# Spacetime interval and basic properties of light

• I
• HansH
In summary: Well, that's a long and complicated story. It is a consequence of the invariance of light speed, but it is not easy to see why. In summary, the space time interval is defined as ds^2=(cdt)^2-(dx^2+dy^2+dz^2), and it comes from the Lorentz Transformation and the invariance of the speed of light. This is a fundamental concept in relativity, and understanding its derivation can help deepen our understanding of the theory. Some sources start with the interval and derive other properties, while others start with the invariance of light speed. Either way, understanding why something is defined a certain way is important in fully grasping a concept.
I think Carroll means the surface of a hyper-cylinder (or whatever the name for it is), where every ##\mathbb{S}^2## is identical and stacked one on top of another, not ever-smaller spheres nested one inside the other, which is what I think you mean. Every point is identical (diffeomorphic to any other), but not every direction.

Ibix said:
I think Carroll means the surface of a hyper-cylinder (or whatever the name for it is), where every ##\mathbb{S}^2## is identical and stacked one on top of another, not ever-smaller spheres nested one inside the other, which is what I think you mean. Every point is identical (diffeomorphic to any other), but not every direction.
Yes, if we take in account (as required) the metric. However --from just a topological point of view-- I believe the 'infinite spherical onion with internal hole' might make sense though.

Ah ok, on the hypercylinder surface the homogeneity is 'encoded' by the diffeomorphism between every pair of points on it.

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