Does the Earth's Rotation Affect Time Dilation in Einstein's Twin Paradox?

[poeteye]

I know that the frame of reference determines that it is the stay-at-home twin who ages faster than the twin traveling at some fraction of light speed. Some years back, an experiment in which one of a pair of synchronized atomic clocks was placed on a commercial airliner and flown around the world, resulted in a discrepancy between the two clocks (a matter of tiny fractions of a second) precisely confirming the time dilation predicted by special relativity.
However, I have another experiment to propose, along the lines of the twin atomic clock experiment. This time, the stay-at-home twin with one clock is positioned on the surface of the Earth at the equator somewhere. At this point, viewed from the perspective of Polaris, the pole star, due to the rotation of the Earth on its axis, the twin and his clock are moving counterclockwise in a circle with a radius of about 8000 miles (the radius of the earth). Now, from the same point, the traveling twin takes off in an airplane and flies westbound (that is, counter to the rotation of the earth) at a speed of about 2000 miles per hour--exactly twice the rotational speed of the Earth at the equator. Viewed from Polaris, with a mighty telescope, the traveling twin moves in a path that is the mirror image of his stay-at home twin -- a circular arc that is 8000 miles in diameter.
Also, since he flies counter to the direction of the Earth's rotation, the Earth's velocity (of which the Polaris viewer is presumed to be ignorant) must be subtracted from his apparent velocity. His speed, as seen from Polaris, is exactly the same as his stay at home twin.
Now, one twin with an atomic clock stays put on the surface of the Earth for twelve hours, while his brother with a synchronized clock flies completely around the globe and returns to the same spot twelve hours later. They have moved with respect to one another at a velocity of some 2000 miles per hour--which should produce a detectable time dilation along the lines of the experiment I described from many years ago. Yet, to the Polaris observer, the two brothers followed two identical, mirror-image semi-circular paths at exactly the same speeds.
Which twin ages faster (and why)? You surely can't say that neither one does, since this experiment has actually been performed -- I really do remember hearing about it!

 

[JesseM]

Now, one twin with an atomic clock stays put on the surface of the Earth for twelve hours, while his brother with a synchronized clock flies completely around the globe and returns to the same spot twelve hours later. They have moved with respect to one another at a velocity of some 2000 miles per hour--which should produce a detectable time dilation along the lines of the experiment I described from many years ago.

Why do you think it should produce a detectable difference in time dilation? Neglecting gravitational time dilation effects and imagining the Earth as a massless sphere (which I think is OK since gravitational time dilation effects would be almost exactly the same for the two twins), if the Polaris observer is inertial, then in his frame both twins have the same speed at every moment, so their clocks are always ticking at the same rate, so they'll have aged the same amount when they reunite (of course, they'll both have aged less than the Polaris observer has according to his clock). Do you understand the distinction between inertial and non-inertial frames in SR, and the fact that the standard time dilation equation used in inertial frames (where time dilation is just a function of velocity in that frame) doesn't work in non-inertial frames, like the rest frames of either of the twins in your example?

 

[matheinste]

Hello poeteye

Winning or losing a vote does not make a physical process right or wrong.

I cannot vote because i have not read your post. If i had read your post i may have formed an opinion but, like many other readers, if i had read it i may have reached the wrong conclusion.

Matheinste.

 

[truhaht]

I'm not going to vote because the issue is so complicated and you supposedly already know the correct answer. Obviously, if two twins were each coasting in space without engines, orbiting around a central gravitational body but in opposite directions, then their clocks would agree each time they meet up. Each ascribes time dilation to the other's clock, but the paths are not strictly inertial and so the existence of certain pseudo-forces will affect the reckoning of a distant clock in subtle ways that require 'difficult' math. In that situation I've described, each twin experiences zero G's, but he must correct for the G-forces by which the opposite twin is clearly being diverted from straightness. According to the Polaris observer also, their clocks will agree at each reunion.

But in the scenario described in the opening post, one twin is being propelled by the Earth's surface while the other is being propelled by a different (more artificial) impetus, and this might very well lead to an accrued discrepancy at each reunion. 'Not sure.

Let's hear more

 

[Ich]

The answer is quite simple, JesseM gave it, and there are definitely no experiments that say otherwise. Maybe poeteye refers to the famous Hafele-Keating experiment, where a discrepancy was measured because the planes had different speeds in the polaris system.

 

[poeteye]

I'm sorry, but the experiment has been performed more than once. I've heard that it has been conducted on the space shuttle. There are not two planes involved. One twin stays put on the surface of the earth, the other twin flies around the Earth at 2000 miles an hour (in my version of the experiment). There can be no doubt that the two twins experience a definite difference in their velocities. And in this situation, a detectable difference in time dilation has been detected--with the traveling twin aging more slowly. Perhaps the experiment with an airplane used a plane that traveled eastbound (in the direction of the Earth's rotation), as the space shuttle certainly did, since our rockets do not yet have enough delta vee to launch a payload around the Earth the other way without looping around the moon--but how can this make a difference? The traveling twin in my experiment travels at twice the rotational velocity of the earth. Even if the Earth's rotation is subtracted, there is still a significant difference in the velocities of the two twins, and there should be a measurable time dilation for traveling west if there is one for traveling east--shouldn't there?

I concocted this thought experiment because of an answer I received to an earlier question. I asked why, in the traditional version of the twin paradox, it was not possible to consider that the rocket bound twin is "stationary" while the Earth and the stay-at-home twin zooms off at near the speed of light and returns to find his brother has aged tremendously. It seems that there must be a preferred frame of reference--that of the the universe at large. I then wanted to come up with a thought experiment in which the universe at large had a difference with both twins.

 

[JesseM]

I'm sorry, but the experiment has been performed more than once. I've heard that it has been conducted on the space shuttle.

You're sure it has been conducted in exactly the way you describe, where both clocks would have precisely the same speed at all times in the Polaris frame?

Even if the Earth's rotation is subtracted, there is still a significant difference in the velocities of the two twins, and there should be a measurable time dilation for traveling west if there is one for traveling east--shouldn't there?

I don't understand what you mean by "even if the Earth's rotation is subtracted". Didn't you specify that both twins had the same speed at all times in the Polaris frame? If so, then assuming the Polaris frame is inertial, that's enough to guarantee that the twins will have aged the same amount as one another when they reunite.

I asked why, in the traditional version of the twin paradox, it was not possible to consider that the rocket bound twin is "stationary" while the Earth and the stay-at-home twin zooms off at near the speed of light and returns to find his brother has aged tremendously. It seems that there must be a preferred frame of reference--that of the the universe at large. I then wanted to come up with a thought experiment in which the universe at large had a difference with both twins.

No. The answer to your question is that the traditional time dilation formula only works in inertial frames, and if you use a frame where the traveling twin was stationary throughout the trip (even when he was firing his rockets to turn around and feeling G-forces), then this is not an inertial frame and you can't assume the normal time dilation rules apply. This is why I asked you if you understood the difference between inertial and non-inertial frames, but you didn't answer. "Inertial" in SR just means moving at constant velocity (note that 'velocity' refers to both speed and direction, so moving in a circle at constant speed is not moving at a constant velocity), and consequently feeling weightless (there's no gravity in SR); a non-inertial observer will know he's moving non-inertially (accelerating) because he feels G-forces.

So there's no preferred inertial frame, though inertial frames do work differently than (and thus are in a sense 'preferred' over) non-inertial ones. Suppose the twin flies away from Earth inertially, but doesn't turn around, instead the Earth itself is accelerated by giant rockets to catch up with the traveling twin (again, we're assuming an SR scenario where there's no gravity, so assume the Earth is something like a hollow sphere of negligible mass). In this case it will be the Earth twin who has aged less when they reunite, because the Earth twin was the one who accelerated while the traveling twin moved inertially throughout the journey. You would reach this conclusion no matter what inertial frame you used to do your calculations--all frames always agree on local predictions like what two clocks will read when they are next to one another.

 

[truhaht]

I asked why, in the traditional version of the twin paradox, it was not possible to consider that the rocket bound twin is "stationary" while the Earth and the stay-at-home twin zooms off at near the speed of light and returns

Either twin can be deemed the stationary one but only the astronaut twin can unequivocally be said to have willfully altered course. That might sound a tiny tad self-contradictory but no.

 

[DrGreg]

I'm sorry, but the experiment has been performed more than once. I've heard that it has been conducted on the space shuttle. There are not two planes involved. One twin stays put on the surface of the earth, the other twin flies around the Earth at 2000 miles an hour (in my version of the experiment). There can be no doubt that the two twins experience a definite difference in their velocities. And in this situation, a detectable difference in time dilation has been detected--with the traveling twin aging more slowly.

I think you are getting confused with the experiment that was actually performed and your own requested version.

In the original version, summarised here (section 5), an eastbound plane lost 59 ns and a westbound plane gained 273 ns. Taking into account the speeds of the planes, the rotation of the Earth and gravitational effects due to altitude changes, these results were entirely in line with what relativity predicted. A more recent version is described here.

In your version of the experiment, the theory predicts both clocks have identical dilation relative to the polar observer. If you are aware of a version of the experiment that contradicts this, please give us a reference for it. Such an experimental result would disprove the theory of relativity and I think we would have heard about it!

 

[yuiop]

...Yet, to the Polaris observer, the two brothers followed two identical, mirror-image semi-circular paths at exactly the same speeds.

I agree with DRGreg, JesseM and Ich. The clocks would age at the same rate in the hypothetical experiment you describe. The clock that is flown around the world at 2000 mph westward would show the same total elapsed time as the clock that stayed on the ground (using a rounded figure of 1000mph for the velocity of the Earth's surface at the equator). The proper time shown by the clocks when they are compared on the ground at the end of the experiment is in exact agreement with the Polaris observers conclusions. This invalidates your poll because you have not included the correct answer as an option!

...
Which twin ages faster (and why)? You surely can't say that neither one does, since this experiment has actually been performed -- I really do remember hearing about it!

Your experiment has not actually been performed (although a similar one has - http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/airtim.html#c3")

In the linked experiment the speed of the Westbound aircraft was about 520 mph (25,000 miles / 48.6 hours) which is far short of 2000 mph required for your experiment. Even Concorde could only manage 1348 mph.

In the Hafele-Keating experiment the average speeds of the aircraft clocks relative to the ground clock were

Westbound clock: 520 mph clockwise.
Eastbound clock: 607 mph anti-clockwise.

From the Polaris observer's point of view the average speeds of the clocks relative to his none rotating frame are,

Westbound clock: 517 mph anti-clockwise.
Earthbase clock: 1037 mph anti-clockwise.
Eastbound clock: 1644 mph anti-clockwise.

From the above list it is easy to see that the Eastbound clock should age the least, the Westbound clock should age the most and the Earthbase clock should age somewhere in-between. That is exactly what was measured in the actual experiment.

If you understand the principles involved, you should also be able to convince yourself that if the aircraft flew at 1000 mph westwards, the clock on the aircraft would age MORE than the clock that remains on the ground and not less as your initial instincts might suggest. Something you have to keep in mind is that rotation has an absolute nature. For example if there were no stars or any other bodies in the universe other than the Earth, there would still be many ways to determine whether or not the Earth was rotating.