# Does time move slower at the equator than at the poles?

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#### John Chapman

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##### Does time move slower at the equator than at the poles?
« on: 25/07/2009 19:55:46 »
According to Einstein if one object travels at greater speed than another object then time for the first object will progress slower. This is the basis of the ‘Twin Paradox’. Actually I never really understood this because, in space, who is to say which twin is travelling and which is stationary. Speed is a relative concept. Anyway...

I’m sure I once heard Dave Ansell say that this has actually been tested. He said (at least I think it was him) that an atomic clock had been placed on a military jet that was then repeatedly zoomed around at about 1,000 mph, or some such speed, and then eventually compared with another atomic clock. The clock in the jet had fractionally lost time. Voila!

So my question is this:

Since the world is spinning, clocks at the equator are travelling at about 1,000 mph faster than those near to the poles. So do clocks around the world have to be constantly adjusted to take into consideration that those near the equator lag behind those at greater latitude?

#### exton

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##### Does time move slower at the equator than at the poles?
« Reply #1 on: 25/07/2009 22:29:50 »
The time difference is small enough that you'd never notice; your average computer or pocket watch doesn't keep time accurately enough for it to matter.

The "official" time on earth is just the average of the times kept by a number of atomic clocks around the world; i don't think anyone worries about relativistic adjustments based on their latitudes, although i could be wrong.

#### John Chapman

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##### Does time move slower at the equator than at the poles?
« Reply #2 on: 25/07/2009 23:19:58 »
Thanks for that exton

Any idea who manages world time? Presumably there must be a single clock somewhere which is recognised internationally. I don't know what part of the world you are in but here in Britain we have one hours time difference between us and France. The French and British are both so belligerent that I can't imagine either backing down if our clocks differed by a couple of seconds each year!

Are you able to quantify the effects of latitude on time keeping because of special relativity? Do you know if we talking about seconds per month or seconds per millennium?

#### lyner

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##### Does time move slower at the equator than at the poles?
« Reply #3 on: 25/07/2009 23:33:34 »
There are TWO effects to consider, here. You are moving faster at the equator BUT your weight is greater - (the gravitational field is higher because the Earth is slightly Oblate, so you are nearer to the Earth's centre of mass at the Poles) - at the poles. So both Special and General Relativity need to be considered if you want the right answer.
afaik, the difference in g makes more difference than the difference in speed so time should go a bit slower at the Poles.
If you were in Low Earth Orbit around the Equator, things might be different.

#### John Chapman

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##### Does time move slower at the equator than at the poles?
« Reply #4 on: 25/07/2009 23:48:52 »
That was very interesting. Thank you sophiecentaur. I'll try to read up some more on that. I have an 'Idiots Guide' to relativity. I'm afraid I found even that challenging to understand!  []

#### LeeE

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##### Does time move slower at the equator than at the poles?
« Reply #5 on: 26/07/2009 10:53:59 »
Time dilation has been tested using aircraft, but not supersonic ones.  The first test I'm aware of used a propeller driven aircraft, which just flew circuits around a race-track pattern and although I've got some photos in a book of both the aircraft and the tracks it flew I couldn't find any references after a quick search on the web.  The web does have numerous mentions of a later experiment though, where two commercial airliners were flown around the world, in opposite directions.

The first experiment was testing both relativistic and gravitational time dilation, as not only were the clocks (the aircraft carried a number of clocks, not just one) moving but they were also running in a different gravitational potential to the reference clocks on the ground.  IIRC, the second experiment was just testing relativistic time dilation as the clocks in each aircraft were in the same gravitational potential i.e. flying at the same height, and were then compared with each other.
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#### Bored chemist

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##### Does time move slower at the equator than at the poles?
« Reply #6 on: 26/07/2009 21:40:07 »
Thanks for that exton

Any idea who manages world time? Presumably there must be a single clock somewhere which is recognised internationally. I don't know what part of the world you are in but here in Britain we have one hours time difference between us and France. The French and British are both so belligerent that I can't imagine either backing down if our clocks differed by a couple of seconds each year!

Are you able to quantify the effects of latitude on time keeping because of special relativity? Do you know if we talking about seconds per month or seconds per millennium?

There isn't one big clock that is internationally accepted. What they use is a weighted average of a number of clocks.
There's some stuff about it here
http://en.wikipedia.org/wiki/Coordinated_Universal_Time
but it's really rather complicated.
The effects are tiny but the next generation of atomic clocks might need to take account of which floor of the building they are installed on.

#### John Chapman

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##### Does time move slower at the equator than at the poles?
« Reply #7 on: 26/07/2009 21:59:06 »
Which floor of the building..... Wow!

Knowing you to be a chemist who is bored I am not sure whether you were being mischievous or meant that literally. If that really is true it is quite amazing.

#### lyner

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##### Does time move slower at the equator than at the poles?
« Reply #8 on: 27/07/2009 11:59:45 »
Time is much more 'fragile' than one would think. I remember a second year experiment at Uni in 1966. The Mossbauer effect shows how the times of oscillation / resonance of steel atoms is affected when they are moving at a few mm per second or just separated by a few metres vertically.

#### Mike Gale

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##### Re: Does time move slower at the equator than at the poles?
« Reply #9 on: 04/11/2016 02:31:06 »
According to Einstein if one object travels at greater speed than another object then time for the first object will progress slower. This is the basis of the ‘Twin Paradox’. Actually I never really understood this because, in space, who is to say which twin is travelling and which is stationary. Speed is a relative concept. Anyway...

I’m sure I once heard Dave Ansell say that this has actually been tested. He said (at least I think it was him) that an atomic clock had been placed on a military jet that was then repeatedly zoomed around at about 1,000 mph, or some such speed, and then eventually compared with another atomic clock. The clock in the jet had fractionally lost time. Voila!

So my question is this:

Since the world is spinning, clocks at the equator are travelling at about 1,000 mph faster than those near to the poles. So do clocks around the world have to be constantly adjusted to take into consideration that those near the equator lag behind those at greater latitude?

The answer is no. The distance between stationary observers on the surface of the Earth is constant so their clocks run at the same speed if they experience the same gravity. They will disagree about the time of day at the center of the Earth though and any other point in space that is not co-rotating. A deeper question is how they perceive the distance to the center of rotation. There's no consensus on that because the relativistic effects of orbital motion are not well understood.

#### Janus

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##### Re: Does time move slower at the equator than at the poles?
« Reply #10 on: 04/11/2016 05:08:20 »

So my question is this:

Since the world is spinning, clocks at the equator are travelling at about 1,000 mph faster than those near to the poles. So do clocks around the world have to be constantly adjusted to take into consideration that those near the equator lag behind those at greater latitude?

Assuming that you are at mean sea level, clocks at the Poles and equator wll run at the same rate.  Here's why:
There are, as mentioned two things to consider here, the clock's position in the Earth's gravity field and the speed it is moving relative to the Earth's axis.

Gravitational time dilation is tied to the difference in gravitational potential( not the local gravitational strength) So on a non-rotating Earth, a clock on the Equator due to the Earth being an oblate spheroid, is further from the center of the Earth and at a higher potential and would run faster than at either Pole.

But why is the Earth an oblate spheroid? Because it is spinning.  If you ignore the Earth's gravity when considering a clock on the Equator and just deal with its circular motion or as , It can also be treated like it has a potential with respect to the Earth's axis. In this case, the higher potential is at the center of the Rotating frame. (just like it takes energy to lift a mass against gravity away from the center of the Earth, it takes energy to move a mass inward towards the center of a rotating frame.

The shape of the Earth is determined by the balancing of these two potentials so that when combined, all points of the Earth's surface at sea level are at an equal potential.  The clocks can't tell the difference between the two potentials (Equivalence principle), so they will run the at the same rate.

#### evan_au

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##### Re: Does time move slower at the equator than at the poles?
« Reply #11 on: 04/11/2016 09:56:35 »
The best everyday test of General Relativity is the GPS satellites.

They are traveling at orbital speed around the Earth, and also at a fair altitude out of Earth's gravitational field.
These effects oppose each other; after all relativistic corrections are combined, the clocks are intentionally configured to be "fast" on the ground (by 38 microseconds per day) so they produce the "right" time when they are in their planned orbit.

(The orbit doesn't stay exactly as originally planned, so the satellite also continuously transmits corrections which are used by your GPS receiver.)

See: https://en.wikipedia.org/wiki/Global_Positioning_System#History

#### Mike Gale

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##### Re: Does time move slower at the equator than at the poles?
« Reply #12 on: 04/11/2016 23:36:08 »
The shape of the Earth is determined by the balancing of these two potentials so that when combined, all points of the Earth's surface at sea level are at an equal potential.  The clocks can't tell the difference between the two potentials (Equivalence principle), so they will run the at the same rate.

Quite right. Centripetal force cancels oblateness. But an equatorial observer travels through space at a significant speed. A polar observer just spins in place. It is therefore tempting to invoke special relativity, but that's a mistake because the observers are actually in the same (rotating) reference frame.

#### yor_on

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##### Re: Does time move slower at the equator than at the poles?
« Reply #13 on: 05/11/2016 00:03:35 »
Welcome to TNS Mike
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#### Mike Gale

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##### Re: Does time move slower at the equator than at the poles?
« Reply #14 on: 05/11/2016 15:44:52 »
I should clarify my "deeper" question because observers in the same reference frame do not disagree about distances. I was thinking about an observer at the center of the Earth or a surface dweller observing a satellite in orbit.

#### Janus

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##### Re: Does time move slower at the equator than at the poles?
« Reply #15 on: 05/11/2016 17:06:38 »
The shape of the Earth is determined by the balancing of these two potentials so that when combined, all points of the Earth's surface at sea level are at an equal potential.  The clocks can't tell the difference between the two potentials (Equivalence principle), so they will run the at the same rate.

Quite right. Centripetal force cancels oblateness. But an equatorial observer travels through space at a significant speed. A polar observer just spins in place. It is therefore tempting to invoke special relativity, but that's a mistake because the observers are actually in the same (rotating) reference frame.

Clocks in the same inertial frame will always agree, but this is not the case with all clocks in an non-inertial frame such as a rotating one.  Let's take an example.  The equation of time dilation for an orbiting clock is
javascript:replaceText('$$\\sqrt{1-3GM/rc^2}$$',%20document.postmodify.message);

If we put a clock at geosynchronous orbit over the equator, r will be equal to 42164 km And the time dilation factor will be 0.9999999998424.

The time dilation for a clock resting on the Equator will be 0.9999999993042.   The clock in orbit will run 1.0000000005382 times faster than the surface clock.

Now since these clocks share the same rotating frame, then if we assume that clocks in the same rotating frame run at the same rate, then the only difference between these two clock rates should be due to gravitational time dilation. But if you compare the clock rates by just this factor, it turns out that the orbital clock should run 1.0000000005895 times faster than the surface clock.  A small difference from that gotten above, but a definite difference. This difference is due to the two clocks being at different radii from the axis of a rotating frame.

Or we can look at it like this:  Set up a line of clocks in a string such that one clock is at the center of the rotating frame with others at various distances from the axis of rotation.  Add another clock on the axis that does not share in the rotation of the other frames.   According to this clock, all the clocks in the rotating frame are moving at different speeds relative to it and ticking at different rates, with the the one sharing the axis with it showing no time dilation.   But this is in contradiction to the idea that all the clocks in the rotating frame run at the same rate.

#### Mike Gale

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##### Re: Does time move slower at the equator than at the poles?
« Reply #16 on: 05/11/2016 23:32:08 »
An observer in geosynchronous orbit experiences no gravity so that reference frame is different from the one on the ground, even though they rotate at the same rate. You're thinking of GPS satellites, which are not geosynchronous. In that case, the satellite moves with respect to the ground so you have to account for special relativity as well as gravity. All terrestrial observers at the same altitude are in the same reference frame so they experience time (and space) in the same way. It doesn't matter how that reference frame moves with respect to any other, even if it is accelerating.

Each of the clocks in your string occupies a different reference frame. I presume there is no gravity in your scenario so the clocks must have rocket engines to maintain their orbits. It is the acceleration of the rocket engines that distorts time.

#### Mike Gale

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##### Re: Does time move slower at the equator than at the poles?
« Reply #17 on: 07/11/2016 03:38:26 »
I stand corrected. The Schwartzchild solution actually contains a factor that depends on latitude. Equitorial speed is 463 m/s so equatorial time runs slower than polar time by 1-sqrt(1-(463/c)^2)=0.1 usec/day.

#### jeffreyH

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##### Re: Does time move slower at the equator than at the poles?
« Reply #18 on: 07/11/2016 04:23:32 »
I stand corrected. The Schwartzchild solution actually contains a factor that depends on latitude. Equitorial speed is 463 m/s so equatorial time runs slower than polar time by 1-sqrt(1-(463/c)^2)=0.1 usec/day.

The Schwarzschild solution is non rotating so cannot change with latitude. The Kerr solution rotates and contains an ergosphere. This does vary with latitude.
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#### Mike Gale

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##### Re: Does time move slower at the equator than at the poles?
« Reply #19 on: 07/11/2016 23:42:02 »
The formula is on Wikipedia. Search for Schwartzchild solution.
Plug in dr=0, dtheta=0, theta=0 deg for polar time and 90 deg for equator time,  and v=r*dphi/dt. The result looks exactly like the special relativity time transform with no spatial displacement.

#### CPT ArkAngel

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##### Re: Does time move slower at the equator than at the poles?
« Reply #20 on: 08/11/2016 02:48:06 »
The Earth is not spherical. The radius of the Earth at sea level (on average) compensates for the rotational speed. So time dilation is the same everywhere. You must calculate the time dilation due to the speed and the gravitational potential. At the equator, the higher radius lowers the gravitational potential and thus lowers time dilation by the same amount as the increase in time dilation due to the rotational speed.

#### Mike Gale

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##### Re: Does time move slower at the equator than at the poles?
« Reply #21 on: 08/11/2016 03:19:39 »
Kerr solution won't make any difference in this use case because we assume the observers experience the same gravity. All it will tell you is that the polar observer is closer to the centre of rotation than the equatorial observer.

#### Jeremy Fiennes

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##### Re: Does time move slower at the equator than at the poles?
« Reply #22 on: 09/11/2016 10:37:34 »
In his 1905 paper Einstein says that an equator clock will run more slowly. But the distance between the clocks being constant, their relative speed is zero, and according to SR they should show the same time. Einstein puts in his paper a statement, which if true, would refute his own theory.

#### Janus

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##### Re: Does time move slower at the equator than at the poles?
« Reply #23 on: 09/11/2016 17:40:34 »
An observer in geosynchronous orbit experiences no gravity so that reference frame is different from the one on the ground, even though they rotate at the same rate.
The fact that such an observer feels no g-forces is irrelevant.  Local g-forces are not responsible for gravitational time dilation, difference in gravitational potential is.  To illustrate the difference compare the calculated gravitational time dilation for the surface of the Earth to that for Uranus and then compare their respective surface gravity
Using a radius of 6378 km for the Earth's radius and 6e24 kg for its mass, you get a surface gravity of 9.842 m/s^2.  Using a radius of 25559 km for the radius and 8.68e25 kg for its mass, you get a surface gravity of 8.867 m/s.  using the same numbers to calculate the time dilation at these two surfaces you get 0.9999999993025 for the Earth and 0.9999999974820 for Uranus.  The clock on Uranus will run slower even it will experience less gravity.
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You're thinking of GPS satellites, which are not geosynchronous. In that case, the satellite moves with respect to the ground so you have to account for special relativity as well as gravity.
No, I choose to use geosynchronous orbit for a particular reason. The formula I gave is the accepted equation for any clock in a circular orbit.  Since a geosynchronous orbit is in the same rotational frame as a clock on the surface of the Earth, then if your argument that clocks in the same rotational frame run at the same rate, then the only factor that should cause a difference between the two clocks would be the gravitational time dilation. The math does not bear this out.
Quote
All terrestrial observers at the same altitude are in the same reference frame so they experience time (and space) in the same way. It doesn't matter how that reference frame moves with respect to any other, even if it is accelerating.
Clocks at the same altitude (relative to mean sea level) do all run at the same rate, just not for the reason you give.
Quote

Each of the clocks in your string occupies a different reference frame. I presume there is no gravity in your scenario so the clocks must have rocket engines to maintain their orbits. It is the acceleration of the rocket engines that distorts time.
The acceleration felt by these clocks have no effect on them, this is the basis of the Clock Postulate.  The clock postulate has been tested by putting radio-isotopes in high speed centrifuges and measuring how much they decay.  By using different radii and speeds you can test a number of different scenarios.  You can expose samples to exactly the same centripetal acceleration/g-force but at different tangential velocities,  or traveling at the same speed while experiencing different centripetal accelerations/g-forces.  The results were that even at 1000's of gs, the acceleration had no effect on the decay rate. It only depended on the tangential velocity.  A clock traveling at 0.866c in circle with a radius of 100 light years would only experience 0.007275 g,  but would still run 1/2 as fast as one at the center of the circle.  A clock traveling at that same speed at a radius of 1 AU would experience 46038 g, and would again run 1/2 as fast as the one at the center.

#### CPT ArkAngel

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##### Re: Does time move slower at the equator than at the poles?
« Reply #24 on: 09/11/2016 21:27:50 »
In his 1905 paper Einstein says that an equator clock will run more slowly. But the distance between the clocks being constant, their relative speed is zero, and according to SR they should show the same time. Einstein puts in his paper a statement, which if true, would refute his own theory.

General relativity was published in 1915 and it added gravitational time dilation.

#### Mike Gale

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##### Re: Does time move slower at the equator than at the poles?
« Reply #25 on: 24/11/2016 23:43:58 »
I think Janus is right. The coordinates in the Schwartzchild line element are earth-cantered, earth-fixed (ECEF) so the equatorial observer is not orbiting the polar observer. If both experience the same gravity, they will experience time in the same way.