- #1

sophiatev

- 39

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- TL;DR Summary
- Just want to confirm my understanding that the length of four-vectors, and their dot products, have the same value in all frames, not just inertial ones.

I know that the mathematical form of the line element of spacetime is invariant in all inertial reference frames, namely

$$ds^2 = -(cdt^2) + dx^2 + dy^2 + dz^2$$

From what I understand, the actual spacetime distance between two events is the same numerical quantity in all reference frames, inertial or not. It is not however the case that the line element describing distances will have this mathematical form in all frames.

Going off of this, if we have a four-vector connecting two events in spacetime, then its length is the spacetime distance between those events. Its length should be the same as calculated in all frames, not just inertial ones. And similarly, since the vector describes a physically immutable concept, operations such as scaling it, calculating its length, or adding and dotting it with other vectors should have the same result in all frames, right? The reason I ask is that in Hartle's

$$ds^2 = -(cdt^2) + dx^2 + dy^2 + dz^2$$

From what I understand, the actual spacetime distance between two events is the same numerical quantity in all reference frames, inertial or not. It is not however the case that the line element describing distances will have this mathematical form in all frames.

Going off of this, if we have a four-vector connecting two events in spacetime, then its length is the spacetime distance between those events. Its length should be the same as calculated in all frames, not just inertial ones. And similarly, since the vector describes a physically immutable concept, operations such as scaling it, calculating its length, or adding and dotting it with other vectors should have the same result in all frames, right? The reason I ask is that in Hartle's

*Gravity: An Introduction to Einstein's General Relativity,*when he introduces four vectors in chapter 5, he claims that these operations are invariant in inertial reference frames, but does not speak more generally about all reference frames. He claims that the scalar product is the same in all inertial frames, that vectors orthogonal in one inertial frame are in all, etc. I just want to confirm that I'm correct in assuming these statements are true in every frame, inertial or not.