Re: Space-time interval...

Le 13/08/2024 à 13:38, Mikko a écrit :

In an orthogonal isometric coordinate system ds² = dt²-dx²-dy²-dz².

 Not really.
 ds² = dx²+dy²+dz²-dt²
But this formulation has little interest in special relativity, and I find it useless to teach it as is to students and high school students.
 Physicists start from this formula, which is a little more complex than Hachel's, which is:
To²=Tr²+Et²
There is little more to do than to place the units of measurement, and the whole theory holds up much more easily than the dogma of "the invariance of the space-time interval".
Hachel replaces with "invariance of proper time", which is pure evidence, like a swallow is a swallow.
This does not even need to be discussed or demonstrated.
We can have fun (uselessly) by posing:
To²=Tr²+Et²
then Tr²=To²-Et²
then if Tr²=-ds²/c then ds²=dl²-To²c²
But all this complicates things.
Let's just talk about the invariance of proper time (which cannot be unique, we cannot have two proper times) and all this will be much simpler and much truer.
Simpler, you understand. No more need for a puzzle with complex notions (i²=-1).
But above all, more fair in the end.
You have noticed that Paul B. Andersen has carried out a small study on the Tau Ceti traveler, where Bella is an astronaut who evolves in accelerated mode (10m/s²) on the 12 ly to cross.
Paul easily finds (a=1.052ly/y²) that in the terrestrial frame of reference Bella will take 12.9156 years since:
To=(x/c).sqrt(1+2c²/ax).
He is right.
But then he gets bogged down in complex considerations, and proposes a proper time for the rocket of about 3.5 years while Tr=4.776 years.
The complexity of the concepts then leads to dramatically false results.
 R.H.