Luca Bombelli

Testing general relativity with present and future astrophysical observations

One century after its formulation, Einsteins general relativity (GR) has made remarkable predictions and turned out to be compatible with all experimental tests. Most of these tests probe the theory in the weak-field regime, and there are theoretical and experimental reasons to believe that GR should be modified when gravitational fields are strong and spacetime curvature is large. The best astrophysical laboratories to probe strong-field gravity are black holes and neutron stars, whether isolated or in binary systems. We review the motivations to consider extensions of GR. We present a (necessarily incomplete) catalog of modified theories of gravity for which strong-field predictions have been computed and contrasted to Einsteins theory, and we summarize our current understanding of the structure and dynamics of compact objects in these theories. We discuss current bounds on modified gravity from binary pulsar and cosmological observations, and we highlight the potential of future gravitational wave measurements to inform us on the behavior of gravity in the strong-field regime.

THE COVARIANT PHASE SPACE OF ASYMPTOTICALLY FLAT GRAVITATIONAL FIELDS

Publisher Summary This chapter discusses the covariant phase space of asymptotically flat gravitational fields and covariant constructions for field theories. In the case of general relativity, boundary conditions play a critical role and must be adjusted carefully for the symplectic structure to be finite and for the framework to be well-defined. The chapter presents a new application: the derivation of the expression of energy-momentum of an isolated gravitating system at null infinity. This derivation makes a crucial use of the covariant construction and cannot be carried out within the familiar, 3+1 phase space frameworks. The chapter presents a summarization of the basic ideas of the covariant procedure and the general framework for field theories on a background space-time. It reviews the covariant Hamiltonian description of gravitational fields in general relativity, which are asymptotically flat at spatial infinity. The chapter shows that the ADM 4-momentum is the generator of the asymptotic translation group, which arises from the boundary conditions. The chapter also discusses space-times that are asymptotically flat at null infinity.

Semiclassical states for constrained systems

The notion of semiclassical states is first sharpened by clarifying two issues that appear to have been overlooked in the literature. Systems with linear and quadratic constraints are then considered and the group averaging procedure is applied to kinematical coherent states to obtain physical semiclassical states. In the specific examples considered, the technique turns out to be surprisingly efficient, suggesting that it may well be possible to use kinematical structures to analyze the semiclassical behavior of physical states of an interesting class of constrained systems.

The Origin of Lorentzian Geometry

Abstract We investigate how a Lorentzian manifold can be reconstructed from a more fundamental structure, in the context of a proposal for a quantum theory of gravity. We first show that some collections of countable sets of points of the manifold, together with their causal relations, contain all the information of the geometrical structure of the manifold. We theb define a notion of closeness between Lorentzian metrics, and formulate a conjecture regarding how a Lorentzian manifold can be approximated by a locally finite subset of points.

Chaos around a black hole

The authors apply the Melnikov method for identifying chaos in near integrable systems to relativistic particle motion around a Schwarzschild black hole. They start by giving a self-contained introduction to the Melnikov method together with some relevant background on dynamical systems. Then they show that a relativistic particle was unstable circular orbits around a Schwarzschild black hole, and that each one of these gives rise to a homoclinic orbit in phase space, which tends to the unstable one for t to +or- infinity . Finally, the authors use the Melnikov method to conclude that, under most periodic perturbations of the black-hole metric, the homoclinic orbit becomes chaotic.

Statistical geometry of random weave states

I describe the first steps in the construction of semiclassical states for non-perturbative canonical quantum gravity using ideas from classical, Riemannian statistical geometry and results from quantum geometry of spin network states. In particular, I concentrate on how those techniques are applied to the construction of random spin networks, and the calculation of their contribution to areas and volumes.

Semiclassical Quantum Gravity: Statistics of Combinatorial Riemannian Geometries

This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at “quantum scales” and continuum, classical geometries at large scales. Such a correspondence can be meaningfully established when one has a “semiclassical” state in the underlying quantum gravity theory, and the uncertainties in the correspondence arise both from quantum fluctuations in this state and from the kinematical procedure of matching a smooth geometry to a discrete one. We focus on the latter type of uncertainty, and suggest the use of statistical geometry as a way to quantify it. With a cell complex as an example of discrete structure, we discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. We also comment briefly on how to combine our results with uncertainties in the underlying quantum state, and on their use when considering phenomenological aspects of quantum gravity.

Statistical Lorentzian geometry and the closeness of Lorentzian manifolds

I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encoded in the partial order that the causal structure induces among those points; one can then define closeness between Lorentzian geometries by comparing the sets of probabilities they give for obtaining the same posets. If the density of points is finite, one gets a pseudo-distance, which only compares the manifolds down to a finite volume scale, as illustrated here by a fully worked out example of two two-dimensional manifolds of different topology; if the density is allowed to become infinite, a true distance can be defined on the space of all Lorentzian geometries. The introductory and concluding sections include some remarks on the motivation for this definition and its applications to quantum gravity.

Gravity and Matter in Causal Set Theory

The goal of this paper is to propose an approach to the formulation of dynamics for causal sets and coupled matter fields. We start from the continuum version of the action for a Klein–Gordon field coupled to gravity, and rewrite it first using quantities that have a direct correspondent in the case of a causal set, namely volumes, causal relations and timelike lengths, as variables to describe the geometry. In this step, the local Lagrangian density for a set of fields f is recast into a quasilocal expression that depends on pairs of causally related points pq and is a function of the values of f in the Alexandrov set defined by those points, and whose limit as p and q approach a common point x is . We then describe how to discretize and use it to define a causal-set-based action.

On energy in 5-dimensional gravity and the mass of the Kaluza-Klein monopole

Abstract We discuss the concept of energy in higher-dimensional gravity, with special attention given to the problem of the choice of a background. Three different approaches to the calculation of energy for solutions of the 5-dimensional Einstein equation are considered. They are then shown to be equivalent and applied to the calculation of the mass of the Kaluza-Klein monopole. The question of the reduction of a 5-dimensional theory to an effective 4-dimensional one in the presence of a Killing vector field along the compact dimension is then discussed, in particular with regard to the conformal ambiguity in the definition of the metric in the reduced theory. We show that there exists an essentially unique choice of reduction for which the 4-dimensional energy has the usual general relativistic form, and that for the Kaluza-Klein monopole this is the reduction that will make the “naive” 4-dimensional energy additive.