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Doesn't the Lorentz invariant already pretty much take care of the relativity of time? As long as we're using the Lorentz invariant, we're free to reparameterize the universe any way we want, and our description will be the same. So I don't see what this Barbour guy is going on about, it seems like standard physics. Whether you write your function f(x,t) or f(y) where y = g(x,t) or even just f(x) where t = h(x) is totally irrelevant to the universe. It's just another coordinate transformation just like translating the whole universe by ten meters to the left.
Now, if you have a new invariant to propose, THAT would amount to an actual change in the laws of physics.
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