Gravitation: SL(2,C) Gauge Theory and Conservation Laws

The following sections are included:

• PREFACE

• ACKNOWLEDGEMENTS

• TABLE OF CONTENTS

• LIST OF SYMBOLS

Einstein's general relativity is probably the most ingenious theory invented by man so far¹. Gravitation became one of the most important topics in theoretical physics because of its implications both on particle physics and field theory as well as on astrophysics. Of special importance is the formulation of Einstein's original theory as a gauge theory, whereby gravitation is cast as the other physical fields all of which, as is well-known, are described nowadays as gauge fields. The conservation laws in general relativity, from the viewpoint of both Einstein's original classical theory and the more newly developed SL(2, C) gauge theory of gravitation, is also of great importance since in addition to playing a central role in these theories they bear on the rest of physics…

The following sections are included:

• Preliminaries

• Spinor Space
  • Spinor algebra
  • Spinor analysis
  • Relations between Spinors and Tensors

• Gravitation as an SL(2,C) Gauge Theory
  • The gauge potential
  • The gauge field
  • The Riemann tensor

• Field Equations
  • Free-field equations
  • Coupling with matter

• Relation to the Newman-Penrose Formalism

• Derivation by Palatini's Method

The following sections are included:

• Linear and Quadratic Lagrangians

• Hilbert's Lagrangian

• The Quadratic Invariants

• Carmeli's Lagrangian

• The Lagrangian Family

• The Tensorial Quadratic Lagrangian
  • General properties
  • The Variation of ℒ
  • The Variation of C_αβγδ and T_αβ
  • The variation of the metric tensor

• Summary

The following sections are included:

• Introduction

• The Problem

• Einstein's Solution

• Landau-Lifshitz Pseudo-Tensor

• Moller's Approach

• Conserved Tensor Density—Gauge Formulation Approach

• Conclusions

The following sections are included:

• Symmetry and Conservation Laws

• The Set of Six Currents

• The SL(2,C) Symmetry

• The Gravitational Current

• Example: The Tolman Metric

• Summary

The following sections are included:

• Preliminaries

• The Nonrotating Coordinate Systems

• The Internal Group

• Inertial Coordinates

• Example

• Conclusions and Remarks

The following sections are included:

• REFERENCES

• SUBJECT INDEX