Speed Difference of Time on Poles vs Equator of Earth and Other Planets

[some bloke]

TL;DR Summary

Is there a reason that the bulge of the earth due to rotation, the dilation of time due to gravity and that of the speed of the surface of the equator all cancel out to mean time is the same all over the earth?

I had the wonder of whether, due to the speed difference, clocks ran at a different speed on the equator than on the poles. I then researched this, and found that, due to being closer to the centre of the Earth at the poles due to the bulging of the Earth due to spinning, the two essentially cancel out.

I'm struggling to find the needed equations, velocity time dilation I found OK but gravitational is eluding me. I have worked out that, purely by relative velocity, a second is 1.00000000000671 different between the poles and the equator (taking the time for a rotation or Earth as 36484s, or 23hrs 56 mins and 4 seconds). I'm looking for the equations for the effects of gravity on time.

My curiosity is this: Is there any particular relevance to the fact that these two cancel out, and is this the case on all planets?

I would like to work out the relative time-speed (I'm sure there's a better term) for the poles & equators of all the planets & the sun, and establish if this is a consistent effect across everything (though time will likely be different between planets) and if this has any relevance to the spin speed of planets at all.

I doubt it'll be ground-breaking or anything, but I am curious, and that's step one. Any help gratefully appreciated!

 

[vanhees71]

Since the "geoid" is to a good approximation a equipotential surface of the effective gravitational field (including the effect of the Earth's rotation around its axis) to a good approximation the clock at the equator runs at the same rate as the one at the pole. In 1905 Einstein didn't know this, and that's why there's a wrong statement in his famous SRT paper ;-).

 

[Janus]

@vanhees71 mentions the equipotential surface. To understand the importance of this you need to consider just what gravitational time dilation is related to. It is the difference in gravitational potential. Or, in other words, it is related to the energy per unit mass needed to move from one position in the field to another.
A clock on a mountain top runs faster than one at sea level because it take energy to move from sea level to the mountain top. The fact that gravity is slightly weaker at the mountain peak has no direct effect.

Now consider a clock on the end of a spinning rod. It runs slow compared to a clock on the spin axis, and we can figure this out by its speed. But there is another way to look at it, from the rotating frame of the arm itself. In this frame, it seems like there is a force pushing outward from the axis. And if you were to try a move a mass inward from the end of the arm to the axis, you would have to expend energy to do so. It would almost like lifting it against gravity. In the rotating arm frame, it is this difference in energy potential between arm end and axis that accounts for the clock on the end of the arm running slow.

So when we think of the clock on the equator running slow due to the spin of the Earth, if we look at it from the rotating Earth frame, we can consider it from the potential energy aspect. So we have a potential difference due to gravity in one direction, and a potential energy due to rotation in the other.

So now think about why the spinning Earth has the shape it does. Fluids (and the interior of the Earth is mostly fluid) tend to follow a surface of equal potential. Again, we have two "potentials" determining this surface for the Earth, gravity, and the "centrifugal" potential. At the surface, the net sum of the two always give the same answer in net potential at all points. But the combination of these two potentials are also what determine the net time dilation measured by a clock on the surface. So you get the same net potential all over the surface resulting in the same time dilation over the surface.

As long as the planet/body is free to have its shape determined by the combination of gravity and spin, clocks on its surface will run at the same speed. ( Of course, the Earth's crust is not fluid, which is why we can have mountains and valleys between which the net potential can differ.)

 

[some bloke]

That is a very helpful reply and has really helped me to visualise and connect the two. Thankyou!

So, the more potential energy within a system that a clock has (the more energy that has been expended to move it against the field), the faster it runs, am I right in establishing?

IE
if a clock is raised against gravity, it runs faster than one which was not.
if a clock is pulled inwards on a rotating arm, it runs faster than one which was not.

As with everything, this has given me more questions:

Is this exclusively true for gravitational potential and rotational forces? Or is it true of any force-field?

What happens when a second gravitational field is involved? For example, that of the moon? If a clock were held at a height and the moon passed overhead, would the gravity from the moon (in what would certainly be a barely perceptible way) affect the clocks speed compared with one on the ground?

Is it directly related to the energy expended - if the clock was raised as the moon passed over, would it experience the same change as if It were raised when the moon wasn't overhead? would that persist when the moon had passed?

Sorry for the deluge of questions. I know it's too big a field to understand without proper study, I'm just too interested not to ask!

 

[Ibix]

So, the more potential energy within a system that a clock has (the more energy that has been expended to move it against the field), the faster it runs, am I right in establishing?

Gravitational potential energy, where that can be defined, yes.

if a clock is raised against gravity, it runs faster than one which was not.

All other things being equal, and making some reasonable assumptions about the gravity field, yes.

if a clock is pulled inwards on a rotating arm, it runs faster than one which was not.

No - at least, not in general. I'm not quite sure what system you are envisaging.

Is this exclusively true for gravitational potential and rotational forces?

It's only true of gravitational potential. Where the rotational motion comes in is that a clock at rest on the surface of the Earth is moving with respect to a clock floating in space, so it ticks slightly slower compared to the floating clock. The amount a clock ticks slower due to moving and the amount it ticks faster due to being higher up exactly cancel on the geoid, so all clocks at rest on the surface of the Earth tick at the same rate (to the extent that the surface of the Earth exactly matches the geoid, anyway).

At least, that's one way to look at it.

What happens when a second gravitational field is involved? For example, that of the moon? If a clock were held at a height and the moon passed overhead, would the gravity from the moon (in what would certainly be a barely perceptible way) affect the clocks speed compared with one on the ground?

Strictly speaking, you have a non-stationary system there, and time dilation cannot be rigorously defined. However, in this kind of weak-field, low-velocity situation, I think you can get away with just adding the potentials and their effects, yes.

Is it directly related to the energy expended - if the clock was raised as the moon passed over, would it experience the same change as if It were raised when the moon wasn't overhead? would that persist when the moon had passed?

It isn't really anything to do with energy expended - it's to do with the gravitational potential difference between the two clocks you are comparing. So if some particular pair of clocks have some particular ratio of rates, and you close your eyes and stuff happens, and you open your eyes again and everything looks the same then the rate ratio will be the same. You might notice that the offset between the clocks has changed more (or less) than you expect during the period your eyes were shut, though, as a result of the stuff that happened.

 

[some bloke]

A clock on a mountain top runs faster than one at sea level because it take energy to move from sea level to the mountain top. The fact that gravity is slightly weaker at the mountain peak has no direct effect.

Now consider a clock on the end of a spinning rod. It runs slow compared to a clock on the spin axis, and we can figure this out by its speed. But there is another way to look at it, from the rotating frame of the arm itself. In this frame, it seems like there is a force pushing outward from the axis. And if you were to try a move a mass inward from the end of the arm to the axis, you would have to expend energy to do so. It would almost like lifting it against gravity. In the rotating arm frame, it is this difference in energy potential between arm end and axis that accounts for the clock on the end of the arm running slow.

This was where I was getting the example of the spinning rod from, and the idea of expended energy.

Strictly speaking, you have a non-stationary system there, and time dilation cannot be rigorously defined. However, in this kind of weak-field, low-velocity situation, I think you can get away with just adding the potentials and their effects, yes.
It isn't really anything to do with energy expended - it's to do with the gravitational potential difference between the two clocks you are comparing. So if some particular pair of clocks have some particular ratio of rates, and you close your eyes and stuff happens, and you open your eyes again and everything looks the same then the rate ratio will be the same. You might notice that the offset between the clocks has changed more (or less) than you expect during the period your eyes were shut, though, as a result of the stuff that happened.

Is the direction of the gravitational potential key, or is it simply any amount of gravitational potential? I understand that gravity essentially has an infinite reach, but becomes insignificant after a point. With the universe being generally uniform in all directions, the net gravity of distant bodies is taken as 0, am I right?

If a pair of clocks were suspended between 2 identical gravitational bodies, at equal distances (so 2 bodies with clocks at 1/3 and 2/3 of the space between), with the clocks held in place by thrusters and whatever else is needed to achieve this.
Would one run faster than the other from the perspective of one of the gravitational bodies but the other from the other body, or would they run the same as their net gravitational potential (regardless of direction) is identical? Or would you have to take the 2 planets as a single entity, and the gravitational potential of the pair?

If it is simply potential, which is related to height and gravitation, does a clock on Earth go a lot faster than one on the sun, as it is so far from the sun? Or would that only be from the perspective of the sun?

Again, sorry for the deluge of questions. Curiosity, and all that.

 

[Nugatory]

Is the direction of the gravitational potential key, or is it simply any amount of gravitational potential?

Potential is a scalar (a number at every point) not a vector, so it doesn’t have a direction. It’s the difference in potential that matters: take the potential at point A and the potential at point B; if they’re different the clock at the point with higher potential will run faster.

If a pair of clocks were suspended between 2 identical gravitational bodies, at equal distances (so 2 bodies with clocks at 1/3 and 2/3 of the space between), with the clocks held in place by thrusters and whatever else is needed to achieve this...

In this case the potential is the same at both points, so they run at the same rate. (But always be cautious about extending single-body solutions to multi-body GR problems, and remember that “potential” is only meaningful in restricted situations.)

Would one run faster than the other from the perspective of one of the gravitational bodies but the other from the other body...

Gravitational time dilation is not relative the way the velocity-based time dilation of special relativity is.

At some point you will want to look at exactly what it really means to say that one clock is running faster than another. That is, exactly what procedure would you use to compare the clocks, and what assumptions does this procedure require?

 

[Ibix]

This was where I was getting the example of the spinning rod fro

I see - @Janus is working in a rotating frame and I wasn't. We are just taking two different ways to describe the same thing. As far as I'm aware, any constant-magnitude acceleration will cause time dilation effects if treated in the appropriate accelerating frame.

 

[Janus]

I see - @Janus is working in a rotating frame and I wasn't. We are just taking two different ways to describe the same thing. As far as I'm aware, any constant-magnitude acceleration will cause time dilation effects if treated in the appropriate accelerating frame.

Right. In the lab frame, the time dilation is just due to the relative speed of the clock on the end of the arm.
In the rotating frame of the arm, it can be considered as due to the difference in potential caused by the pseudo-gravitational force due to the centrifugal effects.
Sometimes, this confuses people and they try to compound the two effects, but as you said, they are just two ways of looking at the same effect.

And as you mention, you don't need rotational motion. Linear acceleration works too. Accelerate a rocket along a straight line and a clock in the nose will tick faster than one at the tail, even for someone riding along in the rocket.

 

[some bloke]

Right. In the lab frame, the time dilation is just due to the relative speed of the clock on the end of the arm.
In the rotating frame of the arm, it can be considered as due to the difference in potential caused by the pseudo-gravitational force due to the centrifugal effects.
Sometimes, this confuses people and they try to compound the two effects, but as you said, they are just two ways of looking at the same effect.

And as you mention, you don't need rotational motion. Linear acceleration works too. Accelerate a rocket along a straight line and a clock in the nose will tick faster than one at the tail, even for someone riding along in the rocket.

Thankyou all for your responses.

So the effect of time-dilation due to the centrifugal pseudo-gravitation in the frame of the spinning arm is identical to that of in the frame of the lab itself caused by the relative speeds of the clock in the centre and the one on the end of the arm, because it is the same effect which is causing it?

I'm afraid I'm lost again on the rocket front & rear clocks, I would have thought that as they are accelerating at the same rate and not moving relative to one another that, b any reference frame, they would tick identically to one another. Why would the one at the front be different to the one at the back?

If rotation-speed time dilation and rotation-pseudogravity time dilation are identical effects calculated differently, where does linear speed come into play? EG a rocket traveling away from Earth in a straight line, having achieved a speed and maintained it, so far as to have negligible increase in potential energy from earth, with a clock compared to one on earth? What about the same ship, same distance, but describing a circular orbit of earth? Or does the increase in potential of ship 1 equal the pseudo-gravitation of ship 2?

Thankyou all for your time helping me with this!

 

[Ibix]

I'm afraid I'm lost again on the rocket front & rear clocks, I would have thought that as they are accelerating at the same rate and not moving relative to one another that, b any reference frame, they would tick identically to one another. Why would the one at the front be

In the lab frame, the rocket gets more and more length contracted as it accelerates. Nose and tail can't be accelerating at the same rate, or the rocket would stay the same length. The explanation in the rocket frame is trickier, partly because you first have to define exactly what you mean by "the rocket frame", but the result is the same.

 

[some bloke]

In the lab frame, the rocket gets more and more length contracted as it accelerates. Nose and tail can't be accelerating at the same rate, or the rocket would stay the same length. The explanation in the rocket frame is trickier, partly because you first have to define exactly what you mean by "the rocket frame", but the result is the same.

OK, I get that now. I was thinking of the rocket as being rigid and incompressible.

Are you referring to constant acceleration or an increasing acceleration? If the former, is the compression due to the ever-increasing force needed to maintain said acceleration as you reach relativistic speeds? If so, I am under the impression that this is due to the mass of the rocket increasing and so the force required to accelerate it increases. If this is the case, what would cause it to compress, as wouldn't this affect the front & rear uniformly?

 

[Nugatory]

I'm afraid I'm lost again on the rocket front & rear clocks, I would have thought that as they are accelerating at the same rate and not moving relative to one another that, b any reference frame, they would tick identically to one another. Why would the one at the front be different to the one at the back?

Suppose we place identically constructed radio transmitters and receivers at each end of the ship, both emitting at some fixed frequency (measured using an inertial frame in which the transmitter is at rest). One cycle represents one tick of a clock.

It takes some time for their signals to travel the length of the ship and the ship is accelerating during this time; the the speed of the receiver when the signal arrives is different from the speed of the transmitter when the signal is emitted. This causes the forward-moving signal to be red-shifted while the rearwards-moving one is blue-shifted. So we have two identically constructed clocks ticking at different rates, with the one at the front faster than the one at the rear.

In fact, this thought experiment is how gravitational time dilation was first predicted. By the equivalence principle, the accelerating ship is equivalent to one at rest in a gravitational field, so if clocks tick at different rates at different positions in the accelerating ship they will also do so at different heights in a gravitational field.

 

[Nugatory]

Are you referring to constant acceleration or an increasing acceleration? If the former, is the compression due to the ever-increasing force needed to maintain said acceleration as you reach relativistic speeds?

Constant acceleration, and no , the compression is not due to any force on the ship.

You are interpreting constant acceleration as implying that all parts of the ship are changing speed by the same amount at the same time (this is how a classical rigid body behaves)... but whenever you see the words “at the same time” you’ve just been suckered by the relativity of simultaneity. If all parts of the ship are changing speed by the same amount at the same time in one frame, then they are changing speed by the same amount at different times in all other frames - there is no such thing as classical rigid body acceleration in relativity.

Google for “Bell’s spaceship paradox” and “Born rigidity” for more. You will also want to get clear in your mind the difference between proper acceleration and coordinate acceleration, the best resource for this will be threads here.

 

[Ibix]

Constant acceleration,

Constant proper acceleration (i.e.the people in the ship feel constant weight due to constant force). That means decreasing coordinate acceleration as seen by the lab frame.

 

[jartsa]

if a clock is pulled inwards on a rotating arm, it runs faster than one which was not.

The inwards pulling results in less motion, which results in less time dilation.

The pulling causes the clock to think: "Something is lifting me, which will cause me to be less gravitationally time dilated". Or maybe it's better to say that an intelligent ant on the arm thinks that the clock is being lifted and becomes less gravitationally time dilated.
As $E' = \gamma E$ and $t' = \frac{t}{\gamma}$ , we can see that the change of clock rate is the same as the change of energy. (E and E' mean total energies) I mean when we changed the speed of the clock, we changed the gamma, and when the gamma changed, energy and time changed the same way. Except that energy decreased and clock rate increased. I guess my point is that if we know the factor by which the total energy of a clock changes, then we can easily calculate the factor by which the ticking rate of the clock changes.If gravity field in a laboratory can not be distinguished from being in a rocket that circles around in a large circle, then calculations of gravitational time dilations in that lab can be done by figuring out the motion time dilations in that circling rocket.

As the lab and the rocket can not be distinguished, the energy needed to move a clock is always the same in both places. So the ticking rate change factor is the same as the energy change factor in the lab too. At least if we naively assume that a lifted clock's energy changes by the amount of energy that the lifter uses to lift the clock.

 

[jartsa]

Nobody gave both of the inertial observer¨´´ s explanations for the fact that a person in an accelerating rocket sees a lower clock to run slower. Well, they are:

1: How light travels from the lower clock to observer​

​

2: Small extra motion time dilation of the lower clock​

Those are not alternatives. Both exist simultaneously.