Re: relativistic ordering lemmas

["John S. Denker" <jsd@AV8N.COM>]

On 11/14/2003 11:10 PM, Stephen Speicher wrote:
> In special relativity _all_ inertial observers will agree on the
> temporal ordering of events A and B _if_ those two events are
> causally related in _any_ inertial frame.

Yes.

> This can be proved with mathematical rigor

Yes.

> using a partial order relation and the Collinear Set theorem. See
> "Independent axioms for Minkowski space-time," John W. Schutz, Pitman
> Research Notes in Mathematics Series, _Longman_, 1997.

That makes it seem 100 times harder than it really is.

You can derive the key results in your head in less time
than it takes to tell about it.

I remember the boost transformation as

       [ t' ]   [ cosh(rho)  sinh(rho) ]  [ t ]
       [    ] = [                      ]  [   ]   (1)
       [ x' ]   [ sinh(rho)  cosh(rho) ]  [ x ]

so all you need to know are two unsurprising lemmas:
  a) cosh > 0      for all rho
  b) cosh > |sinh| for all rho

and you need to recall that the length of a four-vector is
preserved by Lorentz transformations (boosts and rotations).
This should be a memory item, since the whole concept of
vectors doesn't make sense without it.  And/or if needed
it can be rederived from equation (1).

Then the results follow by inspection in D=1+1 and the
generalization to D=1+3 is pretty obvious.
 -- The interior of the future light cone is closed under LTs.
 -- The surface of the future light cone is closed under LTs.
 -- The interior of the past light cone is closed under LTs.
 -- The surface of the past light cone is closed under LTs.
 -- The spacelike region is closed under LTs.
 -- The point of the cone i.e. the [0,0] vector is preserved by LTs.