Can anyone clarify the relativistic twin paradox for me?

[Dale]

Forgive the length of this but I think it's a clear argument.

You don’t need to be presenting clear arguments. You need to be asking clear questions that will help you learn how the universe actually works.

One big problem is that your description is not at all clear as you assume it is. In many cases what you state is unclear because you state a relative quantity without stating what frame it is relative to.

For example, in relativity there are two kinds of time: proper time and coordinate time. Coordinate time is relative meaning it is the time in a specific reference frame. It is defined in the entire reference frame. In contrast, proper time is invariant, it is the time that is physically read on a single clock and is only defined where the clock is located.

Because proper times are only defined where the clocks are located, proper times can only be compared when the clocks are at the same location. When comparing distant clocks, one of the times is a coordinate time, not a proper time. The comparison is relative to the frame whose coordinate time is used.

Similarly with acceleration. Proper acceleration is the physical acceleration measured by an accelerometer and it is invariant. Coordinate acceleration is the second derivative of position in a given frame. It is relative to the given frame.

Energy is also a relative quantity. Different frames will disagree on the energy, so the frame must be specified.

With this understanding, let’s look back and see exactly how unclear your argument is:

the frequency of pulses allows a comparison of clock speeds

In which frame? The frequency comparison only gives the Doppler shift, not a comparison of clock speeds without specifying the frame.

So the length of B's second is 5/3 times A's second

In which frame?

When a time of one week has passed on A's spacecraft then 3/5 weeks has passed on B's.

In which frame? Which is the coordinate time and which is the proper time?

It occurred for B while being an inertial frame.

B’s frame is non inertial. The frame includes time.

while B is accelerating then A is too (in the sense of B measuring a change in velocity in A)

Which of these is proper acceleration and which is coordinate acceleration? For the coordinate acceleration, relative to which frame is the coordinate acceleration measured?

One could define acceleration as that which is measure by an accelerometer but what is really meant by this is that one type of acceleration (B) involves a gain or loss of energy while the other (A) does not.

This is simply wrong. Proper acceleration is absolute, gain or loss of energy is relative.

What this all points to for me is that a gain or loss of energy PHYSICALLY CHANGES a clock.

Gainnor loss of energy is a relative quantity, so relative to which frame is which clock physically changing?

And this change persists when B reverts to being an inertial frame at 0.8c after its one minute of acceleration

Since frames include time, it makes no sense to speak of reverting to be an inertial frame.

 

[Sagittarius A-Star]

Then there is no reason why one clock should run slower than another. There isn't an explanation in terms of structural change.

As I wrote in posting #7, you can replace the travelling clock by 2 clocks. One is moving inertially out-bound, the other inertially in-bound. Together they count a round-trip proper time, that leads also to the twin paradox in combination with the earth-clock. But none of the 2 travelling clocks is or was accelerated.

 

[Nugatory]

No one can sensibly refer to non-inertial frames and claim that the time loss of 2/5 weeks occurred during these two minutes of being a non-inertial frame

Part of the problem here may be that you seem to be misunderstanding what a frame, whether inertial or not, is. A frame is a convention for assigning coordinates (time and space) to events so there as no such thing as "two minutes of being a non-inertial frame" - what we're doing is using one convention for a while, then using another. It is not at all surprising to find that when the traveller changes the convention they use to assign times to the earth clock, the time they've assigned to the earth clock changes. There is no physical change associated with either clock.

 

[ersmith]

Forgive the length of this but I think it's a clear argument.

Suppose the history of acceleration is unknown. Two spaceships A and B pass each other in opposite directions at 0.8c in field free space. They each then start to send the other a signal at one second intervals. That way, the frequency of pulses allows a comparison of clock speeds. Their clock rates should be the same. But they find that they are not.

Indeed. After correcting for light travel time, each finds the other's clock to be ticking more slowly. The geometric reason for this is explained in earlier messages, and in particular in Orodruin's insight at https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

What this all points to for me is that a gain or loss of energy PHYSICALLY CHANGES a clock. And this change persists when B reverts to being an inertial frame at 0.8c after its one minute of acceleration.

Let's consider an analogy. Suppose car A and car B are in New York and have identical odometers, both set to 0. Car A travels to Boston in as straight a line as possible. Car B also goes to Boston, but takes a windy path. Would you explain the difference in their final odometer readings by attempting to construct a theory of how turning the steering wheel in the car imparts energy to the odometer, causing it to tick faster? This *might* be possible in some sense, but it would require a very complicated theory to cover all possible cases. The far simpler explanation is just that the odometer measures distance traveled, and car B traveled a longer distance.

Clocks are odometers in spacetime, and time has the opposite sign in the metric to space (so a straight line is the *longest* path in spacetime, rather than shortest). Acceleration corresponds to turning or bending the clock's worldline. Otherwise the analogy is quite exact.

 

[Nugatory]

No one can sensibly refer to non-inertial frames and claim that the time loss of 2/5 weeks occurred during these two minutes of being a non-inertial frame

You may want to go back and review post #17 by @jbriggs444 in his thread. Keep it in mind as you study the "time gap" section of the twin paradox FAQ that we've already referred you to.

 

[Dale]

Then there is no reason why one clock should run slower than another. There isn't an explanation in terms of structural change.

I agree. There isn’t an explanation in terms of structural change.

I do not agree that that means there isn’t a reason for time dilation. There are good reasons, but they are geometrical, not structural or mechanical.

 

[PeterDonis]

Then there is no reason why one clock should run slower than another. There isn't an explanation in terms of structural change.

That's right, because one clock does not "run slower than another" in any invariant sense. As far as invariants are concerned, all clocks tick at exactly the same rate: one second per second of arc length along their worldlines. There is no "structural change" anywhere.

To put this another way: "time dilation" is not the result of doing anything to a clock. It's a result of looking at the clock in a different frame. It's the hyperbolic geometry analogue of the apparent size of an object changing when you view it from a different angle: you changing your viewing angle doesn't do anything to the object itself. There is no "structural change" involved.

 

[robphy]

Here is a visualization of mine that might help.

It displays the proper time along piecewise-inertial worldlines
using a ticking light clock, which provides a relativity-friendly mechanism for the wristwatch's behavior.

It uses a "circular light clock" because it was based on generalizing the Michelson-Morley apparatus.

From the relativity-principle and the speed-of-light principle,
the visualization displays the resulting effects of
time-dilation, length-contraction, and the relativity-of-simultaneity

VisualizingProperTime - the Clock Effect / Twin Paradox with Circular Light Clocks (alert: it has sound effects)


The spacetime-volume enclosed in the causal-diamonds
(the intersection of the future light-cone of one tick with the past-light-cone of the next tick)
are equal. (The volume is related to the square-interval between the tick-events.)

(In the (1+1) case, the equality of areas can be visualized on "rotated graph paper".
See https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/)

(To see that the causal diamonds along different piecewise-inertial worldlines are related by a boost.
visit
robphy-CircularLightClocks-VisualizingProperTimeInSpecialRelativity
https://www.geogebra.org/m/pr63mk3j .)

The ticking light-clock mechanism is purely kinematic.
As others have said, it's really about the spacetime geometry.
(Between two timelike-related events, A and Z,
the arc-length [using the spacetime metric] along a worldline from A to Z
depends on the worldline.)

 

[Orodruin]

It's the hyperbolic geometry analogue

Minkowski geometry.