- #26

Hurkyl

Staff Emeritus

Science Advisor

Gold Member

- 14,916

- 19

(1) The most commonly used is that at each individual point, the metric has signature +---. This means the tangent space at a point has the same geometry as (the tangent space to) Minkowski spacetime. However, any region of space-time may exhibit (second-order) deviations from the predictions of special relativity.

(2) The other reasonable notion is that each point in space-time is contained in a region that has the same geometry as Minkowski spacetime. Within such regions, there is absolutely no deviation from the predictions of special relativity.

(3) The inverse notion is that of being "globally Minkowski" -- the entirety of spacetime has the geometry (and topology) of Minkowski spacetime.

I am pretty sure that most people mean (3) when they talk about special relativity. I would like to emphasize that (2) is more general than three, even in 'completed' spacetimes: for example, you can have cylinder-like and torus-like spatial slices. In such spacetimes, you get additional

*global*phenomena that arise from the topology.

Forgetting about these global phenomena that distinguish case (2) from case (3) can lead to paradoxes, such as the cosmological twin paradox.