Re: special relativity: accelerated frames

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

At 11:39 AM 4/27/01 -0600, Larry Smith asked whether:

> > ... General Relativity is a superset of Special
> > Relativity; i.e., can all the results of SR be derived as special cases
> > from GR?  After all, aren't inertial reference frames just special cases of
> > accelerated reference frames (where a = 0)?

Those are two very separate questions.

At 03:41 PM 4/27/01 -0400, David Bowman answered in part:

> ... whether or not a frame is accelerated or not has nothing
> necessarily to do with whether GR is necessary or not.  The criterion
> for the necessity of GR is the presence or absense of spacetime
> curvature, not the presence or absense of an inertial coordinate system.

David got this exactly right.  Let me just amplify this a little.

Let me coin a scale of complexity or sophistication:
 Level 1 := Newtonian mechanics
 Level 2 := Special relativity, as it is usually presented.
 Level 3 := General relativity

Then I would say that accelerated reference frames are at level 2.1 or some
such -- more sophisticated than the usual introductory SR discussion, but
certainly not requiring the heavy-duty machinery of GR.

I remember being bothered by this for a day or so, back when I was a
student.  At first, nobody offered an explanation of why SR should work for
accelerated reference frames.  The discussion consisted of asking "why the
heck shouldn't it work" and deciding that there's no reason why it shouldn't.

Later, the outline of an explanation emerged:  The trick is to arrange for
a succession of ___instantaneously comoving___ unaccelerated observers.  We
know SR works for each observer separately.  We then arrange to have enough
observers at the right places at the right velocities, so that they can
observe the action.  Afterwards, we collect all their observations and
integrate them.

[By the correspondence principle, everything (including a modest
acceleration) will look Newtonian to the instantaneously comoving
observers, so life is easy for them.]

The canonical homework exercise that goes with this bit of physics is as
follows:

   Suppose an interstellar spaceship starts from rest, and
   accelerates such that the passengers feel one Gee (980 gal)
   for one year.  How fast are they going at the end of the year?

Huge hint:  V=tanh(rho) where rho is called the "rapidity".  Find rho.

The answer is cute and worth remembering.