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A theory that describes how matter interacts dynamically with 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.

A theory that describes how matter (in this context, the ) interacts dynamically with the geometry of space and time, as described by the . 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 and , more specifically (pseudo-)Riemannian geometry, as it models as the of . Test particles will move along .

Equations of motion

The equations of motion are the Einstein field equations, sc. $$G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu} $$ where $G$ is the so-called Einstein tensor, $\Lambda$ is the Cosmological constant, $\kappa$ is a coupling constant, and $T$ is the Stress-energy tensor.

These equations can be derived from the so-called Einstein-Hilbert Action, $$S_H=\int \mathcal L\sqrt{-g}\mbox{ d}^4x,\qquad \mathcal L = \frac1{2\kappa} (R-2\Lambda)$$ by means of the stationary action principle.

To be consistent with Newtonian gravity, $$\kappa=\frac{8\pi G}{c^4}$$ as can be derived from the non-relativistic limit of the Einstein field equations.

The EFE is a second order hyperbolic system of differential equations in the metric tensor. Some prominent examples of solutions include the Schwarzschild metric, the , the , and the , all of which are able to describe to varying degrees of precision. Other solutions typically include propagating modes, which describe . A more trivial solution is the , which describes space-time in the absence of gravitation, where general relativity becomes .

The cosmological constant $\Lambda$ is typically negligible at planetary and galactic scales. It is only noticeable at cosmological scales.

Cosmology

The Einstein field equations are the basic principle behind , which is the study of the universe as a whole. The latter is typically assumed to be isotropic and homogeneous, in which case the solution of the field equations is the Friedmann–Lemaître–Robertson–Walker metric, which describes, for example, .

In the context of the $\Lambda$CDM model, the has been measured to represent the 70% of the energy density of the current universe.

Quantum Gravity

As of today, there is no consensus about what is the best way to combine gravity and quantum mechanics, i.e., about what a possible theory of should look like. The most promising candidate is , although other alternatives are under study (such as ).

An intermediate result towards a complete theory of quantum gravity is , which studies the dynamics of quantum mechanical fields when they are immersed in a classical gravitating background, such as AdS. These kind theories have inspired a lot of interesting conjectures and partial results, such as the or other dualities.

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, Electrodynamics.

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