# Frame-switching puzzler in the twin paradox

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Short version: Can a space traveler ever observe Earth time appearing to go backward? I claim "no" but under that claim I keep coming around to an inconsistency where more distant things will age more than nearer things. Where am I going wrong?

Long version:

I'm trying to figure out what is observed by the traveling twin during an extremely fast deceleration + return acceleration phase in the twin paradox. This is also described as the rocket undergoing a frame switch. According to my understanding of what I've read, the traveling twin will see the Earth twin age a large amount in that very short period of rocket time.

What happens if the rocket "frame-switches" several times while far from Earth, by coming to a stop and accelerating toward Earth, then stopping and accelerating away from Earth again (involves multiple switches between 2 frames: outbound, and return)? What happens if it repeats this, "shaking" back and forth, reaching high velocity each time, over a very little duration of rocket time?

Solution 1 (no good): My calculations show that the Earth twin will continue to age rapidly during these frame switches (specifically, she will age much as length contraction takes effect when accelerating in each direction, and age not at all as length contraction is released when decelerating). However, it also is apparent that the distance that the rocket is from Earth will determine how much the Earth twin ages when the rocket does this little trick. This leads to inconsistency...

Suppose the rocket has traveled to Planet X which is stationary relative to Earth, and then "shakes" for awhile. The Earth twin will age a lot relative to the rocket twin, but a Planet X twin will age only slightly faster than the rocket twin. This makes no sense because the Earth twin and Planet X twin should not age differently relative to each other.

Solution 2: When the rocket switches from outbound to return frame, the Earth twin will age relatively fast, but when switching from return to outbound frame, the aging difference will be undone. One way for this to happen is for one twin to age fast and then the other twin to age fast. But if the rocket can shake many times in a short period of time, it should age only that short period of time. So if the Earth twin ages a great amount during one frame switch, it must un-age on the other frame switch. This means the rocket can observe earth time going backwards.

I hope that this is NOT the case, because it punches a huge hole in my theory of how time works, and my understanding of observable reality.

Solution 3: The time periods in which the Earth twin seems to age greatly actually overlap, so that if the rocket shakes for awhile, the rocket twin observes only one aging period on Earth (possibly fluctuating between fast and slow aging as the rocket shakes?).

Solution 4: Not all frame switches have the same observed relative aging?

Solution 5: Something I've missed? Some way in which time dilation compensates? Or a maximum possible acceleration rate?

Edited by md65536
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Solution 4. The amount of aging that manifests itself depends on the amount of travel prior to the frame switch, not the frame switch itself (the effects of which are often ignored, because they tend to be small). If we have one scenario where the moving twin ages X years and the earth twin ages Y, with only the acceleration at turnaround, we aren't going to change that if the moving twin undergoes other course-changing (but not speed changing) accelerations along the path. What will change is the rate of observed aging, (i.e how much happened, and when) but not the total amount. Changing speed will affect the amount of dilation that is observed, but that can be directly calculated.

If the earthbound twin sends out a light pulse at regular intervals, there is nothing the other twin can do to make one pulse pass another, which is what would be required to have time going backwards.

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Solution 4. The amount of aging that manifests itself depends on the amount of travel prior to the frame switch, not the frame switch itself (the effects of which are often ignored, because they tend to be small). If we have one scenario where the moving twin ages X years and the earth twin ages Y, with only the acceleration at turnaround, we aren't going to change that if the moving twin undergoes other course-changing (but not speed changing) accelerations along the path. What will change is the rate of observed aging, (i.e how much happened, and when) but not the total amount. Changing speed will affect the amount of dilation that is observed, but that can be directly calculated.

If the earthbound twin sends out a light pulse at regular intervals, there is nothing the other twin can do to make one pulse pass another, which is what would be required to have time going backwards.

Suppose rocket twin is at rest 4 light years away and Earth twin is sending a pulse every year. You might have a situation where there are 4 pulses "en route" that till take respectively 4, 3, 2, and 1 years to reach rocket twin. Then suppose rocket twin accelerates toward Earth such that gamma = 2 for a negligible duration. The space "occupied" by the pulses contracts, so the pulses are now .5 light years apart, and will take 2, 1.5, 1, and 0.5 years to reach the rocket. If the rocket returns to rest the pulses will return to taking 4, 3, 2, 1 years to reach the rocket.

Is this correct?

I'd somehow assumed that invariance of c would mean that the light pulses would remain the same distance apart (oops) and that they'd be 2, 1, 0, and 0 years away (the last 2 compressed into a "burst" of aging the Earth appears to experience).

If the rocket is at rest 4 light years from Earth and instantly accelerates so that gamma = 2, then without the rocket having to move anywhere yet, it is now 2 years away by light signal. So the Earth must have gone through 2 years of aging during that rocket-time. I'd (incorrectly I guess) assumed that the rocket would observe that aging during the instant acceleration. If I now understand correctly, we might say that the 2 years of aging applies to Earth's present ("now" on Earth according to the rocket is 2 years away whereas it was 4 years away only moments ago), which the rocket won't observe for some time. I see now why wikipedia says the frame switch is more of an update to simultaneity than a literal aging. If the rocket remains at that velocity (about 0.866c so that gamma = 2), then it will observe those 2 years of Earth's "extra" aging spread over some time as it makes its way back.

Is this also correct?

Further, if the rocket is 2 light years away and goes from gamma = 2 to gamma = 1, it is now 4 light years away, and the "update to simultaneity" means Earth's present (according to the rocket) is earlier than it was a moment ago, but no "negative time" will be observed because the rocket is no longer traveling. The expected observation that was less than 2 years away a moment ago is now 4 light years away.

I will have to update my calculations and see what I can salvage from them. When I feel I understand some part of it, the twin paradox seems like the greatest math and logic puzzle I've ever attempted. The other 95% of the time it's the worst.

Edited by md65536
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I find that the most understandable explanation of what goes on in the various frames of reference in the twins paradox is to consider the effects of time dilation and the Dopper Effect. I suggest you look at the link below.

and click on:

Its Relative

Archives

(Sorry but a direct link is having problems for some reason)

Hope this helps.

Edited by I ME
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Suppose rocket twin is at rest 4 light years away and Earth twin is sending a pulse every year. You might have a situation where there are 4 pulses "en route" that till take respectively 4, 3, 2, and 1 years to reach rocket twin. Then suppose rocket twin accelerates toward Earth such that gamma = 2 for a negligible duration. The space "occupied" by the pulses contracts, so the pulses are now .5 light years apart, and will take 2, 1.5, 1, and 0.5 years to reach the rocket. If the rocket returns to rest the pulses will return to taking 4, 3, 2, 1 years to reach the rocket.

Is this correct?

Looks OK so far, with the caveat that the time for the pulses to reach will be reduced by whatever time passes. But the spacing would again be 1 year.

I'd somehow assumed that invariance of c would mean that the light pulses would remain the same distance apart (oops) and that they'd be 2, 1, 0, and 0 years away (the last 2 compressed into a "burst" of aging the Earth appears to experience).

If the rocket is at rest 4 light years from Earth and instantly accelerates so that gamma = 2, then without the rocket having to move anywhere yet, it is now 2 years away by light signal. So the Earth must have gone through 2 years of aging during that rocket-time. I'd (incorrectly I guess) assumed that the rocket would observe that aging during the instant acceleration. If I now understand correctly, we might say that the 2 years of aging applies to Earth's present ("now" on Earth according to the rocket is 2 years away whereas it was 4 years away only moments ago), which the rocket won't observe for some time. I see now why wikipedia says the frame switch is more of an update to simultaneity than a literal aging. If the rocket remains at that velocity (about 0.866c so that gamma = 2), then it will observe those 2 years of Earth's "extra" aging spread over some time as it makes its way back.

Is this also correct?

The earth will not instantly age. 2 earth years will not have passed until the moving twin gets two light pulses. The wikipedia part is right.

Further, if the rocket is 2 light years away and goes from gamma = 2 to gamma = 1, it is now 4 light years away, and the "update to simultaneity" means Earth's present (according to the rocket) is earlier than it was a moment ago, but no "negative time" will be observed because the rocket is no longer traveling. The expected observation that was less than 2 years away a moment ago is now 4 light years away.

I will have to update my calculations and see what I can salvage from them. When I feel I understand some part of it, the twin paradox seems like the greatest math and logic puzzle I've ever attempted. The other 95% of the time it's the worst.

Don't get too down. This is not intuitive, which is why it is (incorrectly) called a paradox.

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If the rocket is at rest 4 light years from Earth and instantly accelerates so that gamma = 2, then without the rocket having to move anywhere yet, it is now 2 years away by light signal. So the Earth must have gone through 2 years of aging during that rocket-time.

The earth will not instantly age. 2 earth years will not have passed until the moving twin gets two light pulses. The wikipedia part is right.

Then I should change the wording... something like:

So there must be 2 years of Earth aging corresponding to the contraction in distance between Earth and rocket, but the full 2 years of aging will only be observed over time as the rocket moves, and any yet-unobserved portion of that expected aging can disappear (or be wiped out by another simultaneity correction or something) if the rocket doesn't maintain its velocity.

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Age is the accumulated time, so it's not right to look at it that way. The clock rates change when you undergo an acceleration — you lose the symmetry of having two inertial frames. So the space twin starts aging more slowly when he undergoes an acceleration, meaning his clock rate slows down.

It's similar to (classical) kinematics, where time is the analog to distance. If you briefly accelerate toward something, you don't jump toward it and jump back when the acceleration reverses. You get there faster, i.e. the speed increases, and the distance gets smaller, at a rate given by your speed.

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