Connections between Weyl geometry, quantum potential and quantum entanglement

The Weyl geometry promises potential applications in gravity and quantum mechanics. We study the relationships between the Weyl geometry, quantum entropy and quantum entanglement based on the Weyl geometry endowing the Euclidean metric. We give the formulation of the Weyl Ricci curvature and Weyl scalar curvature in the n-dimensional system. The Weyl scalar field plays a bridge role to connect the Weyl scalar curvature, quantum potential and quantum entanglement. We also give the Einstein–Weyl tensor and the generalized field equation in 3D vacuum case, which reveals the relationship between Weyl geometry and quantum potential. Particularly, we find that the correspondence between the Weyl scalar curvature and quantum potential is dimension-dependent and works only for the 3D space, which reveals a clue to quantize gravity and an understanding why our space must be 3D if quantum gravity is compatible with quantum mechanics. We analyze numerically a typical example of two orthogonal oscillators to reveal the relationships between the Weyl scalar curvature, quantum potential and quantum entanglement based on this formulation. We find that the Weyl scalar curvature shows a negative dip peak for separate state but becomes a positive peak for the entangled state near original point region, which can be regarded as a geometric signal to detect quantum entanglement.

Gravitational Decoupling in Higher Order Theories

Gravitational decoupling via the Minimal Geometric Deformation (MGD) approach has been used extensively in General Relativity (GR), mainly as a simple method for generating exact anisotropic solutions from perfect fluid seed solutions. Recently this method has also been used to generate exact spherically symmetric solutions of the Einstein-scalar system from the Schwarzschild vacuum metric. This was then used to investigate the effect of scalar fields on the Schwarzschild black hole solution. We show that this method can be extended to higher order theories. In particular, we consider fourth order Einstein–Weyl gravity, and in this case by using the Schwarzschild metric as a seed solution to the associated vacuum field equations, we apply the MGD method to generate a solution to the Einstein–Weyl scalar theory representing a hairy black hole solution. This solution is expressed in terms of a series using the Homotopy Analysis Method (HAM).

Gravitational and Electromagnetic Perturbations of a Charged Black Hole in a General Gauge Condition

We derive a set of coupled equations for the gravitational and electromagnetic perturbation in the Reissner–Nordström geometry using the Newman–Penrose formalism. We show that the information of the physical gravitational signal is contained in the Weyl scalar function Ψ4, as is well known, but for the electromagnetic signal, the information is encoded in the function χ, which relates the perturbations of the radiative Maxwell scalars φ2 and the Weyl scalar Ψ3. In deriving the perturbation equations, we do not impose any gauge condition and as a limiting case, our analysis contains previously obtained results, for instance, those from Chandrashekhar’s book. In our analysis, we also include the sources for the perturbations and focus on a dust-like charged fluid distribution falling radially into the black hole. Finally, by writing the functions on the basis of spin-weighted spherical harmonics and the Reissner–Nordström spacetime in Kerr–Schild type coordinates, a hyperbolic system of coupled partial differential equations is presented and numerically solved. In this way, we completely solve a system that generates a gravitational signal as well as an electromagnetic/gravitational one, which sets the basis to find correlations between them and thus facilitates gravitational wave detection via electromagnetic signals.

Dynamics of perfect fluid collapse in f(𝒢,T) gravity

This paper investigates the dynamics of perfect fluid spherical collapse in curvature-matter coupled gravity. Using Darmois junction conditions, we derive smooth matching of both interior and exterior regions. The dynamical equations are formulated through Misner–Sharp formalism that give the collapse rate for both general and constant curvature terms. Finally, we formulate a relationship between correction terms, Weyl scalar and matter variables. For constant value of [Formula: see text], it is found that the metric is conformally flat if and only if energy density of the collapsing system is homogeneous while the positive correction terms decrease the collapse rate.

Variations in the expansion and shear scalars for dissipative fluids

This work is devoted to the study of some dynamical features of spherical relativistic locally anisotropic stellar geometry in f(R) gravity. In this paper, a specific configuration of tanh f(R) cosmic model has been taken into account. The mass function through technique introduced by Misner–Sharp has been formulated and with the help of it, various fruitful relations are derived. After orthogonal decomposition of the Riemann tensor, the tanh modified structure scalars are calculated. The role of these tanh modified structure scalars (MSS) has been discussed through shear, expansion as well as Weyl scalar differential equations. The inhomogeneity factor has also been explored for the case of radiating viscous locally anisotropic spherical system and spherical dust cloud with and without constant Ricci scalar corrections.

Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton–Jacobi theory

Classical field theory is considered as a theory of unparametrized surfaces embedded in a configuration space, which accommodates, in a symmetric way, spacetime positions and field values. Dynamics is defined by a (Hamiltonian) constraint between multivector-valued generalized momenta, and points in the configuration space. Starting from a variational principle, we derive local equations of motion, that is, differential equations that determine classical surfaces and momenta. A local Hamilton–Jacobi equation applicable in the field theory then follows readily. The general method is illustrated with three examples: non-relativistic Hamiltonian mechanics, De Donder–Weyl scalar field theory, and string theory.

Pure geometric branes and mass hierarchy

We consider a toy model with flat thin branes embedded in a five-dimensional Weyl integrable manifold, where the geometric Weyl scalar provides the material that constitute the brane configurations. The brane configuration is similar to the Randall–Sundrum model. However, it is found that the massless graviton is localized on the brane with negative tension. So, in order to solve the gauge hierarchy problem, our world should be confined on the positive tension brane, and this is crucial to reproduce a correct Friedmann-like equation on the brane. The spacings of graviton mass spectrum are very tiny, but these massive gravitons are hidden in low energy experiments because they are weakly coupled with matter on the brane.

NONUNIFORM BRANEWORLD STARS: AN EXACT SOLUTION

In this paper the first exact interior solution to Einstein's field equations for a static and nonuniform braneworld star with local and nonlocal bulk terms is presented. It is shown that the bulk Weyl scalar [Formula: see text] is always negative inside the stellar distribution, and in consequence it reduces both the effective density and the effective pressure. It is found that the anisotropy generated by bulk gravity effect has an acceptable physical behavior inside the distribution. Using a Reissner–Nördstrom-like exterior solution, the effects of bulk gravity on pressure and density are found through matching conditions.

ON WEYL COSMOLOGY IN FIVE DIMENSIONS AND THE COSMOLOGICAL CONSTANT

In this talk notes we expose the possibility to induce the cosmological constant from extra dimensions from a geometrical framework where our four-dimensional Riemannian spacetime is embedded into a five-dimensional Weyl integrable space. In particular following the approach of the induced matter theory (IMT) we show that when we go down from five to four dimensions we may recover the induced energy momentum tensor of the IMT plus a cosmological constant term that is determined by the presence of the Weyl scalar field on the bulk.

On the dispersion of cylindrical impulsive gravitational waves

An exact solution, describing the dispersion of a wave packet of gravita­tional radiation, having initially (at time t = 0) an impulsive character, is analysed. The impulsive character of the wave-packet derives from the space-time being flat, except at a radial distance ϖ = ϖ 1 (say) at t = 0, and the time-derivative of the Weyl scalars exhibiting δ-function singu­larities at ϖ = ϖ 1 , when t → 0. The principal feature of the dispersion is the development of a singularity of the metric function, v , and of the Weyl scalar, ψ 2 , when the wave, after reflection at the centre, collides with the still incoming waves. The evolution of the metric functions and of the Weyl scalars, as the dispersion progresses, is illustrated graphically.