Suekdccia said:
Summary: Are there any types of spacetime where no symmetries, not matter how fundamental, would be valid?
We derive the most basic laws of physics from several fundamental symmetries (those from Noether's theorems, gauge symmetries, Lorentz symmetry...). But are there any types of spacetime where no symmetries, no matter how fundamental, would be valid? Any special metric, geometry or shape?
It depends to some extent of what boundaries you place on the definition of "spacetime".
You can create a mathematical structure that has or lacks any symmetries you want and call it a "spacetime".
But, if you want your mathematical structure to have any resemblance whatsoever to what we usually mean when we are talking about "spacetime" then the outer boundaries of what is necessary to be consistent with that understanding of what a "spacetime" is will set limits on what properties your mathematical description of spacetime may have and still be consistent with that definition.
For example, any reasonable definition of spacetime really ought to encompass the idea that it can contain a particles or objects that can move from one place in the spacetime to another place in the spacetime according to well defined physical laws.
Most of the other posts in this thread have implicitly assumed boundaries on what could legitimately be called a spacetime that are quite a bit more demanding and restrictive than what I believe an ordinary common sense understanding of the word spacetime could plausibly include.
Certainly, it is absolutely possible to have a theoretical construct that fairly fits a common sense understanding of what a "spacetime" is that could have symmetries which are different in some way from the symmetries that we attribute to the real world in general relativity and the Standard Model. The core theories of physics are not unique in provide potentially rigorous models of the way that any conceivably imaginable possible universe would have to be.
For example, one could easily devise a hypothetical spacetime in which general and special relativity do not exist and there is no fixed speed of light, that is otherwise fairly familiar. We call that spacetime a real four dimensional Euclidian space in which Newtonian mechanics applies.
Likewise, one would imagine a theoretical construction that has symmetries different from the commonplace gauge symmetries that we believe really govern the physical world, that is still consistent with our common sense understanding of what a "spacetime" is.
Noether's theorem is another matter, however, because that isn't a symmetry itself. Instead, Noether's theorem is a meta theory about symmetries. Noether's theorem basically states in a very general sense that has very few conditions that must be satisfied for it to apply, to oversimplify, that in a law of physics, a conserved quantity always has a corresponding symmetry, and that a symmetry always has a corresponding conserved quantity. So, what Noether's theorem means is that if you have a spacetime in which there are no symmetries, then you must also have a spacetime in which there are no conserved quantities.
It would stretch the common sense meaning of the concept of "spacetime" to include within that definition, a class of possible theories that has both no symmetries of any kind, and as a consequence of Noether's theorem, no conserved quantities either.
While I am not able to rigorous prove that it is impossible to do so (indeed, you can't even begin to rigorously come to any conclusion until you pin down a definition of what must be true for something to be a spacetime), I have a very hard time imagining a mathematical construct of that kind that could reasonably be described as a spacetime. I would be willing to entertain a suggestion, from someone more clever than I, about how that could be possible, but I just don't see it.
