exact part of the Fadeev-Popov determinant. This makes the unphysical character of the divergence obvious, at least in perturbation theory.
The relevant operators are then CC and G. For a real metric, the operator CC is positive semi-definite, as we have shown above. Once its zero modes are projected out, it contributes a positive factor to the final result. Physical instabilities are only possible if there are negative eigenvalues of the operatorG,
(GhT T)ab=λhT Tab , (3.33)
in which case the eigenmodeshT Tab are called negative modes. Notice that, forλ6= 0, there is no gauge ambiguity (3.12) becauseGαV = 0.
3.3. NEGATIVE MODES AND LOCAL THERMODYNAMIC STABILITY 59 Consider a black hole instanton B(x) uniquely specified by parameters xα. Let T(x), Ωi(x), M(x), etc. denote the temperature, angular velocities, mass, etc. of this solution. We can construct an off-shell generalizationB(x,x), specified by parameters ˆˆ xα as follows [154, 155]. Assume that the Killing isometries ∂τ and ∂φi, where φi are the rotation angles, of B(x) are preserved. Perform an ADM decomposition of the metric, using the imaginary time coordinate τ. Take the spatial geometry of B(x,x) to be theˆ same as that of B(ˆx). Now choose the lapse function and shift vector so that (i)B(x,x)ˆ has the same asymptotics asB(x); (ii)B(x,x) is regular everywhere, in particular at theˆ bolt, subject to the identifications (τ, φi) ∼ (τ, φi+ 2π) ∼ (τ +β(x), φi −i Ωi(x)β(x));
(iii)B(x, x) =B(x). Note that (ii) implies thatB(x,x) is a configuration in the Euclideanˆ path integral defined for temperature T(x) and angular velocities Ωi(x), for which the saddle point is B(x). Calculating the Euclidean action of B(x,x) using the Hamiltonianˆ formalism gives
I(x,x) =ˆ β(x)M(ˆx)−S(ˆx)−β(x)Ωi(x)Ji(ˆx). (3.34) Condition (iii) implies that the geometry with ˆx=x satisfies the equations of motion and hence the first derivative of the action with respect to ˆxα must vanish for ˆxα =xα. This is a consequence of the fact that the black hole satisfies the first law of thermodynamics,
dM =T dS+ ΩiJi. (3.35)
The second derivative of the action, i.e. the Hessian of the action, now reduces to ∂2I
∂ˆxα∂ˆxβ
ˆ x=x
=
β ∂2M
∂xα∂xβ − ∂2S
∂xα∂xβ −βΩi ∂2Ji
∂xα∂xβ
, (3.36)
where the right-hand side is evaluated atx. If the chargesM andJiuniquely parameterize the solution, we can choosexα= (M, Ji). We then have
∂2I
∂ˆxα∂xˆβ
y=x
=−Sαβ(M, J). (3.37)
Therefore, if−Sαβ fails to be positive definite for a black hole (i.e. if condition (2.14) for
local thermodynamic stability fails), then the Euclidean action decreases in some direction and hence the black hole must admit a negative mode.6 Given that our off-shell geometries are stationary, this negative mode must also be stationary.7 In general, there must be at least as many negative modes as there are negative eigenvalues of this Hessian.
We shall refer to a negative mode whose existence is predicted by this thermody-namic argument as a thermodynamic negative mode. We shall see in Chapters 6–8 that there are some negative modes whose existence cannot be predicted in this way. The latter modes found in [68, 72] are relevant not just for the quantum stability of the black hole but also for its classical stability.
6Note that we have not constructed the negative mode explicitly by this argument: the linearisation of B(x,x) around ˆˆ x=xwill give a superposition of eigenfunctions ofG. The point is that, since the action decreases in some direction, this must involve a negative mode.
7The operatorGcommutes with∂τ so one can work with simultaneous eigenfunctions of these operators.
Eigenfunctions with different eigenvalues of the latter will be orthogonal.
Chapter 4
Kerr-AdS negative mode
In this Chapter, based on [156], we will analyse the semiclassical stability of Kerr-AdS black holes. In particular, we will see that a stationary and axisymmetric negative mode exists only when local thermodynamic stability fails. This test of the gravitational partition function is remarkable also because rotating black holes have complex instanton metrics, whose subtleties we briefly discuss.
The lack of symmetry of the Kerr-AdS solution makes the problem much harder to solve than the spherically symmetric cases. We address this by applying a spectral numerical method to solve linear coupled partial differential equations. The method is reviewed in the Appendix at the end of the thesis.
4.1 Quasi-Euclidean instantons
The gravitational partition function is defined as the path integral (3.1), a sum over geometries with imaginary timeτ = it. However, while static geometries remain real for this analytical continuation, the same does not hold for stationary non-static geometries.
In the canonical formalism, whereγij is the metric on a constant time slice,N is the lapse 61
function and Ni is the shift vector required for rotating spacetimes, we have
ds2=N2dτ2+γij(dxi−iNidτ)(dxj−iNjdτ). (4.1) Regularity at the bolt (instanton horizon) requires the periodic identifications of imaginary time and rotation angles: (τ, φi) ∼ (τ, φi+ 2π) ∼ (τ +β, φi−i Ωiβ). These geometries have been called quasi-Euclidean.1 This seems to pose a difficulty for the path integral because we expect physical quantities to be real. Notice also that the procedure applied in the last Chapter to decompose the metric perturbations and deal with the conformal factor problem assumed the instanton to be (real) Euclidean.
Nevertheless, we share the view of [154, 155] that quasi-Euclidean instantons pose no problem of principle. The instanton action is real and gives the physical free energy (divided by the temperature). Notice that, while one might be tempted to take imaginary lapse functions, which would make the line element (4.1) real, the resulting geometry would bear no relation to the Lorentzian black hole, e.g. the bolt radius would be different from the event horizon radius and there would be no ergosphere. Furthermore, although this can be done for Myers-Perry black holes (1.5) by analytically continuing the rotation parameters ai, it is impossible for black rings because not all conical singularities can be removed in the would-be real instanton [157, 158].
The treatment of quantum corrections about a quasi-Euclidean instanton is more subtle. However, Refs. [154,155] show that the action is real not just for the instanton, but also for a family of off-shell geometries – the one used in Section 3.3 to connect negative modes to local thermodynamic stability. The problem of thermodynamic negative modes should then be posed in terms of real quantities.
The procedure leading to the expression (3.32) for the one-loop quantum corrections assumed a Euclidean instanton, while we now want to consider the quasi-Euclidean case.
However, that expression may still hold for an appropriate complex contour of integration
1Note that, although the metric is complex, the manifold is real. Real coordinates can be defined by setting ˜φi=φi+ i Ωiτ, so the identifications are (τ,φ˜i)∼(τ,φ˜i+ 2π)∼(τ+β,φ˜i).
4.1. QUASI-EUCLIDEAN INSTANTONS 63 in the space of perturbations (and ghosts), specified as usual by the steepest descent method. This is our assumption. It would be important to construct such a contour explicitly.
The numerical technique applied here differs from the analytical but perhaps sim-plistic first approach to the problem in [159], which could only account for the effect of a single direction in the perturbation space. That single direction was provided by an easily constructed traceless-transverse perturbation, which kept the second order action real but was not an eigenmode. The difficulty with that approach, which may explain the small discrepancy in the final result for the Kerr-AdS negative mode, is that it is not clear if such a direction lies on the steepest descent path, despite the second order action being real. The steepest descent path is here infinite-dimensional, spanned by the normalised eigenmodes ofG, which we can only determine numerically.
The family of off-shell geometries related to thermodynamic stability in Section 3.3 preserves the Killing isometries ∂τ and ∂φi, where φi are the rotation angles. Therefore, we will focus on stationary and axisymmetric negative modes (3.33) of the Kerr-AdS instanton. Since this problem is independent of the time and the rotation angle, which are responsible for the instanton metric being non-real, it is equivalent to the Lorentzian problem, and it reduces to a set of explicitly real differential equations.
In the next Section, we review the Kerr-AdS solution and its thermodynamic stabil-ity. In Section 4.3, we outline our implementation of the eigenvalue problem for negative modes. The results are presented and discussed in Section 4.4.