I'm teaching myself a little about holonomy groups in the context of general relativity. This paper by Hall and Lonie classifies a lot of the possibilities for simply connected spacetimes in 3+1 dimensions. It helps my intuition to look at examples, and there are only a few in the Hall paper:

  • Minkowski space: the trivial group
  • FRW spacetimes: the full Lorentz group
  • gravitational plane wave: $R_8$ (see below about notation)

What about the Schwarzschild spacetime? My guess, without formal proof, is that its holonomy group is the full Lorentz group, since we have the geodetic effect (giving rotations), and there can be conjugate points (which seems like it would give boosts). (If the part about boosts is correct, then we automatically get the rotations as well, since the composition of two boosts will in general include a rotation.)

Are there any other good examples of fundamental physical interest? It seems like nearly all examples have the full Lorentz group as their holonomy group.

A side issue is the interpretation of the $R_i$. Hall gives a brief description in terms of bivectors in a certain basis, with a reference to a paywalled 1962 paper by Schell. Wikipedia gives a different set of notations, presumably describing the same groups, in terms of some generators $X_i$, referencing a more recent book by Hall.


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