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.
The science of Time is a noble science, that reveals the secret of Eternity. Only the Elites of Sages may ever come to know this secret. It is called the First Age, or the Age of ages, from which time is emerging.
