Einstein–Hilbert action

Abstract: A general ansatz for gravitational entropy can be provided using the criterion that any patch of area which acts as a horizon for a suitably defined accelerated observer must have an entropy proportional to its area. After providing a brief justification for this ansatz, several consequences are derived. (i) In any static spacetime with a horizon and associated temperature β−1, this entropy satisfies the relation S = (1/2)βE where E is the energy source for gravitational acceleration, obtained as an integral of (Tab − (1/2)Tgab)uaub. (ii) With this ansatz of S, the minimization of Einstein–Hilbert action is equivalent to minimizing the free energy F with βF = βU − S where U is the integral of Tabuaub. We discuss the conditions under which these results imply S ∝ E2 and/or S ∝ U2 thereby generalizing the results known for black holes. This approach links with several other known results, especially the holographic views of spacetime.

Abstract: Constructing a well-posed variational principle is a non-trivial issue in general relativity. For spacelike and timelike boundaries, one knows that the addition of the Gibbons–Hawking–York (GHY) counter-term will make the variational principle well-defined. This result, however, does not directly generalize to null boundaries on which the 3-metric becomes degenerate. In this work, we address the following question: What is the counter-term that may be added on a null boundary to make the variational principle well-defined? We propose the boundary integral of $$2 \sqrt{-g} \left( \Theta +\kappa \right) $$ as an appropriate counter-term for a null boundary. We also conduct a preliminary analysis of the variations of the metric on the null boundary and conclude that isolating the degrees of freedom that may be fixed for a well-posed variational principle requires a deeper investigation.

Abstract: We study spherically symmetric solutions in f(R) theories and its compatibility with local tests of gravity. We start by clarifying the range of validity of the weak field expansion and show that, for many models proposed to address the dark energy problem, this expansion breaks down in realistic situations. This invalidates the conclusions of several papers that make inappropriate use of this expansion. For the stable models that modify gravity only at small curvatures we find that when the asymptotic background curvature is large we approximately recover the solutions of Einstein gravity through the so-called Chameleon mechanism, as a result of the non-linear dynamics of the extra scalar degree of freedom contained in the metric. In these models one would observe a transition from Einstein to scalar-tensor gravity as the Universe expands and the background curvature diminishes. Assuming an adiabatic evolution we estimate the redshift at which this transition would take place for a source with given mass and radius. We also show that models of dynamical dark energy claimed to be compatible with tests of gravity because the mass of the scalar is large in vacuum (e.g. those that also include R2 corrections in the action) are not viable.

Abstract: We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator $D$ on an $n$ dimensional compact Riemannian manifold with $n\geq 4$, $n$ even, the Wodzicki residue Res$(D^{-n+2})$ is the integral of the second coefficient of the heat kernel expansion of $D^{2}$. We use this result to derive a gravity action for commutative geometry which is the usual Einstein Hilbert action and we also apply our results to a non-commutative extension which, is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.