# Is General Theory of Relativity a Gauge Theory? Trajectories of Diffeomorphism.

Historically the problem of observables in classical and quantum gravity is closely related to the so-called Einstein hole problem, i.e. to some of the consequences of general covariance in general relativity (GTR).

The central question is the physical meaning of the points of the event manifold underlying GTR. In contrast to pure mathematics this is a non-trivial point in physics. While in pure differential geometry one simply decrees the existence of, for example, a (pseudo-) Riemannian manifold with a differentiable structure (i.e., an appropriate cover with coordinate patches) plus a (pseudo-) Riemannian metric, g, the relation to physics is not simply one-one. In popular textbooks about GTR, it is frequently stated that all diffeomorphic (space-time) manifolds, M are physically indistinguishable. Put differently:

S − T = Riem/Diff —– (1)

This becomes particularly virulent in the Einstein hole problem. i.e., assuming that we have a region of space-time, free of matter, we can apply a local diffeomorphism which only acts within this hole, letting the exterior invariant. We get thus in general two different metric tensors

g(x) , g′(x) := Φ ◦ g(x) —– (2)

in the hole while certain inital conditions lying outside of the hole are unchanged, thus yielding two different solutions of the Einstein field equations.

Many physicists consider this to be a violation of determinism (which it is not!) and hence argue that the class of observable quantities have to be drastically reduced in (quantum) gravity theory. They follow the line of reasoning developed by Dirac in the context of gauge theory, thus implying that GTR is essentially also a gauge theory. This then winds up to the conclusion:

Dirac observables in quantum gravity are quantities which are diffeomorphism invariant with the diffeomorphism group, Diff acting from M to M, i.e.

Φ : M → M —– (3)

One should note that with respect to physical observations there is no violation of determinism. An observer can never really observe two different metric fields on one and the same space-time manifold. This can only happen on the mathematical paper. He will use a fixed measurement protocol, using rods and clocks in e.g. a local inertial frame where special relativity locally applies and then extend the results to general coordinate frames.

We get a certain orbit under Diff if we start from a particular manifold M with a metric tensor g and take the orbit

{M, Φ ◦g} —– (4)

In general we have additional fields and matter distributions on M which are transformd accordingly.

Note that not even scalars are invariant in general in the above sense, i.e., not even the Ricci scalar is observable in the Dirac sense:

R(x) ≠ Φ ◦ R(x) —– (5)

in the generic case. Thus, this would imply that the class of admissible observables can be pretty small (even empty!). Furthermore, it follows that points of M are not a priori distinguishable. On the other hand, many consider the Ricci scalar at a point to be an observable quantity.

This winds up to the question whether GTR is a true gauge theory or perhaps only apparently so at a first glance, while on a more fundamental level it is something different. In the words of Kuchar (What is observable..),

Quantities non-invariant under the full diffeomorphism group are observable in gravity.

The reason for these apparently diverging opinions stems from the role reference systems are assumed to play in GTR with some arguing that the gauge property of general coordinate invariance is only of a formal nature.

In the hole argument it is for example argued that it is important to add some particle trajectories which cross each other, thus generating concrete events on M. As these point events transform accordingly under a diffeomorphism, the distance between the corresponding coordinates x, y equals the distance between the transformed points Φ(x), Φ(y), thus being a Dirac observable. On the other hand, the coordinates x or y are not observable.

One should note that this observation is somewhat tautological in the realm of Riemannian geometry as the metric is an absolute quantity, put differently (and somewhat sloppily), ds2 is invariant under passive and by the same token active coordinate transformation (diffeomorphisms) because, while conceptually different, the transformation properties under the latter operations are defined as in the passive case. In the case of GTR this absolute quantity enters via the equivalence principle i.e., distances are measured for example in a local inertial frame (LIF) where special relativity holds and are then generalized to arbitrary coordinate systems.