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Register a new userIntroduction to Regge Calculus for Gravitation
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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.
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In Regge calculus the space-time manifold is approximated by certain abstract simplicial complex, called a pseudo-manifold, and the metric is approximated by an assignment of a length to each 1-simplex. In this paper for each pseudomanifold we construct a smooth manifold which we call a manifold with defects. This manifold emerges from the purely combinatorial simplicial complex as a result of gluing geometric realizations of its n-simplices followed by removing the simplices of dimension n-2. The Regge geometry is encoded in a boundary data of a BF-theory on this manifold. We construct an action functional which coincides with the standard BF action for suitably regular manifolds with defects and fields. On the other hand, the action evaluated at solutions of the field equations satisfying certain boundary conditions coincides with an evaluation of the Regge action at Regge geometries defined by the boundary data. As a result we trade the degrees of freedom of Regge calculus for discrete degrees of freedom of topological BF theory.
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.
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.
A summary is given of recent exact results concerning the functional integration measure in Regge gravity.
We discuss the evolution of the universe in the context of the second law of thermodynamics from its early stages to the far future. Cosmological observations suggest that most matter and radiation will be absorbed by the cosmological horizon. On the local scale, the matter that is not ejected from our supercluster will collapse to a supermassive black hole and then slowly evaporate. The history of the universe is that of an approach to the equilibrium state of the gravitational field.
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