About

A theory that describes how matter (in this context, the ) interacts dynamically with the geometry of space and time. It was first published by Albert Einstein in 1915 and is currently used to study the structure and evolution of the universe, as well as having practical applications like GPS (see here for an introduction to GPS and GR).

General relativity employs , specifically Riemannian geometry, as it models gravity as the curvature of spacetime.

General relativity can be derived either from the Einstein-Hilbert Action (EHA):

$$S=\int R\sqrt{-g}\mbox{ d}^4x$$

$$\mathcal L = \lambda R$$

Or from the Einstein Field Equation (EFE):

$$G_{\mu\nu}=\kappa T_{\mu\nu} $$

Both of these can be deduced from each other along with the relation $\lambda=\frac1{2\kappa} $. Einstein's original approach was the EFE, but Hilbert's was the EHA. To be consistent with Newtonian gravity,

$$\kappa=\frac{8\pi G}{c^4}$$

The EFE is second order hyperbolic differential equation in the . The solution of the field equation is the metric tensor itself. Some prominent examples of solutions include the Schwarzschild metric, the Reissner-Nordström metric, the Kerr metric, and the Kerr-Newman metric.

Introductory Resources

Carroll's online introduction

Zee's "Nutshell" introduction (includes a full treatment of )

Carroll's graduate level introduction

Prerequisites

Mathematics: Vector Calculus, Calculus of Variations , Linear Algebra, Multilinear Algebra, Differential Geometry, Riemannian Geometry, Differential Topology.

Physics: Lagrangian Mechanics, Special Relativity, Electromagnetism.

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