General Relativity & Quantum Cosmology

I will discuss the contribution to black hole thermodynamics from a variation in the cosmological constant. The description of black hole with a cosmological constant is facilitated by introducing a two-form potential for the static Killing field. The resulting Smarr formula then includes a term proportional to the cosmological constant times an effective volume, which arises as the difference between the Killing potential on the horizon and the boundary at infinity. This volume is shown to be equal to the difference between the (infinite) volume of AdS and the (infinite) volume outside the black hole horizon of AdS containing a black hole--and so can be interpreted as the volume occupied by the black hole. I will outline the derivation for the first law for AdS black holes including a variation in the cosmological constant. This yields a new work term, the change in the cosmological constant times the effective volume. Hence this is analogous to a "volume times change in pressure" work term in classical thermodynamics. This suggests that the usual change in mass term is better interpreted as a change in the enthalpy, the mass plus the pressure times the volume. In the AdS/CFT correspondence a change in the cosmological constant corresponds to a change in the t'Hooft coupling, for example, a change in the number of degrees of freedom. The effective volume multiplier then looks like a chemical potential. Members of the audience will be asked to contribute their own interpretations at this point.

As a necessary step towards the extraction of realistic results from Loop Quantum Cosmology, we analyze the physical consequences of including inhomogeneities. We consider a gravitational model in vacuo which possesses local degrees of freedom, namely, the linearly polarized Gowdy cosmologies. We carry out a hybrid quantization which combines loop and Fock techniques. We discuss the main results of this hybrid quantization, which include the resolution of the cosmological singularity, the construction of the Hilbert space of physical states, and the recovery of a conventional quantization for the inhomogeneities. In addition, an analysis of the model at the effective level confirms the robustness of the Big Bounce scenario, with preservation -or partial amplification- of the amplitudes of the inhomogeneous modes through the bounce in a statistical average.

I will give account of a work in progress in which I attempt to modify the metric-manifold structure of GR in the infra-red. The proposed modification does not contain any massive parameter as it is effective at length scales comparable with the inverse (extrinsic) curvature. The guiding line for this modification is an "ultra-strong" equivalence principle, according to which even semi-classical gravitational effects (i.e. particle production) are definitely banned from a sufficiently small free-falling elevator. Some cosmological consequences of this modification will be discussed.

The currently accelerating Hubble expansion is in accord with the old heuristic prediction, from causal set theory, of a fluctuating and ever-present cosmological "constant''. More recently, a phenomenological model based on certain of the ideas behind the prediction has been devised, but it remains incomplete. I will review these developments and also mention a possible consequence for the dimensionality of spacetime.

Normalizing the Einstein-Hilbert action by the volume functional makes the theory invariant under constant shifts in the Lagrangian. The field equations then resemble unimodular gravity whose otherwise arbitrary cosmological constant is now determined as a Machian universal average. We first prove that an empty space-time is necessarily flat, and then demonstrate, by invoking the scalar field paradigm, that the cosmological constant is actually zero. Normalized general relativistic cosmology is then discussed at the mini-superspace level, where we confirm the generic zero cosmological constant result, and furthermore prove that if matter is attractive then the Universe cannot be closed. The only non-vanishing cosmological constant solutions are associated with static Einstein-like closed universes.