Trying to derive the infinitesimal time dilation relation , where is the proper time, 𝑡 the coordinate time, and the time dependent Lorentz factor. The derivation is trivial if one starts by considering the invariant interval , but it should be possible to obtain the result considering only Lorentz transformations. So, in my approach I am using two different reference frames will denote an intertial laboratory frame while will be the set of all inertial frames momentarily coinciding with the observed particle, i.e. the rest frame of the particle. These frames are related by
where is some nonconstant (i.e. time dependent) parameter which is, hopefully, the velocity of the particle in the laboratory frame. Treating , and as independent variables (for now) and taking the differential of the above relations,
Imposing either the definition of the rest frame or (what should be equivalent) , the only way in which i obtain is if . So, the derivation breaks badly at some point or I must be wrong in using some of the above equations. Which one is it?