Is it possible that the geometry of spacetime isn't Minkowskian, if we only take special relativity into account?

Question

If you don't consider general relativity, could you say that the geometry of spacetime might be different from how it's specified by the Minkowski metric? Or is Minkowski spacetime the only geometry that can fully portray the fundamentals of SR?

Answer 1

The question isn't very well defined. You can interpret special relativity in an obtuse way, if you desire, by saying that the special relativity doesn't deal with the geometry of spacetime. It's called Lorentz Ether Theory, and since it makes all the same predictions of special relativity it's not worth arguing about!

Other than that, any theory of special relativity will have $(ct)^2-x^2$ conserved, will have $E^2-(pc)^2$ conserved, etc., giving you all the same things corresponding to the Minkowski metric. The book Spacetime Physics by Taylor and Wheeler actually opens with a discussion about the invariant interval $(ct)^2-x^2$ discussed with a parable. It's not just like Minkowski spacetime portrays special relativity, it really is special relativity.

Answer 2

The Minkowski metric is precisely the metric that is left invariant be the Lorentz group $\mathrm{SO}(1,3)$, so the Minkowskian nature of spacetime is indeed a feature of Special Relativity alone.

Answer 3

When Maxwell discovered the equations for electromagnetic waves it was noticed that their symmetry group was the Lorentz group. It has always fascinated me the fact that if you "dualise" the d'Alembert operator $$\square = \frac1{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2$$ one gets precisely the Minkowski metric $$\text ds^2 = c^2\text dt^2 - \text dx^2 - \text dy^2 - \text dz^2.$$ Observe that the speed of light can't change because otherwise Maxwell's equations would change, so the Lorentz group has to preserve this, which is the same as to require that the metric must be preserved, and by definition the (full) Lorentz group is then $O(1,3)$. So we started from the mathematical description of electromagnetic phenomena and we ended up with (part of the) symmetry group of space-time (the other part is the semidirect product with the translations, which of course leave the metric invariant, and hence one ends up with the full Poincaré group $O(1,3)\ltimes\mathbb R^4$).