COLLIDING PLANE WAVES IN GENERAL RELATIVITY
CONTENTS INTRODUCTION WHY CONSIDER WAVE INTERACTIONS? SIMPLIFYING ASSUMPTIONS ELEMENTS OF GENERAL RELATIVITY BASIC NOTATION COMPONENTS OF THE CURVATURE TENSOR SPIN COE?CIENTS EINSTEIN–MAXWELL ?ELDS COLLIDING IMPULSIVE GRAVITATIONAL WAVES THE APPROACHING WAVES THE SOLUTION DESCRIBING THE INTERACTION THE STRUCTURE OF THE SOLUTION PLANE WAVES THE CLASS OF PP-WAVES THE CLASS OF PLANE WAVES PARTICULAR CASES GLOBAL PROPERTIES GEOMETRICAL CONSIDERATIONS THE FOCUSING OF CONGRUENCES GENERAL THEOREMS COLLIDING WAVES THE FIELD EQUATIONS THE COORDINATE SYSTEM THE DERIVATION OF THE ?ELD EQUATIONS THE EINSTEIN AND EINSTEIN–MAXWELL EQUATIONS INTEGRATING THE ?ELD EQUATIONS BOUNDARY CONDITIONS GENERAL DISCUSSION JUNCTION CONDITIONS FOR COLLIDING PLANE WAVES SINGULARITY STRUCTURE SINGULARITIES THE SINGULARITY IN REGION IV THE KHAN–PENROSE SOLUTION THE STRUCTURE OF OTHER SOLUTIONS THE SZEKERES CLASS OF VACUUM SOLUTIONS THE SOLUTION IN REGION IV THE SINGULARITY STRUCTURE OTHER VACUUM SOLUTIONS WITH ALIGNED POLARIZATION 10 1 A GENERAL METHOD 10 2 THE NON-SINGULAR ‘SOLUTION’ OF STOYANOV 10 3 THE SOLUTION OF FERRARI AND IBA˜NEZ AND GRI?THS 10 4 THE SOLITON SOLUTION OF FERRARI AND IBA˜NEZ 10 5 THE DEGENERATE FERRARI–IBA˜NEZ SOLUTIONS 10 6 AN ODD ORDER SOLUTION 10 7 THE SECOND YURTSEVER AND THE FEINSTEIN–IBA˜NEZ SOLUTIONS 10 8 THE ?RST YURTSEVER SOLUTIONS 10 9 FURTHER EXPLICIT SOLUTIONS ERNST’S EQUATION FOR COLLIDING GRAVITATIONAL WAVES 11 1 A DERIVATION OF THE ERNST EQUATION 11 2 BOUNDARY CONDITIONS 11 3 COLINEAR SOLUTIONS SOLUTION-GENERATING TECHNIQUES 12 1 THE COLINEAR CASE 12 2 ROTATIONS AND EHLERS TRANSFORMATIONS 12 3 GEROCH TRANSFORMATIONS 12 4 THE NEUGEBAUER–KRAMER INVOLUTION 12 5 A COMBINED TRANSFORMATION 12 6 OTHER METHODS VACUUM SOLUTIONS WITH NON-ALIGNED POLARIZATION 13 1 THE NUTKU–HALIL SOLUTION 13 2 THE PANOV SOLUTION 13 3 THE CHANDRASEKHAR–XANTHOPOULOS SOLUTION 13 4 OTHER SOLUTIONS THE INITIAL VALUE PROBLEM 14 1 THE INITIAL DATA 14 2 THE COLINEAR CASE 14 3 THE NON-COLINEAR CASE COLLIDING ELECTROMAGNETIC WAVES: THE BELL–SZEKERES SOLUTION 15 1 THE BELL–SZEKERES SOLUTION 15 2 THE STRUCTURE OF THE SOLUTION 15 3 EXTENSIONS OF THE SOLUTION 15 4 A NON-COLINEAR COLLISION ERNST’S EQUATION FOR COLLIDING ELECTROMAGNETIC WAVES 16 1 THE ?ELD EQUATIONS 16 2 A SIMPLE CLASS OF SOLUTIONS 16 3 THE BELL–SZEKERES SOLUTION COLLIDING ELECTROMAGNETIC WAVES: EXACT SOLUTIONS 17 1 A TECHNIQUE OF CHANDRASEKHAR AND XANTHOPOULOS 17 2 TWO PARTICULAR EXAMPLES 17 3 ANOTHER TYPE D SOLUTION 17 4 A TECHNIQUE OF HALILSOY 17 5 OTHER SOLUTIONS COLLIDING ELECTROMAGNETIC WAVES: DIAGONAL SOLUTIONS 18 1 THE GENERATION TECHNIQUE OF PANOV 18 2 AN ALTERNATIVE APPROACH 18 3 ELECTROMAGNETIC GOWDY COSMOLOGIES 18 4 OTHER METHODS ELECTROMAGNETIC WAVES COLLIDING WITH GRAVITATIONAL WAVES 19 1 A SIMPLE EXAMPLE 19 2 GENERAL INITIAL DATA 19 3 A GENERAL CLASS OF SOLUTIONS OTHER SOURCES 20 1 SCALAR ?ELDS 20 2 PERFECT ?UID SOLUTIONS 20 3 NULL ?UIDS AND THE UNIQUENESS PROBLEM 20 4 PLANE SHELLS OF MATTER 20 5 NEUTRINO ?ELDS 20 6 OTHER NULL ?ELDS RELATED RESULTS 21 1 OTHER WAVE INTERACTIONS 21 2 COLLISIONS IN NON-?AT BACKGROUNDS 21 3 SOLITONS 21 4 ALTERNATIVE GRAVITATIONAL THEORIES 21 5 NUMERICAL TECHNIQUES CONCLUSIONS AND PROSPECTS 22 1 GENERAL CONCLUSIONS 22 2 PROSPECTS FOR FURTHER WORK