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Here is an explanation of why the spacetime interval $c^2t^2-x^2-y^2-z^2$ is invariant under Lorentz transformation in Wikipedia. Link of the proof. I kind of understand it, but the explanation does not mention anything specific about the speed of light $c$. No properties of $c$ are used explicitly. I am confused by this because the proof still seems to work even if I replace $c$ with any other constant.

Which steps in the proof use the property of $c$ ? If I am not clear enough, Why the proof become invalid if I replace $c$ with another constant in the proof?

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Well you should notice that you never see a $c_2$ or $c'$ floating around. It's just $c$. i.e. it doesn't depend on the reference frame. This is a property of the speed of light we see in the universe.

I am confused by this because the proof still seems to work even if I replace c with any other constant.

The Lorentz transformation and invariance of the space-time interval have nothing to do with the specific value of the speed of light. It's valid mathematically for any constant speed $c$. The only reason we would want to use the right numerical value for $c$ is so that we can actually use the math to describe the universe we find ourselves to be in.

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  • $\begingroup$ Thank you a lot! But I have one more concern. The proof assumes homogeneity of spacetime and isotropy of $ds$. It is reasonable to require "physics laws" to satisfy homogeneity of spacetime, but how can I know that $ds$ is a "physics law"? I mean, we cannot show that $dx$ is invariant since it is NOT a "law of physics". $\endgroup$ – Ma Joad Feb 19 at 11:06
  • $\begingroup$ @MaJoad I'm unsure what you mean. We define the Lorentz transformation. We define what $\text ds$ is in a given frame. Based on these two things we can show that $\text d s$ between two events does not depend on the reference frame. It is an assumption of physics that the laws of physics are true at all points in the universe (since no experiment to date has shown otherwise). $\endgroup$ – Aaron Stevens Feb 19 at 17:01

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