Re: Special Relativity Question

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Fri, 4 Feb 2000, Abul Kalam wrote:

> (1)    My colleague, who is a mathematician by vocation but quite
> knowlegeable about and interested in much of physics, is intrigued by a
> Special Relativity question : A fast moving object in space (v ~ c )
> would see celestial objects crowding in one direction over another, such
> as a driver in a rain would see the rain come down at an angle.
> According to Special Theory, laws of physics are the same in all
> inertial frames, but in this case, the property of isotropy of free
> space is changed for some inertial frames at higher velocities.  Does
> this apparent incongruity violates the first postulate of Special
> Relativity ?  I suggested that in Doppler Shift, a similar thing is
> noticed when various frequencies are heard, but that, by no means,
> violates the first postulate of special relativity.  What do you
> suggest?

Special relativity describes flat space time so it should not be
surprising that it fails when applied to the universe.  The anisotropy of
the 3 degree background radiation is evidence of this same sort that we
are moving in some absolute sense relative to a preferred frame in which
there would be no anisotropy.

> (2) His second special relativity question :  Two objects, A and B, move
> near the speed of light along x and y-directions, as those speeds are
> measured by an observer, C, at the origin of co-ordinates.  Then,
> according to observer C, the objects A and B are moving away from each
> other with a relative velocity of  1.414c.  Is Pythagorean theorem
> applicable in this case ?

Yes. *C* says A and B move apart at a relative velocity of v*sqrt(2)
assuming that both move at speed v relative to C.  But the relativistic
velocity addition formula shows that *they* will consider themselves to be
moving apart at v*sqrt(2-(v/c)^2) which is always less than c for v < c.

I'm not sure what your colleague is concerned by here. If it is the fact
that the relative speed observed by C is greater than c, then why make the
problem so difficult? Why not let A and B move in opposite directions
relative to C.  Then their relative velocity is 2*v, but again there is no
problem since *they* will find that they move apart at 2*v/(1+(v/c)^2)
which is, again, always less than c for v < c.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm