The spacetime interval between two events
and
is defined as
If
is in reference frame
, then
is in reference frame
moving at a velocity
along
the x-axis. Therefore, to show that the spacetime interval is invariant under a Lorentz transformation we must show
with the reference frames related by The Lorentz transformation
The change in coordinates between events in the frame is then given by
Squaring the terms yield
Substituting these terms into the spacetime interval gives
Adding the first two terms with common denominators together yields
Pulling out a
Factoring out a
in the numerator
Finally, canceling terms gives
Hence, the spacetime interval is invariant under a Lorentz transformation.
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