Gravitational duals to the grand canonical ensemble abhor Cauchy horizons

The gravitational dual to the grand canonical ensemble of a large $N$ holographic theory is a charged black hole. These spacetimes -- for example Reissner-Nordstr\"om-AdS -- can have Cauchy horizons that render the classical gravitational dynamics of the black hole interior incomplete. We show that a (spatially uniform) deformation of the CFT by a neutral scalar operator generically leads to a black hole with no inner horizon. There is instead a spacelike Kasner singularity in the interior. For relevant deformations, Cauchy horizons never form. For certain irrelevant deformations, Cauchy horizons can exist at one specific temperature. We show that the scalar field triggers a rapid collapse of the Einstein-Rosen bridge at the would-be Cauchy horizon. Finally, we make some observations on the interior of charged dilatonic black holes where the Kasner exponent at the singularity exhibits an attractor mechanism in the low temperature limit.


Introduction
Black hole interiors present many theoretical challenges, at both a classical and quantum level. One of these challenges is the singularity at which spacetime ends [1]. The classical approach to generic singularities is expected to be very complicated [2], while the classical description itself eventually breaks down as curvatures become large. Another challenge is the possible presence of Cauchy horizons, at which the predictability of the classical dynamics breaks down, even away from regions with large curvature [3]. The strong cosmic censorship conjecture posits that such Cauchy horizons are artifacts of some highly symmetric solutions that are known analytically, and do not arise from generic initial data [4].
In holographic duality, eternal black holes in asymptotically AdS spacetimes arise as thermofield double states in a large N CFT [5]. This fact has led to rigorous boundary probes of the black hole interior using e.g. entanglement entropy [6]. So far, probes of the region close to spacelike singularities have required analytic continuation of boundary correlation functions [7] and do not appear to directly access Cauchy horizons [8]. Holographic arguments suggest that in general, Cauchy horizons do not survive in the full quantum gravity theory [9,10]. (The case of three dimensional BTZ black holes appears to be an exception to this fact, as first suggested in [11].) With ongoing interest in probing the interior, it is important not to be led astray by aspects of the spacetime that may be artifacts of the simplest known solutions. The most studied solutions in holography are the Schwarzschild-AdS spacetime as dual to the canonical ensemble [12] and Reissner-Nordström-AdS (RN-AdS) spacetime as dual to the grand canonical ensemble [13]. These have rather particular singularity structures and RN-AdS has an inner Cauchy horizon.
While a fully generic interior will be highly inhomogeneous, a tractable step in the direction of genericity for uncharged black holes was considered in [14], motivated from the dual field theory perspective. The simplest AdS black holes spacetimes are dual to the thermofield double state of a CFT. The CFT itself is often non-generic within the space of field theories in the sense that relevant deformations (such as mass terms) must be tuned to zero to remain at the critical point. To probe more generic thermal states, the relevant operators can be turned on. This can be done with a coupling constant that is uniform in the boundary spacetime. Relevant operators are described in the bulk by scalar fields with negative mass squared (but above the Breitenlohner-Freedman bound [15]). Sourcing such fields at the AdS boundary should be expected to produce more generic black hole solutions.
In [14] it was found that, indeed, these solutions had a more generic behavior at the black hole singularity, with the Schwarzschild singularity arising as a fine-tuned special case. The more generic behavior is described by a one parameter family of homogeneous, anisotropic cosmologies known as Kasner spacetimes. Thus, genericity at the boundary led to genericity at the singularity. In this paper we will ask an analogous and perhaps more consequential question for charged black holes: does turning on a relevant deformation of the boundary theory remove the Cauchy horizon? The answer will be that it does.
The boundary perspective motivates a holographic version of strong cosmic censorship with a slightly different flavor from the conventional one. Usually one asks about the stability of Cauchy horizons in the space of generic initial conditions. Holographically one can ask instead whether a generic time-independent thermal state of the boundary theory leadsin the classical large N limit -to a dual black hole with a Cauchy horizon. As we have explained above, from this boundary perspective RN-AdS is not generic if the CFT has relevant deformations that have been fine tuned to zero. The results below are evidence in favor of such a notion of holographic strong cosmic censorship.
Three comments should be made here. Firstly, since the radial black hole coordinate becomes timelike in the interior, what start off as asymptotic boundary conditions ultimately play the role of initial conditions for the interior. Thus the two formulations of strong cosmic censorship have some overlap. Secondly, a key step in attempting to prove strong cosmic censorship involves establishing that perturbations outside the horizon do not decay too quickly, so that they can build up inside and prevent the formation of a Cauchy horizon [16].
For example, it has recently been shown that the Cauchy horizon is stable for some charged black holes in de Sitter space where the perturbations fall off exponentially fast outside the horizon [17,18]. A source at the boundary that is present for all time clearly helps with this issue and therefore this holographic version is weaker than the conventional one. Thirdly, inhomogeneous deformations of the boundary can induce regions of strong curvature that are directly visible to boundary observers [19,20]. These are violations of weak cosmic censorship, which is not the focus of our present discussion.
We have focused on relevant deformations so far, but irrelevant deformations of the CFT will also be generically present at nonzero temperature if the CFT is obtained as the IR fixed point of some UV completion such as a lattice model. While relevant deformations always remove Cauchy horizons, we will show that certain irrelevant deformations, dual to a bulk scalar with m 2 > 0 (positive mass squared), do allow them. But these Cauchy horizons can only occur at a discrete set of m 2 for each temperature. Irrelevant deformations destroy the asymptotic AdS region, which must either be explicitly cut off or otherwise allowed to flow to some distinct UV fixed point where the operator is relevant. However, our discussion will only require knowledge of the spacetime inside the event horizon.
The fate of the Cauchy horizon is especially dramatic for the case of a small deformation of the AdS boundary. The solution remains close to RN-AdS until one approaches the would-be Cauchy horizon. At that point there is a rapid collapse of the Einstein-Rosen bridge connecting the two asymptotic boundaries. That is, any finite stretch of this bridge rapidly shrinks to an exponentially small size. This is universal behavior that we will see both analytically and numerically. Following this rapid collapse, the solution approaches a spacelike Kasner singularity. These regimes are illustrated in Fig. 1. With a larger source there is a smoother transition between the RN-AdS and Kasner epochs.
Although Schwarzschild-AdS also has a Kasner singularity, we will see that the singularities that arise from deformations of RN-AdS have Kasner exponents that are bounded away from that of Schwarzschild-AdS. For small deformations, the Kasner singularity is almost null and 'bends up' in the Penrose diagram, while for sufficiently large deformations it can 'bend down' like the Schwarzschild-AdS case. Finally, we will also discuss the singularities inside Einstein-Maxwell-dilaton AdS black holes. In these theories, there are analytically

Background and equations
In the grand canonical ensemble the dual field theory is held at a chemical potential µ for some global U (1) symmetry. In the bulk we must correspondingly introduce a Maxwell field A such that A t → µ at the boundary. To deform the boundary theory by a scalar operator O we must introduce a dual scalar field φ in the bulk. The leading asymptotic behavior φ (0) of the scalar field will be the source for operator. A minimal bulk theory that contains these ingredients is We will consider this theory in 3 + 1 bulk dimensions, though as we note below our results hold in higher dimensions also. We have set the AdS radius and the gravitational coupling to one. The mass m will determine the scaling dimension ∆ of the operator O through the usual formula: The Maxwell field strength is F = dA.
We wish to find planar charged black hole solutions to the theory (2.1). We will assume the solutions are static and homogeneous, so they can be written in the form 3) The AdS boundary is at z = 0 and the singularity will be at z → ∞. At a horizon, f = 0.
The scalar field and scalar potential take the form In this gauge regularity requires Φ = 0 at a horizon. As we will be especially interested in the behavior of the solution behind the horizon, we rewrite the metric in ingoing coordinates: The radial functions should obey the following leading asymptotic behavior at the AdS boundary as z → 0 This behavior fixes the normalization of time on the boundary as well as the chemical potential µ and source φ (0) for the dual operator O. Because there is no charged matter in the bulk, it will be convenient to introduce the boundary charge density The bulk equations of motion with the above ansatz are written down as follows. First, the Maxwell equation can be integrated once to give Here ρ is a constant, the boundary charge density (2.7). The remaining minimal set of equations of motion can be taken to be

Horizons
Solutions to the equations of motion with the asymptotics (2.6) will typically have a horizon at z H , with f (z H ) = 0. The temperature of the dual quantum field theory is The infalling coordinates (2.5) continue across the horizon. Our main interest is the interior geometry that is found beyond the horizon.
In the absence of a scalar field, with φ = 0 everywhere, the solution is the Reissner-Nordström-AdS spacetime, with χ = 0 and In addition to the horizon at z = z H , there is an inner horizon at z = z I with This inner horizon is well known to be a Cauchy horizon, leading to the breakdown of predictability in the black hole interior. At high temperatures ρ 2 z 4 H → 0 and in this limit the inner horizon is at z I ≈ 4z H /(ρ 2 z 4 H ) → ∞, although the proper time between the horizons does not become large. At low temperatures z I → z H as the black hole becomes extremal.
We now discuss the effect of a nonzero scalar field on the inner horizon. For the theory with action (2.1) this depends on the sign of the mass squared, which also corresponds to whether the operator is relevant or irrelevant. With a more general potential for the scalar field, however, there need be no connection between relevance or irrelevance near the AdS boundary and the sign of the potential in the black hole interior.

Relevant deformations remove Cauchy horizons
The black hole interior is dramatically changed by a nonzero φ. For m 2 ≤ 0, which corresponds to relevant operators with ∆ ≤ 3 in our theory, we can prove that there is no inner horizon as follows. Suppose that there were two horizons at z H and z I . From Eq. (2.9a): In the first equality we have used the fact that f (z H ) = f (z I ) = 0. In the final expression note that between the two horizons f < 0. If m 2 ≤ 0, the integrand in the final expression is therefore non-positive over the range of integration. Thus, the only way there can be two horizons is if φ = 0 identically. The scalar field necessarily removes the inner horizon. For more general scalar potentials V (φ), the above argument still applies provided φV (φ) < 0.

Irrelevant deformations can have fine-tuned Cauchy horizons
For certain irrelevant deformations, we will see that inner horizons can exist at one specific temperature. Irrelevant operators are dual to bulk fields with m 2 > 0. These grow large towards the AdS boundary, and so cannot be consistently included as sources. Instead they will induce a renormalization group flow up towards a different UV completion. However, our analysis will only depend on the scalar field profile in between the black hole horizon and the Cauchy horizon, and is therefore independent of the UV completion. We will do this in two steps: first we analyse the linear problem, and then we bootstrap the problem non-linearly.
For the linearized problem we look at the scalar field on the Reissner-Nordström background. This amounts to Eq. (2.9a) with χ = 0 and f = f RN as in (3.2): We wish to solve (3.5) for z ∈ (z H , z I ) -where here z H and z I are the outer and inner horizons of RN-AdS -with the regularity conditions that Given a UV completion that restores an asymptotically AdS region, for instance due to a more complicated scalar potential than just m 2 φ 2 , the ratio ξ has the same information as the dimensionless boundary quantity T /µ. (The asymptotic region is necessary to fix the normalization of the time coordinate.) At extremality, ξ = 1.
The linearized eigenvalue problem can be readily solved via the numerical methods detailed in [21]. Alternatively, we can perturbatively solve (3.5) around extremality, using the methods of [22]. As expected, we find an infinite tower of modes, which we label by an integer ≥ 1. For these masses, a regular scalar field configuration exists between the inner and outer horizon. We shall just quote here the result for m 2 to quartic order in (ξ − 1) for generic values of . Once the dust settles, we find: where λ = ( + 1). In Fig. 2 we show the numerically determined values of m 2 as a function of (ξ − 1). The numerical and perturbative results agree for ξ ∼ 1. The disks, squares, diamonds and triangles are the corresponding exact numerical data.
The large ξ behavior shown in Fig. 2 can also be understood analytically. In the strict ξ → ∞ limit, the RN-AdS background becomes Schwarzschild-AdS. Generically, linear massive scalar fields in this spacetime diverge logarithmically near the singularity. (This leads to a change in the Kasner exponents in the full nonlinear solutions, as discussed in [14].) The analog of demanding that the Cauchy horizon remain smooth, is to demand that the scalar field vanish at the singularity. If one imposes this (and regularity at the event horizon), one again obtains an eigenvalue equation for m 2 with eigenvalues: The large ξ results in Fig. 2 indeed asymptote to these values. In fact, one can go a bit further and compute the corrections in ξ −1 using standard perturbation theory. These turn out to be given by is a generalised hypergeometric function.
Since m 2 is a parameter in the bulk action, it is probably more physical to turn Fig. 2 around and interpret it as saying that for certain given m 2 , there can be one value of T /µ for which the inner horizon is not destroyed (at the linearized level).
We now establish that these linearized solutions extend without obstruction to nonlinear solutions with a smooth Cauchy horizon. As noted below (2.9) the equations to be solved are a first order equation for f and a second order equation for φ. There are correspondingly three constants of integration. We can take these to be {ξ, φ H , φ I }.
Here φ H = φ(z H ) and These equations in addition depend on the parameters ρz 2 H and m 2 . A solution can therefore be specified by the five parameters {m 2 , ρz 2 H , ξ, φ H , φ I }. Suppose that we take a solution that is regular at the outer horizon and integrate in, and we take a solution that is regular at the inner horizon and integrate out. These will combine into a solution that is regular everywhere between the horizons if {φ, φ , f } match at some intermediate point.
With five paramaters and three constraints we expect to find a two-parameter family of solutions with a smooth Cauchy horizon. These can be labelled e.g. by {ξ, φ H }. As φ H → 0, m 2 should match the values obtained previously from a linearized analysis in Fig. 2.
We have scanned a large portion of parameter space, and found the above counting picture to be correct. In Fig. 3

Collapse of the Einstein-Rosen bridge
When the inner horizon is absent, the black hole interior ends at a spacelike singularity as z → ∞. We describe the asymptotic near-singularity behavior in the following section. In this section we describe a crossover that occurs at the location z I of the would-be horizon. The crossover is most dramatic when the scalar field is small, and in this limit can be obtained analytically. While the scalar field is small, the spacetime dynamics is highly nonlinear in this regime. We will see that it corresponds to a collapse of the Einstein-Rosen bridge between the two asymptotic boundaries.
The collapse occurs over an extremely short range in the z coordinate, so it is consistent to simply set z → z I in the equations of motion (2.9a) -(2.9c). We can think of the variables f, χ, φ as functions of δz = z − z I . Furthermore, at these values of z it can be verified numerically (or, a posteriori on the solution below) that the mass of the scalar field becomes negligible in (2.9a) and (2.9b), as does the cosmological constant term in (2.9b), which is the factor of −12. With these approximations the equations become The general solution to these equations can be found, starting by integrating the first equation and writing φ = − c 1 z 2 I ρ/ √ 2 e χ/2 /f . Here c 1 is a constant and the normalization is for future convenience. The solution is most nicely expressed in terms of the metric component g tt = −f e −χ /z 2 I . This is found to obey The general solution to this equation takes the form (recall g tt > 0 in the black hole interior) Here c 2 and c 3 are additional constants of integration (again normalized for convenience).
And then, in addition to f = −e χ g tt z 2 I , one finds that The scalar field exhibits the expected logarithmic growth as g tt becomes small close to the would-be inner horizon. The special cases discussed in the previous section where the inner horizon survives will have c 1 = 0.
The first equation in (4.4) suggests that c 2 /c 1 will become large when the boundary source for φ is small. This is because the argument of the logarithm in (4.4) is order one at the end of the crossover region, and in the limit of a small scalar field, the scalar can be Here we see that a linear vanishing of g tt towards the would-be inner horizon is replaced by a rapid collapse to an exponentially small value, while the divergence in the scalar field deriva- In the black hole interior, g tt sets the measure for the spatial t coordinate that runs along the wormhole connecting the two exteriors of the black hole. This is the Einstein-Rosen bridge. The rapid decrease in g tt that we have just described can therefore be thought of as a collapse of the Einstein-Rosen bridge for a fixed coordinate separation ∆t. The collapse to an exponentially small g tt happens over a short proper time ∝ c 3 1 /c 3 2 .

Kasner singularity
After the collapse of the Einstein-Rosen bridge, the spacetime enters an asymptotic regime that tends towards a Kasner singularity. Recall that the Kasner solution is a homogeneous, anisotropic cosmology with power law behavior near the singularity. When the Maxwell flux terms are subleading, the asymptotic solution is given by [23,24] Here c t and c x are constants. The Kasner exponents obey p t + 2p x = 1 and p 2 φ + p 2 t + 2p 2 x = 1. The near-singularity behavior is similar to that of the neutral black holes studied in [14].
We find that as z → ∞, the solutions take the form with α > 1. This restriction on α ensures that the Maxwell flux terms are always unimportant asymptotically. It is easy to see that the metric and scalar are indeed of the Kasner form The lower bound on p t (following from the bound on α) excludes the Schwarzschild nearsingularity behavior which has p t = −1/3.  As the scalar field is turned off at fixed temperature, the Kasner exponent p t → 1.
This is different to the case of neutral black holes, where p t → −1/3 as the deformation is turned off [14]. The difference is easily understood: In the neutral case the solution reverts to the Schwarzschild singularity, while in the charged case the Kasner singularity reverts to the regular inner horizon (which has p t = 1 in Kasner coordinates). An exception to this statement arises at very low temperatures. At sufficiently low temperatures, neutral scalar fields can spontaneously condense in the Reissner-Nordström-AdS background [25]. Below the critical temperature T c , this leads to a Kasner singularity with p t < 1 even in the absence of a source, φ (0) = 0, as shown in Fig. 6. Each spontaneous solution will extend to a family of solutions with nonzero source. For small values of the source, these solutions will compete with the solutions that continue to the trivial solution at φ (0) = 0. As T /µ → 0 for fixed deformation strength φ (0) /µ, again the Kasner exponent p t → 1. This is not completely clear from Fig. 5 but is seen clearly in Fig. 6. We make two further observations. Firstly, in Fig. 6 we see that extending to very low temperatures, the limiting numerical behavior is well fit by α ∝ [− log(T /µ)] 1/2 → ∞ in (5.3). This corresponds to p t → 1. Secondly, in this limit the outer horizon is verified in numerics to be approaching a singular solution first written down in [26]: The series expansion continues in powers of 1/(ρz 2 ) 2 . In addition, there is a nonperturba- The parameter A is fixed by the asymptotic source, φ (0) . In the low temperature limit ρz 2 H → ∞. This allows the expansion above in ρz 2 1 even outside the outer horizon where z < z H . The divergence of α in the low temperature limit is consistent with the scalar field φ crossing over to the stronger than logarithmic growth of (5.4).
Finally, we note that in the discrete cases with m 2 > 0 where an inner horizon survives, as discussed in §3.2, the singularity beyond the inner horizon will be that of the Reissner-Nordström black hole, with the scalar field becoming unimportant towards the singularity (φ ∼ 1/z) and the equations dominated by the flux terms.

Penrose diagrams
Penrose diagrams are convenient ways to picture the global causal structure of a spacetime.
Given a static AdS black hole with a spacelike singularity, one often imagines its Penrose diagram is a square, with singularities on top and bottom. However, as pointed out in [7] there is a conformally invariant distinction between spacelike singularities that bend down toward the event horizon and ones that bend up away from the horizon. In Schwarzschild-AdS, the singularity bends down [7]. For small deformations of the RN-AdS spacetime, the singularity appears close to the would-be inner Cauchy horizon and hence one might expect the singularity to bend upwards in this limit. Let us now discuss this more systematically.
An ingoing radial null geodesic that leaves the boundary at boundary time t = 0 reaches the singularity at a value of the interior spatial coordinate t given by where P V denotes taking the principal value upon crossing the horizon at z = z H . The principal value indicates that the interior spatial t coordinate is naturally related to the boundary time by a constant imaginary shift from the residue −iπe χ(z H )/2 /f (z H ) = i/(4T ), but this shift is unnecessary to understand the bulk Penrose diagram of a purely real spacetime.
Recall that f > 0 outside the horizon and f < 0 inside the horizon, so the integral in (6.1) could have either sign. The direction in which the singularity bends depends on the sign of t . This is because in the black hole interior t = 0 corresponds to the midpoint of the Penrose diagram. If t > 0, for example, then the geodesic has reached the singularity before reaching the midpoint of the diagram, and hence the singularity must have bent down.
Similarly, if t < 0 then the singularity bends up. Both of these possibilities are realized in our solutions and are shown in Fig. 7.
We find that t → −∞ both as we turn off the deformation, φ (0) /µ → 0, and also at low temperatures, T /µ → 0. In these limits the singularity therefore bends up and becomes null. At higher temperatures and larger deformations, the singularity bends down.
These two regimes are shown in Fig. 5. The singularity can bend down more than that of Schwarzschild AdS, but approaches this in the limit T /µ → ∞ at fixed φ

Dilatonic theories: Lifshitz to Kasner
Einstein-Maxwell-dilaton theories have exact black hole solutions with no inner horizons and with Kasner singularities determined by parameters in the action [27]. In this section we describe how these fixed exponents relate to the source-dependent Kasner exponents we have described so far. We will see that, in a holographic context, the known explicit solutions describe a low temperature, near-horizon limit of a class of geometries with more general Kasner exponents. The fixed exponents arise in this limit in a sort of 'attractor mechanism'.
In direct analogy to the nondilatonic case, deformations away from this limit change the exponents by an amount that depends on the deformation.
The simplest holographic setting for the physics we are after is the theory [28,29] There is a single coupling γ in the Lagrangian. With the same ansatz for the fields as we have been considering all along, the equations of motion are now These equations have an exact black hole solution given by [28] f with the constants This solution has the following asymptotics. As z → 0 it tends to a so-called Lifshitz geometry with dynamical scaling exponent As z → ∞ it tends towards a Kasner singularity with Thus in this solution the Kasner exponent is fixed by the parameter γ in the theory. Despite the presence of a Maxwell field, these solutions are best thought of as a one-parameter family generalization of the Schwarzschild-AdS solution, which is recovered in the limit γ → ∞ (wherein z L → 1 and p t → −1/3).
The exact solution discussed above arises as the near-horizon geometry of a near extremal black hole in an asymptotically AdS spacetime [29]. While not strictly necessary, it is clarifying to take this bigger perspective, in which case the geometry is divided into three regions for near-extremal states with T µ (here µ is some UV energy scale, such as the chemical potential, that sets the crossover from AdS to Lifshitz): The important point here is the following. If the Lifshitz geometry is obtained in this way by flowing from some AdS UV, then the solution cannot be pure Lifshitz at any finite z. There must also be irrelevant deformations that decay as z → ∞. These are the deformations that flow the Lifshitz solution back up to AdS. At any nonzero temperature, where z H is finite, these deformations will be nonzero on the horizon. We will see that these deformations on the horizon shift the Kasner exponent away from the value p t (γ) in (7.7). As T → 0, z H → ∞ and the deformation becomes small on the horizon so that p t → p t (γ). This is a transhorizon manifestation of the attractor mechanism -really just an IR fixed point in the RG sense -discussed in [29]. However, at any T > 0 the value of p t is different and depends on the strength of the irrelevant deformation.
The simplest point is to verify that Kasner scalings with more general exponents than (7.7) are consistent asymptotic near-singularity behaviors of the theory. The only constraint on the asymptotic Kasner exponent from the equations of motion is that At γ = 0 this recovers the constraint that p t > 0 that we found in (5.3) for Einstein-Maxwell theory. As γ → ∞, the lower bound goes down to the Schwarzschild value of −1/3, which is also consistent.
To see explicitly how a source shifts the Kasner exponent it is sufficient to work within the Lifshitz IR scaling regime. The irrelevant deformation appears as a source δφ (0) for the scalar field at the Lifshitz boundary (because the mode will be irrelevant and grow towards the UV, the source should be imposed at some small but nonzero cutoff z UV ). This will lead to a linearized perturbation of the bulk fields about the Lifshitz black hole background.
Radial perturbations are easily seen to have the general form where δχ o and δf o are constants and δφ must obey We will focuss on δφ. The constants δχ o and δf o can be chosen to keep either the energy density (sourced by δg tt ) or the temperature constant as we deform by the scalar operator.
This choice does not affect the considerations below.
The scalar equation (7.11) can be solved in terms of Gaussian hypergeometric functions.
Expanding the solution beyond the horizon as z → ∞ we find δφ = 2ε cos π∆ 2 + z L · log z 2+z L + · · · . (7.13) This logarithmic growth towards the singularity amounts to a linearized shift in the Kasner exponent to the value p t = p t (γ) + 32γ ε 8 + 3γ 2 cos π∆γ 2 8 + 3γ 2 . (7.14) The strength ε is given, on dimensional grounds, in terms of the strength δφ (0) of the deformation and the temperature T as For this irrelevant deformation ∆ > 2+z L . Therefore, as we should expect, the shift becomes small as the temperature goes to zero (and hence the perturbative computation is selfconsistent in this limit). That is because the perturbation decays towards the IR in the Lifshitz region outside the horizon. As the horizon goes deeper into the IR, the perturbation on the horizon becomes smaller. Once past the horizon, the perturbation starts to grow logarithmically and shifts the Kasner exponent. This shift of the exponent is therefore smaller at small temperatures as we see in (7.15). The value p t (γ) is achieved in the limit T → 0. Fig. 7  Radial spacelike geodesics in the black hole background can be labelled by a constant 'energy' E. These geodesics fall into the black hole up to a turning point z given by [7,14]:

The Penrose diagram in
Recall that g tt = −f e −χ /z 2 > 0 beyond the horizon. After reaching the turning point, the geodesics emerge on the other side of the Einstein-Rosen bridge. The behavior of the geodesic in the interior depends upon the form of g tt (z). Clearly g tt vanishes on the horizon.
If g tt increases without bound beyond the horizon then z → ∞ as E → ∞. These geodesics can come arbitrarily close to the singularity [7]. However, if g tt has a maximum at some z c beyond the horizon, i.e. with g tt (z c ) = 0, then geodesics anchored at the boundary get 'stuck' at this critical value and do not come closer to the singularity [6].
In the asymptotic Kasner regime g tt ∼ z 1−α 2 ∼ z −4pt/(1−pt) . If p t > 0 then g tt → 0 asymptotically, while if p t < 0 then g tt → ∞. Our charged black holes have p t > 0, and therefore g tt must have a maximum at some intermediate z c . This maximum is visible, for example, in Fig. 4 above. In contrast, neutral black holes deformed by a scalar field source necessarily have p t < 0 and there is no critical radius for real geodesics [14].
It was explained in [6] that if real (as opposed to complex) geodesics get stuck at a critical interior radius z c , then large mass scalar fields in the black hole exterior have an overdamped, non-oscillatory, quasinormal mode. The mode decays as e −Γt with decay rate Γ determined directly from the black hole interior as Γ = M g tt (z c ) . (8.2) mode directly from numerical computation of perturbations in the black hole exterior. More general, oscillating, quasinormal modes are instead related to complex geodesics [6,7,[31][32][33].
In our solutions the maximum of g tt is in between the horizon and the would-be inner horizon, where the ER bridge collapses. Indeed, the maximum exists also for RN-AdS, where g tt vanishes at both horizons and must therefore have a maximum in between. In this sense, we can think of the existence of this maximum (and hence the overdamped mode (8.2)) in our solutions as a remnant of the RN-AdS inner horizon. It is not obvious a priori that the maximum would survive with large boundary deformations, but the fact that p t > 0 implies that it does.

Discussion
We have studied the gravitational dual of the grand canonical ensemble of a CFT deformed by relevant or irrelevant operators. These black hole spacetimes are more generic than the familiar Reissner-Nordström AdS solution, which is the most widely studied dual to the grand canonical ensemble of a CFT. The region of spacetime inside the horizon turns out to be quite different, and has some interesting properties. We have shown that Cauchy horizons never arise for relevant perturbations dual to a bulk scalar with m 2 < 0 (but above the BF bound). Instead, the spacetime ends in a spacelike Kasner singularity. For small deformations, the Kasner phase is preceded by a dramatic collapse of the Einstein-Rosen bridge connecting the two asymptotic regions.
It remains an open question how the experience of an infalling observer is encoded in the dual field theory. Even though we expect the classical description of such an observer to break down near the singularity, we can ensure that quantum and stringy corrections remain small until we are well within the Kasner epoch by taking large N and large coupling in the field theory. By constructing more generic black hole interiors, as we have done, we can start to understand the classical data that is needed to characterize the approach to the singularity. This data -such as the Kasner exponents -must be part of any eventual field theoretic understanding of the fate of infalling observers or of the black hole interior more generally.
We conclude with a few comments that extend some of our results. Firstly, we describe a different setting in which Cauchy horizons can survive scalar field deformations at finetuned values of the parameters. We have seen that Cauchy horizons can exist at a certain discrete set of m 2 > 0 for each temperature. Without changing the scalar potential, these are specific irrelevant deformations which will destroy the asymptotic AdS boundary. It is interesting to note that one can also construct asymptotically AdS solutions with a smooth Cauchy horizon and a simple quadratic potential with negative mass squared, m 2 < 0. This can be achieved with two complex scalar fields φ 1 , φ 2 and a (slightly) inhomogeneous field configuration. Consider the theory (2.1) with two complex scalars with the same m 2 < 0.
Furthermore, it is straightforward to check that the proof of no Cauchy horizons discussed in §3.1 goes through for general dimensions.