There have been persistent suggestions, based on several diverse data sets, that the cosmic expansion is not exactly isotropic. It is not easy to develop a coherent theoretical account of such a “Hubble anisotropy”, for, in standard General Relativity, intuition suggests that it contradicts the predictions of the very successful Inflationary hypothesis. We put this intuition on a firm basis, by proving that if we [a] make use of an Inflationary theory in which Inflation isotropises spatial geometry — this, of course, includes the vast majority of such theories — and if [b] we insist on assuming that spacetime has a strictly metric geometry (one in which the geometry is completely determined by a metric tensor), then indeed all aspects of the “Hubble field” must be isotropic. Conversely, should a Hubble anisotropy be confirmed, then either we must contrive to build anisotropy into Inflation (and into the geometry of space) from the outset, or we will have to accept that spacetime geometry is not strictly metric. We argue that the second option, implemented by allowing spacetime torsion to be non-zero, would be by far the most natural way to accommodate such observations. Such theories can reconcile non-isotropic matter distributions with a perfectly isotropic spatial geometry, and thus are able to reconcile Inflation with possibly observed anisotropies. They also allow us to reconcile the absence of anisotropy in one era (say, that of the CMB) with its presence in another.

Singapore

Brett McInnes

Reconciling inflation with Hubble anisotropies

It has been argued that the rate at which the interior of an AdS black hole evolves is dual to the rate of evolution of the (quantum state of the) strongly coupled matter on the boundary which, according to holography, is dual to the black hole. However, we have shown elsewhere that it seems to be possible, by adjusting the specific angular momentum of an AdS5-Kerr black hole, to reduce this rate to (effectively) zero. We argue that this is unphysical, and that it is prevented by the intervention of a superradiant instability, which causes the black hole to shed angular momentum when the angular velocity exceeds a certain critical value. The precise way in which this works has recently been explained by the “grey galaxy” model of the end state, in which the angular momentum is transferred to a “galactic disc.” Thus, the black hole itself cannot sustain a specific angular momentum beyond a critical value: there is an effective upper bound. The holographic interpretation is that, beyond a certain limiting specific angular momentum, strongly coupled matter (corresponding to the black hole) will spontaneously shed angular momentum to some other, confined, form of matter (corresponding to the disc). This idea is supported by recent numerical work on ultra-vortical plasmas. Such an upper bound on specific angular momentum would prevent arbitrarily small rates of quantum state evolution on the boundary. We give a tentative discussion of the relevant observational data in the case of the vortical Quark-Gluon Plasma, and suggest a way in which such an upper bound might appear in future observations.

Superradiance in the bulk protects quantum state evolution of rapidly rotating matter on the boundary

A “large” AdS black hole can attain equilibrium with its own Hawking radiation, and in that condition it is thought to be dual to a strongly coupled field theory, also at equilibrium. But the interior of the black hole is by no means static: the geometry of spatial sections lying inside the event horizon evolves at some rate. This prompts the obvious question: can this rate possibly have a holographic dual? We present circumstantial evidence that such a dual does exist. We do this by making concrete proposals for two objects: first, a rough measure (already suggested in the literature) of the rate at which the unitary evolution of the state vector describing equilibrated strongly coupled matter evolves; and, second, a rough measure of the rate at which the interior (just under the horizon) evolves (in the case of the AdS$_{5}$-Kerr black hole). We then study how these two very different objects change as two physical parameters describing the exterior of the bulk black hole (the specific angular momentum and the temperature) are varied. We find that they change in remarkably similar ways, as holographic duals should. The results hinge on having a correct identification of the physical mass of rotating AdS black holes, and in fact our results provide very strong support to a recent proposal for that crucial parameter. This may be of general interest in studies of such black holes.

Evidence that the rate of evolution of a black hole interior has a holographic dual

The fact that AdS black hole interior geometries are time-dependent presents two challenges: first, to holographic duality (the boundary matter tends to equilibrate, often very quickly), and, second, to the idea that wormholes can be traversable (the wormhole geometry is dynamic, and the wormhole is apt to collapse too quickly for traversal to be possible). As is well known, the first puzzle can be addressed by considering the quantum circuit complexity of the strongly coupled boundary matter, which can continue to grow long after equilibrium is established. We show that data from a phenomenological model of the Quark-Gluon Plasma indicate the existence of an upper bound on the rate of increase of the (specific) complexity, in agreement with a simple holographic model. We then point out that, in this model, this upper bound becomes stricter if angular momentum is added to the bulk black hole while fixing the temperature (at any value, so the black hole is <italic>not</italic> near-extremal). We show that the dual phenomenon, a dramatic slowing of the black hole interior dynamics at high specific angular momentum, also occurs. We conjecture that sufficiently slow complexification of the field theories dual to rotating black holes is associated with traversability of the bulk wormhole, when quantum effects are taken into account.

Slow complexification

In the standard holographic “dictionary”, the deep infrared of the strongly coupled boundary field theory is studied by examining the bulk region near to the event horizon of a simple AdS-Reissner-Nordström black hole, near to extremality. Recently Horowitz et al. have argued that this is not correct, except in the case of small toroidal black holes, which are therefore revealed to be particularly interesting and important. On the other hand, the Weak Gravity Conjecture postulates that black holes (including toroidal black holes) which are extremely near to extremality spontaneously emit black holes of the same kind. We show that, in the toroidal case, these “emitted” black holes are always small in the sense of Horowitz et al. As an application, we discuss the Grinberg-Maldacena analysis of the way one-point functions, evaluated outside an AdS-Reissner-Nordström black hole, depend on the proper time of fall from the event horizon to the Cauchy horizon. We find that, for emitted toroidal black holes, this dependence effectively drops out.

The special role of toroidal black holes in holography

It is well known that an asymptotically flat four-dimensional Kerr black hole has a smaller (specific) entropy than a Schwarzschild black hole of the same mass. We show here that the same is true if the temperature, rather than the mass, is held fixed; and we also show that an asymptotically AdS5-Kerr black hole has a smaller specific entropy than an AdS5-Schwarzschild black hole of the same temperature, except in a negligibly small class of special examples. The AdS5-Kerr case is particularly interesting, because here the gauge-gravity duality applies; if we further accept that there is a useful analogy between the strongly coupled field theories dual to AdS black holes and the best-understood example of a strongly coupled fluid (the Quark-Gluon Plasma), then we can apply QGP theory to predict the behaviour of black hole entropy in this case. The prediction agrees with our study of AdS5-Kerr entropy. The hope is that such results might lead ultimately to an identification of black hole microstates.

Why is black hole entropy affected by rotation?

Black hole spacetimes asymptotic to five-dimensional anti-de Sitter spacetime are of great interest in connection with the string-gauge duality. In the rotating case, such black holes tend to become unstable, in several different ways, if their specific angular momenta fall in certain ranges. Here we consider the well-known Emparan-Myers fragmentation instability for singly rotating AdS$_{5}$-Kerr black holes, paying particular attention to the case where the specific angular momentum exceeds the asymptotic AdS$_{5}$ curvature length scale.

Fragmentation of AdS -Kerr black holes

One version of the Weak Gravity Conjecture requires that it should be possible for an extremal black hole to emit a smaller black hole: that is, the original black hole bifurcates. For asymptotically flat and asymptotically AdS Reissner-Nordström black holes with spherical event horizons, such a bifurcation reduces the total classical entropy of the system, and so it is apparently forbidden by the second law of thermodynamics. It may well be possible to remedy this by taking other (for example, quantum-gravitational) effects into account, but it is difficult to understand this in a quantitative way. In the case of asymptotically AdS Reissner-Nordström black holes with planar event horizons, however, one can show that bifurcations are definitely compatible with the second law. (Naked singularities, generated by the bifurcation, may play an important role here.) Furthermore, in this case one can exhibit a detailed mechanism explaining precisely why planar black holes must indeed be unstable (through emission of branes) when they are sufficiently close to extremality. Thus planar black holes can improve our understanding of the WGC.

Planar black holes as a route to understanding the Weak Gravity Conjecture

The initial idea underlying the Weak Gravity Conjecture is that extremal black holes must always be “unstable”, in the sense that they should slowly decay by emitting either particles or smaller black holes. Here we show that, when this idea is applied to the planar asymptotically AdS black holes which play a central role in applications of holography, the result, via gauge-gravity duality, is a prediction that there should exist a lower bound on the possible densities of cold strongly coupled matter. Recent observations of neutron stars suggest that, in many cases, even the extreme densities in their cores may not be sufficient to generate quark matter, showing that there is indeed a (very high) lower bound on the possible density of cold quark matter.

Extremal instability for topological black holes

The AdS/CFT correspondence is useful primarily when the number of colours, <math><msub><mrow><mi>N</mi></mrow><mrow><mtext>c</mtext></mrow></msub></math>, characterising the boundary field theory, is “large”, and when the mass of the bulk black hole that is usually present is “large” relative to the bulk Planck mass. But this prompts two questions: first, can these large numbers be estimated, even very approximately, in a given application? Second: if these quantities are themselves computed holographically from physical data constraining the field theory, is this computation self-consistent, in the sense that it actually produces large numbers — an outcome which is far from obvious? Here we consider these questions in the case of the application of holographic techniques to the study of the quark-gluon plasma. We find that holography in this case is able to generate estimates of the dimensionless numbers in question, and, very remarkably, they are indeed large, despite the fact that the dimensionless input data are of order unity.