0.83

  • Due to this difference, in Sec. [ref] we shall see that, when rotation is turned on, not only the exact modes of the Ernst BH become unstable due to superradiance, but also that the instability time scale can be orders of magnitude shorter than that associated to a massive field [ref] – and, consequently, the instability is stronger than that discussed in Ref. [ref] .

    Superradiant instability of black holes immersed in a magnetic field · Relation with previous results in the literature · 0.96

  • The superradiance here observed for the charged non-rotating BS is not linked to the superradiant instability highlighted in [ref] : indeed, the BS lacks the reflection mechanism needed to trap and enhance the outgoing perturbation.

    Charge (in)stability and superradiance of Topological Stars · Amplification factor · 0.91

Cites it as background 31 papers

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