In a paper by Gregory and Laflamme (http://arxiv.org/abs/hep-th/9301052) in 1993, it was demonstrated that black strings and $p$-branes which were solutions to certain low energy string theories were unstable by searching for solutions to the perturbation equations,

$$\delta R_{ab}=-\frac{1}{2}\Delta_L h_{ab}=0$$

where $h_{ab}$ is a linear perturbation, and $\Delta_L$ the Lichnerowicz operator,$^{\dagger}$ computed with respect to the background solution, e.g. a black string. In addition, they formulated an ansatz,

$$h^{\mu\nu}= e^{\Omega t} e^{i\mu_i x^i}\left( \begin{matrix} H^{tt} & H^{tr} & 0 & 0 & \dots \\ H^{tr} & H^{rr} & 0 & 0 & \dots \\ 0 & 0 & K & 0 & \dots \\ 0 & 0 & 0 & K/\sin^2 \theta & \dots \\ \dots & \dots & \dots & \dots & \dots \end{matrix} \right)$$

My questions are the following:

  1. What is the motivation for the ansatz above?
  2. Why are solutions to the perturbation equations which maintain the Ricci curvature (and hence energy-momentum tensor) deemed instabilities?
  3. Is the general approach outlined above applied to most solutions to determine instabilities? Are there alternate methods (excluding the entropic argument by comparing solutions' entropies)?

$\dagger$ In de Donder gauge, the operator is given by $\Delta_L h_{ab}=\left[\delta^c_a \delta^d_b (\nabla_e \nabla^e) +2R^{cd}_{ab}\right]h_{cd}$.

  1. They are looking at s-wave solutions, ie zero angular momentum. They are also looking at modes that are pure tensor with respect to the Schwarzchild geometry (ie they set the vector and scalar parts to zero). So the angular parts of the perturbation are just proportional to the angular parts of the background metric (in a sense the angular parts are proportional to the identity, because the angular momentum is zero). The interesting parts of the perturbation then are only allowed to involve $r$ and $t$, and this is the most general form for the perturbation that mixes $r$ and $t$ but leaves the angular parts alone.

  2. The instability comes from the fact that there are solutions that are regular at infinity that blow up at the horizon. See the discussion after equation 11.

  3. There are many tricks to find instabilities. The methods used in the Gregory-Laflamme paper are certainly valid, but their methods are by no means the only nor the most common ones. Generally speaking, the question is whether the split into background + perturbations is a good one in the sense that the perturbations remain small. The perturbations might fail to remain small for many reasons--you might have a runaway potential that becomes infinitely negative (hamiltonian unbounded below) or at least pushes you away from your chosen background (tachyon instability), you might have negative kinetic energy (ghost instability), you might have negative gradient energy (gradient instability)--there are lots of different ways for instabilities to manifest themselves. Here, the method is essentially brute force: numerically construct solutions and show explicitly that there are valid solutions to the perturbation equations where the perturbations become infinitely large.

  • $\begingroup$ Thank you for your answer. Regarding $\#3$, can you recommend any (preferably free) papers that explore instabilities in GR and string theory? In addition, you mentioned 'many tricks' for finding instabilities, could you elaborate slightly? $\endgroup$
    – JamalS
    May 19 '14 at 13:47
  • $\begingroup$ I don't have time right now to find sources but I'll try to come back to it later. What I had in mind with 'many tricks' is that sometimes you can do things analytically when the perturbation equations are simpler. In that case you are looking for things that I mentioned in my answer. The simplest one is something you learn early in physics--are you expanding around a solution that is a local maximum of a potential? This is unstable because perturbations will run away from the maximum. There's other things you can look for--for example, is the kinetic energy negative? That is a very bad one. $\endgroup$
    – Andrew
    May 19 '14 at 13:55
  • $\begingroup$ Here is another example of an instability: physics.stackexchange.com/questions/73118/…. Admittedly it is not GR or string theory, but honestly looking for instabilities is really about perturbation theory, the same principles apply in classical mechanics as in string theory or GR. The nice thing about this example is that it is not one of the things you usually look for in my experience, so it illustrates that really what you are doing is trying to find some problem, rather than follow a recipe. $\endgroup$
    – Andrew
    May 19 '14 at 13:59

The answer accepted here is, in fact, incorrect in its comments in part 2. This is not at all how the instability arises.

The instability comes from the fact that the solution is of the form $e^{\Omega t}e^{i\omega z}H_{\mu\nu}(r)$ with $\Omega\in \mathbb{R}_+$, i.e. exponentially growing. They numerically solve the decoupled ODEs for $H_{\mu\nu}(r)$ to find a solution that is regular at the horizon and regular at infinity. This is a subtle point: Schwarzschild coordinates do NOT cover the horizon, hence one needs to examine the perturbation at the horizon in coordinates that extend across it, for example, Ingoing Eddington Finkelstein. Now, this allows you to identify the right branch of asymptotic solution at the horizon in the singular Schwarzschild coordinates.


Regarding your second question, I would like to mention a point mentioned in Horowitz's book Higher Dimensional Black Holes. Although the book discusses the Gregory-Laflamme instability in the context of higher-dimensional classical general relativity (with some arguments from semi-classical gravity), I think it is relevant here. I should mention this is not a complete argument but a good enough motivation to expect the instability.

The simple argument is that the black string solutions, despite having the same mass (and thus, energy) as the black hole solutions, these black strings have lesser entropy than the same mass black holes. Thus, as a thermodynamic consequence, they have a tendency to move towards being in the state of a black hole rather than a black string.

  • $\begingroup$ Yes, this is covered in Ruth Gregory's video lectures, and it is included in the original paper. However, this provides a motivation for searching for an instability - it does not provide a motivation for the ansatz itself. $\endgroup$
    – JamalS
    Dec 15 '17 at 16:13
  • $\begingroup$ @JamalS Yes, I was only talking about the motivation behind expecting an instability. That is why I mentioned ``Regarding your second question''. Btw, can you provide a link to the video lectures? $\endgroup$
    – Dvij D.C.
    Dec 15 '17 at 16:15
  • $\begingroup$ Go to pirsa.org, and search for Ruth Gregory - she only does one course, you can pick any version in recent years. $\endgroup$
    – JamalS
    Dec 15 '17 at 16:19

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