I just happened to see this response on Quora today:
=============================
How does one resolve the twin paradox in a toroidal universe?
<https://www.quora.com/How-does-one-resolve-the-twin-paradox-in-a-toroidal-universe>
Viktor T. Toth <https://www.quora.com/profile/Viktor-T-Toth-1>
Viktor T. Toth <https://www.quora.com/profile/Viktor-T-Toth-1>, IT pro,
part-time physicist
Answered 10h ago
<https://www.quora.com/How-does-one-resolve-the-twin-paradox-in-a-toroidal-universe/answer/Viktor-T-Toth-1>
<https://www.quora.com/#>
You resolve the twin “paradox” in any universe the same as always: by
noting that each traveler measures/proper time/, which is the
four-dimensional “length” of their worldline, and that when two
worldlines connect the same two events, they need not be of equal length.
I mean, really. What’s the paradox here? Grab a sheet of paper. Mark two
points. Connect them with a pencil, drawing a fancy curve. Now connect
them again, drawing a fancier curve. Or a less fancy one. whatever. You
will see that some curves are longer than others. But pray tell me, is
this really a paradox?
In the very special case of the flat space spacetime of special
relativity, when two events are connected, there is only
one/geodesic/(that is to say, a straight line in flat spacetime) that
connects the two events. So if the two twins travel different
trajectories, only one can follow a geodesic, that is, move with no
acceleration. The other twin has to accelerate, so his worldline will
not be a geodesic. (Because of the peculiar way time works in
relativistic spacetime, this twin, with the seemingly “longer”
worldline, will be the one who experiences less elapsed time.)
In the case of a generic spacetime, there may be multiple geodesics of
unequal length connecting the same two events.
This has been well known since the 19th century, and has never been
viewed as a real “paradox” by those who first considered it in the
context of relativity theory, merely a peculiar consequence of the
non-intuitive nature of the theory. I suspect it only became a “paradox”
later, mostly as a result of either efforts to “prove” Einstein’s theory
wrong by folks who never understood it in the first place, or by
misguided popularizations, aiming to impress, rather than inform, the
lay reader.
==================================
Brent
On 1/4/2019 1:24 PM, John Clark wrote:
On Fri, Jan 4, 2019 at 3:15 PM Brent Meeker <[email protected]
<mailto:[email protected]>> wrote:
> /You said *t was the proper time for me to take the trip*. /
Yes.
> /But the proper time is what a clock measures /
Yes,
> /and so it depends on the path you took in making the trip. /
Yes. And that is exactly why proper time is *NOT* an invariant, but
the 4D length through spacetime is.
>> I said "proper time is *NOT*an invariant"!
/> You apparently use "invariant" in a strange way. /
I only know how to use the word one way, a invariant is something that
doesn't change with a change in coordinates.
/>Proper time is an invariant length of a path in 4-space. /
No No a thousand times NO!! Proper timeis not the length of ANYTHING
in 4-space and proper time is NOT a invariant, Newton thought it was
but Einstein showed it was not.
> /Invariant means that all different observers agree on it. /
And I and my astronaut twin do not agree on the proper time, if we did
we'd still be the same age when he returned to Earth and we're not.
> /It doesn't mean the length of a path is independent of the path./
It does mean the 4D length between events is independent of the path.
>>No observer agrees on that because no observer knows what the
hell meters minus seconds means. But you did agree above that
proper time is the time measured by a clock along any line
through spacetime, so for my twin that was on a rocket at near
light speed and then returned the proper time is one year, but
for me who stayed on Earth the proper time was 10 years.
Therefore proper time can not be a invariant, therefore the
length of the path throughspacetime can NOT be the proper time
because the length of thepath through spacetime IS an invariant.
/> The length of which path? /
The length of any 4D spacetime path between Event A and Event B that
you care to name. The distance between events is always the same
regardless of the particular path chosen, the distance traveled in the
X,Y and Z directions could all be different, and t could be different
too, but when anybody calculates X^2+y^2 + Z^2 - (ct)^2 they always
get the exact same number because the spacetime distance is an invariant.
> /Every observer can read the clock as it moves along the path and
they will all agree on the length of the path. /
They all agree on the length of the 4D path through spacetime to get
from Event A to Event B but if they took different paths they will
disagree on the distance traveled through space and the time it took
to make it aka the proper time.
>> They don't agree on the spacial interval or the time interval,
but they agree on the proper time the clock measures along the
path. THAT's what "invariant" means.
> /You are using the word "observer" as though it referred to a
traveler, but in relativity it usually means someone measuring a
physical process from a different state of motion. /
If I'm measuring a clock that is in a different state of motion, that
is to say a clock that is not in my reference frame, then what I am
measuring is NOT my proper time.
>>Proper Time is defined as the amount of change an observer has
seen a clock make that is in the same reference frame as the
observer.
/> NO. The definition doesn't require the observer to be moving
with the clock.
/
At this point you're not even trying, if I say X=Y all you can say is
"no, X is not y".
An argument is not just contradiction
<https://www.youtube.com/watch?v=uLlv_aZjHXc>
John K Clark
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