Reflections on the U(1) problem in general relativity

After a brief introduction to classical relativity, we describe how to solve the Cauchy problem in general relativity. In particular, we introduce the notion of gauge source functions and explain how they can be used in order to reduce the problem to that of solving a system of hyperbolic partial differential equations. We then go on to explain how the initial value problem is formulated for the so-called Einstein-Vlasov system and describe a recent future global non-linear stability result in this setting. In particular, this result applies to models of the universe which are consistent with observations.

Tensor calculus serves as the mathematical framework underpinning the theory of General Relativity, bridging the gap between abstract mathematical concepts and their physical applications. This paper explores the fundamental principles of tensor calculus, its critical role in formulating Einstein's field equations, and its impact on our understanding of spacetime and gravity. We delve into key tensor operations, including differentiation and contraction, and examine their applications in solving problems in general relativity. By highlighting both historical developments and modern advancements, this study underscores the profound connection between tensor calculus and theoretical physics, demonstrating its significance in describing the universe's structure and dynamics.

This chapter contains a description of the Cauchy problem in the general theory of relativity. The intended audience consists of physicists with basic knowledge of general relativity. Beyond the formulation, there is a discussion of ways of measuring the distance between initial data sets. Finally the notions of linear and nonlinear stability are compared.

The global existence problem in general relativity does not reduce toa global existence theorem for a solution of the Einstein equations withsome choice of time coordinate. The physical problem is the existenceof spacetimes for an infinite proper time. But proper time depends on theobserver, i.e. on the timelike line along which it is observed. The singularitytheorems of general relativity consider as a singularity the timelike or nullgeodesic incompleteness of a spacetime and are established under variousgeometric hypotheses.On the other hand, existence theorems for a solution of generic Cauchyproblems for the Einstein equations with an infinite proper time of existencefor a family of observers have been obtained but their timelike or null

The notion of mass or energy has been one of the most challenging problems in general relativity since Einstein's time. As is well known from the equivalence principle, there is no well-defined concept of energy density for gravitation. On the other hand, when there is asymptotic symmetry, concepts of total energy and momentum can be defined. This is the ADM energy-momentum and the Bondi energy-momentum when the system is viewed from spatial infinity and null infinity, respectively. These concepts are fundamental in general relativity but there are limitations to such definitions if the physical system is not isolated and cannot quite be viewed from infinity where asymptotic symmetry exists.The positive energy conjecture states that the total energy of a nontrivial isolated physical system must be positive. This conjecture lies in the foundation of general relativity upon which stability of the system rests.This long standing conjecture had attracted many physicists and mathematician,but only very special cases were verified up until the seventies.

We show how approximate solutions of the two-body problem in General Relativity, and the approximate solutions of Einstein’s equations in vacuo can be constructed using small deformations of geodesics and of Einstein space-times embedded into a pseudo-Euclidean flat space of higher dimension. The method consists in using expansions of equations around a given simple solution (a circular orbit in the case of geodesics, and Minkowskian or Schwarzschild space in the case of Einstein’s equations) in a series of powers of small deformation parameter, and then solving by iteration the corresponding linear systems of differential equations.

Coordinate approach to the energy conservation problem in general relativity

We discuss some aspects about the computation of kinematic, spectroscopic, Fermi and astrometric relative velocities that are geometrically defined in general relativity. Mainly, we state that kinematic and spectroscopic relative velocities only depend on the 4-velocities of the observer and the test particle, unlike Fermi and astrometric relative velocities, that also depend on the acceleration of the observer and the corresponding relative position of the test particle, but only at the event of observation and not around it, as it would be deduced, in principle, from the definition of these velocities. Finally, we propose an open problem in general relativity that consists on finding intrinsic expressions for Fermi and astrometric relative velocities avoiding terms that involve the evolution of the relative position of the test particle. For this purpose, the proofs given in this paper can serve as inspiration.

A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely solvable up to gauge solutions, given initial data on a spacelike Cauchy hypersurface. The solution map is an isomorphism between initial data (modulo gauge producing initial data) and solutions (modulo gauge solutions). In the first part of this work, we show that the solution map is actually an isomorphism of locally convex topological vector spaces. This implies that the equivalence class of solutions depends continuously on the equivalence class of initial data. We may therefore conclude well-posedness of the Cauchy problem. In the second part, we show that the linearised constraint equations can always be solved on a closed manifold with vanishing scalar curvature. This generalises the classical notion of TT-tensors on flat space used to produce models of gravitational waves. All our results are proven for smooth and distributional initial data of arbitrary real Sobolev regularity.

The post-Minkowskian expansion of Einstein’s general theory of relativity has received much attention in recent years due to the possibility of harnessing the computational power of modern amplitude calculations in such a classical context. In this brief review, we focus on the post-Minkowskian expansion as applied to the two-body problem in general relativity without spin, and we describe how relativistic quantum field theory can be used to greatly simplify analytical calculations based on the Einstein–Hilbert action. Subtleties related to the extraction of classical physics from such quantum mechanical calculations highlight the care which must be taken when both positive and negative powers of Planck’s constant are at play. In the process of obtaining classical results in both Einstein gravity and supergravity, one learns new aspects of quantum field theory that are obscured when using units in which Planck’s constant is set to unity. The scattering amplitude approach provides a self-contained framework for deriving the two-body scattering valid in all regimes of energy. There is hope that the full impact of amplitude computations in this field may significantly alter the way in which gravitational wave predictions will advance in the coming years.

. Penrose has described a method for computing a solution for the characteristic initial value problem for the spin-2 equation for the Weyl spinor. This method uses the spinorial properties in an essential way. From the symmetrized derivatives of the Weyl spinor which are known from the null datum on a cone one can compute all the derivatives by using the field equation and thus one is able to write down a power series expansion for a solution of the equation. A recursive algorithm for computing the higher terms in the power series is presented and the possibility of its implementation on a computer is discussed.

The effective one-body formalism of the gravitational two-body problem in general relativity is reconsidered in the light of recent scattering amplitude calculations. Based on the kinematic relationship between momenta and the effective potential, we consider an energy-dependent effective metric describing the scattering in terms of an effective one-body problem for the reduced mass. The identification of the effective metric simplifies considerably in isotropic coordinates when combined with a redefined angular momentum map. While the effective energy-dependent metric as expected is not unique, solutions can be chosen perturbatively in the post-Minkowskian expansion without the need to introduce nonmetric corrections. By a canonical transformation, our condition maps to the one based on the standard angular momentum map. Expanding our metric around the Schwarzschild solution we recover the solution based on additional nonmetric contributions.

Note on conformastat vacuum space-times

The current state of the vacuum evolution problem in general relativity is reviewed from a perspective of rigorous mathematical results.

The usual method of solution of problems in General Relativity was to use a local coordinate basis to suit a given problem. However, it has been found useful to employ non-coordinate basis techniques in some problems. This is the tetrad formalism which consists of setting up four linearly independent vector fields called a tetrad basis at each point of a region of spacetime. The relevant equations are written in this basis and if the tetrad is aligned according to the symmetries of the spacetime an easier set of equations may ensue which could be amenable to solution.