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Invariant space-like interval from first principles
Posted:
Mar 30, 2013 11:46 PM
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It is very easy to come up with the expression for an invariant time
like interval.
A light flash at ground level goes up, hits a mirror and comes back to
the source in a time T as seen from the inertial frame of the source. A
fast moving observer passes this light clock and measures
t^2=(vt)^2+T^2=x^2+T^2. So T^2=t^2-x^2.
I have never been able to come up with anything nearly as simple to give
the expression X^2=x^2-t^2, where X is the proper length of a rigid body.
This bugs the daylights out of me. I guess the reason is that the above
example depends on two dimensions of space, so finding a comparable
derivation for the space-like interval fails because of a broken
symmetry between space and time.
Is there a reasonable concise way to arrive at X^2=x^2-t^2? I want to
do this without the use of Lorentz transformations.
The trick is, if you can show both T^2=t^2-x^2 and X^2=x^2-t^2, the
Lorentz transformations are just two derivatives away.
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