Georgina,
I already mentioned that Crothers' sigma=1 corresponds to v=0. It does not matter whether or not Crothers directly grasped what Einstein tacitly did when he applied Poincaré's two-way method of synchronizing that was successfully used in practice on condition v = 0.
When Einstein claimed that this method is generally valid he was evidemtly wrong in case of relative motion (v =|= 0) between A and B. In order to understand this you don't need Galilean and Lorentz transformation, just the simple and plausible redshift/blueshift reasoning by Christian Doppler.
If the clock of A emits a timespan and the distance between A and B increases, then B receives an increased timespan (redshift): In case of decreasing distance, B receives a shorter timespan (blueshift). A and B mutually "see" their clocks run differently plus or minus and by more than "time dilution" amounts.
The reason for Einstein to use Poincaré's two-way (round-trip) synchronization without any reasonable justification was to mathematically arrive at the transformation which was introduced by Lorentz and adopted by Poincaré in order to defend the aether thich was not found in Michelson's (and Michelson with Morley's) experiment. Mathematically seemingly flawless, Einstein's Relativity gave rise to useless controversy up to now.
Unfortunately, what has been called Relativity or invariance theories were confused with the experimentally well confirmed strong effect of Lorentz Gamma on the so called "relativistic" mass. Im that, Klingman is definitely on the right trace.
If you Georgina, are imagining an omnipresent ubiquitous now, then you should be aware that this is inconsistent with Einstein's (itself inconsistent) Relativity which is a tight bundle of useless paradoxical tenets including time dilution, length contraction, etc. Notice, I am capitalizing Relativity as to indicate that it got meanwhile a irrefutable belief.
EB