Historical research guide · 0519 · 1993
Black strings and p-branes are unstable
Ruth Gregory · Raymond Laflamme
In one minute
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.
Connection
Chapter 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.
From the source
Uncharged 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
A 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
The 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’
Compactifying 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
Four-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
The 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
The way in
https://doi.org/10.1103/PhysRevLett.70.2837How to cite it
Ruth Gregory, Raymond Laflamme (1993) Black strings and p-branes are unstable doi:10.1103/PhysRevLett.70.2837