[Phys-L] frame independence

[John Denker <jsd@AV8N.COM>]

I wrote:

> > .... any statement where the meaning depends on what
> > reference frame is assumed is not merely ambiguous; it is defective.  It
> > is unphysical.

Rauber, Joel wrote:
> I'll make the obvious disclaimer that this is true unless you are asking
> a frame dependent question; in which case one should make clear in what
> frame the problem should be analyzed.  (I'm sure John meant to exclude
> such situations in the above comment.)

Actually, I didn't mean to exclude frame-dependent statements;  I meant
to complain about them.

My point will be more comprehensible and vastly more useful if I explain
what _should_ be done, rather than just complaining about what shouldn't
be done ... so here goes:

Rather than asking
      -- What is the X-component of Alice's velocity in Bob's frame?
it would be conceptually preferable to ask
      -- What is V dot Xhat?
where V is Alice's velocity and Xhat is a unit vector aligned with
Bob's X-axis.

The point is that V dot Xhat is manifestly invariant.  It is a scalar,
expressed as the inner product of two vectors.  All observers including
Carol (who is moving relative to both Alice and Bob in some unspecified
way) agree as to the value of V dot Xhat.

As an example where V dot Xhat makes physical sense, imagine V is
the velocity vector associated with a particle, and imagine that Xhat
describes the shape of a target.  In this example, V has physical
meaning, Xhat has physical meaning, and V dot Xhat has physical
meaning.

   Of course, it is proverbial that no matter what you are doing, you
   can always do it wrong.  In that spirit, it is always possible to
   point to situations where V dot Xhat is unphysical.  So you should
   learn to avoid writing down unphysical expressions, and concentrate
   on writing down physically-meaningful expressions.

Everything above applies equally well to Galilean relativity and to
Einsteinian relativity.
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