Wave Equations on Lorentzian Manifolds and Quantization

Abstract

In General Relativity spacetime is described mathematically by a Lorentzian manifold. Gravitation manifests itself as the curvature of this manifold. Physical fields, such as the electromagnetic field, are defined on this manifold and have to satisfy a wave equation. This book provides an introduction to the theory of linear wave equations on Lorentzian manifolds. In contrast to other texts on this topic [Friedlander1975, Günther1988] we develop the global theory. This means, we ask for existence and uniqueness of solutions which are defined on all of the underlying manifold. Such results are of great importance and are already used much in the literature despite the fact that published proofs are missing. Tracing back the references one typically ends at Leray's unpublished lecture notes [Leray1953] or their exposition [Choquet-Bruhat1968].In this text we develop the global theory from scratch in a modern geometric language. In the first chapter we provide basic definitions and facts about distributions on manifolds, Lorentzian geometry, and normally hyperbolic operators. We study the building blocks for local solutions, the Riesz distributions, in some detail. In the second chapter we show how to solve wave equations locally. Using Riesz distributions and a formal recursive procedure one first constructs formal fundamental solutions. These are formal series solving the equations formally but in general they do not converge. Using suitable cut-offs one gets "almost solutions" from these formal solutions. They are well-defined distributions but solve the equation only up to an error term. This is then corrected by some further analysis which yields true local fundamental solutions.This procedure is similar to the construction of the heat kernel for a Laplace type operator on a compact Riemannian manifold. The analogy goes even further. Similar to the short-time asymptotics for the heat kernel, the formal fundamental solution turns out to be an asymptotic expansion of the true fundamental solution. Along the diagonal the coefficients of this asymptotic expansion are given by the same algebraic expression in the curvature of the manifold, the coefficients of the operator, and their derivatives as the heat kernel coefficients.In the third chapter we use the local theory to study global solutions. This means we construct global fundamental solutions, Green's operators, and solutions to the Cauchy problem. This requires assumptions on the geometry of the underlying manifold. In Lorentzian geometry one has to deal with the problem that there is no good analog for the notion of completeness of Riemannian manifolds. In our context globally hyperbolic manifolds turn out to be the right class of manifolds to consider. Most basic models in General Relativity turn out to be globally hyperbolic but there are exceptions such as iv anti-deSitter spacetime. This is why we also include a section in which we study cases where one can guarantee existence (but not uniqueness) of global solutions on certain non-globally hyperbolic...

The Generalised Principle of Perturbative Agreement and the Thermal Mass

“…First of all we recall that ∆ R 2 and ∆ R 1 map smooth functions with past-compact support to smooth functions with past-compact support continuously in the topology of E (M ) [BGP07].…”

“…We shall consider only spacetimes (M , g) which are globally hyperbolic throughout, see e.g. [BGP07] for a definition and properties. Moreover, we set E (M n ) .…”

“…We recall that the causal propagator is uniquely defined as ∆ = ∆ R − ∆ A , the retarded-minus-advanced fundamental solution of the linear part of (6), (see e.g. [BGP07] for their rigorous unique construction in every globally hyperbolic spacetime), whereas ∆ S is non-unique. Moreover, we demand that ∆ + satisfies the Hadamard condition in its microlocal form [Ra96], i.e.…”

“…where F G means that there exists a Cauchy surface (see [BGP07] for a definition) Σ such that supp(F ) ⊆ J + (Σ) and supp(G) ⊆ J − (Σ). We may first consider the time-ordered product on F reg .…”

Local Thermal Equilibrium States and Quantum Energy Inequalities

“…We will consider the case of spacetime dimension equal to 4 with metric signature (+ − −−), but most of our considerations can be readily generalized, with appropriate modifications, to arbitrary spacetime dimensions. We recall that global hyperbolicity means that the spacetime is time-orientable and possesses Cauchy surfaces [2,40]. Our conventions for curvature quantities, like in [14], are those of Birrell and Davies, i.e.…”