# What does it exactly mean for spacetime to have no global symmetries?

Are there spacetimes or metrics with no global symmetries? Spacetimes/metrics with no global Poincaré, Lorentz, diffeomorphism, CPT, translational and gauge invariances?

And if there are, what does it exactly mean?

For example, I know that in certain expanding spacetimes (like the one representing our universe) the global time translation invariance is broken and therefore energy is not conserved (globally, or at cosmological scales)

If there are spacetimes with no global symmetries, would it mean that all these symmetries would break (or be violated) at global scales?

• It is easy to make things without global symmetries from symmetric things, by putting small (smooth) bumps at just a few places. Three suitable bumps on a two-sphere suffice. (Two identical bumps on a two-sphere still admit a finite group of symmetries.) Apr 8 at 20:51
• @paulgarrett so there are spacetimes and metrics without any global symmetries? Can you name a specific example? (Like some type of spacetime, metric or solution that has no global symmetries) Apr 9 at 11:00
• @paulgarrett also, what would this exactly mean? Just as some expanding spaces that do not have global time translational symmetries imply that they do not have energy conservation laws at global/large scales, would a spacetime with no global symmetries imply that all symmetries would be broken or violated also at global/large scales? Apr 9 at 11:04
• I only know some purely mathematical things about "spaces" with and without symmetries, so I'd certainly hesitate to comment on physics interpretations or aspects... Hence, just a comment. As another mathematical example, compact (connected) Riemann surfaces of genus 2 or more have only finitely-many automorphisms (=symmetries). Apr 9 at 15:17
• @vengaq Paul Garrett has given you a wonderful mathematical answer. If you wish to know a cosmologist's perspective, please beware that cosmologists are more likely to be met at StackExchange Physics rather than Astronomy. You may consider placing your question there. Apr 10 at 13:48