# Is General Relativity based on a Symmetry?

In short: Is there any kind of symmetry one can start with to derive general relativity (GR)?

Longer: Einstein had the opinion that GR was the generalisation of special relativity, because instead of inertial frames, all frames are equally allowed in GR. This is commonly named general covariance. Unfortunately, this was soon after found to be wrong, because every theory can be written in a general covariant way and therefore this can not be the defining feature of general relativity.

1) In analogy with SR one can search for the group that leaves a given metric (= a solution of the Einstien equation), describing a given situation, invariant. These are called Killing groups and are only valid for one given situation. For the Minkowski metric one finds the Poincare group (=the symmetry group of special relativity) and for example, for the Schwarzschild metric, the corresponding Killing group is a subgroup of the Poincare group (See http://en.wikipedia.org/wiki/Schwarzschild_metric#Symmetries). Nevertheless, if one searches for the symmetry group that leaves a general solution of the Einstein equation (a general metric) invariant one finds that this group contains only the identity transformation.

2) Alternatively there exists the Anderson approach, that enables one to distinguish between covariance and invariance, by introducing the notion of dynamical and absolute objects of a given theory. The covariance group leaves the space of kinematically allowed models invariant, whereas the invariance group is a subgroup of the covariance group that leaves the absolute objects of the theory invariant. This way its possible to make precise what distinguishes general relativity from all other theories:

An absolute object of special relativity is the Minkowski metric and the corresponding invariance group is the Poincare group. In the standard formulation the corresponding covariance group is the Poincare group, too. Nevertheless, its possible to rewrite the equations of SR, such that the theory becomes general covariant and in other words: the covariance group becomes $Diff(M)$, the group of all diffeomorphisms. Therefore what matters is the invariance group.

Now, GR is the theory with invariance group $Diff(M)$, i.e. all Diffeomorphism leave the absolute objects of the theory invariant. This could be seen as the defining feature of GR. The next step would be to ask: What are the absolute objects of GR? Answer: GR has no absolute objects. Therefore its no wonder that all Diffeomorphisms leave the absolute objects invariant.

The two possible symmetry concepts described above are those I stumbled upon most of the time searching for symmetry and GR. Unfortunately, both do not seem to be useful to mean in the seach for the defining feature of GR. GR, as far as I understand and as it is presented in the books I read, is not based on some kind of symmetry or symmetry idea, but is rather a result of the idea gravity = curvature of spacetime (+ equivalence principle?). This is in stark contrast to all other fundamental theories of physics, which are based on symmetry without an exception.

Is there some kind, maybe in a broader sense, symmetry idea on which General Relativity is based? Any ideas or suggestions would be awesome!

• It is not true that GR has no absolute objects in your sense. For any pseudo-Riemannian manifold, Élie Cartan showed that it is possible to form a complete set of classifying scalar invariants from the frame components of the Riemann tensor and its covariant derivatives. One such invariant is the curvature scalar $R$. With some refinements of the method, a practical algorithm can be formulated: en.wikipedia.org/wiki/Cartan%E2%80%93Karlhede_algorithm There is a lot of literature about this, an it is quite easy to do with computer algebra programs. – Robin Ekman Oct 29 '14 at 17:32
• By the way, GR is not "the" theory invariant under diffeos. All covariant theories are invatiant under diffeos, in particular all topological theories are. – Urs Schreiber Oct 29 '14 at 19:26
• @UrsSchreiber: This is not true. Newtonian mechanics is covariant under the group of space rotations + t′=t but not covariant under general diffeomorphisms. e.g. take $x\rightarrow x'=x+at^3;t\rightarrow t′=t$ which has non-vanishing jacobian but Newtonain mechanics is not form-invariant under this transformation – image Mar 22 '15 at 3:07
• bah, "generally covariant" theories. – Urs Schreiber Mar 24 '15 at 13:33

Note that all invariants of GR (e.g the curvature scalar $R$) are absolute in this sense (as invariants).