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A Thought Experiment - The Earth is Older at the North Pole!?


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This is most probably a bad application of logic on my part, but I'm intrested to see how this is contradicted.

N.B. I do not profess to know much about deep physics, this is just me following something through to it's logical conclusion.

Special Relativity tells us that as something approaches light speed, time slows down so that the speed of light reamins constant. This is shown by the famous "Twins Paradox" and has been proved by high-speed jets carrying atomic clocks. So obviously slower objects (such as the plane) dilate time in a very small way too.

Now, the Earth moves as a unit, so it's revolution around the Sun and the Sun's own movement are of no consequense in the following experiment.

At the Equator, a single point moves at approximatly 1038 miles per hour. At a Pole, the Earth hardly moves at all, we just spin on the spot.

Consider this.

An atomic clock (A) is placed on the north pole, and another identical atomic clock (:D on a point on the equator at the time the Earth gaines it's modern spherical shape.

The Earth spins for 4.5 billion years.

According to special relativity, Atomic Clock B should travel at a speed which dilates time very slightly over a long period of time. The difference may be hardly significant, but Clock A has a higher count than B.

Thus the Earth is older at the poles.

Mental eh?

Now explain to me why this is wrong. :D

Edited by JamesK
Clarification.
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My thoughts are slightly different. JamesK do you presume that the clock at the North Pole would not be moving while the clock at the Equator would be.

This would not be the case as the North Pole does indeed move at the same speed as the Equator. As you say, it spins around it's own axis. (Give or take a small distance for any variations of the Earth's cycle)

Therefore both clocks would be moving at the same speed. Clock A spinning around it's own axis and Clock B in a geo-stationary orbit with the Equator.

Just my ha'penny worth to muddy the waters a little.

Alan.

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SR relies on inertial reference frames...ie a non accelerating frame. However, despite the pole and equator moving at different speeds, the velocity is changing becuase the direction is.

SR is not valid.

also the angular velocity of the pole and the equator is the same, but the tangential velocity is as you say.

maybe this answers the question....

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SR is not valid.

Actually, it's a myth that special relativity can't deal with accelerated motion. :D For example, special relativity can be used to compare quantitatively proper time for a clock undergoing uniform circular motion in an inertial reference frame to coordinate time for the inertial reference.

In the scenario given in the original post, general relativity may, however, play a role. I'm too tired to think right now; I'll make a longer post tomorrow.

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Will any of this make a difference to what time Dr. Who is on?

Probably Not. Unless your TV is controlled by a 4.5 billion year old atomic clock. :D

I probably should have stated I was ignoring actual physical changes to the Earth (Tectonics, Errosion, Atmosphere, Convection Currents etc.), to be honest it was probably a bad example. Just imagine a large solid iron ball rotating! Ignoring rust :D.

I assume what is being said is that as the velocity is changing, the effect does not happen? I was thinking that the speed being constant would create the effect, as in the high speed plane experiments?

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Actually, it's a myth that special relativity can't deal with accelerated motion.
Indeed, you can't even compare the running rate of two clocks in relative motion without accellerating one or other. If you have a clock in a spaceship travelling past you at relativistic velocity, you can detect the Doppler shift but you have no means of knowing the actual rate at which the clock on board the spaceship is running ... that's the true essence of special relativity.

So, in that sense, the question is meaningless. If nothing on a perfectly speherical Earth ever moved, except for a uniform rotation an observer at the pole would see an equatorial clock appearing to run slowly by reason of its relative motion, whilst an observer at the equator would see a polar clock appearing to run slowly for exactly the same reason.

In actual practice, general relativity is much more significant here. The clock at the pole will run more slowly because of the stronger gravitational field caused indirectly by the Earth's rotation - centripetal acceleration deforming the Earth into an oblate spheroid; the poles are ~0.1% closer to the centre of mass than the equator is, therefore the gravitational pull is ~0.2% stronger, even before deducting the contribution of the centripetal acceleration itself to the reduced weight of an object at the equator.

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So, what I am understanding here is:

1) Thinking Theoretically, the rotation of the earth means that Special Relativity is invalid.

2) Thinking Practically, there are so many variables that the question is pretty meaningless.

3) Einstein Physics makes my brain hurt.

P.S. I'm sorry if I'm a bit useless in this department, as I've said I don't know much beyond standard mechanics (Newtonian Type) physics, to be honest I'm not sure I really understand much of that!

Edited by JamesK
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I probably should have stated I was ignoring actual physical changes to the Earth (Tectonics, Errosion, Atmosphere, Convection Currents etc.),

Might as well ignore the atomic clocks as well then........

How about one of those Casio digital watches with the red display from the 1970's :D

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1) Thinking Theoretically, the rotation of the earth means that Special Relativity is invalid.

2) Thinking Practically, there are so many variables that the question is pretty meaningless.

3) Einstein Physics makes my brain hurt.

1) Special Relativity is valid but other factors come into play.

2. Actually it's a very simple problem compared with e.g. the quantum state of an electron.

3. No comment. But if Einstein makes your brain hurt, stay well away from Schroedinger, Heisenberg, Dirac & co.

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I think you're right, a clock at the pole of a "perfectly rigid" sphere will record a different time from one that has always been on the equator, and I see nothing wrong with that.

A very similar problem was considered by Einstein, in the context of a rotating disc.

Imagine the disc to be very large (say Earth-sized) and spinning very fast. Put a ruler on the edge, aligned at a tangent to the disc.

Relative to an observer at the axis (i.e. at the "pole"), the moving ruler will appear shortened (because of "Lorentz-contraction"). The observer at the axis will see all edge measurements shortened, and therefore will see the circumference of the disc as being smaller than an observer on the edge would measure.

However both observers, on the axis or edge, would agree on the radius (or diameter) of the disc.

So if anyone on the edge divides the circumference by the diameter he will get pi, but if the observer at the "pole" does the same calculation he will get a number smaller than pi. He will conclude that the geometry of his disc is non-Euclidean (having also realised that the term "rigid body" is no longer really meaningful).

The problem is said to have been raised initially by Ehrenfest, before Einstein developed general relativity, and so is called the Ehrenfest Paradox. It was allegedly influential in making Einstein realise he would need a non-Euclidean approach. See the biography by Pais, "Subtle Is The Lord".

Your thought-experiment of the two clocks is a nice variant of the same problem, and we would conclude that an observer at the axis or pole will see a clock at the edge or equator run slow relative to his own. That's not a paradox, it's a familiar feature of relativity.

Incidentally, if you have one clock on the ground (anywhere on Earth) and another on a high tower directly above it, the one on the ground will run slower - but this is because of gravitational time dilation (Earth's gravity is stronger at the surface).

There's a Wikipedia article on the Ehrenfest Paradox:

Ehrenfest paradox - Wikipedia, the free encyclopedia

I don't have time to read it right now but I notice it's in need of "expert attention" so maybe it's rubbish. There's also a link here that is from a reputable source so should be good:

The Rigid Rotating Disk in Relativity

Postscript:

Having argued that the Earth is older at the poles, you might then ask how we can meaningfully speak of the age of the Earth, or of the unvierse.

In the case of the Earth, the difference between clocks (or radioactive isotopes) at the pole and equator is going to be tiny, even after billions of years.

In the case of the universe, the age is taken to be what would be measured by a "clock" that has remained fixed relative to the microwave background radiation. There are particles that have been moving close to the speed of light since the Big Bang and as far as they're concerned, they haven't been around for anything like 13 or 14 billion years.

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  • 3 weeks later...
Now explain to me why this is wrong. ;)

Why is it wrong? I believe this experiment has been done and verified more than once (though I can't quote one!)

A disc when coated with a decaying isotope and rotated at high speed is found to decay slower at the edges than near the centre - i.e. time is travelling slower there.

Steve

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