Why does time slow down the faster you go?

[PAllen]

:-) See cross-edited post above.

With regard to your belief that it takes energy to go one way, are you claiming that you could extract energy by going the other way? We are ignoring friction.

And thank you for discussing this in a way that will be productive.

Of course you can extract energy from the Earth's rotation. If you put a giant charged ball on the equator, you could extract energy (slowing down the Earth in the process).

 

[PAllen]

You may think you've described enough, but from where I sit, I have yet to see anything approaching a description of energy conservation violation in one inertial frame.

[edit: and you really do need to talk about what you mean by sending gamma ray between the equator and the pole. It can be impractical, that's fine, e.g. there is a tunnel from equator to pole, and some of the gamma rays will be in the right direction to make it to the pole if the tunnel is not too narrow. If it is too narrow, none will make it because the tunnel is moving non-inertially]. If you don't specify things like this, I may completely mis-interpret you, as has been an issue in this discussion.]

 

[Elroch]

Firstly, if there's no friction you can just nudge a boat South at the North pole and it will eventually reach the equator. No kidding. Any tiny nudge will do, in principle. Now my argument about the properties of sea level mean it has to be going the right speed when it reaches the equator but, to be honest, I am not sure what direction it will be going in. This doesn't matter as you could bounce it elastically off a flat wall to make it go the right way.

Yes, you can extract energy from the Earth's rotation, but if that was possible at our ideal sea level rotating in step with the Earth, the water could adjust to reach a lower energy state by changing the levels. With regard to moving a mass from point (x1, v1) to (x2, v2) where x is a position and v is a velocity, if there is no energy exchange only the start and the end matter, not the path.

 

[Elroch]

For broken energy conservation in a single frame, take another look at my earlier post ending "perpetuum mobile". Each time you go through the loop, you get a bit of spare energy in the inertial frame at the North pole. Everything else stays the same. Of course you can think of everything happening in this frame.

[By the way, after I started taking part in this discussion, I noticed PatrickPowers had already mentioned something about sea level in post #2. I misremembered this as being you, PAllen, which confused me later]

I've just noticed a relevant point (which I didn't have to realize because of my argument). If you nudged a boat South at the North pole in a truly frictionless sea rotating in step with the Earth and it reached the equator still moving South, it would have gathered speed because, to it the sea is downhill! For the liquid helium in the sea rotating in step, it is flat because centrifugal force (excuse me) is increasing. This is how the boat reaches the rotation speed of the Earth by the time it gets to the equator, albeit in the wrong direction.

 

[PAllen]

Firstly, if there's no friction you can just nudge a boat South at the North pole and it will eventually reach the equator. No kidding. Any tiny nudge will do, in principle. Now my argument about the properties of sea level mean it has to be going the right speed when it reaches the equator but, to be honest, I am not sure what direction it will be going in. This doesn't matter as you could bounce it elastically off a flat wall to make it go the right way.

Ok, we have a basic disagreement on physics. I say, if I introduce a boat at the pole and float it to the equator, the ocean does work on it; the Earth excluding the boat loses a tiny amount of angular momentum and rotational KE, which the boat gains. Friction is exactly how the ocean applies force to the boat. If there were no friction, the boat would reach the equator moving 1000 mph relative to the surface of the ocean (assuming my model of truly frictionless surface just above the ocean, but with gravity intact, and this surface is rigid.)

Yes, you can extract energy from the Earth's rotation, but if that was possible at our ideal sea level rotating in step with the Earth, the water could adjust to reach a lower energy state by changing the levels. With regard to moving a mass from point (x1, v1) to (x2, v2) where x is a position and v is a velocity, if there is no energy exchange only the start and the end matter, not the path.

My example would absolutely work at sea level. Float a super charged ball in an insulating sphere on the ocean, and it would radiate. If the ocean suddenly became superfluid, as the ball radiated it would slow down to be inertially motionless, thus moving 1000 mph relative to the superfluid ocean. Without superfluidity, the ocean would continue doing work on it, it would continue radiating, and the Earth would slow down.

 

[PeterDonis]

There is no contradiction because the Corioli's force interfering with the motion of the boats is always perpendicular to their motion. Hence no energy.

You're correct that the Coriolis force does not affect the energy of the boats, but that's not the same as saying it doesn't affect their motion. It does. Launch a boat South from the North Pole on the frictionless rotating Earth-ocean. The boat's initial velocity is straight outward from the rotation axis, therefore the Coriolis acceleration opposes the local direction of rotation. In other words, as the boat moves South, it also acquires a Westward velocity (since the Earth rotates from West to East) relative to an observer who is rotating with the Earth just slightly South of the North Pole. As the boat moves further South, it continues to acquire more Westward velocity, again relative to the observers who are locally rotating with the Earth. By the time the boat gets to the equator, the Coriolis force is zero (because the boat's local velocity is now parallel to the Earth's rotation axis) and the boat is moving at 1000 mph westward relative to an observer rotating with the Earth, just as PAllen says.

 

[Elroch]

You're correct that the Coriolis force does not affect the energy of the boats, but that's not the same as saying it doesn't affect their motion. It does. Launch a boat South from the North Pole on the frictionless rotating Earth-ocean. The boat's initial velocity is straight outward from the rotation axis, therefore the Coriolis acceleration opposes the local direction of rotation. In other words, as the boat moves South, it also acquires a Westward velocity (since the Earth rotates from West to East) relative to an observer who is rotating with the Earth just slightly South of the North Pole. As the boat moves further South, it continues to acquire more Westward velocity, again relative to the observers who are locally rotating with the Earth. By the time the boat gets to the equator, the Coriolis force is zero (because the boat's local velocity is now parallel to the Earth's rotation axis) and the boat is moving at 1000 mph westward relative to an observer rotating with the Earth, just as PAllen says.

As I said, I didn't even know what direction the boat would be going in, but I inferred it would be going the right speed so all would be ok.

Doesn't my energy argument still hold?
(1) Taking a little water from one place in the ocean and adding it at another cannot lower the energy if it is in the lowest energy state with the constraint that it the water rotates with the Earth (no constraint on angular momentum).
(2) This implies there is no net work done moving a mass from one place to the other by any means.

It is better to ignore the method used to move the mass, because we know it is possible, and it doesn't matter at all. Moving the mass could use some external angular momentum, as long as it does not use external energy: this is feasible.

 

[PAllen]

As I said, I didn't even know what direction the boat would be going in, but I inferred it would be going the right speed so all was ok.

But then, Einstein would predict no time dilation (and would be correct ignoring gravitational time dilation). No time dilation, because the boat is still inertial relative to the polar observer if the ocean does not do work on it.

 

[PAllen]

Here is a complete scenario how Einstein might have explained that there simply is no violation of conservation of energy. Let's specify a gamma ray emitting nucleus. Let's specify a magic tunnel such I have described in an earlier post (to ship gammas from equator to pole).

Ship a block gamma ray emitter from the pole to equator. (A tiny amount of energy has been transferred from the Earth to the block, in the process). It emits gamma rays, received at the pole. They will be lower in energy than expected. Then bring the decayed block back. Note, it locally weighs less after gamma emission. It will transfer back less energy to the Earth than it gained on the path down. You then have: lighter block, lower energy gamma rays, but slightly faster earth. This compares to the local reaction: lighter block, normal energy gamma rays, no change to earth.

Thus, as I have said, it is crucial to analyze the whole thing in one inertial frame. As done above, there simply is no energy conservation violation associated with Einstein's prediction, so he would be clearly unmoved by it.

The only problem with his prediction was what he wouldn't suspect for a couple of more years - gravitational time dilation.

[Edit: There is a fatal flaw above. If the lighter block transfers less energy back to the earth, the Earth would be turning slower at the end not faster. There would be an energy imbalance. Hmm. Needs more thought.]

[Edit2: I found the flaw. All energy would be conserved. I am sure Einstein would have a much easier time of it.
In going from pole to equator, gamma emitting block (B) gains KE (earth loses a little). B emits gamma toward the pole. Per time dilation, it is a lower energy gamma than expected, and is received as such at the pole. However, for the emitted gamma to be going poleward, in the B rest frame, it was emitted partly backward at normal energy, and B' has forward momentum in this equator B rest frame. Going back to polar frame, B' has carried away exactly as much extra momentum and KE by its forward emission kick as the photon is lower in energy (all now in polar frame). As B' gets back in equilibrium with the surface, it transfers this extra energy and momentum to the Earth at equator. Then, when brought back to pole, you have B' rest mass, lower gamma, but equator with extra energy. Complete energy balance in the polar frame as a whole. Thus Einstein's answer would simply be there is no violation of conservation, and the prediction stands. Until, a few years later, when he deduced the idea of gravitational time dilation, that is.
]

 

[Elroch]

But then, Einstein would predict no time dilation (and would be correct ignoring gravitational time dilation). No time dilation, because the boat is still inertial relative to the polar observer if the ocean does not do work on it.

But you can then bounce the boat off an elastic wall without changing its energy, to get it in the right direction. (for convenience, ensure the wall is not part of the Earth). Remember this is only an energy argument, we are not worried about other things.

And I am not sure what you mean by "still inertial w.r.t to polar observer". Its velocity has changed radically since it left. It has gained this because the ocean surface has a sort of slope to compensate for its varying rotation speed.

 

[PAllen]

But you can then bounce the boat off an elastic wall without changing its energy, to get it in the right direction. (for convenience, ensure the wall is not part of the Earth). Remember this is only an energy argument, we are not worried about other things.

Au contraire. All energy, of all types, for all objects, must be accounted for in one inertial frame to make an energy conservation argument.

 

[PAllen]

And I am not sure what you mean by "still inertial w.r.t to polar observer". Its velocity has changed radically since it left.

It has not changed at all relative to an inertial polar observer. The Earth is rotating under it. It is stationary relative to the inertial polar observer.

 

[Elroch]

But then, Einstein would predict no time dilation (and would be correct ignoring gravitational time dilation). No time dilation, because the boat is still inertial relative to the polar observer if the ocean does not do work on it.

But Einstein did predict time dilation for exactly the situation of the pole versus corotating equator. I believe there is none at (idealised) sea level. Note that I have no interest in the time dilation of the boat before it is bounced in the right direction, and the point of bouncing it is to get it in the right state without doing work.

 

[Elroch]

It has not changed at all relative to an inertial polar observer. The Earth is rotating under it. It is stationary relative to the inertial polar observer.

No. The boat started virtually at rest and is moving at 1000 mph by the end. I think the easiest way to see how this happens is to compare sea level with and without rotation. The difference is a slope which gives you the 1000mph.

 

[PAllen]

No. The boat started virtually at rest and is moving at 1000 mph by the end. I think the easiest way to see how this happens is to compare sea level with and without rotation. The difference is a slope which gives you the 1000mph.

No. If you agree with Peter Donis (which agrees with mine), if the Earth is rotating 1000 mph at equator, and the frictionless boat at equator is moving 1000 mph the other way - relative to the surface, it is stationary relative to the poll. If it acquired KE without work, that would be a major violation of conservation.

 

[PAllen]

But Einstein did predict time dilation for exactly the situation of the pole versus corotating equator. I believe there is none at (idealised) sea level. Note that I have no interest in the time dilation of the boat before it is bounced in the right direction, and the point of bouncing it is to get it in the right state without doing work.

I said 'inertial' when I meant 'stationary'. If the boat were co-rotating (thus 1000 mph) relative to pole, Einstein predicted time dilation (and without gravity, he would be right). If the boat reached the equator without friction, moving 1000 mph relative to surface, it would be stationary relative to pole, with no time dilation predicted.

 

[Elroch]

No. If you agree with Peter Donis (which agrees with mine), if the Earth is rotating 1000 mph at equator, and the frictionless boat at equator is moving 1000 mph the other way - relative to the surface, it is stationary relative to the poll. If it acquired KE without work, that would be a major violation of conservation.

Surprisingly (to me) ocean depth at the equator would appear to be shallower because of the rotation of the Earth. This is where the 1000mph comes from: simply gravitational potential of the Earth.

Let me clarify the argument:

If ocean depth is a function of latitude chosen to minimise energy when rotating with the Earth, moving a little water from one point (x1, v1) to another (x2, v2), where v is rotating with the Earth, can never lower the energy of the ocean (by definition). Bear in mind we don't give a damn about angular momentum, as long as we look after energy.

If this is correct, the same is true for moving any mass from one point on the surface at sea level, rotating with the Earth (no-one cares whether the mass is water or not).

Specifically, moving a mass from stationary at the pole at sea level to rotating at 1000mph at sea level at the equator requires zero energy.

I repeat my concept that rotating the Earth makes the ocean surface effectively downhill from the poles to the equator (in order to compensate for the speed). This reminds me of the lowering of pressure with speed in fluid mechanics and is probably related.

 

[PAllen]

Surprisingly (to me) ocean depth at the equator would appear to be shallower because of the rotation of the Earth. This is where the 1000mph comes from: simply gravitational potential of the Earth.

Let me clarify the argument:

If ocean depth is a function of latitude chosen to minimise energy when rotating with the Earth, moving a little water from one point (x1, v1) to another (x2, v2), where v is rotating with the Earth, can surely never lower the energy of the ocean? Bear in mind we don't give a damn about angular momentum, as long as we look after energy.

If this is correct, surely the same is true for moving any mass from one point on the surface at sea level, rotating with the Earth?

Specifically, moving a mass from stationary at the pole at sea level to rotating at 1000mph at the equator.

I repeat my view that rotating the Earth makes the ocean go downhill from the poles to the equator (in order to compensate for the speed). This reminds me of the lowering of pressure with speed in fluid mechanics and is probably related.

1) The Earth is oblate. The ocean surface must be further away from the center than the poles for gravity to counteract the rotational time dilation.

2) It is simply a fact that a body moving from the pole to the equator without friction will be stationary relative to the pole and 1000 mph relative to the co-rotating surface.

Other discussions are not relevant unless these are agreed on.

Note, Peter Donis and others would agree on these two points.

 

[Elroch]

Ah. I think I see the light at the end of the tunnel. The ocean is not in its lowest energy state. In order for water to move from one place to another, it would have to transfer some angular momentum to the rest of the system, specifically the solid Earth. But this would change the energy in the Earth, so it is necessary to minimise the total energy of Earth plus ocean, not just the ocean.

In addition, given that about 10km would be necessary to compensate for 1000 mph, this empirically disproves my assertion, as the polar oceans are not even that deep, never mind with the 6km of oblateness as well at the equator.

Apologies for being quite so certain I was right.

In hindsight, my error of thought was in thinking of the ocean like a hydrostatic system. The rotating frame makes it critically different.

I need to find a better application for what was a pretty argument applied to the wrong problem.

Hat's off to you both, PAllen and PeterDonis, and thanks for my enlightenment, even if I was trying to enlighten you.

 

[PAllen]

Note, I added a big edit to #69, showing there is no conservation violation if all things are accounted for in the polar inertial frame. Thus the argument would have no weight since Einstein would just have considered it wrong (with much less effort than I took, I am sure).

 

[ThomasT]

To Elroch and PAllen,
I'm not sure what your argument has to do with the OP's questiion, but it seems to have emerged as one of those more or less epic PF arguments. So, for us laypersons, might one, or both, of you summarize the points of contention of your argument in a readily understandable form. Just asking. Thanks.

EDIT: I want you to condense/simplify your arguments to the essential conflicting contentions ... and then proceed. Thanks.

 

[ghwellsjr]

1) The Earth is oblate. The ocean surface must be further away from the center than the poles for gravity to counteract the rotational time dilation.

2) It is simply a fact that a body moving from the pole to the equator without friction will be stationary relative to the pole and 1000 mph relative to the co-rotating surface.

Other discussions are not relevant unless these are agreed on.

Note, Peter Donis and others would agree on these two points.

For someone who claims Einstein made an incorrect prediction about the time dilation difference between the poles and the equator, I wonder why you excuse yourself for ignoring the effect of the moon's gravity in your statement about the ocean surface in relation to the center of the earth. I forgive you, but I can't speak for Einstein. (And don't forget about the sun.)

 

[ThomasT]

For someone who claims Einstein made an incorrect prediction about the time dilation difference between the poles and the equator, I wonder why you excuse yourself for ignoring the effect of the moon's gravity in your statement about the ocean surface in relation to the center of the earth. I forgive you, but I can't speak for Einstein. (And don't forget about the sun.)

Glad to see you're online. Has the OP question been satisfactorily answered (I think so). Then, under what thread title might this discussion between Epoch and PAllen be subsumed?

 

[PAllen]

For someone who claims Einstein made an incorrect prediction about the time dilation difference between the poles and the equator, I wonder why you excuse yourself for ignoring the effect of the moon's gravity in your statement about the ocean surface in relation to the center of the earth. I forgive you, but I can't speak for Einstein. (And don't forget about the sun.)

Ours is a quibble of terminology. A prediction can be logically correct based on known theory and yet be a mistaken prediction about the world we live in. That is how science advances. That is the terminology used with all similar cases. I remain truly puzzled why you want to use unique, exceptional, terminology with Einstein.

[Einstein didn't ignore gravity as a simplification. He had no clue it was relevant in any way. Of course, he was the first, just a few years later, to begin to understand its relevance.]

 

[PeterDonis]

Hat's off to you both, PAllen and PeterDonis, and thanks for my enlightenment, even if I was trying to enlighten you.

Elroch, no problem, your posts made me think and that's always good. (Though perhaps not all who know me would agree. )

However, I still have a question I would like to pose, even though it may mess things up again. Consider the three observers on the Earth that we've been discussing (by "Earth" I mean the idealized, exactly oblate spheroidal Earth with the frictionless ocean surface that is *exactly* an equipotential surface in terms of "rate of time flow"). Observer A is at the North Pole. Observer B is at the equator and got there by sliding slowly Southward, with zero energy change, from the North Pole; by my earlier argument, he is therefore still at rest in an inertial frame (not rotating with the Earth), so he is moving at 1000 mph westward relative to Observer C, who is at the equator and rotating with the Earth.

Here's the question: for each of these observers, how much energy would a rocket need to expend to launch them into a marginal escape trajectory, i.e., a trajectory that just barely escapes "to infinity" from the Earth's gravity? (Assume no other bodies in the universe.) I'm not so much interested in exact numbers as in the relationship between the numbers for A, B, and C.

Thoughts?

 

[PAllen]

Elroch, no problem, your posts made me think and that's always good. (Though perhaps not all who know me would agree. )

However, I still have a question I would like to pose, even though it may mess things up again. Consider the three observers on the Earth that we've been discussing (by "Earth" I mean the idealized, exactly oblate spheroidal Earth with the frictionless ocean surface that is *exactly* an equipotential surface in terms of "rate of time flow"). Observer A is at the North Pole. Observer B is at the equator and got there by sliding slowly Southward, with zero energy change, from the North Pole; by my earlier argument, he is therefore still at rest in an inertial frame (not rotating with the Earth), so he is moving at 1000 mph westward relative to Observer C, who is at the equator and rotating with the Earth.

Here's the question: for each of these observers, how much energy would a rocket need to expend to launch them into a marginal escape trajectory, i.e., a trajectory that just barely escapes "to infinity" from the Earth's gravity? (Assume no other bodies in the universe.) I'm not so much interested in exact numbers as in the relationship between the numbers for A, B, and C.

Thoughts?

That's fun, I'll bite.

First, an observation that has occurred to me in the last few hours. To get B from the the pole to its final position requires a swift kick. Otherwise, B will just oscillate partway to the equator, back to the pole, up the other way, and back. B (by definition) is not picking up any angular momentum. With just the right swift kick, B will come to rest on the equator (about as likely as balancing a ceramic knife on edge on a ceramic surface).

Secondly, consider time flow. By construction, A and C clocks are in synch. B clock will run faster (consistent with prior observation - higher in potential well).

Finally, a clarification: I assume we can kick each whatever direction we want to escape to infinity in any desired path. In particular, there is an obvious better kick direction for C.

So, I claim least kick needed for C, to produce hyperbolic escape tangent to C's rotation. Next large kick needed for B, and most for A.

Now I am a little queasy that these are not correlated to the time flow rates. Yet I find any other conclusion hard to accept.

 

[Elroch]

Elroch, no problem, your posts made me think and that's always good. (Though perhaps not all who know me would agree. )

However, I still have a question I would like to pose, even though it may mess things up again. Consider the three observers on the Earth that we've been discussing (by "Earth" I mean the idealized, exactly oblate spheroidal Earth with the frictionless ocean surface that is *exactly* an equipotential surface in terms of "rate of time flow"). Observer A is at the North Pole. Observer B is at the equator and got there by sliding slowly Southward, with zero energy change, from the North Pole; by my earlier argument, he is therefore still at rest in an inertial frame (not rotating with the Earth), so he is moving at 1000 mph westward relative to Observer C, who is at the equator and rotating with the Earth.

Here's the question: for each of these observers, how much energy would a rocket need to expend to launch them into a marginal escape trajectory, i.e., a trajectory that just barely escapes "to infinity" from the Earth's gravity? (Assume no other bodies in the universe.) I'm not so much interested in exact numbers as in the relationship between the numbers for A, B, and C.

Thoughts?

My immediate thought is that this consideration should have told me that someone co-rotating at the equator has a different energy to one stationary at the pole! When launching rockets into orbit, less energy is required if you start near the equator (C). If I recall, even Jules Verne took account of this in "from the Earth to the moon", and it is taken advantage of by the US launch sites, amongst others.

The other two sites (A and B) require the same energy if they are at the same gravitational potential but are stationary w.r.t. the centre of the Earth.

 

[PatrickPowers]

That's fun, I'll bite.

First, an observation that has occurred to me in the last few hours. To get B from the the pole to its final position requires a swift kick. Otherwise, B will just oscillate partway to the equator, back to the pole, up the other way, and back. B (by definition) is not picking up any angular momentum. With just the right swift kick, B will come to rest on the equator (about as likely as balancing a ceramic knife on edge on a ceramic surface).

Secondly, consider time flow. By construction, A and C clocks are in synch. B clock will run faster (consistent with prior observation - higher in potential well).

Finally, a clarification: I assume we can kick each whatever direction we want to escape to infinity in any desired path. In particular, there is an obvious better kick direction for C.

So, I claim least kick needed for C, to produce hyperbolic escape tangent to C's rotation. Next large kick needed for B, and most for A.

Now I am a little queasy that these are not correlated to the time flow rates. Yet I find any other conclusion hard to accept.

I get the same result. B's oscillation confused me for a minute: it is because gravity is greater at the poles.

This is why orbital rocket launch points are close to the tropics. 1500km/hour, that alone is enough to carry one to the Moon in ten days. (Yes, I know that it isn't that simple.)

 

[Elroch]

To Elroch and PAllen,
I'm not sure what your argument has to do with the OP's questiion, but it seems to have emerged as one of those more or less epic PF arguments. So, for us laypersons, might one, or both, of you summarize the points of contention of your argument in a readily understandable form. Just asking. Thanks.

EDIT: I want you to condense/simplify your arguments to the essential conflicting contentions ... and then proceed. Thanks.

Well, I would generalise the central issue like this:

Suppose you have a static gravitational field. If there are two points in this gravitational field, with a relative velocity that is perpendicular to the line between the two points, then I gave an argument that the time dilation between the two points is related in a very simple way to W, the work done moving a mass from one point with one velocity to the other point with the other velocity. I still believe this.

However, I further claimed that two such points co-rotating with the Earth at sea level make W zero. This was wrong because points at the equator are both higher and faster so have more gravitational and kinetic energy (as a result I infer that, in principle, you can generate power by moving something from the equatorial ocean to the pole).

If I understand correctly, the opposing position implied that sea level is the same as if it was frictionless and not moving with the Earth at all (hence the same as if the Earth was not rotating). In this case moving a mass from the pole at rest at sea level to the equator at rest (not rotating with the Earth) at sea level without friction would take no energy (PeterDonis described it moving in a way that exactly canceled out the rotation of the Earth - frankly, if there is no friction, you can consider the Earth and its surface to be stationary), but the co-rotating point at the equator has extra kinetic energy, hence work is done to join it.

But isn't there a problem with this position? If the Earth rotated at an appropriate speed, the equator would become weightless, and worse. Surely the sea bulges, as well as the Earth? So sea level is not the same as if the Earth was not rotating. If so, I believe both positions were incorrect.

This is why orbital rocket launch points are close to the tropics. 1500km/hour, that alone is enough to carry one to the Moon in ten days. (Yes, I know that it isn't that simple.)

Yes, in fact it only saves a small fraction of the energy needed to get there (because the escape speed is nearly 12000km/h). However, because of the roughly exponential relationship between final speed (ignoring gravity) and initial mass ratio, this is significant.

 

[PAllen]

Thanks for trying to summarize all this. I few little thoughts below...

Well, I would generalise the central issue like this:

Suppose you have a static gravitational field. If there are two points in this gravitational field, with a relative velocity that is perpendicular to the line between the two points, then I gave an argument that the time dilation between the two points is related in a very simple way to W, the work done moving a mass from one point with one velocity to the other point with the other velocity. I still believe this.

A few caveats. First, even in a strictly conservative field, work to go from A to B is path independent only if you apply minimum force at all times. For arbitrary motion, where you are free to add and subtract KE at will, you can do any amount of work above the minimum to get from A to B. More substantively, not all work (even assuming paths that use minimum work) has the same impact on time dilation. This is shown clearly using Peter's nomenclature for B versus C. Moving from state A to state B requires some work against gravity. The result is time for B runs faster than A. To get from B state to C state, you have to do more work to add KE (1000 mph in polar inertial frame). However, this KE, leaving potential unchanged, but adding velocity, makes time slow down. This disproves your simple model.

However, I further claimed that two such points co-rotating with the Earth at sea level make W zero. This was wrong because points at the equator are both higher and faster so have more gravitational and kinetic energy (as a result I infer that, in principle, you can generate power by moving something from the equatorial ocean to the pole).

If I understand correctly, the opposing position implied that sea level is the same as if it was frictionless and not moving with the Earth at all (hence the same as if the Earth was not rotating).

I don't think I (or Peter) ever suggested this frictionless surface modeled anything (even ideally) about the real earth. I introduced this to clarify that the real ocean is inherently doing work via friction on a boat moving from the pole to the equator.

In this case moving a mass from the pole at rest at sea level to the equator at rest (not rotating with the Earth) at sea level without friction would take no energy (PeterDonis described it moving in a way that exactly canceled out the rotation of the Earth - frankly, if there is no friction, you can consider the Earth and its surface to be stationary), but the co-rotating point at the equator has extra kinetic energy, hence work is done to join it.

Actually, the point of the frictionless surface was to show the rotational KE of a comoving equatorial boat had to come from somewhere. There may have been a few slopply implications that A to B was zero work, but that wasn't germane to the main discussion. Peter's latest scenario now makes very clear (see my answer to it) that A to B involves substantial work. Then B to C involves more work. These two components of work happen to cancel as to time flow.

But isn't there a problem with this position? If the Earth rotated at an appropriate speed, the equator would become weightless, and worse. Surely the sea bulges, as well as the Earth? So sea level is not the same as if the Earth was not rotating. If so, I believe both positions were incorrect.

See above. Note, I always insisted that the earth, including the sea, was oblate due to rotation. I said this a number of times, arguing that if you got to the equator on a frictionless surface just above the sea, time would be slower than at the pole. I may not have always been clear about the implication that this implies significant work is needed to get from the pole to the equator even on a frictionless surface.