Re: Proper Time v. Absolute Motion.

sal wrote:

You're apparently asking for some way to look at this which might
help.  You seem not to be interested in a rehash of the math which
leads to the conclusion.  Right?


On Fri, 01 Jul 2005 11:08:09 -0700, Daniel Weston wrote:


Deep in intergalactic space, 2 rocket pilots, (twins) have met and
had a drink and a bite to eat.  They then assume their respective
positions as pilots, and each blast off on their respective random
galactic tours. No person or equipment is keeping track of their
movements.  They happen to meet ~10 years later.

After a snack they start wondering as to which twin had traveled the
furthest.  It occurred to them that they might resolve this inquiry
by comparing their watches.  Twin A shows 200 days less elapsed
time. It is therefore concluded that A traveled the furthest.

Problems: What caused the clock of A to run slower than B, and have
the difference in time proportional to the distance (or speed)
traveled.  Quite apart from unknown causality, is the issue of
absolute comparative motion.  The motion between 2 bodies is
relative,



WHOA Slow down.  This is an overstatement.  "Velocity" between two
bodies is relative.  But acceleration, which is also a form of
"motion", is absolute.  You don't need to look out the porthole to
know if you're accelerating.



so how do we conclude that totally relative motion is
consistent with absolute time differences in clocks?  How can
relative motion be used to prove that one of the parties traveled
differently than the other?  To a novice, there is needed some
enlightening information and reasoning to put this apparent
incongruity into resolution.  Will anybody help?



You have identified the wrong paradox.  The difference in ages of the
twins is not a paradox.  Here is the paradox:

Acceleration does not affect the rate at which a clock runs.

Yet, acceleration is responsible for the difference in their ages.

That's a paradox (or at any rate it's awfully confusing).
Furthermore, I think it's part of the key to any "intuitive"
understanding of this problem.

You have left out some items from the picture you drew, and putting
them in, explicitly, may help with seeing what happened.

(0) Just as background, let's make this explicit: No gravity.  Stars
    are made of something very light (like hydrogen? <g>), and planets
    are all made of styrofoam, and the g fields are so weak that we
    can assume space is flat.  Curvature adds complexity without
    affecting the basic problem, so dispense with it.


(1) Acceleration is absolute, and can be measured by a single
    observer.  Some information was thrown away in the construction of
    this problem, which doesn't require any additional observers to
    obtain.  Let's add in an explicit measurement of that extra
    information: Give each twin an integrating accelerometer, so that
    positions and velocities at each moment can be computed by dead
    reckoning.  Now each twin can tell _exactly_ how much he or she
    deviated from pure linear motion, without asking the other twin
    anything.


(2) There's a THIRD frame of reference, which is inertial, and which
    you didn't mention: Just before the twins blast off, they share an
    inertial frame.  Call the point they leave from, in that shared
    inertial frame, the Origin.  The Origin continues to exist
    throughout the problem, though you didn't mention it again.  By
    consulting their inertial navigation systems, the twins can
    determine where the Origin is relative to their final meeting
    point.  Once they meet, let's have them journey -- together! -- to
    the Origin and stop there (relative to the Origin).  Since this
    last, added, leg of the journey is done together, it won't affect
    the absolute difference in their ages, nor their conclusion as to
    who traveled farther.


Now, look:  We've got the same problem, but I've reframed it in terms
of an inertial coordinate system in which two travelers leave, move
around, and return to the same spatial point at a later moment in
time.  At this point we're comparing the proper lengths of three
curves all of which intersect at two points.  The one which deviates
most strongly from a geodesic will be the "shortest"; the one which
follows a geodesic will be the "longest".  The degree of deviation
from geodesic motion corresponds to the amount of accelerating done by
each twin.

Final point: You can DISPENSE with one of the twins.  Just have ONE
traveler, with an integrating accelerometer.  He or she can tell just
by looking at the output of the accelerometer exactly how much he's
deviated from geodesic motion.  He (or she or it or whatever) can also
find his way back to his starting point in the inertial frame he began
in by dead reckoning, and he can tell, from the record of his
(absolute!)  acceleration, how much "life-line time" he's "saved" by
comparison with how much time would have passed for him if he'd never
started his engine and had simply waited until he got to this event in
the original inertial frame.

Does this help at all?

IMO it's wrong to attribute the age differences to differences in acceleration. Here's why.

Consider twin1 and twin2 in an Inertial reference frame. Let twin1 accelerate away from twin2 until reaching a specified velocity wrt the original frame. Also, sometime after twin1 leaves, let twin2 undergo the same acceleration twin1 did. Both twins will then be in the same Inertial frame, and both will have experienced identical accelerations, but according to SR, they'll no longer be the same age*.

As to the question of what causes the age difference, it's really a question of what "breaks the symmetry" between the twins. In the above example (and in general) the symmetry breaker isn't acceleration per se, but rather the fact that the twins have travelled at different velocities for different proper times in relation to the reference frame in which their ages were ultimately compared.


* Since the twins aren't co-located, not all observers (e.g. their stay-at-home parents) will agree on the difference in their ages, but the twins will agree.
.

Relevant Pages

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  • Re: Proper Time v. Absolute Motion.
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