Black strings and p-branes are unstable
Ruth Gregory · Raymond Laflamme
Abstract and summary · read the original at the source
In one page
Ruth Gregory and Raymond Laflamme asked a plain question with a surprising answer. Take a black hole and stretch it into a line — a black string, or in more dimensions a black p-brane — and does it hold its shape? In four dimensions a black hole is stable: once it forms it settles down and stays. The stretched versions, they show, do not. They perturb the metric of an uncharged black brane, reduce the perturbation equations to a single ordinary differential equation, and integrate it inward from infinity. Growing modes turn up over a whole range of frequency and wavelength along the extra dimensions, in every case from four to nine dimensions. A short entropy argument says why to expect it: a long stretch of black string carries far less entropy than the same mass gathered into one hyperspherical black hole, so a long horizon is under pressure to break up. The instability is not tucked away inside the horizon either — it lives in the exterior spacetime, where it could fragment the horizon itself.
Why it matters hereChapter 4 is about spacetime geometries you intend to build and hold, and this is the classic demonstration that solving Einstein’s equations is only half the job: a metric that exists on paper still has to survive a nudge, and the higher-dimensional stretched black holes do not. That is a chapter 1 lesson about the evidence ladder as much as a chapter 4 one — existence is a lower rung than stability, and Gregory and Laflamme show how you climb from the first to the second with a mode decomposition, a boundary condition chosen on the future horizon, and a numerical integration. The paper has no direct bearing on vacuum energy density or on drawing energy from the vacuum, so the skeleton’s chapter 2 and chapter 6 tags are not carried forward.
What it claims
01Uncharged black strings and black p-branes, the extended low-energy string-theory solutions of Horowitz and Strominger, are classically unstable: regular growing modes exist for a range of frequency and transverse wavenumber, and were found numerically for every spacetime dimension from four to nine.Stability analysis, the reduced equation for H(tr) and Figure 2
Published and peer-reviewed02A short entropy argument predicts the result before any calculation: a length L of five-dimensional black string of mass M carries entropy proportional to M squared over L, while a five-dimensional black hole of the same mass carries entropy proportional to M to the three halves, so for long horizons the same mass would be far better off as a hyperspherical hole — which is what makes long-wavelength perturbations along the extra dimension suspect.Introduction, the heuristic entropy argument
Published and peer-reviewed03The instability is a real physical feature of the exterior spacetime, not a gauge artefact and not the hidden inner-horizon instability of the Reissner–Nordström solution; the growing perturbation cannot be written as a pure coordinate change, because a pure gauge perturbation is a zero mode of the D-dimensional Lichnerowicz operator and these modes are not.Discussion following the mode decomposition; the paragraph beginning ‘The significance of these results’
Published and peer-reviewed04Compactifying the extra dimensions below the shortest unstable wavelength quantises the transverse wavenumbers and removes every unstable mode, so the result does not threaten large astrophysical black holes in four dimensions.Abstract; closing discussion on compactification
Published and peer-reviewed05Four-dimensional intuition does not transfer: the stability of the four-dimensional Schwarzschild black hole says nothing about five-dimensional black strings or ten-dimensional black branes, and the authors state the moral as the domain of validity of four-dimensional Einstein gravity being four-dimensional Einstein gravity.Final paragraph
Published and peer-reviewed06The endpoint is the open question the authors name: a linear analysis cannot fix where an unstable evolution lands, but the entropy argument tempts them toward fragmentation into a line of black holes, which would expose a singularity and bear on the Cosmic Censorship Hypothesis — and the charged case, which they expect to be unstable too, was under investigation as they wrote.Closing discussion; the paragraphs on charged and axionic black branes
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Abstract
We investigate the evolution of small perturbations around black strings and branes which are low energy solutions of string theory. For simplicity we focus attention on the zero charge case and show that there are unstable modes for a range of time frequency and wavelength in the extra 10 minus D dimensions. These perturbations can be stabilized if the extra dimensions are compactified to a scale smaller than the minimum wavelength for which instability occurs and thus will not affect large astrophysical black holes in four dimensions. We comment on the implications of this result for the Cosmic Censorship Hypothesis.
Ruth Gregory, Enrico Fermi Institute, University of Chicago; Raymond Laflamme, Theoretical Astrophysics T-6, Los Alamos National Laboratory. Physical Review Letters 70, 2837 (1993); preprint EFI-93-02, January 1993.
(Abstract only. The complete paper is free to read at https://arxiv.org/abs/hep-th/9301052 and via https://doi.org/10.1103/PhysRevLett.70.2837 — see the rights note for why the full text is not reproduced here.)
The way in
https://doi.org/10.1103/PhysRevLett.70.2837TITLE. The skeleton title arrived from Crossref with the publisher’s markup still in it — line breaks and an italic tag around the p of p-branes. The title is restored here from the paper’s own first page, ‘Black strings and p-branes are unstable’, Physical Review Letters 70, 2837 (1993), preprint EFI-93-02, January 1993. LICENCE. Checked directly rather than taken from an aggregator label: the journal version carries the APS default licence, and the green open-access copy, arXiv hep-th/9301052v2 of 15 January 1993, is filed under arXiv’s assumed licence for 1991 to 2003 submissions, which grants arXiv distribution rights only and no Creative Commons re-use. So this page carries the summary, the claims and the authors’ own abstract, and sends the reader to the full text at the source.
How to cite it
Ruth Gregory, Raymond Laflamme (1993) Black strings and p-branes are unstable. doi:10.1103/PhysRevLett.70.2837
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