Huygens' Principle and Hyperbolic Equations is devoted to certain mathematical aspects of wave propagation in curved space-times.

The book aims to present special nontrivial Huygens' operators and to describe their individual properties and to characterize these examples of Huygens' operators within certain more or less comprehensive classes of general hyperbolic operators. The materials covered in the book include a treatment of the wave equation for p-forms over a space of constant sectional curvature, the Riesz distributions, the Euler-Poisson-Darboux-equations over a Riemannian manifold, and plane wave manifolds.

Physicists will find the book invaluable.

Gunther Paul is an Ergonomist and James Cook University Principal Research Fellow for Occupational Health and Safety at the Australian Institute for Tropical Health and Medicine (AITHM), and the Mackay Institute for Research and Innovation (MIRI). He holds a PhD in Ergonomics and MPhil in Control Engineering from Darmstadt University of Technology. His research focuses on complex work system related issues, such as health systems, respiratory health, human-in-the-loop modelling, or musculoskeletal disorders. Gunther has been the Chief Investigator in 17 research projects. He is the Editor-In-Chief of the International Journal of Human Factors Modelling and Simulation, and a reviewer for over 20 international journals. He chairs the International Ergonomics Association Technical Committee on Human Simulation and Virtual Environments, and is a Member of the Queensland Government Safety Leadership at Work Expert Reference Group, Member of the Commonwealth Department of Employment Research and Evaluation Services Panel, and Member of the Panel of Assessors, Queensland Civil and Administrative Tribunal (QCAT). Gunther is also the Ambassador of the Foundation for Professional Ergonomics in Australia. He has published over 100 journal articles, books and book chapters, and has been regularly presenting and chairing sessions at International conferences over the last 25 years. In his most recent previous employments, Gunther led the Health Safety Environment Discipline in the School of Public Health and Social Work at QUT, and before that he was Director of Ergolab at UniSA. In his 10 year industrial career, he worked as Project Manager for Ford, Daimler, and Faurecia.

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Acknowledgements


Introduction


Chapter I


Normal Domains


The Causal Structure of Space-Times


Vector Bundles


The Wave Equations for Differential Forms in Non-Euclidean Spaces


A Spinor Calculus


Chapter II Riesz Distributions


The Riesz Distributions in the Minkowski Space


The Riesz Distributions in Curved Space-Times


Some Generalizations


Chapter III The Fundamental Solutions


The Hadamard Coefficients


B-Series


The Fundamental Solutions


Applications of the Fundamental Solutions


The Cauchy Problem


Chapter IV Huygens' Operators


Hadamard's Criterion


Huygens' Triples


Diversors. General Wave Families


Maxwell's Equations. Dirac's Equations


Chapter V The Euler-Poisson-Darboux Equation


An Application of the Method of Descent


The Singular Cauchy Problem


Huygens' Principle for the EPD-Equation


Stellmacher's Equations


Elliptic Operators with Vanishing First Hadamard Coefficient


Appendix


Relations to Spectral Geometry


Chapter VI Transformation Theory


The Bundle Connection Associated to an Operator P


A Property of the Hadamard Coefficients


Conformal Gauge Transformations of an Operator P


Tensors with Simple Transformation Law


The Moments of a Normal Hyperbolic Operator (Even Dimension)


The Moments for Maxwell's Equations


Chapter VII Some Theorems on Huygens' Operators Over Four-Dimensional Space-Times


Some Preparatory Transformations


The Moments of Order = 3


Applications to Huygens' Operators in a Four-Dimensional Space-Time


The Case of Conformally Flat Metrics


Chapter VIII Plane Wave Manifolds and Huygens' Principle


Introduction. Results


pp- and Plane Wave Manifolds


Huygens' Principle for Plane Wave Manifolds


A Characterization of Plane Wave Manifolds


Some Conformally Invariant Tensors


Testing Coefficients by pp-Metrics


Testing Coefficients by Metrics of Constant Curvature


Table I Identities for the Weyl Tensor


Table II Moments of Order = 4 in Four Dimensions


Table III Some Formulas for pp-Metrics


Table IV Some Formulas for Plane Wave Metrics


Appendix I Metric and Curvature in Normal Coordinates


Appendix II Weak Huygens' Operators by V. Wsch


Appendix III Huygens' Principle for Spin Tensor Equations by V. Wsch


Index


Bibliography