About

A theory that describes how matter (in this context, the ) produces and responds to the geometry of space and time. It was first published by Einstein in 1915 and is currently used to study the structure and evolution of the universe, as well as having practical applications like GPS (without General Relativity, the GPS would be inaccurate to the degree that it would be useless.).

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):

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

$$\mathcal L = \lambda R$$

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,

$$\kappa=\frac{8\pi G}{c_0^4}$$

The Einstein Field Equation (EFE) is a differential, polynomial equation in the . Solutions to the EFE would mean solutions for the . Some prominent examples of solutions include the schwarzschild metric, the reisnerr nordstorm metric, the kerr metric, and the kerr-newman metric.

Introductory Resources

Ludvigsen

Wikipedia on the EH Action

Wald (for a mathematically rigorous approach.)

(Einstein, 1915) "On the Foundation of the General Theory of Relativity" (for a historical treatment)

Prerequisites

Mathematics: Calculus (and all its prerequisites), Linear Algebra (and all its prerequisites), Matrix Algebra, Exterior Algebra, Clifford Algebra, Tensor Algebra, Vector Calculus, Matrix Calculus, Tensor Calculus, Differential Geometry, Riemannian Geometry, Calculus of Variations

Physics: Lagrangian Mechanics (and all its Mathematics/Physics prerequisites), Hamiltonian Mechanics, Special Relativity,

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