spacetime interval is invariant for a Lorentz transformation
(Result)
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.
References
[1]Carroll, Bradley, Ostlie, Dale, An Introduction to Modern Astrophysics.
Addison-Wesley Publishing Company, Reading, Massachusetts, 1996.
[2]Cheng, Ta-Pei, Relativity, Gravitation and Cosmology. Oxford University Press,
Oxford, 2005.
[3]Einstein, Albert, Relativity: The Special and General Theory. 1916.
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