Targeting the learning material stimulus

whereY y∗ must be sufficiently large. Let us rescale the perturbation functions too,

δµ0(r, θ) = q1(y, x)

y(1−x²), (4.16a)

δχ(r, θ) = q₂(y, x)

1−x² , (4.16b)

δµ₁(r, θ) = q₃(y, x)

y(1−x²), (4.16c)

so that the boundary conditions discussed before are simplyqi = 0 (i= 1,2,3) along the edgesx=±1 andy= 0, Y. It is convenient to combine our previous equationsrr,rθand θθ, now in terms of the redefined quantities, into the form

∂_y²qi(y, x) +. . .+λ∗f(y, x;y∗, `∗)qi(y, x) = 0, (4.17) which turns out to be possible. Notice that the last term above gives the only dependence of the equations onλ∗. We are now ready to implement the spectral numerical method [160], which is briefly described in the Appendix at the end of this thesis.

4.4 Results

Let us first consider the Kerr case, i.e. the limit` → ∞. The results are represented in Fig. 4.2. We find that there is a single negative mode for |a| ≤ r0/2, monotonically in-creasing in magnitude with the angular momentum. Surprisingly, our probe perturbation method [159] approached the value of the negative eigenvalue now found to within 10%.

One way to understand the increase in magnitude is to recall the connection between the black hole thermodynamic negative mode and the classical Gregory-Laflamme instabil-ity of the respective black string/brane [137], which we shall review in greater detail in Chapter 6. The threshold wavenumber k = (~k·~k)^1/2 for the Gregory-Laflamme insta-bility corresponds to the four dimensional stationary solution of Gh^{T T} = −k²h^{T T}, with the appropriate boundary conditions [100]. This is exactly the problem addressed here

0.0 0.1 0.2 0.3 0.4 0.5 -10

-8 -6 -4 -2

ar₀ Λr₀²

Figure 4.2: For the Kerr instanton, λ∗ is negative, decreasing monotonically away from a= 0 and evaluating to a finite value at extremality |a|=r₀/2.

if we identify λ = −k². The fact that we are dealing with a quasi-Euclidean geometry rather than a Lorentzian geometry is irrelevant since time plays no role in the solutions to the perturbation functions defined in (4.11). The curve in Fig. 4.2 thus implies that the Gregory-Laflamme instability of the Kerr string persists up to extremality. The larger in magnitude is the negative mode, the smaller is the threshold length scale k⁻¹ for the instability. We expect on physical grounds that the centrifugal force caused by the rota-tion will favour the instability of ripples along the string (Ref. [110] presents a fluid dual analogy) thus decreasing their threshold length scale and explaining the stronger negative mode of the black hole. See [68, 72, 108] for analogous results in higher dimensions.

We now turn to the Kerr-AdS case. If we look at Fig. 4.3, the agreement between the existence of a negative mode and the condition (4.9) is striking. We find a negative mode only whenC_Ω is negative. Unfortunately the numerical method does not allow us to safely zoom in the line of critical stability. Nevertheless, it is clear that the gravitational partition function reproduces the thermodynamics of the system beyond the instanton approximation, even for this non-static black hole with a quasi-Euclidean instanton.

4.4. RESULTS 71

0 1 2 3 4

0.1 0.2 0.3 0.4 0.5

r

 { a  r

Figure 4.3: Phase diagram of the Kerr-AdS black hole. The points represent the parameter region where we find a negative mode and the line represents the change of sign of C_Ω, which is negative in the Kerr limit`→ ∞.

Chapter 5

Reissner-Nordstr¨ om negative mode

In this Chapter, based on [161], we analyse the problem of thermodynamic negative modes of the Reissner-Nordstr¨om black hole in four dimensions. We find analytically that a negative mode disappears when the specific heat at constant charge becomes positive.

The sector of perturbations analysed here is included in the canonical partition function of the magnetically charged black hole. The result obeys the usual rule that the partition function is only well-defined when there is local thermodynamical equilibrium.

We emphasise the challenge of quantising Einstein-Maxwell theory, even as a low energy effective theory, because the conformal factor problem is more intricate than in the vacuum case, which was discussed in Section 3.2.2. We circumvent this difficulty by considering a dimensional reduction from five to four dimensions. The method allows us to decouple the divergent gauge volume and treat the metric perturbations sector in a gauge-invariant way.

73

5.1 Introduction

The inclusion of matter presents considerable challenges to the study of black hole negative modes. This is because the standard decomposition of the perturbations [130, 131], re-viewed in Section 3.2.2, which singles out the unphysical conformal modes, was performed for the Einstein theory only (with or without cosmological constant). In particular, the presence of a background gauge field invalidates the procedure even for metric perturba-tions only, i.e. when the gauge field perturbaperturba-tions can be consistently set to zero.

One way out to address this problem is suggested by the alternative procedure of Gratton, Lewis and Turok [162, 163] for cosmological instantons. They look for well-behaved gauge-invariant perturbations which allow for a decoupling in the action between the relevant modes and the unphysical ones, which are integrated out. See Kol [164] for a general and systematic discussion of the same basic idea, which he calls “power of the ac-tion,” following its initial application to the Schwarzschild black hole and black string [138].

This corresponds to the procedure we adopt in this work. There is a further complication though. The problem of radial perturbations requires two gauge conditions (as the trace-less and transverse conditions in pure gravity) for the metric perturbations. The method is inconsistent because the radial ansatz for the perturbation of the action only has a radial diffeomorphism invariance (one gauge choice), and including time does not allow for the construction of gauge-invariant quantities. The work of Kol [138] on the Schwarzschild black hole gives a solution to this problem. Considering a lift to one higher dimension, along which there is translational invariance, it is possible to construct gauge-invariant quantities, i.e. perturbations which are invariant for infinitesimal diffeomorphisms along the radial direction and along the extra dimension. If the action for the zero modes (infinite wavelength) along the extra-dimension reproduces the lower-dimensional action for perturbations, then the higher-dimensional action can be decomposed, and the long wavelength limit can be taken when a simple reduced action is available.

Using this Kaluza-Klein method, we are able to study the four-dimensional magnetic Reissner-Nordstr¨om black hole in a sector of perturbations corresponding to the canonical

5.1. INTRODUCTION 75 ensemble,i.e. fixed chargeQ. We find that one negative mode still exists if the charge is small compared with the massM, as expected, but disappears for|Q| ≥√

3M/2, exactly when the specific heat becomes positive. This supports the validity of the canonical partition function of Euclidean quantum gravity [129] even when gauge fields are present.

One might be worried by the fact that such a theory is not renormalisable at one loop, which is indeed the case of Einstein-Maxwell theory [165], so that we cannot compute quantum corrections. However, we can take an effective field theory approach [166], where Einstein-Maxwell theory is regarded as the low energy limit of an underlying fundamental theory. The effective theory is valid up to a cut-off scale, after which ultraviolet completion effects become important. As long as the energies of the fields involved are nowhere near that scale, the perturbative quantisation of the non-renormalisable effective theory is meaningful. It may help to recall that Einstein-Maxwell theory corresponds to the bosonic sector of four-dimensionalN = 2 supergravity, which can be embedded into string theory.

We mentioned that Einstein-Maxwell theory had not been considered for the per-turbative path integral quantisation on black hole backgrounds. Miyamoto and Ku-doh [167, 168] have analysed the classical stability of magnetically charged branes and verified that they are stable when there is local thermodynamic stability. This result is related to the Gubser-Mitra conjecture mentioned in Sections 2.2.2 and 6.1.1. The issue we address here is whether the partition function at one loop, defined as a saddle point approximation to a Euclidean path integral, conforms to the usual thermodynamic stabil-ity criterion. It is also worth mentioning that Einstein-Maxwell theory in four dimensions cannot be the result of dimensionally reducing Einstein-Maxwell theory in five dimensions, since there is then an extra scalar field. The fact that the Reissner-Nordstr¨om black hole is not straightforwardly related to a string will force us to be careful in our Kaluza-Klein action method, so that we make sure the correct quantum theory is obtained.

This Chapter is organised as follows. In Section 5.2, we discuss the problems with quantising Einstein-Maxwell theory, stressing the differences with the pure Einstein case, and recall the relation between the boundary conditions of the path integral and the

cor-responding thermodynamic ensemble. In Section 5.3, we explain the Kaluza-Klein action method to analyse the second-order action. In Section 5.4, we describe the application of the method to the magnetic Reissner-Nordstr¨om black hole. We start by presenting an appropriate lift to five dimensions. We then construct the gauge-invariant quantities and obtain the reduced action. We find that the action possesses a negative mode when the specific heat at constant charge is negative. Finally, in Section 5.5, we present the conclusions.