The Duality of Time Theory

Einstein Field Equations are a set of ten equations that describes the curvature of space-time due to the presence of mass and energy, and they are written in tensor form that simply equates the curvature with the energy and momentum:

The tensors here are written in abstract index notation using the two indexesandthat represent the four dimensions of space-time. The tensoris the Ricci curvature tensor,is the scaler curvature,is the metric tensor, andis the stress-energy tensor. The constantis the cosmological constant which was added later as we shall mention below, and we shall discuss it further in section 9.21. The constantsand, on the right side of the equation, are Newton’s gravitational constant, and the speed of light in vacuum, respectively. The left side of the equation is usually defined as Einstein tensor:, which is a symmetric second-rank tensor that is a function of the metric.

This relationship between the Einstein tensor and the metric tensor allows the equations to be written as a set of non-linear partial differential equations, whose solutions are the components of the metric tensor. However, exact solutions can not be found without further assumptions to simplify the analysis. Various solutions are tailored to model many gravitational phenomena, such as rotating black holes, expanding Universe, and gravitational waves.

The derivation of Einstein’s equations from an action allows for easy unification of General Relativity with other classical field theories, such as Maxwell theory of Electromagnetism, which are also formulated in terms of an action. In the process, the derivation from an action identifies a natural candidate for the source term coupling the metric to matter fields. Moreover, the action allows for the easy identification of conserved quantities through Noether’s theorem by studying the symmetries of the action.

In General Relativity, the action is usually assumed to be a functional of the metric and matter fields, and the connection is given by the Levi-Civita connection. The Palatini formulation of General Relativity assumes the metric and connection to be independent, which makes it possible to include fermionic matter fields with non-integral spin.

The Einstein equations in the presence of matter are given by adding the matter action to the Einstein-Hilbert action.