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Re: Special Relativity Question



Regarding Abby Kalam's questions:

(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,

This is true.

but in this case, the property of isotropy of free
space is changed for some inertial frames at higher velocities.

No it is not. The *laws* of nature *are* the same in both inertial
frames. The *observations* between the two cases are different though.

Does
this apparent incongruity violates the first postulate of Special
Relativity ?

Not at all. If in one frame a collection of objects happen to be
isotropically distributed about some point in space, then there is no
reason to suppose that this isotropy in the distribution ought to be
present in any other frame moving wrt the first one. The original
isotropy in the original distribution is an artifact of the particular
preparation of the state of that surrounding matter. That matter could
have (in principle, if not necessarily in practice) been distributed in
an infinite number of other manners *including* in such a particular
anisotropic way that when the distribution was viewed from the second
'moving' frame the distribution ended up being isotropic in that other
frame instead. Just because the laws of nature have a given symmetry,
that is no reason to suppose that an observed state of nature ought to
respect that symmetry as well. The observed state depends on initial/
boundary conditions *as well as* on the laws of nature. Those initial/
boundary conditions do not have to be as symmetric as the laws of
nature are, and their broken symmetries will be reflected in the observed
states of nature. In fact, there is even such a thing as spontaneous
symmetry breaking that can happen where the observed state has a broken
symmetry that is obeyed by *both* the laws of nature and by the
initial/boundary conditions.

(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 ?
- Abby

Suppose A and B both move with the same speed v respectively along the
x & y axis directions in C's frame. Then the speed of A relative to B
(and vice versa) is v*sqrt(2 - (v/c)^2). In the limit of v --> c this
result approaches c not c*sqrt(2). But it is *true* that in *C's* frame
the distance between A and B does become larger at a rate of v*sqrt(2).
But this rate of increase in the spatial separation between two moving
objects does *not* represent any actual relative velocity between any two
objects, and no causally informative signal propagates anywhere at any
speed greater than c, in any frame, in spite of this fast separation
rate. The same goes for the situation where A and B each move with
speed v along oppositely directed paths along the - & + x-axis
respectively relative to C. In this later case the spatial separation
between A and B in C's frame increases at the rate 2*v. But the relative
speed of A wrt B (and B wrt A) is 2*v/(1 + (v/c)^2) which is always
less than c, and only approachs c as v --> c. Relativity doesn't forbid
the spatial separation between *2 different* moving objects from
increasing at a rate faster than c. Rather, it forbids any causally
informative influence from propagating at a speed faster than c. This
means (as a consequence) that no *single* material object can move in
any frame faster than c (otherwise the moving object itself could be
used to carry a causally informative influence).

David Bowman
David_Bowman@georgetowncollege.edu