Re: [Phys-l] how to explain relativity

[John Denker <jsd@av8n.com>]

On 06/19/2010 07:06 PM, John Denker wrote:
> > > Also, which clock will be ahead of the other?
> The clock that is higher in the potential will gain time
> relative to the lower one.
>
> It may help to visualize photons falling the potential,
> down from the upper clock to the lower clock.  Each 
> such photon will be blue-shifted.  If the photon stream 
> is modulated to indicate the upper clock's notion of 
> time, à la WWV, then the lower observer will conclude 
> that /everything/ happening upstairs is happening faster.

That's the explanation in general relativity terms.

I should add that the result depends on acceleration,
and *any* acceleration will do, not just gravitational
acceleration.

I will now restate the answer in non-GR terms, i.e. 
for ships that accelerate in flat spacetime.

In the diagram below, the worldline of ship A goes 
from point A to point A'.  Similarly ship B goes from 
B to B'.  The two worldlines are the same except for 
a rigid translation in the X direction.  For each 
ship, the acceleration profile a(τ) can be divided 
into a roll-in phase, an acceration phase, and a 
roll-out phase.  During roll-in and roll-out, the 
acceleration is zero.

The principle of translation invariance tells us 
that each observer keeps proper time in the same way, 
so long as each observer worries only about his own 
/local/ proper time (as defined by his own clock)
and not the other guy's clock.  In particular the 
A--B line is a contour of constant lab-frame time 
and also a contour of constant /local/ proper time 
... and similarly the A'--B' line is a contour of 
constant lab-frame time and also a contour of 
constant /local/ proper time.

Ship B emits a steady beam of light.  At a certain
time (in the lab frame) the beam is modulated with
marker 3.  Later it is modulated with marker 4.
Still later it is modulated with marker 5.  The
markers of course propagate along with the light,
at the speed of light.  The underlying frequency of
the light, i.e. the carrier frequency, is closely
controlled ... perhaps by an atomic transition of
the kind that defines the SI unit of time:
  http://www.bipm.org/en/CGPM/db/13/1/
  http://www.bipm.org/en/si/si_brochure/chapter2/2-1/second.html
and cross-checked against numerous very good clocks,
all of which agree.

                    A'         B'
               5   /          /
                5
          4      5/          / 
           4      5
            4    / 5        /
             4      5
              4 /    5     /
               4      5
          3    /4      5  /
           3     4      5
            3 /   4      /  roll-out
             3     4
             /3     4   /   
               3     4      acceleration
            |   3     4|
                 3
            |     3    |
                   3
            |       3  |
                     3
            |         3|   roll-in

            |          |
            A          B


At some later time ship B encounters the beam of light,
including the markers.

The interesting thing is that all the light received 
between marker 3 and marker 4 will be blueshifted.  
We can see this must be the case, because it was 
emitted (by B) with the roll-in velocity and received 
(by A) with the roll-out velocity.  This can be 
understood in terms of an ordinary relativistic 
Doppler shift, taking into account the conditions
when/where the light was emitted and when/where it
was received.

Similarly the elapsed proper time between marker 3
and marker 4 will be shorter according to A than
according to B.  This must be so, since the number
of light wave cycles between the markers is an
invariant scalar, so you can't blueshift the light
without changing the timescale for everything else.

Recall that A's worldline is the same as B's worldline
except for a rigid translation in the X direction.
The laws of physics are invariant with respect to
any such translation.  The laws are also invariant
with respect to rotation, which means we could flip
the diagram by rotating it 180 degrees in the XY 
plane, thereby putting B on the left and A on the
right.

So ... we must make sure our explanation of the
peculiar timekeeping is translation invariant and 
rotation invariant.  This can be done in terms of
acceleration dot separation, i.e. (a • ΔX), which
is an invariant scalar.  It tells us that the guy 
who is ahead _in the direction of the acceleration_
will appear blueshifted when observed by the pursuer.

  However, be careful, because the effect of the
  acceleration is somewhat peculiar:  the light 
  between marker 3 and marker 4 is blueshifted even 
  though the emitter was not accelerating when the 
  light was emitted, and the receiver was not 
  accelerating when the light was received.

  The key question is, how much did you accelerate
  while the light was enroute?  That's what controls
  the physics, i.e. the amount of blueshift, i.e.
  the amount of time-distortion.

This is consistent with the following fact:  Suppose
we ask a ship (any ship) in the roll-out phase to
construct a contour of constant coordinate time (t)
according to his frame of reference.  This contour
is not plotted in the diagram given above, but if
we did plot it, it would slope from southwest to 
northeast, since the diagram is aligned with the
lab frame.  So in the roll-out phase, the contours 
of constant t do not align with the contours of 
constant piecewise local proper time (τ).  For 
example, observer B thinks event A' happens late
compared to B' ... and by the same token observer
A thinks event B' happens early compared to A'
... even though both A' and B' happen at the same
/local/ proper time, as defined by each ship's
own clock.