Ibix said:
Are you sure about that? I would take that to mean that you want to replace the (cdt)2 term in the interval with a (Vdt)2 term, which wouldn't be invariant.
The point of the conversion constant is so that you can
"add" space-components and time-components...
so use "x and Ct" or "x/C and t" [or even "Ax and Bt" so that Ax and Bt have the same units].
When I said use "any speed", i mean a constant-value with units of speed, like the "speed of light", or "speed-of-sound" or even "10m/s".
If you choose "speed of light", then your equations look simple...
if you choose any other, then your equations carry a dimensionless constant [which doesn't transform] like "speed of light"/"speed of sound",
which will be an annoyance but not incorrect.
So, as you wrote, "Vdt" would be bad because it transforms... but "(10m/s) t" is fine but annoying.
I raise this issue because when I teach relativity, using Galilean relativity, as bridge from Euclidean geometry to Minkowski geometry,
I express my distances in units of "x/c" and and times in "t", where c is the speed "3e8 m/s"... even though c has no real importance in Galilean relativity.
In other words, instead of using "meters", I use "light-seconds"...even in Galilean relativity.
(Note: I can't use the invariant speed of infinity in Galilean relativity.)
Then, in my method, I can more easily make the analogies between the three geometries.