# Properties of space-time intervals derived by symmetry principles

In many books about special relativity I found the following arguments:

1) because the transformations between two inertial reference frames $K$ and $K'$ are linear and because if $\Delta s^2 = 0$ we find also $\Delta s'^2=0$ then it has to be $\Delta s^2 = \lambda \Delta s'^2$

2) because space-time is uniform and space is isotropic the constant $\lambda$ can depend at most on $|\vec v|$, where $\vec v$ is the velocity of $K'$ with respect to $K$

I know that the consequences of those reasoning are right but I can't see how they can be demonstrated and if, stated in that way, they are undoubtly right.

• +1 because this seems like a reasonable question, but I don't understand why these books (which ones?) are approaching this in such a strange way. When you apply the metric to a vector, you get a Lorentz scalar, which can't depend on what coordinate system you chose. That's the definition of a scalar. To me that seems like a very basic requirement, more basic than any other requirement I can think of. – Ben Crowell Feb 16 '18 at 0:32
• Physically, the intervals tells us, for a timelike world-line, the proper time of a clock that moves along that world-line. That clearly can't depend on the coordinates we choose. If we constructed some coordinate expression for the interval, and it turned out to be coordinate-dependent, then that would just be an indication that our expression was a mistake. – Ben Crowell Feb 16 '18 at 0:34
• Thank you Ben, as a reference look at the chapter on special relativity by Jackson's Classical Electrodynamics. I believe the reason why they are approaching the problem this way is that they are trying to show, using only symmetry principles, that the transformations between different inertial frames have to preserve spacetime intervals. – L.R. Feb 16 '18 at 7:08