mhill said:
There was a rumour in scientific community that Poincare actually knew that space and time were relative , and derived the Lorentz transforms and Hamiltonian before Einstein.
I'm not sure what you mean by "Hamiltonian" in this context, but the Lorentz transforms were surely derived -- or at least written down -- first by Lorentz. That's why they're called "Lorentz" transforms, rather than "Einstein" transforms.
Einstein doesn't name the transformation in the main text of his 1905 electrodynamics paper, as far as I can recall, but in the footnotes, which he added in 1923, he calls it the "Lorentz transformation".
Certainly, relativity of simultaneity is mentioned in the Poincaré paper from 1900. However, that paper as a whole does not present a clear, coherent picture of how it all should hang together; it stands in stark contrast to Einstein's paper of 5 years later, which presents a solid edifice in which everything fits. In 1900, Poincaré was still trying to see how to fit the pieces into an aether theory. (Poincaré's 1905 paper, which I've only just started looking at, seems to take it a lot farther, but since it was apparently only published in 1906 it's hard to see how anyone could claim it predates Einstein's 1905 SR paper.)
My general impression is that the thing Einstein "brought to the table" was a powerful intuition which allowed him to see what the mathematics
meant. The mathematical machinery for relativity was already largely in place by 1900 or so, but it was Einstein who first realized what to do with it. This seems to be particularly true in the case of general relativity, where Riemannian geometry had been known since -- well, since Riemann invented it -- but what had been lacking was the insight to see how to apply it to the problem of gravitation. It's often mentioned that Hilbert derived the field equations about the same time as Einstein, or perhaps slightly sooner, and did it more elegantly than Einstein, but what isn't always mentioned is that Hilbert got the inspiration to
look for the equations from
Einstein. In other words, it was Einstein who first stated the problem clearly enough so that others could see where to look for the solution. Or at least that's what I've read. (Sorry, can't cite a reference on it.)