Einstein–cartan Theory

Standard notation and terminology of differential geometry and general relativity are used in this article. All considerations are local so that the four-dimensional space-time M is assumed to be a smooth manifold diffeomorphic to R4. It is endowed with a metric tensor g of signature (1, 3) and a linear connection defining the covariant differentiation of tensor fields. Greek indices range from 0 to 3 and refer to space-time. Given a field of frames (eμ) on M , and the dual field of coframes (θμ), one can write the metric tensor as g = gμνθθ , where gμν = g(eμ, eν) and Einstein’s summation convention is assumed to hold. Tensor indices are lowered with gμν and raised with its inverse gμν . General-relativistic units are used so that both Newton’s constant of gravitation and the speed of light are 1. This implies ~ = `2, where ` ≈ 10−33 cm is the Planck length. Both mass and energy are measured in centimeters.