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1 - Introduction to general relativity 1: Kinematics and Einstein equations
from Part I - Formalism
Horatiu Nastase, Universidade Estadual Paulista, São Paulo
Book:

Introduction to Supergravity and its Applications

Published online:

14 November 2024

Print publication:

21 November 2024, pp 5-17
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Summary

After an introduction to general relativity and supersymmetry, the formalism of supergravity is defined, on-shell, off-shell, and in superspace, using coset theory and local superspace. Higher dimensions, extended susy, and KK reduction are also defined. Then, various applications are described: dualities and solution-generating techniques, solutions and their susy algebra, gravity duals and deformations, supergravity on the string worldsheet and superembeddings, cosmological inflation, no-go theorems and Witten’s positive energy theorem, compactification of low-energy string theory and toward embedding the Standard Model using supergravity, susy breaking and minimal supergravity.

12 - Gravity as a quantum effect on quantum spacetime
from Part IV - Matrix theory and gravity
Harold C. Steinacker, Universität Wien, Austria
Book:

Quantum Geometry, Matrix Theory, and Gravity

Published online:

04 April 2024

Print publication:

11 April 2024, pp 342-362
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Summary

In this core chapter, the one-loop effective action for Matrix theory on 3 + 1 dimensional branes is elaborated, and the Einstein–Hilbert term is obtained in the presence of fuzzy extra dimensions. Some justification for the stability of the background is given.

39 - The Einstein Action and the Einstein Equation
from Part III - Other Spins or Statistics; General Relativity
Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
Book:

Classical Field Theory

Published online:

04 March 2019

Print publication:

14 March 2019, pp 367-377
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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.