Resolving the Twin Paradox: A Mathematical Approach

[rczmiller]

I am not sure if this is the right place to ask this question. I am an amateur but I enjoy reading and trying to understand relativity. If anyone is aware of an article that addresses this question, I would appreciate the reference so that I can read up on it. Unfortunately, there are a lot of websites out there that claims to have the answer, but some of the info seems unfounded. If someone can recommend a better site that I can ask questions to try to clarify my thoughts/readings/understands, I would also welcome the info. I don’t want to waste the experts’ time dealing with my simple questions.

Anyway – it is my understanding that relativity and the twin paradox addresses time passage for 2 objects when one object speeds away from the other then returns. The object that accelerates away will experience time passage “slower” then the other “stationary” object. Since space/time is relative, it does not matter what direction the moving object speeds off. Time relative to the stationary object will slow down.

I have been playing with a thought experiment using 3 objects and I am not sure of the results. From Earth (object #1) a spacecraft (object #2) speeds away at near light speeds. Due to relativity, time would slow down for the spacecraft so that the speed of light would remain constant. What if at some point early in this trip, the spacecraft launched a probe (object #3) which accelerates back towards Earth at the same speed the spacecraft was traveling away from the Earth. The spacecraft would see the object speed away from it at near light speed, but the Earth would view the probe as slowing down to a standstill relative to the Earth. Does time slow down for the probe as it relates to the spacecraft , or will the probe’s time begin to keep step with the Earth clocks?

To finish up this thought experiment, say that the spacecraft travels around the nearest star and returns back to Earth. However, before the spacecraft returns home, the probe accelerates again towards Earth so that the spacecraft can capture the probe. When the spacecraft and probe return to Earth, what would the clocks read? It is my understanding that since the probe accelerated away for the relative clock on the spacecraft , the probe’s clock would be slower than the spacecraft ’s clock and the spacecraft ’s clock would be slower than the Earth’s clock.

Here’s my first problem, what if a second spacecraft was launched from Earth at a speed well below light and retrieved the probe and returned it back to Earth long before the original spacecraft returned to Earth. It seems to me that at some point, the probe’s clock should begin to keep time with the Earth. So if the probe accelerated away from the first spacecraft so that it “hung” over the Earth, why would the time passage for the probe speed up to match Earth’s clock if it was accelerated at near light speed after it’s original burst to move away from the first spacecraft ?

These thoughts lead me to the subject of my title. If the universe started with a “Big Bang” and the galaxies are all moving away from each other (for the most part), what if the origin point of the universe was object #1 and the Earth was considered object #2 and we launched a spacecraft (object #3) back toward the universe’s origin point. If the spacecraft ever returned to this point, what would the clocks read for the spacecraft relative to the starting point and relative to the Earth?

Can anyone recommend a presentation for a layman on the above? I am sure I am not the first to ask this question, so if there is a paper out there on this subject, I would love to read it.

Now off to more reading...thank you to all!

 

[JesseM]

Anyway – it is my understanding that relativity and the twin paradox addresses time passage for 2 objects when one object speeds away from the other then returns. The object that accelerates away will experience time passage “slower” then the other “stationary” object. Since space/time is relative, it does not matter what direction the moving object speeds off. Time relative to the stationary object will slow down.

It doesn't actually matter who accelerates initially, although it does matter which twin accelerates to turn around and return to the other one. But no matter who accelerates, you're free to analyze the problem from the point of view of any inertial (non-accelerating) reference frame, and in any given frame, whichever of two clocks is moving faster at a given moment will be ticking slower.

I have been playing with a thought experiment using 3 objects and I am not sure of the results. From Earth (object #1) a spacecraft (object #2) speeds away at near light speeds. Due to relativity, time would slow down for the spacecraft so that the speed of light would remain constant.

In the Earth's frame, yes (although the fact that the speed of light is measured to be constant is not just due to time dilation, it's also due to length contraction and the relativity of simultaneity). But now look at it in the frame of an inertial observer who sees the Earth moving at near light speed, and then sees the spacecraft come to rest after it accelerates. In his frame, it is the Earth's clock that is ticking slower.

What if at some point early in this trip, the spacecraft launched a probe (object #3) which accelerates back towards Earth at the same speed the spacecraft was traveling away from the Earth. The spacecraft would see the object speed away from it at near light speed, but the Earth would view the probe as slowing down to a standstill relative to the Earth. Does time slow down for the probe as it relates to the spacecraft , or will the probe’s time begin to keep step with the Earth clocks?

If the probe is at rest relative to the earth, then in the Earth's frame the probe's clock will be ticking at the normal rate.

To finish up this thought experiment, say that the spacecraft travels around the nearest star and returns back to Earth. However, before the spacecraft returns home, the probe accelerates again towards Earth so that the spacecraft can capture the probe.

You mean that at the moment the spacecraft passes the probe on its way back to earth, the probe accelerates so it is at rest relative to the spacecraft (ie traveling alongside it back to earth)?

When the spacecraft and probe return to Earth, what would the clocks read?

We'd need distances and times to have specific numbers, but the Earth's clock will have ticked the greatest time, the probe will have ticked the second-greatest, and the spacecraft will have ticked the least time.

It is my understanding that since the probe accelerated away for the relative clock on the spacecraft , the probe’s clock would be slower than the spacecraft ’s clock and the spacecraft ’s clock would be slower than the Earth’s clock.

It's only velocity that matters, not acceleration. The rate of ticking at any given time is dependent only on the velocity at that moment, and the total elapsed time can be found by integrating the rate of ticking over the entire trip. In the Earth's frame, the Earth clock was at rest 100% of the time, the spacecraft 's clock was moving at relativistic velocity 100% of the time, and the probe's clock was at rest part of the time and moving at relativistic velocity part of the time, so the spacecraft 's clock will have ticked the least time and the probe's clock will have ticked a time midway between the spacecraft 's and the earth's. You could also analyze things from some other inertial frame and you'd get the same answer about what they read when they meet, but it might be a little more complicated.

Here’s my first problem, what if a second spacecraft was launched from Earth at a speed well below light and retrieved the probe and returned it back to Earth long before the original spacecraft returned to Earth. It seems to me that at some point, the probe’s clock should begin to keep time with the Earth. So if the probe accelerated away from the first spacecraft so that it “hung” over the Earth, why would the time passage for the probe speed up to match Earth’s clock if it was accelerated at near light speed after it’s original burst to move away from the first spacecraft ?

Again, the rate the probe's clock ticks at a given moment in Earth's frame depends only on its velocity in Earth's frame at that moment. At any given moment, if the probe's velocity is $$v$$, then it ticks at $$\sqrt{1 - v^2/c^2}$$ the normal rate at that moment, in the Earth's frame.

These thoughts lead me to the subject of my title. If the universe started with a “Big Bang” and the galaxies are all moving away from each other (for the most part), what if the origin point of the universe was object #1 and the Earth was considered object #2 and we launched a spacecraft (object #3) back toward the universe’s origin point. If the spacecraft ever returned to this point, what would the clocks read for the spacecraft relative to the starting point and relative to the Earth?

In GR, the Big Bang did not happen at a distinct point in a preexisting space, it's an expansion of space itself, so there is no "origin point of the universe". Check out the article http://www.sciam.com/article.cfm?articleID=0009F0CA-C523-1213-852383414B7F0147 on differential geometry, the mathematical basis for general relativity, which talks about the difference between intrinsic and extrinsic descriptions of curvature).

For a universe with zero curvature, picture an infinite chessboard in which all the squares are growing at the same rate, while the pieces at the center of each square remain unchanged in size. If you play the movie backwards, the distance between any two squares approaches zero as you approach the moment of the big bang, which means the density of the matter on the squares (represented by the chess pieces) approaches infinity as it gets smushed together more and more tightly. A universe with negative curvature would be something like an infinite saddle-shape which is a little harder to picture expanding, but if you can picture the other two you get the basic idea. From Ned Wright's Cosmology Tutorial, a graphic showing the 2D analogues of the three types of spatial curvature, negative, zero, and positive:

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[Perspicacious]

There are subtleties in the twin paradox that can't be avoided unless you're competent in high school algebra and can follow carefully stated mathematical reasoning, line by line. I only know of one paper to resolve your question. It comes highly recommended. It is carefully detailed and is written by a mathematician. Let me know if you have any questions. Click here.

 

[JesseM]

There are subtleties in the twin paradox that can't be avoided unless you're competent in high school algebra and can follow carefully stated mathematical reasoning, line by line. I only know of one paper to resolve your question. It comes highly recommended. It is carefully detailed and is written by a mathematician. Let me know if you have any questions. Click here.

Just so rczmiller knows, this paper contains a lot of non-mainstream ideas...and aren't you, Perspicacious, the author of it?