Once I had a long email discussion with a professor of logic who claimed to have found logical mistake in the deduction of time dilation formula. It was easy to find that he thought in terms of Newtonian space-time and this was of course in conflict with relativistic view. The logical error was his, not Einstein's. I tried to tell this. In vain again.
At this time I was demanded to explain why the 2 page article of Stephen Crothers (see this). This article was a good example of own logical error projected to that of Einstein. The author assumed besides the basic formulas for Lorentz transformation also synchronization of clocks so that they show the same time everywhere (about how this is achieved see this).
Even more: Crothers assumes that Einstein assumed that this synchronization is Lorentz invariant. Lorentz invariant synchronization of clocks is not however possible for the linear time coordinate of Minkowski space as also Crothers demonstrates. Einstein was wrong! Or was he? No!: Einstein of course did not assume Lorentz invariant synchronization!
The assumption that the synchronization of clock network is invariant under Lorentz transformations is of course in conflict with SR. In Lorentz boosted system the clocks are not in synchrony. This expresses just Einstein's basic idea about the relativity of simultaneity. Basic message of Einstein is misunderstood! The Newtonian notion of absolute time again!
The basic predictions of SR - time dilation and Lorentz contraction - do not depend on the model of synchronization of clocks. Time dilation and Lorentz contraction follow from basic geometry of Minkowskian space-time extremely easily.
Draw system K and K' moving with constant velocity with respect to K. The t' and x' axis of K' have angle smaller than π/2 and are in first quadrant.
- Assume first that K corresponds to the rest system of particle. You see that the projection of segment=(0,t') t'-axis to t-axis is shorter than the segment (0,t'): time dilation.
- Take K to be the system of stationary observer. Project the segment L=(0,x') to segment on x axis. It is shorter than L: Lorentz contraction.
This however raises a question. Is it possible to find a system in which synchronization is possible in Lorentz invariant manner? The quantity a2=t2-x2 defines proper time coordinate a along time like geodesics as Lorentz invariant time coordinate of light-one. a = constant hyper-surfaces are now hyperboloids. If you have a synchronized network of clocks, its Lorentz boost is also synchronized. General coordinate invariance of course allows this choice of time coordinate.
For Robertson-Walker cosmologies with sub-critical mass time coordinate a is Lorenz invariant so that one can have Lorentz invariant synchronization of clocks. General Coordinate Invariance allows infinitely many choices of time coordinate and the condition of Lorentz invariant synchronization fixes the time coordinate to cosmic time (or its function to be precise). To my opinion this is rather intesting fact.
What about TGD? In TGD space-time is 4-D surface in H=M4×CP2. a2= t2-r2 defines Lorentz invariant time coordinate a in future light-cone M4+ ⊂ M4 which can be used as time-coordinate also for space-time surfaces.
Robertson-Walker cosmologies can be imbedded as 4-surfaces to H=M4×CP2. The empty cosmology would be just the lightcone M4+ imbedded in H by putting CP2 coordinates constant. If CP2 coordinates depend on M4+ proper time a, one obtains more general expanding RW cosmologies. One can have also sub-critical and critical cosmologies for which Lorentz transformations are not isometries of a= constant section. Also in this case clocks are synchronized in Lorentz invariant manner. The duration of these cosmologies is finite: the mass density diverges after finite time.
For a summary of earlier postings see Latest progress in TGD.