Relativistic Twins

[davar55]

Ripping my posts or monograph to shreds
line-by-line with some negative comment
for just about every line isn't possible to
reply to. I generally write and think in
paragraphs, the sentences are just
components of which to for but.

I think, but I'm not sure, that your
objection is to my basic use of linear
displacement (a vector) and velocity
(also a vector) and acceleration (also
a vector, btw) instead of distance (a
scalar out of which displacement is
defined and measured) and speed
relative to light (a scalar from which
the velocity vector is composed) if
or and to from.

Particle kinematics (and that's what
we're discussing when the forces
causing acceleration are ignored, i.e.
not dynamics), is based on distance and
speed, but in the 3-d world these must
be elevated into dispacement and
velocity, of by in from out.

Perhaps you are objecting to the
conclusions to be drawn from zero
displacement of a RT, and not to
the fact that I brought the vectors
into the discussion of the Twin Puzzle?
I didn't draw any conclusions myself
in this thread which how when why the.

[cheesehead]

I'm accusing you not of failing to understand the Lorentz factor, but of overlooking the implication of the squaring of v in it.

Simple idealized example:

Alice travels straight away from Earth at speed 0.99c for a year (Earth time), then travels back straight to Earth at speed 0.99c for a year (Earth time), arriving where she had left. (Assume that Alice is unharmed by the accelerations involved.)

According to your logic, her total displacement is zero, and thus her total velocity is zero.

but even though -0.99c is negative, (-0.99)² is nonnegative, and thus produces the same (positive) Lorentz factor as (0.99)².

Her time/space dilations do NOT cancel out to zero, and all your arguments about "zero-total-force", "total velocity" and "total displacement" are irrelevant to the relativistic time dilation at the heart of the twin paradox.

[xilman]

Originally Posted by cheesehead

I'm accusing you not of failing to understand the Lorentz factor, but of overlooking the implication of the squaring of v in it.

Simple idealized example:

Alice travels straight away from Earth at speed 0.99c for a year (Earth time), then travels back straight to Earth at speed 0.99c for a year (Earth time), arriving where she had left. (Assume that Alice is unharmed by the accelerations involved.)

According to your logic, her total displacement is zero, and thus her total velocity is zero.

but even though -0.99c is negative, (-0.99)² is nonnegative, and thus produces the same (positive) Lorentz factor as (0.99)².

Her time/space dilations do NOT cancel out to zero, and all your arguments about "zero-total-force", "total velocity" and "total displacement" are irrelevant to the relativistic time dilation at the heart of the twin paradox.

Good post.

I'm still thinking about the case in which the universe is closed, whatever the topology. It could be a 4-ball at simplest or have any of a number of different topologies. Davar55's thesis doesn't distinguish between them.

[chris2be8]

In the closed universe case, what happens if the universe is expanding? The "easy" case is when the expansion is large enough to make it impossible to circumnavigate the universe (this allows you to duck all the interesting questions). If not would the traveler pass their starting point at the same speed as they left it (assuming they don't accelerate or decelerate in flight)?

Chris

[ewmayer]

Originally Posted by xilman

I'm still thinking about the case in which the universe is closed, whatever the topology.

A question which I believe may help focus on the core issue here: In my previous post I used the example of a freely moving test particle to illustrate that spacetime curved by the presence of mass/energy is inherently a non-IFR; even absent any knowledge of the mass/energy content of the local neighborhood one could easily infer non-IFR-ness from a test particle moving in a self-crossing loop, or more generally, multiple TPs which at some "initial" time point are moving on parallel-but-not-collinear paths at identical speeds having nonconstant spacetime separation along their respective trajectories as time advances. Now to the

Question: Is "structural acceleration" due to cosmological exapnsion/contraction/distortion of the very spacetime fabric itself - that is, not induced by the presence of mass/energy - immune to this kind of IFR-ness test? For example, owing to the global spacetime expansion (purportedly) described by the BBC [the cosmological body of theory and observation, not the broadcasting network] we infer acceleration-away-from-us of distant objects based on their redshift^* and assume ourselves part of this same globally-expansding spacetime fabric even though none of us "feels" any acceleration. If that indeed connotes non-IFR-ness, then is stands to reason that any global-spacetime topology which "curves back on itself" must similarly be non-inertial, hence any twin-paradox-style scenario would need to use the more general theory of relativity, not just the special.

(I'm pretty sure I can look up the answer to the above question, but for now I found it a fun brain workout to just explore the issue without resorting to thought-displacing search tactics.)

-----------

^* Edit: Not merely based on the fact of redshifts alone but rather in the sense that redshifts, on average, increase with distance on cosmological scales.

[davar55]

Originally Posted by xilman

Good post.

I'm still thinking about the case in which the universe is closed, whatever the topology. It could be a 4-ball at simplest or have any of a number of different topologies. Davar55's thesis doesn't distinguish between them.

Given convexity, continuity (of each of the four spatial dimensions),
and closedness as youall called it (no boundary and light would
return through the other "side"), then the cosmological principle
implies isotropy, hence I conclude the 3-d space is an essentially
perfect "sphere" with a fixed fiinite radius R, and the skin (fourth
spatial dimension) is much much thinner than R, so a 4-ball only
works if the fourth dimension is weighted differently. But ignoring
that detail, is there really another topology? I also wrote that
3-d space is everywhere Locally Euclidean, so the simplest toplogy
is also the best description.

[xilman]

Originally Posted by davar55

Given convexity, continuity (of each of the four spatial dimensions),
and closedness as youall called it (no boundary and light would
return through the other "side"), then the cosmological principle
implies isotropy, hence I conclude the 3-d space is an essentially
perfect "sphere" with a fixed fiinite radius R, and the skin (fourth
spatial dimension) is much much thinner than R, so a 4-ball only
works if the fourth dimension is weighted differently. But ignoring
that detail, is there really another topology? I also wrote that
3-d space is everywhere Locally Euclidean, so the simplest toplogy
is also the best description.

How about a 3-torus or a twisted space?

If it helps you visualize what I mean, drop a dimension and think of a doughnut or a Klein bottle.

[Mr. P-1]

Originally Posted by ewmayer

even absent any knowledge of the mass/energy content of the local neighborhood one could easily infer non-IFR-ness from a test particle moving in a self-crossing loop

What is a self crossing loop? If you mean a trajectory which returns the particle to the same place, you would need to define "same place".

[davar55]

Originally Posted by xilman

How about a 3-torus or a twisted space?

If it helps you visualize what I mean, drop a dimension and think of a doughnut or a Klein bottle.

By my interpretation of Occam's Razor, the simplest shape that
satisfies all physical phenomena and theory is most likely the
correct description of the shape of the Universe.

So we rule out the 4-d analog of a Klein bottle (what you called
twisted space), and rule out the 4-d analog of a doughnut or
torus (what you called a 3-torus), and taake the simplest shape
that does the job, a 4-sphere (which within my explanation is just
a 3-d sphere or 3-sphere Riemann-folded everywhere, and which
because of the time dimension I labeled a super-hyper-sphere).

[davar55]

I understand Einstein envisioned a "cylindrical space-time". This was
I think only a 3-d conceptualization of 4-d space-time, with time
being the axis of the cylinder. Do we know what 3-d spatial shape
this was supposed to represent? Not a sphere, and not a cylinder.

[CRGreathouse]

I think the signature and topology are much more important than the shape. Key to Einsteinian physics is the use of the Minkowski metric, with signature (-1, +1, +1, +1) or the like. (Actually how this is represented seems to be something of a holy war which I don't want to enter, but the point is that one coordinate differs from the others in sign.)