Leo wrote: really appreciate the effort you put in your explanations, Mowgli. But I suppose I didn’t understand some bit, because I merrily do integration on a Minkowski diagram, watching the transformed frame go through all kinds of states depending on the velocity of the observed object, and summing up the results onto the “fixed frame” – or back to the deforming frame. I do it visually and it doesn’t hurt my logic – at least the Minkowskian spacetime logic, which is but one interpretation of the Lorentz transforms. The fact that I understand it visually doesn’t necessarily mean it is possible mathematically, of course. I’ll have to re-read your posts several more times.
The reason why you feel it’s okay to integrate over time is what I meant by the semantic context of time. You kind of visualize that at every instant of “time,” there is a particular velocity and a particular set of LT equations, kind of ignoring the fact that time is being transformed. May be you could consider a situation where there is no acceleration, but the velocity is a function of x. Would you still consider it okay to do the integration?
CCC observed that the rotation analogy is perhaps not too accurate because of the periodicity of the x-dependent transformation. You could consider the small angle approximation (cos -> 1 and sin -> angle) to take out the periodicity. The warping of the transformed xy plane doesn’t go away.
And cincirob argues that the equation for time interval may be immune to the warping of the underlying geometry. But time interval is just the difference of two time coordinates in the transformed space. The point really is that coordinates do not exist once the transformation is non-linear. Only some affine connections describing the metric tensor exist. Since the time dilation equation (as applied to a time interval) comes from a “coordinate” transformation that’s no longer valid, the only sensible thing to do is what Einstein did – discard SR when there is acceleration and go for a more general theory.
I wish I could explain this better, but unfortunately, I’m pretty close to the limits of my knowledge of GR May be someone who has solved some problems in GR and has an intuitive understanding of non-linear geometry can step in and say more?