Deriving the Spacetime Interval

[Anonymous No.9827121]

How does one prove the invariance of the spacetime interval in flat spacetime?

I see many proofs for its invariance in one spacial dimension that rely on some constant value in another spatial dimension (pic related), but what about proving it for three spacial dimensions, where you can no longer use some spatial constant.

Is there a clean geometric proof or something similar? I would prefer if it could be proven without the Lorentz transformations because I feel that would be less intuitive.

[Anonymous No.9827134]

I don't understand what you want to prove, if you are asking for an explicit computation showing that the the interval is invariany after a lorentz tranaformation, then just plug and chug a general lorentz transformation and compute the interval.

[Anonymous No.9827152]

>>9827134
That is what I'm meaning to prove.
I'm teaching this to some kids who wanted to learn, but I'm deeply unsatisfied with how I learned about the spacetime interval (in the pic) and "plug and chug" sounds less enlightening.
If I must do it that way then I guess I'm stuck with it.

[Anonymous No.9827165]

>>9827121
Any reference frame in 3+1 dimensions can be reduced to the 1+1 dimensional case through a spatial rotation.

Essentially, even though the velocity vector is 3 dimensional, it can be made to point along the x-axis by a change of basis.

[Anonymous No.9827170]

>>9827165
Wow, I'm dumb.

I think I was stuck on the idea the motion could be some curve that required all spacial coordinates to be included, but I realize now that would require acceleration and therefore not be included in special relativity.

[Anonymous No.9827186]

>>9827165
How though can this be generalized to work for something moving at any speed?
This visual thought experiment is reliant on a specific instance of light itself moving, right?