Scalar curvature

About: Scalar curvature is a research topic. Over the lifetime, 12701 publications have been published within this topic receiving 296040 citations. The topic is also known as: Ricci scalar.

Gauge fields and extrapotentials in constrained quantum systems

Abstract: We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the extrinsic curvature of the constraint manifold, the curvature of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential appears as a gauge potential in the constrained Hamiltonian. This gauge potential is closely related to the geometric phase originally discussed by Berry. The constrained Hamiltonian also contains an effective potential depending on the extrinsic curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The general nature of our analysis allows applications to a wide variety of problems, such as rigid molecules, the evolution of molecular systems along reaction paths, and quantum strip waveguides.

Nearly Round Spheres Look Convex

Abstract: We prove that a Riemannian manifold (M, g), close enough to the round sphere in the C4 topology, has uniformly convex injectivity domains so M appears uniformly convex in any exponential chart. The proof is based on the Ma-Trudinger-Wang nonlocal curvature tensor, which originates from the regularity theory of optimal transport.

Palatini formulation of f( R, T) gravity theory, and its cosmological implications

Abstract: We consider the Palatini formulation of f(R, T) gravity theory, in which a non-minimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similar to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the f(R, T) gravity in the Palatini formulation. Cosmological models with Lagrangians of the type $$f=R-\alpha ^2/R+g(T)$$ and $$f=R+\alpha ^2R^2+g(T)$$ are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.

The AdS/CFT Correspondence Conjecture and Topological Censorship

Abstract: In gr-qc/9902061 it was shown that (n+1)-dimensional asymptotically anti-de-Sitter spacetimes obeying natural causality conditions exhibit topological censorship. We use this fact in this paper to derive in arbitrary dimension relations between the topology of the timelike boundary-at-infinity, $\scri$, and that of the spacetime interior to this boundary. We prove as a simple corollary of topological censorship that any asymptotically anti-de Sitter spacetime with a disconnected boundary-at-infinity necessarily contains black hole horizons which screen the boundary components from each other. This corollary may be viewed as a Lorentzian analog of the Witten and Yau result hep-th/9910245, but is independent of the scalar curvature of $\scri$. Furthermore, the topology of V', the Cauchy surface (as defined for asymptotically anti-de Sitter spacetime with boundary-at-infinity) for regions exterior to event horizons, is constrained by that of $\scri$. In this paper, we prove a generalization of the homology results in gr-qc/9902061 in arbitrary dimension, that H_{n-1}(V;Z)=Z^k where V is the closure of V' and k is the number of boundaries $\Sigma_i$ interior to $\Sigma_0$. As a consequence, V does not contain any wormholes or other compact, non-simply connected topological structures. Finally, for the case of n=2, we show that these constraints and the onto homomorphism of the fundamental groups from which they follow are sufficient to limit the topology of interior of V to either B^2 or $I\times S^1$.