Re: special relativity: accelerated frames

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

> >     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?


At 03:42 AM 5/3/01 -0700, Michael Bowen wrote:

> My conceptual understanding of rapidity  is that it "adds" the way that
> velocity does in Newtonian physics.

Yes, rapidities add (in D=1; see below).  The Newtonian
addition-of-velocities law is the first-order approximation to the
relativistic addition-of-rapidities law.

> (This is
> an alternative approach to the addition-of-velocity formula; its
> validity may be thought of as arising from the structural similarity
> of the a-of-v formula with that for the tanh of a sum.)

The agreement between
 -- the formula for compounding velocities, and
 -- the formula for the tanh of a sum of rapidities
is much more than a mere structural similarity!  You can't have one without
the other.

Consider the analogy to rotations in a plane.
 1) Start by rotating the hand of a clock by an angle theta_1 relative to
the clock.  This causes some X and Y displacement of the point at the end
of the hand.
 2) Then rotate the whole clock by an angle theta_2.  This causes further
X and Y displacements of the point.

 *) We know that the angles add linearly.  Given the total angle (and the
radius) we can find the total displacements.
 *) The displacements do not add linearly.  Given the displacements due to
theta_1 alone, and the displacements due to theta_2 alone, we need to use a
nonlinear formula to find the total displacement.

I don't know the nonlinear formula.  If I ever wanted it, I would work it
out by adding the angles and taking the sines and cosines.

There is a profound analogy between a rotation (parameterized by an angle)
and a boost (parameterized by a rapidity).  I don't know the formula for
addition of velocities.  If I ever wanted it, I would work it out by adding
the rapidities and taking the tanh.

If we consider more than one dimension, we can have rotations in the XY
plane, the YZ plane, and the ZX plane.  Boosts are the corresponding
operations in the TX plane, the TY plane, and the TZ plane.  Just as an
angle is related to a slope (dY/dX), a rapidity is related to a velocity
(dX/dT).

The unified view of all this is called the Lorentz group.  This has various
subgroups, including
 -- rotations in the XY plane
 -- rotations in XYZ space
 -- boosts in the TX plane
 -- boosts and rotations in TXY spacetime
 -- etc.

Rapidities in different directions do not simply add.  That is, large
boosts in different directions don't commute -- just like large rotations
about different axes don't commute.  (Small ones do commute.)  In an
elementary course, you should stick to straight-line accelerations.

> works out numerically to be very close to unity (within about 3
> percent; is this the cute part?).

Yes.  Earth gravity is about one c per year.  The value is a mere
numerological coincidence.  It is, however, a handy way to remember the
radius of the curvature that the Earth's mass imparts to the local
spacetime.  But we are digressing beyond SR into GR here.....

> I assumed that the year of elapsed time was spaceship time.

Yes, that was the intent of the question.

> Note also that I used the term "nearest neighbor observers" in a
> semantically peculiar way to mean observers with consecutive values of
> k. In this context, there is no implication that "nearest neighbor
> observers" are (or were) located physically close to each other, only
> that their relative velocity is v.

It is important that they be neighbors (in space) _and_ that they have the
appropriate velocity.  That is, you want instantaneously comoving colocated
observers.