Stability of black holes

1.7 Stability of black holes

As interest in higher-dimensional holes has developed, so has interest in their classical stability. Most of the work so far relies heavily on numerics. In Chapters 4 and 5 we will take a new approach, and see how much progress can be made analytically.

The standard approach to perturbation theory in GR (see e.g. [17]) begins by making a linearized metric perturbation of the form

𝑔𝜇𝜈 7→𝑔𝜇𝜈+ℎ𝜇𝜈. (1.29)

Care is needed though, since many choices ofℎ𝜇𝜈 give a perturbed metric that is related to𝑔𝜇𝜈 by an infinitesimal general coordinate transformation. To eliminate some of this gauge freedom, one can choose a traceless, transverse gauge, fixingℎ_𝜇^𝜇= 0 and∇^𝜇ℎ𝜇𝜈 = 0 (where indices are raised and lowered with 𝑔_𝜇𝜈). Given this, the Einstein equations (linearized in ℎ), reduce to

ΔLℎ𝜇𝜈 = 2Λℎ𝜇𝜈 (1.30)

where ΔL is the Lichnerowicz operator, defined in the case of Einstein spacetimes (1.8) by

Δ_Lℎ_𝜇𝜈 =−∇^𝜌∇_𝜌ℎ_𝜇𝜈 −2𝑅_{𝜇 𝜈}^{𝜌 𝜎}ℎ_𝜌𝜎+ 2Λℎ_𝜇𝜈. (1.31) In this chapter, we are mainly interested in the classical stability of asymptotically flat, and asymptotically𝐴𝑑𝑆black holes. However, it is useful to first recall an important result of Gregory & Laflamme [93] regarding the stability of black strings and black branes. They studied a𝑑-dimensional spacetime constructed from adding 𝑛=𝑑−𝐷flat directions to a 𝐷-dimensional Schwarzchild black hole, with a metric of the form

𝑑𝑠² =−𝑉(𝑟)𝑑𝑡²+ 𝑑𝑟²

𝑉(𝑟) +𝑟²𝑑Ω²_𝐷−2 +∑^𝑛

𝑖=1

𝑑𝑧_𝑖𝑑𝑧_𝑖, 𝑉(𝑟) = 1−(𝑟₀/𝑟)^𝐷−3 (1.32) Consider Fourier mode solutionsℎ𝜇𝜈 ∝𝑒^{Ω𝑇+𝑖𝑚𝑖𝑧𝑖} to (1.30) (in the case Λ = 0), subject to boundary conditions imposing that modes are regular at the event horizon and outgoing at null infinity. Any mode with Re(Ω)>0 grows exponentially, and is interpreted as an instability. Ref. [93] showed numerically that such unstable modes do exist for all such black strings and black branes; corresponding to long wavelength perturbations along the flat directions (i.e. those with small∑

𝑚²_𝑖). This is interpreted as showing that black strings and black branes are classically unstable. It was speculated that the endpoint of this instability is a chain of localized black holes. This was investigated in recent numerical work by Lehner & Pretorius [94]. They showed that the perturbed string evolves first to a sequence of black holes connected by increasingly thin black string

sections. However, the radius of the connecting strings becomes zero in finite asymptotic time, exposing a naked singularity.

The link with asymptotically flat black holes is as follows. Recall from the introduc- tion that, in six or more dimensions, singly-spinning Myers-Perry black holes can have arbitrarily large angular momentum. Suchultra-spinning black holes have ‘pancake-like’

horizons, with two separate lengthscales corresponding to the thickness of the horizon, and its width. Emparan & Myers [95] argue that, close to the axis of rotation, such horizons are well approximated by black branes. Hence, they suffer from the Gregory- Laflamme instability, and are unstable.

Shortly before this, it was established by Ishibashi & Kodama [96] that the higher- dimensional Schwarzschild solution is stable against linearized gravitational perturba- tions for all 𝑑 >4. From this, it seems reasonable to conjecture that Myers-Perry black holes will be stable provided they are sufficiently slowly rotating. If slowly rotating MP black holes are stable, and rapidly rotating ones are unstable, then there must exist some critical value of angular momentum where an instability appears. Can this value be identified?

A conjecture regarding this can be made by studying the thermodynamics of black hole horizons. It is known that the area of the black hole horizon(s) in any spacetime is always non-decreasing [19]. This is reminiscent of the second law of thermodynamics, and for this reason (and others), the entropy of a black hole horizon can be identified as proportional to its area [97]. Hence, given two black hole solutions to the Einstein equa- tions (possibly with multiple disconnected horizons), with the same asymptotic mass and angular momenta, the solution with the highest entropy seems to be ‘thermodynamically preferred’. Based on this, it seems reasonable to conjecture that when there exist two black hole solutions with the same asymptotic mass and angular momenta, but different entropy, that the solution with lower entropy is likely to be unstable. Arguments along these lines have been used to make conjectures about the phase space of vacuum black hole solutions in higher dimensions [98, 99], leading on to the recent development of the so-called blackfold approach [100, 101].

The intuition about links between different kinds of instability was formalised by a conjecture of Gubser & Mitra [102, 103], who suggest that a black brane with trans- lational symmetry is classically unstable if and only if it is locally thermodynamically unstable. This conjecture was proved for a particular class of black brane solutions by Reall [104].

These ideas were linked to asymptotically flat black holes by Monteiroet al.[105, 106].

They demonstrate, for several examples of rotating black holes (including singly-spinning black rings), that in the semi-classical approximation the gravitational partition function

1.7. STABILITY OF BLACK HOLES 21 admits a negative mode precisely when they are locally thermodynamically unstable.

More precisely, for a family of black holes with entropy𝑆, labelled by angular momenta 𝐽_𝑖, one can define the (reduced) Hessian

𝐻_𝑖𝑗 =

( ∂²𝑆

∂𝐽_𝑖∂𝐽_𝑗 )

𝑀

. (1.33)

In this paper, they consider a black hole to belocally thermodynamically unstable if 𝐻𝑖𝑗

is not negative definite. This negative mode appears for all black holes with angular momentum parameter 𝑎 larger than some critical value 𝑎₀. This critical value can be used to give a precise definition of an ultra-spinning black hole; i.e. all MP black holes with 𝑎 > 𝑎0 are ultraspinning.

This represents further evidence that locally thermodynamically unstable black holes are classically unstable, but does not prove it. For a proof, one needs to exhibit an explicit linearized instability of a black hole spacetime.

Dias et al. [107, 108] made progress towards this goal by studying the case of a singly-spinning, asymptotically flat MP black hole in dimensions 𝑑 = 7,8,9. Rather than working with the black hole spacetime directly, they construct a𝑑+ 1 dimensional black string by adding a single flat direction𝑑𝑧, and consider the eigenvalue problem

ΔLℎ𝜇𝜈 =−𝑘²ℎ𝜇𝜈, (1.34)

subject to particular boundary conditions, with a Fourier mode ansatz of the formℎ𝜇𝜈 ∝ 𝑒^𝑖𝑘𝑧˜ℎ𝜇𝜈. This is useful because there exist powerful numerical techniques allowing them to find these eigenvalues 𝑘 with relative ease, for given mass and angular momentum parameter 𝑎. Their approach is to start with a particular (small) value of 𝑎, and find the corresponding eigenvalues 𝑘. They then increase 𝑎 until they find a critical value where 𝑘 = 0. Such a mode is independent of the string direction 𝑧, and hence can be interpreted as a stationary perturbation mode of the black hole spacetime. It was argued that this corresponds to the threshold of instability, that is, black holes with larger angular momentum are unstable.

This work motivated the construction of the first explicit example of a linearized instability of an asymptotically flat black hole [109], for the cohomogeneity-1 MP black hole. They demonstrated that, for sufficiently large angular momentum, there exist certain gravitational perturbation modes that grow exponentially with time. The insta- bilities found appear at a slightly larger value of angular momentum predicted by the thermodynamic arguments discussed above.

Instabilities of singly-spinning MP black holes have also been found via nonlinear numerical evolution of a perturbed black hole in five [110] and higher [111] dimensions.

These instabilities are of a qualitatively different nature to those found in [109], appearing at a lower value of angular momentum, and breaking more of the symmetry of the original solution.

Despite this recent progress, performing an analysis of the linearized stability of general Myers-Perry black holes seems to be extremely difficult. Though the principles of doing this are well understood, doing it in practice is not easy. The equations of motion involved in these perturbations are extremely complicated, which hinders attempts to extract information from them analytically, whilst the large parameter space makes numerical approaches time consuming. Things are even worse in the case of black rings [68, 69, 70], for which there are physical arguments for various kinds of instabilities [112, 113] but little in the way of concrete results.