For a contravariant 4-metric which changes signature from Lorentzian to Riemannian across a spatial hypersurface, the mixed Einstein tensor is manifestly nonsingular. In Gaussian normal coordinates, the metric contains a step function and the Einstein tensor contains the Dirac delta function with support at the junction. The coefficient of the Dirac function is a linear combination of the secon...

We consider the quasi-de Sitter geometry of the inflationary universe. We calculate the energy flux of the slowly rolling background scalar field through the quasi-de Sitter apparent horizon and set it equal to the change of the entropy (1/4 of the area) multiplied by the temperature, dE = TdS. Remarkably, this thermodynamic law reproduces the Friedmann equation for the rolling scalar field. Th...

We show that a Minkowski phase space endowed with bracket relatively to conformable differential realizes Poisson algebra, confering bi-Hamiltonian structure the resulting manifold. infer related Hamiltonian vector field is an infinitesimal Noether symmetry, and compute corresponding deformed recursion operator. Besides, using Hamiltonian–Jacobi separability, we construct operators for fields i...

The aim of this paper is to characterize $3$-dimensional $N(k)$-paracontact metric manifolds satisfying certain curvature conditions. We prove that a $3$-dimensional $N(k)$-paracontact metric manifold $M$ admits a Ricci soliton whose potential vector field is the Reeb vector field $xi$ if and only if the manifold is a paraSasaki-Einstein manifold. Several consequences of this result are discuss...

Brane-world models, where observers are restricted to a brane in a higher dimensional spacetime, offer a novel perspective on cosmology. I discuss some approaches to cosmology in extra dimensions and some interesting aspects of gravity and cosmology in brane-world models. COSMOLOGY AFTER EINSTEIN A century after Einstein first proposed his theory of relativity, it has become a cornerstone of th...

We investigated the effects of bulk scalar field in the braneworld cosmological scenario. The Friedmann equations and acceleration condition in presence of the bulk scalar field for a zero tension brane and cosmological constant are studied. In DGP model the effective Einstein equation on the brane is obtained with bulk scalar field. The rescaled bulk scalar field on the brane in the DGP model ...

A link between the semiclassical Einstein equation and a maximal vacuum entanglement hypothesis is established. The hypothesis asserts that entanglement entropy in small geodesic balls is maximized at fixed volume in a locally maximally symmetric vacuum state of geometry and quantum fields. A qualitative argument suggests that the Einstein equation implies the validity of the hypothesis. A more...

We study the spectrum of the scalar Laplacian on the five-dimensional toric Sasaki-Einstein manifolds Y . The eigenvalue equation reduces to Heun’s equation, which is a Fuchsian equation with four regular singularities. We show that the ground states, which are given by constant solutions of Heun’s equation, are identified with BPS states corresponding to the chiral primary operators in the dua...

We study the Einstein-Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold M admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics η on M × R such that (M × R, η) admit Einstein spinors (resp. weak Killing spinors). To prove the result we split the Einstein-Dirac equ...