This book discusses some of the open questions addressed by researchers in general relativity. Photons and particles play important roles in the theoretical framework, since they are involved in analyzing and measuring gravitational fields and in constructing mathematical models of gravitational fields of various types. The authors highlight this aspect covering topics such as the construction of models of Bateman electromagnetic waves and analogous gravitational waves, the studies of gravitational radiation in presence of a cosmological constant and the gravitational compass or clock compass for providing an operational way of measuring a gravitational field. The book is meant for advanced students and young researchers in general relativity, who look for an updated text which covers in depth the calculations and, equally, takes on new challenges. The reader, along the learning path, is stimulated by provocative examples interspersed in the text that help to find novel representations of the uses of particles and photons.
Author(s): Peter A. Hogan, Dirk Puetzfeld
Series: Lecture Notes in Physics
Publisher: Springer
Year: 2021
Language: English
Pages: 250
Preface
Contents
About the Authors
1 Congruences of World Lines
1.1 Time-Like Conguences
1.2 Null Geodesic Congruences
1.3 The World Function and Deviation Equations
References
2 Bivector Formalism in General Relativity
2.1 Bivectors and Electromagnetic Fields
2.2 Electromagnetic Radiation
2.3 Bivectors and Gravitational Fields
2.4 The Kerr Space-Time
2.5 Passage to Charged Kerr Space-Time
2.6 Using the Bianchi Identities
References
3 Hypothetical Objects in Electromagnetism and Gravity
3.1 Part I: A Light-Like Charge
3.2 Geometry Based on a Non-geodesic Null World Line
3.3 Maxwell Field of a Charge with Non-geodesic World Line
3.4 Part II: A Kerr Black Hole and Light-Like Matter
3.5 Axial Symmetry
3.6 Energy-Momentum-Stress Tensor
3.7 Two Conservation Laws
References
4 Bateman Waves
4.1 Electromagnetic Radiation
4.2 Bateman Electromagnetic Waves
4.3 Some `Spherical' Electromagnetic Waves
4.4 Gravitational Radiation
4.5 Bateman Gravitational Waves
4.6 Some `Spherical' Gravitational Waves
References
5 Gravitational (Clock) Compass
5.1 Determination of the Gravitational Field by Means of Test Bodies
5.2 Gravitational Compass
General Solution
Vacuum Solution
Summary
5.3 Determination of the Gravitational Field by Means of Clocks
Frequency Ratio in Flat Space-Time
Frequency Ratio in Curved Space-Time
5.4 Gravitational Clock Compass
General Solution
Vacuum Solution
Special Spacetimes
Plane Gravitational Waves
Waves Radial Relative to r=0
Summary
References
6 de Sitter Cosmology
6.1 Null Hyperplanes in Space-Times of Constant Curvature
6.2 Intersecting Null Hyperplanes
6.3 Generalized Kerr–Schild Space-Times and Gravitational Waves
6.4 de Sitter Space-Time Revisited
6.5 Collision of Gravitational Waves and ≠0
6.6 Post Collision Physical Properties
References
7 Small Magnetic Black Hole
7.1 Magnetic Poles
7.2 Background Space-Time/External Fields
7.3 Magnetic Black Hole Perturbation of Background
7.4 Equations of Motion in First Approximation
7.5 Equations of Motion in Second Approximation
7.6 Review of Approximations
References
8 Run-Away Reissner–Nordström Particle
8.1 Robinson–Trautman Solutions of the Einstein–Maxwell Equations
8.2 Accelerating Reissner–Nordström Particle
8.3 Perturbations of a Reissner–Nordström Particle
8.4 Equations of Motion of a Reissner–Nordström Particle
8.5 Energy Radiation Rate and Run-Away Motion
8.6 Run-Away Magnetic Reissner–Nordström Particle
References
A Congruences of World Lines
A.1 Properties of ηabcd
B Bateman Waves in the Linear Approximation
B.1 Covariant Treatment on General Space-Time
B.2 Ricci Identities and Bianchi Identities
References
C Gravitational (Clock) Compass
C.1 Coordinates for Minkowskian Space-Time
C.2 Plane Gravitational Waves I
C.3 Plane Gravitational Waves II
C.4 Waves Moving Radially Relative to r=0
References
D de Sitter Cosmology
D.1 Properties of Generalised Kerr–Schild Metrics
Reference
E Small Magnetic Black Hole
E.1 The Electromagnetic Field
Index
