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We need a way to move from our coordinates (x, t) to those of one of those moving clocks (x’, t’), such that the invariance of proper time τ is maintained. The so-called Lorentz transformations are just what we will need. It is helpful to recall that in classical physics and in ordinary experience we rely on the so-called Galilean transformations between the coordinates (x, t) and (x’, t’) of two inertial observers O and O’ moving at constant relative velocity v: Galilean transformations: x’ = x - vt and t’ = t. Notice that the universal, global present is represented by the second equation or, equivalently, all clocks tick at the same rate. But we know this transformation is wrong empirically. Instead we will use the Lorentzian transformations between the same coordinates (x, t) and (x’, t’) of two inertial observers O and O’ moving at constant relative velocity v: Lorentz transformations: x’ = γ (x - vt) and t’ = γ (t - vx) where γ = (1 - v2/c2) . Using a bit of algebra, we can show that these Lorentz transformations manifest relativistic invariance: spacetime interval τ between two events in spacetime will be measured the same by all inertial observers, as Einstein requires and as we have seen before to be true experimentally, even though the individual space and time measurements will differ(as is routine, set c=1 by redefining scales for convenience): τ2 = t’2 - x’2 = γ2 (t - vx)2 - γ2 (x - vt)2 = γ2 (t2 - 2vxt + v2x2) - γ2 (x2 - 2vxt + v2t2) = γ2 (t2 - 2vxt + v2x2 - x2 + 2vxt - v2t2) = γ2 (t2 - v2t2 + v2x2 - x2 ) = γ2 (1- v2) (t2 - x2 ) = t2 - x2 .

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