starship-design: The "So-called" Twin Paradox, cont'd

[Steve VanDevender <stevev@efn.org>]

DotarSojat@aol.com writes:
 > >The difference in trip time is ultimately the result of different
 > >paths through spacetime.
 > 
 > The difference in trip time could also be attributed to different
 > paths through *space*, but relativists don't like absolutes.

Relativity has its absolutes, just not the absolutes of Newtonian
physics.  For example, all observers will agree on the spacetime
interval between two points, no matter what reference frame they measure
from.  So invariance of the spacetime interval between events is an
absolute in relativistic physics.

Furthermore, the distinction between space and spacetime in relativity
is quite important.  Look at the contrail of a jet in the sky.  The
contrail tells you its path through space.  But from that path in space,
you cannot tell how fast the jet was moving at any moment or the amount
of acceleration it experienced to cause bends in the contrail.
Relativity uses spacetime because intervals must be measured in
spacetime.  A path through spacetime tells you everything about the
motion state of the particle at all times along the path.  Placing that
path in different reference frames may result in different coordinate
measurements, but all observers will agree on the proper time
experienced by the particle at any point along the path relative to any
reference event.

 > (I see the idea of an absolute frame of reference in Timothy's
 > comment, in his email of 1/26 at 10:12 EST, "So to find the
 > symmetry breaking factor in the Twin Paradox you *need* to look at
 > a global (non-homogeneous) environment.")  But, one relativist,
 > Einstein himself (on p. 56 of his "The Meaning of Relativity,"
 > Fifth Edition, Princeton, 1956), raises the possibility of an
 > absolute frame by citing "Mach's Principle," which postulates that
 > "a material particle [moves] relatively to the centre of mass of
 > the universe."  [See also Holstein and Swift, Amer. J. Phys. 40,
 > p. 749 (1972).]

As Timothy has already pointed out, Mach's principle may define a
reference frame that everyone can agree on, but there is nothing
privileged about that frame; the laws of physics will be the same in all
other frames.  Measurements in the Mach frame may differ from
measurements in some other frame, but will tell you nothing different
than those other measurements.

 > >In a sense, Rex's formulation only puts blinders on Peter and
 > >Paul, preventing them from making the observations that they
 > >could make entirely between themselves to determine who should
 > >age less.  If we remove these unreasonable restrictions then they
 > >could easily determine between themselves who should have aged
 > >least when they meet again.
 > 
 > However, if the "observations...entirely between themselves" are
 > composed of Doppler-shift measurements by each of light from the
 > other, they both measure exactly the same shift (as the Special
 > Theory stipulates) and can't tell which one is "moving."  But they
 > *can* determine which one is moving by looking at the Doppler
 > shift of light from the "fixed stars."  That was my point that I
 > may have oversimplified a bit in my 11/29/96 response to Nick.

Measurements of doppler shift may not tell you anything, but something
very simple will.  Allow Peter and Paul to have radios and clocks that
they carry with them, and to send each other messages through the radios
(which, through a strictly-enforced FCC regulation, are on a frequency
band allocated only Peter and Paul and which no one else in the universe
is allowed to transmit on, preventing Peter and Paul from communicating
with anyone but each other -- assume the band is also wide enough to
deal with any doppler-shifting they may experience from their relative
motion).

Peter and Paul use a protocol where each message they send is identified
by their name and their locally-measured elapsed time since separation,
and their reply to a message includes the identification of the
original, as in "Peter at 20 days, 1 hour replying to Paul's message of
19 days, 15 hours".

This is sufficient to allow Peter and Paul to determine which twin is
experiencing acceleration, by measuring round-trip times for
acknowledgment of their own messages as well as the elapsed time on the
other's clock at which the other received a message and sent the
acknowledgment.

Because it's late (and because it will take me time to work this out to
a blatantly obvious level) I'll leave this as an exercise for the more
dedicated readers.