General Relativity

General vs. Special Relativity

While SR is a theory of spacetime, eponymous with Minkowski space an Lorentz transforms, GR is a theory of gravitation. In the limit of velocities small compared to the speed of light in vacuum, SR becomes classical mechanics with the Galilei transform. GR, for small masses, becomes Newronian mechanics, while for large masses gravitation deforms spacetime, leading to phenomena like gravitational lensing and black holes.

Field Equation

The field equation, as a generalization of Newton's law of gravitation, has the following form..
$$ \large{ R_{\mu\nu} - \frac{1}{2}\,R\,g_{\mu\nu} + \Lambda\,g_{\mu\nu} = \kappa\,T_{\mu\nu}} $$

Cosmological Constant

The cosmological constant $\Lambda$ in the field equation describes whether a universe subject to the equation is contracting, static or expanding. Observations by Hubble showed that our universe is expanding. Observations give the value of $\Lambda$ as
$$\Lambda = 3\,\frac{H_0^2}{c^2}\,\Omega_\Lambda$$
where $H_0 = 66.67\,m\,s^{-1}\,kpc^{-1}$ is the Hubble constant and $\Omega_\Lambda\approx 0.69$ is the fractional amount of dark energy present. Measurements by the Planck collaboration, published in 2018, lead to a value for $\Lambda$ of
$$\Lambda = 1.1055\times 10^{-52} m^{-2}$$