Tilting Pisa

Matt Seaborn wrote:
 ...if these two ships take positions some distance apart, match their 
 clocks and accelerate identically to relativistic speeds towards each 
 others then the statement in the previous paragraph becomes true.  If 
 these ships then decelerate so that they are in the same frame how do 
 their clocks compare, as each has supposedly been going slower than the 
 other?

Maybe some pictures would help.  (Use a fixed-pitch font to look at 
these.)  Initially the two ships are at rest relative to each other,
and not accelerating relative to the (assumed flat) spacetime metric 
in this region [note 1].  This condition is illustrated below:


           t_a |                         t_b |
               |                             |
               |                             |
               |                             |
               |                             |
               |                             |
               |                             |
               |                             |
               |                             |
   ............A............      ...........B............
               |        x_a                  |         x_b
               |                             |

The symbols t_* and x_* denote the time and space axes, respectively, of 
observers A and B in this initial condition.  Note that their lines of 
simultaneity (i.e., the dotted "space" axes) are identical.

Now suppose observers A and B abruptly accelerate toward each other
by the same amount, so they now have high constant velocities, with A 
moving to the right and B to the left.  Their space and time axes now 
look like this:

                      t_b \     / t_a
                           \   /
                            \ /
                             x
                            / \
                           /   \
           .              /     \              .
         x_b ' .         /       \         . '  x_a
                 ' .    /         \    . '
                     ' r           s '
                      /  ' .   . '  \
                     /     . ' .     \
                    /  . '       ' .  \
                   A '               ' B
               . '/                     \' .
           . '   /                       \   ' .
         '      /                         \      '


The fact that the time axes tilt in the direction of motion should 
be familiar to you, because they do the same thing in the classical 
Galilean theory of relativity.  Regardless of your state of motion, 
you can take your worldline as your (proper) time axis.  However,
you may be a little surprised to see that the SPACE axes of the two
observers have also been "tilted".  In Galilean relativity the space
axis (representing a line of simultaneity) never tilts; it is always
flat and the same for all observers.  In Einsteinian relativity 
the space and time axes both tilt symmetrically about the "null" 
worldlines, which are the paths that light rays follow in a vacuum.

Before explaining WHY they tilt this way, let's first observe that
this tilt answers your question about how time passes for each 
observer relative to the other observer's frame of reference.  
As observer A abruptly accelerates, his space axis shifts from 
horizontal in the first figure to an upward tilt in the second 
figure, so initially he was simultaneous with the point on B's 
worldline labeled "B", whereas following the acceleration he is 
now simultaneous with the point on B's worldline labeled "s".  Thus, 
according to A's frame of reference, observer B has advanced from 
the location marked "B" to the location marked "s" in however long 
it took for A to accelerate - and we can assume the acceleration 
occurs over as short a time span as we like.  This implies that 
B's clock is advancing VERY rapidly according to A's reference 
frame as A accelerates.

Likewise, according to B's reference frame, observer A jumps from
the point labelled "A" to the point labeled "r" as B accelerates.
Then, once they have both accelerated and are moving at constant
speed, each clock is advancing slow relative to the other observer's 
frame of reference, until they smash into each other at point x,
at which point they will both show the same total elapsed time on
their clocks, because first they saw each other jump ahead (during 
the accelerations), and then they saw each other going slow (while 
travelling at constant speed).

If you're new at this, you'll probably want to dream up all sorts
of other scenarios with various initial conditions and patterns 
of accelerations, etc., and you'll find that they all work out
consistently.  Then, if you get over your aversion to math, you
would notice (as Minkowski did) that there is a very simply way
of encapsulating Einsteinian relativity in the form of a metric,
from which it is self-evident that you must always get consistent
answers.

However, the more interesting question is why we need to tilt the
space axes for different frames of reference.  What's wrong with the 
Galilean notion of leaving the space axes unaffected?  Well, if you 
think about it for a minute, you can see that Galilean relativity 
doesn't make perfect sense, because on one hand it clearly implies 
a transmutability between space and time, as shown by the tilting of 
the time axes in the spacetime plane due to motion, but on the other 
hand it does not allow for any absolute measure of the complete 
spatio-temporal separation between two events.  Let me try to 
explain this problem.

Given two arbitrary events at two different locations and times, 
Galilean relativity tells us that the temporal separation between 
those two events is some absolute constant relative to any and 
every inertial frame, but the SPATIAL distance between those two 
events is indefinite, and can range anywhere from zero to infinity,
depending on the choice of reference frame.

If we want an absolute measure of the spatio-temporal separation 
between two events (meaning a measure that is applicable to any 
and every reference frame), we need a function S of the individual 
space and time separations del_x and del_t with respect to a given 
reference frame, such that S returns the same absolute separation 
for those two specific events, regardless of the frame of reference.
But this is clearly a problem for Galilean relativity, because for 
two specific events we have del_t = constant for all frames and 
del_x varies from 0 to infinity.  The only way to get an invariant 
function of del_t and del_x in these circumstances is to simply 
IGNORE del_x and define the measure S(del_x,del_t) as something 
like (del_t)^2.

This is not very satisfactory, because there surely seems to be a
physically significant distinction between our separation from the 
Sun tomorrow and our separation from the Andromeda galaxy tomorrow,
even though according to Galilean relativity they are both just 
one day away, which has absolute significance, whereas their 
spatial distances have no absolute significance, and we could just 
as well be spatially "close" to Andromeda and "far" from the Sun.
This occurs because, according to Galilean relativity, we can
use motion (which translates between space and time) to create
or annihilate spatial separation, but we cannot affect temporal
separation at all, and *no quantity involving spatial separation 
is conserved from one inertial reference frame to another*.

Thus, motion within the context of Galilean relativity is really 
very peculiar, and doesn't make a lot of sense from a purely 
abstract point of view.  In order to artificially impose some 
"sense" on it, people like Newton had to introduce the problematical 
concept of absolute space to give some ontological significance to 
spatial separation, but this move was contrary to the essential 
content of Galilean relativity, as pointed out by Huygens, Berkeley, 
Leibniz, Mach, and so on.  The fundamental problems with the naive 
concept of motion in the context of Galilean relativity can even be 
seen in Zeno's arguments on motion circa 500 BC (particularly in 
his argument known as "The Stade").

In contrast, Einsteinian relativity gives a logically coherent
absolute measure of spatio-temporal separation for any two events,
namely, S(delta_x,delta_t) = (delta_t)^2 - (delta_x/c)^2 for some 
constant c.  (The hyperbolic form of this function is strongly 
suggested by the uni-directionality of time as distinct from 
space, but I won't go into that.)  This quantity is invariant 
when evaluated with respect to any and every inertial frame of 
reference.  Of course, if c is fairly large in terms of ordinary 
units, the second term becomes negligible, and we have something 
that could empirically be mistaken for Galilean relativity, where 
S equals simply (delta_t)^2.  However, from a logical standpoint, 
Einsteinian relativity with some finite value of c is far more 
intelligible.  As Minkowski mused

  "...it looks as though the thought might have struck some
   mathematician, fancy free, that natural phenomena do not 
   possess an invariance with the group G_inf, but rather 
   with a group G_c, c being finite and determinate, but
   in ordinary units of measure extremely great.  Such a 
   premonition would have been an extraordinary triumph 
   for pure mathematics."

Unfortunately, none of the critics of Galilean invariance, from
Zeno to Mach, had this premonition, so we needed to wait until
an accumulation of experimental results forced the idea upon us.

Note 1: Please do not ask what determines the metric in this region
        of spacetime, because that would lead to a consideration of
        cosmology and general relativity, whereas this newsgroup 
        is devoted exclusively to *special* relativity, i.e., the
        epistemologically crippled theory that Einstein abandoned 
        in 1911.

Matt Seaborn wrote:
 A second question is about acceleration.  If acceleration is the same 
 as being in a gravitational field, does it also affect space time in 
 the same manner.  So in the above example would the fact that they 
 accelerate affect the time flow *in addition* to their relative 
 speeds?  If so what about the fact that general relativity (?) puts 
 no upper limit on acceleration?

Now you've done it.  As soon as you introduce the Equivalence 
Principle you undermine the intelligibility of special relativity,
as Einstein clearly perceived around 1907.  This was Einstein's best 
moment, having won attention and honors for his theory of special 
relativity, he was immediately willing to challenge and deconstruct 
his own theory, discarding global Lorentz invariance, to seek 
whatever was necessary to make a theory consistent with the 
Equivalence Principle.  (As Wilde said, "The witty contradict 
other people; the wise contradict themselves".)

But seriously, if you want to gain a mature understanding of the
modern theory of relativity (viz, general relativity), you will need
to exit from this newsgroup, which is filled with roving packs of
specially-trained Pavlovian dogs who begin to hyper-ventilate at 
the mere mention of the words "general relativity" and "twins 
paradox" in the same post.  For a sensible discussion of the twins 
paradox that doesn't duck the Equivalence Principle, see Max Born's 
excellent "Einstein's Theory of Relativity", in which you'll find a
quantitative analysis of the twins situation with the accelerations 
treated as gravitational fields and vice versa.  You won't find an 
explanation of that kind in this newsgroup (at least not often), 
because it traces the asymmetry of the twins back to a cosmological 
source.  This is actually a superior explanation from an epistemological 
standpoint, but it's more fun to taunt mathematically-averse people 
about not being able to grasp the workings of  ds^2 = dt^2 - dx^2  
than it is to give philosophically meaningful answers.