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Register a new userEnergy in Gravitation and Noethers Theorems
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I exhibit the conflicting roles of Noethers two great theorems in defining conserved quantities, especially Energy in General Relativity and its extensions: It is the breaking of coordinate invariance through boundary conditions that removes the barrier her second theorem otherwise poses to the applicability of her first. There is nothing new here, except the emphasis that General must be broken down to Special Relativity in a special, but physically natural, way in order for the Poincare or other global groups such as (A)dS to re-emerge.
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Taking up four model universes we study the behaviour and contribution of dark energy to the accelerated expansion of the universe, in the modified scale covariant theory of gravitation. Here, it is seen that though this modified theory may be a cause of the accelerated expansion it cannot totally outcast the contribution of dark energy in causing the accelerated expansion. In one case the dark energy is found to be the sole cause of the accelerated expansion. The dark energy contained in these models come out to be of the $Lambda$CDM type and quintessence type comparable to the modern observations. Some of the models originated with a big bang, the dark energy being prevalent inside the universe before the evolution of this era. One of the models predicts big rip singularity, though at a very distant future. It is interestingly found that the interaction between the dark energy and the other part of the universe containing different matters is enticed and enhanced by the gauge function $phi(t)$ here.
With the theory of general relativity, Einstein abolished the interpretation of gravitation as a force and associated it to the curvature of spacetime. Tensorial calculus and differential geometry are the mathematical resources necessary to study the spacetime manifold in the context of Einsteins theory. In 1961, Tullio Regge published a work on which he uses the old idea of triangulation of surfaces aiming the description of curvature, and, therefore, gravitation, through the use of discrete calculus. In this paper, we approach Regge Calculus pedagogically, as well as the main results towards a discretized version of Einsteins theory of gravitation.
We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments either survives with probability $p$ or disappears with probability $1!-!p$. It describes a stochastic dyadic Cantor set that evolves in time, and eventually becomes a fractal. We investigate this phenomenon, through analytical methods and Monte Carlo simulation, for a generic class of models, where segment breakup points follow a symmetric beta distribution with shape parameter $alpha$, which also determines the fragmentation rate. For a fractal dimension $d_f$, we find that the $d_f$-th moment $M_{d_f}$ is a conserved quantity, independent of $p$ and $alpha$. We use the idea of data collapse -- a consequence of dynamical scaling symmetry -- to demonstrate that the system exhibits self-similarity. In an attempt to connect the symmetry with the conserved quantity, we reinterpret the fragmentation equation as the continuity equation of a Euclidean quantum-mechanical system. Surprisingly, the Noether charge corresponding to dynamical scaling is trivial, while $M_{d_f}$ relates to a purely mathematical symmetry: quantum-mechanical phase rotation in Euclidean time.
We propose a new class of vector fields to construct a conserved charge in a general field theory whose energy momentum tensor is covariantly conserved. We show that there always exists such a vector field in a given field theory even without global symmetry. We also argue that the conserved current constructed from the (asymptotically) time-like vector field can be identified with the entropy current of the system. As a piece of evidence we show that the conserved charge defined therefrom satisfies the first law of thermodynamics for an isotropic system with a suitable definition of temperature. We apply our formulation to several gravitational systems such as the expanding universe, Schwarzschild and BTZ black holes, and gravitational plane waves. We confirm the conservation of the proposed entropy density under any homogeneous and isotropic expansion of the universe, the precise reproduction of the Bekenstein-Hawking entropy incorporating the first law of thermodynamics, and the existence of gravitational plane wave carrying no charge, respectively. We also comment on the energy conservation during gravitational collapse in simple models.
A general approach to find out exact cosmological solutions in f(R)-gravity is discussed. Instead of taking into account phenomenological models, we assume, as a physical criterium, the existence of Noether symmetries in the cosmological f(R) Lagrangian. As a result, the presence of such symmetries selects viable models and allow to solve the equations of motion. We discuss also the case in which no Noether charge is present but general criteria can be used to achieve solutions.
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