“Relativistic Investigations of Laser-Plasma Interactions and Electrodynamics in a Medium.” by Terrence Pryce Rowlands A thesis submitted for the degree of Doctor of Philosophy at the University of New South Wales Sydney, October 1996 U N S W 1 9 JUM 1997 LIBRARY Statement of Originality I hereby declare that this submission is my own work and that, to the best of my knowledge and belief, it contains no material previously published or written by another person nor material which to a substantial extent has been accepted for the award of any other degree or diploma of a university or other institute of higher learning, except where due acknowledgment is made in the text. I also declare that the intellectual content of this thesis is the product of my own work, even though I may have received assistance from others on style, presentation and language expression. Terrence P. Rowlands B.Sc.(Hons) Sydney, October 1996. page -ii- Acknowledgments I am indebted to Professor H. Hora for suggesting the initial research topic. I am especially grateful to him for maintaining his enthusiasm and assistance after his retirement. I would also like to thank Associate Professor R. Stening for taking over the mantle of supervision after Professor Hora’s retirement. I am indebted to the late Associate Professor (retired) S. Prokhovnik for his mathematical and relativistic insights. Several individuals at the University of New South Wales were very helpful; Dr. John D’Ambra, Associate Professor M. O’Connor, Mrs. M. Ackerman, and others who have provided encouragement and support. Lastly, I would like to extend my gratitude to my family and friends for their understanding during this thesis. In particular I would like to express my unbounded gratitude toward Miss T. Evans. Without her encouragement, support and selflessness, this thesis would never have come to fruition. page -iii- Dedication This thesis is dedicated to the memory of Associate Professor (retired) Simon Prokhovnik whose insight, experience and encouragement was invaluable. page -iv- Abstract The interaction of electromagnetic radiation with plasmas and other media is studied in a relativistic four-vector formalism. A gauge and Lorentz invariant ponderomotive four-force is derived from the time-dependent nonlinear three-force of H. Hora (Phys. Fluids, 12, 182 [1985] ). This four-force, due to it’s Lorentz invariance, contains new magnetic field terms. It is shown to have a symmetric stress-energy tensor. A new gauge and Lorentz invariant model of the response of a plasma to electromagnetic radiation is then devised. An expression for the dispersion relation is obtained from this model. It is then proved that the magnetic permeability of a plasma is unity for a general reference frame. This is an important result since it has been previously assumed in many plasma models. Gauge and Lorentz invariant electrodynamics in a medium is investigated utilizing fourth rank tensors and a redefinition of the Minkowski six-vectors. Initially a general isotropic medium is investigated. The permeability and permittivity are encapsulated within a fourth rank ‘media’ tensor. The fundamental electromagnetic relations are concisely expressed within this fourth rank tensor framework. The refractive index is recovered as an invariant of the fourth rank media tensor. page -v- The generalization of the fourth rank tensor model to encompass anisotropic media is achieved by associating a direction with each tensor component. The fourth rank tensorial relations developed for isotropic media are still valid. The refractive index expression appropriate for the anisotropic medium is then recovered. page -vi- Table of Contents Chapter 1. Introduction........................................................................................... 1 1.1 Overview............................................................................................................ 1 1.2 Introductory Concepts....................................................................................... 3 1.3 Chapter Synopsis............................................................................................... 9 1.3.1 Plasma Response to Electromagnetic Radiation......................................... 9 1.3.2 A Formulation of Electromagnetics in the General Medium.................... 10 1.4 Units and Conventions............................................................... 11 1.5 Bibliography.....................................................................................................12 Chapter 2. Stress-Energy Tensors and Nonlinear Plasma Theory..................... 14 2.1 Introduction......................................................................................................14 2.2 The Abraham-Minkowski Problem...................................................................15 2.2.1 Introduction...............................................................................................15 2.2.2 Asymmetric Stress-Energy Tensors.......................................................... 17 2.2.3 Canonical Stress-Energy Tensors............................................................. 22 2.2.4 More on the Abraham and Minkowski Stress Tensors.............................24 2.3 Plasma Theory and Ponderomotive Forces......................................................26 page -vii- ^ 26 2.3.2 The Time-Independent Ponderomotive Force.............................................. 27 2.3.3 The Time-Independent Nonlinear Ponderomotive Force............................ 29 2.4 The Nonlinear Transient Ponderomotive 4-Force............................................... 33 2.4.1 Introduction.......................................................................................................33 2.4.2 Derivation of the Ponderomotive 4-Force and its Stress-Energy Tensor .34 2.4.3 Gauge Invariance of the Ponderomotive 4-Force.........................................37 2.4.4 Lorentz Invariance of the Ponderomotive 4-Force...................................... 38 2.4.5 An Example for a Specific Current................................................................ 38 2.4.6 The Symmetry of the Nonlinear Ponderomotive Stress Energy Tensor ...39 2.4.7 The Trace of the Nonlinear Ponderomotive Stress Energy Tensor...........41 2.4.8 The Time-Independent Nonlinear Ponderomotive 4-Force.........................44 2.5 Discussion.................................................................................................................46 2.6 Bibliography............................................................................................................. 48 Chapter 3. Relativistic Plasma Theory and the Plasma Magnetic Response.... 50 3.1 Introduction...............................................................................................................50 3.2 Relativistic Plasmas and Electromagnetic waves..................................................51 page -viii- 3.2.1 Introduction 51 3.2.2 Constitutive Equation Methods......................................................................52 3.2.3 Boltzmann-Vlasov methods.............................................................................55 3.2.4 Other approaches..............................................................................................58 3.2.5 Overview........................................................................................................... 63 3.3 The Magnetic Response of a Plasma.....................................................................63 3.3.1 Introduction.......................................................................................................63 3.3.2 Theoretical Tokamak investigations..............................................................64 3.3.3 Quantum theory applied to the Magnetic Response.................................... 67 3.3.4 Overview........................................................................................................... 69 3.4 Relativistic derivation of a value for the Magnetic Permeability....................... 70 3.4.1 Introduction.......................................................................................................70 3.4.2 Considerations.................................................................................................. 70 3.4.3 The Lorentz Force Density..............................................................................72 3.4.4 The Charged Fluid Force Density...................................................................73 3.4.5 Equating the Forces.........................................................................................74 3.4.6 The 4-Current Wave Equation.......................................................................75 page -ix- 3.4.7 A solution to the Wave equation. 76 3.4.8 The Dispersion Relation for the 4-Current Wave Equation.......................77 3.4.9 The Electric and Magnetic response of the Charged Fluid.........................77 3.4.10 The derivation of the Magnetic permeability of a Plasma.........................79 3.4.11 A pressure based modification to the model...............................................83 3.5 Discussion.................................................................................................................85 3.6 Bibliography..............................................................................................................86 Chapter 4. Electrodynamics of Relativistic Isotropic Media..............................