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“It would be permissible to look upon the Hamiltonian form as the fundamental one, and there would then be no fundamental four-dimensional symmetry in the theory. One would have a Hamiltonian built up from four weekly [sic] vanishing functions, given by [the Hamiltonian and diffeomorphism constraints]. The usual requirement of four-dimensional symmetry in physical laws would then get replaced by the requirement that the functions have weakly vanishing P.B.'s, so that they can be provided with arbitrary coefficients in the equations of motion, corresponding to an arbitrary motion of the surface on which the state is defined.” P. A. M. Dirac, in “The theory of gravitation in Hamiltonian form,” Proc. Roy. Soc. A 246 (1958) 333–43.
In its different incarnations, quantum gravity must face a diverse set of fascinating problems and difficulties, a set of issues best seen as both challenges and opportunities. One of the main problems in canonical approaches, for instance, is the issue of anomalies in the gauge algebra underlying space-time covariance. Classically, the gauge generators, given by constraints, have weakly vanishing Poisson brackets with each other: they vanish when the constraints are satisfied. After quantization, the same behavior must be realized for commutators of quantum constraints (or for Poisson brackets of effective constraints), or else the theory becomes inconsistent due to gauge anomalies. If and how canonical quantum gravity can be obtained in an anomaly-free way is an important question, not yet convincingly addressed in full generality.
As we will see throughout this book, Hamiltonian formulations provide important insights, especially for gauge theories such as general relativity with its underlying symmetry principle of general covariance. Canonical structures play a role for a general analysis of the systems of dynamical equations encountered in this setting, for the issue of observables, for the specific types of equation as they occur in cosmology or the physics of black holes, for a numerical investigation of solutions, and, last but not least, for diverse sets of issues forming the basis of quantum gravity.
Several different Hamiltonian formulations of general relativity exist. In his comprehensive analysis, Dirac (1969), based on Dirac (1958a) and Dirac (1958b) and in parallel with Anderson and Bergmann (1951), developed much of the general framework of constrained systems as they are realized for gauge theories. (Earlier versions of Hamiltonian equations for gravity were developed by Pirani and Schild (1950) and Pirani et al. (1953). In many of these papers, the canonical analysis is presented as a mere prelude to canonical quantization. It is now clear that quantum gravity entails much more, as indicated in Chapter 6, but also that a Hamiltonian formulation of gravity has its own merits for classical purposes.) The most widely used canonical formulation in metric variables is named after Arnowitt, Deser and Misner (Arnowitt et al. (1962)) who first undertook the lengthy derivations in coordinate-independent form.
In general relativity, the space-time metric provides the physical field of gravity and is subject to dynamical laws. For a complete and uniform fundamental description of nature, the gravitational force, and thus space-time, is to be quantized by implementing the usual features of quantum states, endowing it with quantum fluctuations and imposing the superposition principle. Only then do we obtain a fully consistent description of nature, since matter as well as the non-gravitational forces are quantum, described by quantum stress-energy which can couple to gravity only via some quantum version of the Einstein tensor.
An implementation of this program requires a clear distinction of the different concepts used in general relativity. One normally works with the line element for metric purposes, but this is a combination of metric tensor components and coordinate differentials (separating events from each other). Only the geometry is dynamical, not the coordinates. After quantization, we may have a representation for geometrical observables such as the sizes of physically characterized regions, but not for coordinates or distances between mathematical points as mere auxiliary ingredients. Metrics or other tensors may arise in an effective form from quantum gravity, but they are not the basic object. One has to dig deeper, similarly to hydrodynamics where the continuous fluid flow is not suitable for a fundamental quantum theory, which must, rather, be based on an atomic picture.
Canonical methods are a powerful mathematical tool within the field of gravitational research, both theoretical and experimental, and have contributed to a number of recent developments in physics. Providing mathematical foundations as well as physical applications, this is the first systematic explanation of canonical methods in gravity. The book discusses the mathematical and geometrical notions underlying canonical tools, highlighting their applications in all aspects of gravitational research from advanced mathematical foundations to modern applications in cosmology and black hole physics. The main canonical formulations, including the Arnowitt-Deser-Misner (ADM) formalism and Ashtekar variables, are derived and discussed. Ideal for both graduate students and researchers, this book provides a link between standard introductions to general relativity and advanced expositions of black hole physics, theoretical cosmology or quantum gravity.
Hyperbolicity, as verified for general relativity in Chapter 3.4.2, guarantees the existence of local solutions in terms of initial data, but not the existence of global ones at all times. When evolving for long time intervals, singularities can develop in the solution and prevent it from being extendable further. We have already seen examples in homogeneous solutions of Bianchi models and the simpler isotropic solutions of FLRW models. In these cases, for matter satisfying the strong energy condition, there were always reasonable initial values which led to solutions with a diverging Hubble parameter and expansion rate at some time in the future or the past. The Hubble parameter corresponds to the expansion of the family of timelike geodesics followed by comoving Eulerian observers, a concept which presents a useful perspective in the context of singularities. When this expansion parameter diverges, the geodesic family no longer defines a smooth submanifold of space-time but develops a caustic or focal point where, in a homogeneous geometry, all geodesics intersect. For every spatial point on a non-singular slice, there is exactly one such geodesic intersecting it; when all these geodesics simultaneously intersect at the singular slice, it means that the whole space collapses into a single point after a finite amount of proper time. At this time, the initial value problem breaks down and space-time cannot be extended further.