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 socalled 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 socalled 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  v^{2}/c^{2})^{½} . 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): τ^{2} = t^{2}  x^{2}
= γ^{2} (t  vx)^{2}
 γ^{2} (x  vt)^{2}
= γ^{2} (t^{2}
 2vxt + v^{2}x^{2})  γ^{2} (x^{2}  2vxt
+ v^{2}t^{2}) = γ^{2}
(t^{2}  2vxt + v^{2}x^{2}  x^{2} + 2vxt  v^{2}t^{2}) =
γ^{2} (t^{2}  v^{2}t^{2 }+ v^{2}x^{2}
 x^{2} ) = γ^{2}
(1 v^{2}) (t^{2}  x^{2}
) = t^{2}  x^{2} .
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