Covariance in special and general relativity

Question

I am self-studying SR and GR and need to make sense of the covariance principle. I understand the idea that physical principles should have no preference in coordinates and therefore must be expressed in a way such that it continues to be valid under a coordinate transformation from the 'covariance group'. In SR, the covariance group is given by the Poincare Group (Lorentz transformation, rotations) and in GR, the covariance group consists of all differentiable and invertible coordinate transformations.

Why is it that the physical laws of SR do not have as big of a covariance group as GR, as SR should be a local version of GR? I feel like I might be mixing up the two covariance principles but cannot figure out how.

Answer 1

The "principle of general covariance" as it is usually stated is a pretty vacuous statement.

Basically any theory can be made generally covariant. The biggest hurdle is that special relativity uses an affine space (Minkowski spacetime is basically a special pseudo-Euclidean affine space) to model spacetime, and affine spaces admit position vectors separating two distant points. Differentiable manifolds are only infinitesimally affine (they admit position vectors separating infinitesimally close points), so if you wish to provide a formulation of SR which is generally covariant, you need to get rid of all objects that rely on the affine structure. This is pretty much always possible however, although the formulation can be complicated.

The primary difference between SR and GR is not really in the allowed group of transformations, since SR can be made diffeomorphism-invariant. The difference is the existence of background structures.

In SR, one is always given the Minkowksi-metric $\eta$ as an eternal and unchanging background structure, therefore physics in SR can be described in a way that is adapted to this background structure. Hence the Poincaré group which is the isometry group of the background metric. If one so desires, using the Poincaré group as a symmetry group instead of the diffeomorphism group can be seen essentially as a gauge-fixing adapted to this preferred background structure. Since the background structure is known, we know that such a gauge-fixing is possible.

On the other hand in GR, the metric $g$ is a dynamical object one has to solve a complicated nonlinear partial differential equation for. The local geometry of the manifold is not fixed from the onset, and is only given once the Einstein Field Equations are solved. Moreover, different solutions to the EFE have widely different isometry groups and some do not have any at all (aside from the trivial group consisting of the identity element).

This means that in GR, such a simplification from the onset is not possible, and one has to use a language that can accomodate any local geometry. In SR, the local (and also global) geometry is known beforehand, therefore it is possible to use a language specifically molded to this background geometry.

But the real difference between the two is not "general covariance" or any other vacuous, nearly-content-free statement, but the existence or lack of thereof of "God-given" background structures.

Answer 2

Special relativity actually holds locally in GR (i.e. on length scales much smaller than $R^{-1/2}$ where $R$ is the Ricci scalar or any of the components of the Riemann curvature tensor in an orthonormal basis).

In particular, special relativity says that the metric and all the equations physics will be the same in any Lorentz frame. Imposing it locally is a very special restriction on any spacetime. It implies causality, the constancy of the speed of light, and other important properties. To be more precise, the proper time of an observer and proper lengths are things that are physically measurable, using clocks and meter sticks. Special relativity says that observers moving on different worldlines can use the same equations in terms of their proper coordinates.

On the other hand, general covariance just allows you to choose any coordinates for your manifold. It gives the laws of physics a geometric interpretation without reference to coordinates. These coordinates do not have to correspond to proper coordinates of any observer. Nevertheless, the local proper time and proper distance is still a thing that can physically defined for any local observer moving on a timelike worldline. But general covariance on its does not on its own imply that the equations will be the same in terms of these proper coordinates.

Answer 3

Why is it that the physical laws of SR do not have as big of a covariance group as GR

Simply put, SR is a symmetry breaking phase of GR. Thus there is less symmetry for SR compared with GR.

A close analogy is the Higgs mechanism. The metric $g_{\mu\nu}$ plays the role of the Higgs doublet field $H$.

In the non-symmetry breaking phase, i.e. GR, the metric $g_{\mu\nu}$ is not fixed, any value is allowed, just as the Higgs field $H$ in the non-symmetry breaking phase.

In the symmetry breaking phase, the metric $g_{\mu\nu}$ is fixed to the Minkowsky metric $g_{0\mu\nu}=\eta_{\mu\nu}$, just like the Higgs field $H$ aquires a VEV $H_0$. In the Higgs case, the original symmetry of $$ SU(2) * U_Y(1) $$ is broken down to $$ U_{EM}(1). $$ In the GR/SR case, the original diffeomorphism and local Lorentz symmetries are broken down to the global Poincare symmetry.