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 stress-energy-tensor) interacts dynamically with the geometry of space and time, as described by the metric-tensor. 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 tensor-calculus and differential-geometry, more specifically (pseudo-)Riemannian geometry, as it models gravity as the curvature of spacetime. Test particles will move along geodesics.
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 using the so-called Einstein-Hilbert Action, $$S_{EH}= \frac1{2\kappa} \int (R-2\Lambda)\sqrt{-g}\mbox{ d}^4x,$$ by means of the stationary action principle. The Einstein–Hilbert action yields the left-hand side of the EFE, while the right-hand side (the stress tensor) is obtained from a matter action. The complete action looks like $$S = S_{EH} + S_M,$$ where $S_M$ is the matter action.
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 reissner-nordstrom-metric, the kerr-metric, and the kerr-newman-metric, all of which are able to describe black-holes to varying degrees of precision. Other solutions typically include propagating modes, which describe gravitational-waves. A more trivial solution is the minkowski-space, which describes space-time in the absence of gravitation, where general relativity becomes special-relativity.
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 cosmology, 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, space-expansion.
In the context of the $\Lambda$CDM model, the cosmological-constant 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 quantum-gravity should look like. Some candidate theories are string-theory, loop-quantum-gravity, asymptotically safe quantum gravity, causal dynamical triangulation, among many others.
An intermediate result towards a complete theory of quantum gravity is qft-in-curved-spacetime (QFTCS), which studies the dynamics of quantum mechanical fields when they are immersed in a classical gravitating background. One possible application is to consider non-inertial observers on minkowski-space, where QFTCS leads to the unruh-effect prediction. Applications on spacetimes containing black-holes led to the prediction of hawking-radiation. Another possibility could be, for example, anti-de-sitter-spacetime, where this kind of theory has inspired a lot of interesting conjectures and partial results, such as the holographic-principle or other ads-cft dualities.
Introductory Resources
Zee's "Nutshell" introduction (includes a full treatment of special-relativity)
Carroll's graduate-level introduction
Prerequisites
Mathematics: Vector Calculus, Calculus of Variations, Linear Algebra, Multilinear Algebra, Differential Geometry, Riemannian Geometry, Differential Topology.
Most courses and references in general relativity will discuss the necessary topics in multilinear algebra, differential geometry, (pseudo-)Riemannian geometry, and differential topology.
Physics: Lagrangian Mechanics, Special Relativity, Electrodynamics.
