Re: Space-time interval...

On 2024-08-13 13:10:33 +0000, Richard Hachel said:

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

Both sign conventions are used. It doesn't matter as long as one knows
which one is used. The information content is the same anyway.
If you represent vectors and position defferences with quaternions then
the real part of the square of the quaternion is ds² according to the
sign convetion that I used. But quaternions are rarely used in this
context so that is not important.

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.

The useful concepts are proper duration and proper distance. They are
related to ds², which therefore is at least interesting.

Physicists start from this formula, which is a little more complex than Hachel's, which is:
To²=Tr²+Et²

Physicists don't start. They started when they were students. Now they
continue from what they and others have already achieved.

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".

Units of measurement are not needed for the theory.
In the usual formulation the invariance of the space-time interval is proven
from empirically validated postulates.

Hachel replaces with "invariance of proper time", which is pure evidence, like a swallow is a swallow.

It is not really a replacement. Proper duration and proper distance are
just terms used in certain situations for the more general concept.
--
Mikko