f(R,T)-Gravity Model with Perfect Fluid Admitting Einstein Solitons

f(R,T)-gravity is a generalization of Einstein’s field equations (EFEs) and f(R)-gravity. In this research article, we demonstrate the virtues of the f(R,T)-gravity model with Einstein solitons (ES) and gradient Einstein solitons (GES). We acquire the equation of state of f(R,T)-gravity, provided the matter of f(R,T)-gravity is perfect fluid. In this series, we give a clue to determine pressure and density in radiation and phantom barrier era, respectively. It is proved that if a f(R,T)-gravity filled with perfect fluid admits an Einstein soliton (g,ρ,λ) and the Einstein soliton vector field ρ of (g,ρ,λ) is Killing, then the scalar curvature is constant and the Ricci tensor is proportional to the metric tensor. We also establish the Liouville’s equation in the f(R,T)-gravity model. Next, we prove that if a f(R,T)-gravity filled with perfect fluid admits a gradient Einstein soliton, then the potential function of gradient Einstein soliton satisfies Poisson equation. We also establish some physical properties of the f(R,T)-gravity model together with gradient Einstein soliton.

An Energy Conservative hp-method for Liouville’s Equation of Geometrical Optics

Liouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation. When dealing with optical interfaces in phase space, non-local boundary conditions arise. Besides being a difficulty in itself, these non-local boundary conditions must also satisfy energy conservation constraints. To this end, we introduce an energy conservative treatment of optical interfaces. Numerical experiments are performed to prove that the method obeys energy conservation. Furthermore, the method is compared to the industry standard ray tracing. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing by reducing the computation time by up to three orders of magnitude for an error of $$10^{-6}$$ 10 - 6 .

Hyperbolic three-string vertex

We begin developing tools to compute off-shell string amplitudes with the recently proposed hyperbolic string vertices of Costello and Zwiebach. Exploiting the relation between a boundary value problem for Liouville’s equation and a monodromy problem for a Fuchsian equation, we construct the local coordinates around the punctures for the generalized hyperbolic three-string vertex and investigate their various limits. This vertex corresponds to the general pants diagram with three boundary geodesics of unequal lengths. We derive the conservation laws associated with such vertex and perform sample computations. We note the relevance of our construction to the calculations of the higher-order string vertices using the pants decomposition of hyperbolic Riemann surfaces.

An energy conservative hp-scheme for light propagation using Liouville’s equation for geometrical optics

In this contribution an alternative method to standard forward ray-tracing is briefly outlined. The method is based on a phase-space description of light propagating through an optical system. The propagation of light rays are governed by Hamilton’s equations. Conservation of energy and étendue for a beam of light, allow us to derive a Liouville’s equation for the energy propagation through an optical system. Liouville’s equation is solved numerically using an hp-adaptive scheme, which for a smooth refractive index field is energy conservative. A proper treatment of optical interfaces ensures that the scheme is energy conservative over the full domain.

Quantum Mechanics and Liouville's Equation

In non-relativistic quantum mechanics, the absolute square of Schrödinger's wave function for a particle in a potential determines the probability of finding it either at a position or momentum at a given time. In classical mechanics the corresponding problem is determined by the solution of Liouville's equation for the probability density of finding the joint position and momentum of the particle at a given time. Integrating this classical solution over either one of these two variables can then be compared with the probability in quantum mechanics. For the special case that the force is a constant, it is shown analytically that for an initial Gaussian probability distribution, the solution of Liouville's integrated over momentum is equal to Schrödinger's probability function in coordinate space, provided the coordinate and momentum initial widths of this classical solution satisfy the minimal Heisenberg uncertainty relation. Likewise, integrating Lioville's solution over position is equal to Schrödinger's probability function in momentum space.Quanta 2017; 6: 53–56.