Gauge × gauge = gravity on homogeneous spaces using tensor convolutions

A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of ‘gravity = gauge × gauge’. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the ‘gauge × gauge’ convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.

When Painlevé–Gullstrand coordinates fail

Painlevé–Gullstrand coordinates, a very useful tool in spherical horizon thermodynamics, fail in anti-de Sitter space and in the inner region of Reissner–Nordström. We predict this breakdown to occur in any region containing negative Misner–Sharp–Hernandez quasilocal mass because of repulsive gravity stopping the motion of PG observers, which are in radial free fall with zero initial velocity. PG coordinates break down also in the static Einstein universe for completely different reasons. The more general Martel-Poisson family of charts, which normally has PG coordinates as a limit, is reported for static cosmologies (de Sitter, anti-de Sitter and the static Einstein universe).

STATIC EINSTEIN'S UNIVERSE AS A QUANTUM SOLUTION OF CAUSAL QUANTUM GRAVITY

We shall in the framework of Bohmian quantum gravity show that it is possible to find a pure quantum state which leads to the static Einstein universe whose classical counterpart is flat space–time. We obtain the solution in the long-wavelength approximation. At the end an exact solution is found.

DYNAMICAL MASS GENERATION AND CRITICAL CURVATURE IN THE O(N) NONLINEAR SIGMA MODEL

The large-N nonlinear O(N) sigma model with the curvature coupled term ξRn2 is examined on a spacetime of R1×S2 topology (three-dimensional static Einstein universe). Making use of the cutoff method, we find the renormalized effective potential which shows that, for ξ>1/8, there is a second-order phase transition. Above the critical curvature, the dynamical mass generation does not take place even in the strong-coupled regime. The phase structure of the model on S2 is also discussed.