III.1.2 Symmetry in General Relativity
In moving from Special to General Relativity, Einstein observed the equivalence between the gravitational force and the inertial force experienced by an observer in a non-inertial frame of reference. This is the same as the equivalence between active gravitational mass and passive inertial mass, which has been later accurately tested in many experiments. However, apart from the famous spacecraft accelerator thought experiment which relies on induction, there is no direct mathematical derivation of this crucial principle on which the theory of General Relativity is based. For this reason, symmetry is not so evident between non-inertial frames, unless we impose some particular symmetrical situations. There is no unique group for General Relativity similar to the Poincar group of Special Relativity.
When Einstein combined the equivalence principle with the other two principles of Special Relativity, he was able to predict the curved geometry of space-time, which is directly related to its contents of energy and momentum of matter and radiation, through a system of partial differential equations, known as Einstein Field Equations, and they are normally written in tensor form that simply equates the curvature with energy and momentum. However, no exact solutions of these equations can be found without further assumptions to simplify the analysis. Therefore, various solutions are tailored to model many gravitational phenomena, such as rotating black holes, expanding Universe, and gravitational waves.
When Hilbert applied the principle of least action he obtained these equations, thus the action is called the Einstein-Hilbert action. This allowed for easy unification of General Relativity with other classical field theories, such as Maxwell s theory of Electromagnetism, which are also formulated in terms of an action. The action also allows for the easy identification of conserved quantities, through Noether s theorem, by studying the symmetries of the action, which is usually assumed to be a functional of the metric and matter fields.
However, the concept of symmetry in General Relativity is rather vacuous, because all transformations leave the action invariant. In contrary, isolated systems, such as a localized gravitational source surrounded by empty space, exhibit asymptotic symmetries with transformations preserving the isolated character of the system. Those so-called super-translations can be used to extract information about gravitational radiation. The curvature tensor describing the gravitational field includes the matter part and the free gravitational part, and the construction of gravitational potentials satisfying Einstein s field equations is often achieved by imposing symmetries on the geometry compatible with the dynamics of the chosen distribution of matter. The geometrical symmetries of the space-time are expressible through the vanishing of the Lie derivative of certain tensors with respect to a vector that can be time-like, space-like or null.