Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Introduction
- Part I General Properties of Fields; Scalars and Gauge Fields
- Part II Solitons and Topology; Non-Abelian Theory
- Part III Other Spins or Statistics; General Relativity
- 33 Chern–Simons Terms: Emergent Gauge Fields, the Quantum Hall Effect (Integer and Fractional), Anyonic statistics
- 34 Chern–Simons and Self-Duality in Odd Dimensions, Its Duality to Topologically Massive Theory and Dualities in General
- 35 Particle–Vortex Duality in Three Dimensions, Particle–String Duality in Four Dimensions, and p-Form Fields in Four Dimensions
- 36 Fermions and Dirac Spinors
- 37 The Dirac Equation and Its Solutions
- 38 General Relativity: Metric and General Coordinate Invariance
- 39 The Einstein Action and the Einstein Equation
- 40 Perturbative Gravity: Fierz-Pauli Action, de Donder Gauge and Other Gauges, Gravitational Waves
- 41 Nonperturbative Gravity: The Vacuum Schwarzschild Solution
- 42 Deflection of Light by the Sun and Comparison with General Relativity
- 43 Fully Linear Gravity: Parallel Plane (pp) Waves and Gravitational Shock Wave Solutions
- 44 Dimensional Reduction: The Domain Wall, Cosmic String, and BTZ Black Hole Solutions
- 45 Time-Dependent Gravity: The Friedmann-Lemaître-Robertson-Walker (FLRW) Cosmological Solution
- 46 Vielbein-Spin Connection Formulation of General Relativity and Gravitational Instantons
- References
- Index
39 - The Einstein Action and the Einstein Equation
from Part III - Other Spins or Statistics; General Relativity
Published online by Cambridge University Press: 04 March 2019
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Introduction
- Part I General Properties of Fields; Scalars and Gauge Fields
- Part II Solitons and Topology; Non-Abelian Theory
- Part III Other Spins or Statistics; General Relativity
- 33 Chern–Simons Terms: Emergent Gauge Fields, the Quantum Hall Effect (Integer and Fractional), Anyonic statistics
- 34 Chern–Simons and Self-Duality in Odd Dimensions, Its Duality to Topologically Massive Theory and Dualities in General
- 35 Particle–Vortex Duality in Three Dimensions, Particle–String Duality in Four Dimensions, and p-Form Fields in Four Dimensions
- 36 Fermions and Dirac Spinors
- 37 The Dirac Equation and Its Solutions
- 38 General Relativity: Metric and General Coordinate Invariance
- 39 The Einstein Action and the Einstein Equation
- 40 Perturbative Gravity: Fierz-Pauli Action, de Donder Gauge and Other Gauges, Gravitational Waves
- 41 Nonperturbative Gravity: The Vacuum Schwarzschild Solution
- 42 Deflection of Light by the Sun and Comparison with General Relativity
- 43 Fully Linear Gravity: Parallel Plane (pp) Waves and Gravitational Shock Wave Solutions
- 44 Dimensional Reduction: The Domain Wall, Cosmic String, and BTZ Black Hole Solutions
- 45 Time-Dependent Gravity: The Friedmann-Lemaître-Robertson-Walker (FLRW) Cosmological Solution
- 46 Vielbein-Spin Connection Formulation of General Relativity and Gravitational Instantons
- References
- Index
Summary
We start by defining the notion of Riemann tensor and curvature, and positive and negative curvature spaces. We then show how to turn a special relativistic invariant theory into a general relativistic invariant one and write down the Einstein–Hilbert action for gravity, based on Einstein's principles and on matching with experiment. We then derive its equations of motion, Einstein's equations. We give examples of usual energy–momentum tensors in curved space and end by interpreting the Einstein's equations.
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- Classical Field Theory , pp. 367 - 377Publisher: Cambridge University PressPrint publication year: 2019