Abstract
We give a novel definition of gravitational energy for an arbitrary theory of gravity including quadratic-curvature corrections to Einstein equations. We focus on the theory in four dimensions, in presence of negative cosmological constant, and with asymptotically anti-de Sitter (AdS) boundary conditions. As a first example, we compute the gravitational energy and angular momentum of Schwarzschild-AdS black holes, for which we obtain results consistent with previous computations performed using different methods. However, our method differs qualitatively from other ones in the feature of being intrinsically non-linear. It relies on the idea of adding to the gravity action topological invariant terms which suffice to regularize the Noether charges and render the variational problem well-posed. This is an idea that has been previously considered in the case of second-order theories, such as general relativity and which, as shown here, extends to higher-derivative theories. Besides black holes, we consider other solutions, such as gravitational waves in AdS, for which we also find results in agreement. This enables us to investigate the consistency of this approach in the non-Einstein sector of the theory.
Highlights
Topology plays a fundamental role in theoretical physics, from the theory of fundamental particles to the study of topological phases of matter
It is a central notion in the study of dynamical systems and, from there, it extends to practically every area of physics
In the case of the theory of gravity, being a theory of the geometry of the spacetime itself, the role played by topological invariants is important
Summary
Topology plays a fundamental role in theoretical physics, from the theory of fundamental particles to the study of topological phases of matter. Following an idea previously explored by one of the authors and collaborators in the case of second-order theories such as general relativity [1], we propose a novel use of topological invariants in the context of gravity, namely, as regulators of the Noether-charge computation in theories that include higher-derivative terms This is a very important problem, as higher-derivative corrections to Einstein’s theory naturally arise when quantum effects are taken into account, and having a method to compute the observables in those cases is crucial. For theories of this sort, we show that the Euler characteristic—which in virtue of the ChernWeil-Gauss-Bonnet theorem is expressed in terms of the Riemann tensor—can be used to regularize the infrared divergences that typically appear when trying to compute the conserved Noether charges in asymptotically, locally maximally symmetric spaces. Having such a nonlinear method to compute charges in theories with actions that are quadratic in the curvature tensor and in the presence of a cosmological constant is very important, as such theories are known to suffer from linear instabilities that cause the linear analysis to break down
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