Proving that timelike and spacelike spacetime intervals are invariant across inertial frames

I'm trying to understand the justification for using the Minkowski metric. It's clear to me that it's the natural choice of metric given that spacetime separations denoted by $$(-c^2\Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2)$$ are invariant across inertial frames of reference. So the next step is to show that spacetime interval is invariant.

It's also clear that lightlike spacetime intervals are all $$0$$ and hence invariant. In other words, if $$\Delta s^2=0$$ in one frame, then it's $$0$$ in all the other IRFs as well.

But it's not obvious to me how to show that spacelike and timelike spacetime intervals are invariant as well (without assuming Lorentz transformation). I've looked at Why does Minkowski space provide an accurate description of flat spacetime? and Physical reasons for metric definition in special relativity, and as good the answers are to those questions, I still didn't find what I was looking for.

Would appreciate any guidance how to prove that spacelike and timelike spacetime intervals must be invariant. [Tried googling for it myself as well, but no luck]

I think you are asking for too much here. You say that you want to demonstrate that $$-c^2\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$ is invariant, but invariance is a meaningless label unless you specify what kinds of transformations you are considering.

From a mathematical standpoint, there would be nothing inconsistent about taking the group of symmetry transformations between reference frames to be $$SO(4)$$, and taking the spacetime metric to be $$\operatorname{diag}(1,1,1,1)$$. To recognize that this is not a good model for the universe we inhabit, we need physical input.

That input comes in the form of Lorentz symmetry. You mention in a comment

But I'm looking for a different approach in which we don't use the LT. As far as I know, it should be possible. Using isotropy, homogeneity and the principle of relativity, we can conclude that transformation between IRFs is Galilean or Lorentz. To finally conclude that it's indeed Lorentz, I need to show that the metric is Minkowski, and for that I need to show interval invariance.

A Galilean transformation does not generically preserve the light-like interval, meaning that a light ray moving in one frame has a different speed from a light ray moving in another. In order for you to have an invariant speed, one must choose the Lorentz transformations, in which the invariant speed enters as a free parameter.

This is the physical input needed. Since we observe that light travels at an invariant speed, then we can immediately conclude (a) that there is an invariant speed in the first place, so the proper symmetry transformations are Lorentz, and (b) that the parameter which appears in the Lorentz transformation is $$c$$. From here, the metric must be covariant under Lorentz transformations, which leads you to the Minkowski metric.

• Thanks so much! That's a good explanation, even though I totally disagree on that I'm "asking for too much". Physical input was exactly what I'm looking for and I know that it's not possible to prove invariance in a strict mathematical sense (without assuming a transformation first). I guess my wording in the question wasn't clear - I apologize if it wasn't. I do think it's possible to prove invariance with the help of physical considerations even if an explicit transformation isn't specified. Jun 11 '20 at 17:26
• @ShirishKulhari When I say "asking for too much", I mean that you appeared to be asking for a mathematical justification for the Minkowski metric without talking about the Lorentz transformations. There is no such justification because the universe could in principle have a different structure - it's just observed not to. Jun 11 '20 at 17:29
• Ah I see. My bad for misinterpreting your comment. Mathematical justification for the Minkowski metric certainly isn't possible without knowing about LT first. But maybe I could argue there's a physical justification. If I know that the quantity that we call spacetime interval is invariant, then it makes sense to choose Minkowski metric (even though that's far from a rigorous justification). Choosing any other metric would be silly. Even given the LT, we're still implicitly assuming that all IRFs are characterized by the same metric and we arrive at Minkowski since... (cont'd) Jun 11 '20 at 17:32
• (cont'd)... that's the metric preserved by LTs. Nothing's stopping me from choosing some arbitrary metric that changes from frame to frame, but that'd be downright silly. Jun 11 '20 at 17:33

To perform such a calculation, you have to think on how you can relate the spatial and time coordinates from one reference frame to the other. The answer is here. Try to think before clicking the link.

• Thanks! So I'm aware of the Lorentz transformation. It can be derived from the postulate that the speed of light is constant, and I can show the invariance of the interval using the LT. That's one approach. But I'm looking for a different approach in which we don't use the LT. As far as I know, it should be possible. Using isotropy, homogeneity and the principle of relativity, we can conclude that transformation between IRFs is Galilean or Lorentz. To finally conclude that it's indeed Lorentz, I need to show that the metric is Minkowski, and for that I need to show interval invariance. Jun 11 '20 at 17:01
• Okay. So by writing a spacetime interval in the form $\Delta s^2=-c^2\Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2$ you are already assuming that the Minkowski metric is being used, in which the line element has a value of $ds=-cdt+dx+dy+dz$. Jun 11 '20 at 17:05
• My bad, I should've been clearer. I have edited the question accordingly. Jun 11 '20 at 17:05
• Fair point - I see how that can be misleading. Let's not call it "spacetime interval" at this juncture then. Let's just say that I want to prove that $-c^2\Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2$ is invariant. And let's say I'm not claiming that it's a "distance" of any sort - it's just some quantity. Jun 11 '20 at 17:07
• I still don't understand why you would want to prove this sort of thing in a different way, because as you said LTs arise naturally when assuming the relativity principle. Maybe I still don't understand your question well. But coming back to your first comment, if you assume the principle of relativity there is no other transformation than LT to be made, for Galileo's transformation adds velocities without regard of an absolute upper limit. Jun 11 '20 at 17:14