Why does time slow down the faster you go?

[PAllen]

You do not appear to be disagreeing with what I wrote, PAllen, but with something entirely different. Let me summarise:

(1) I understood that Einstein believed that a point at sea level on the equator would experience time dilation compared with a point at sea level at the pole. He said "Thence we conclude that a balance-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions"
(2) I gave an argument why this was not so, which in no way relied on the effect of height on time dilation or general relativity.

I invite you to review my previous posts (#27 and #30) and see if my argument is flawed or say why Einstein could not have come to the same conclusion as I did from this argument.

I believe your argument would be considered irrelevant in 1906 by Einstein and everyone else. I say this because they would believe the same consequences were true irrespective of sea level.

 

[Elroch]

If Einstein would accept my argument, he would not expect a difference in clocks between points at sea level at the equator and the pole. So if you wish to not accept my point your choices are:

(1) my argument is wrong
(2) there was some reason why Einstein would not believe it

Which is it?

Please be specific in either case, as to which part of the argument.

 

[PAllen]

If Einstein would accept my argument, he would not expect a difference in clocks between points at sea level at the equator and the pole. So you still need to say either my argument is wrong, or there was some reason why Einstein would not believe it. You need to do one or the other or accept my point.

No I don't. You argument is not wrong, but I do not accept your point. I claim your argument is irrelevant to the 1906 understanding. I believe Einstein would, despite your argument, believe there would be adifference at the equator and pole - precisely because he would view the issue of sea level and conservation of energy as irrelevant.

[As with the other circular argument on this thread, have the last word on this if desired - I have no more to say. For some reason, our back and forth is not facilitating mutual understanding. Maybe someone else can help.]

 

[Elroch]

PAllen, the idea that Einstein would consider conservation of energy as "irrelevant" seems rather outrageous. The role of sea level is rather subtle, as it automatically incorporates two factors.

I would have thought Einstein would instantly see the point that transporting masses at sea level requires no energy. The more demanding point (to me) is that a difference in time dilation between two points would result in a change in the energy in electromagnetic radiation transmitted between the two points (by shifting the frequency). Would this be a conclusion he could reach from his understanding of the photon?

 

[Elroch]

I will explain why PAllen's curt dismissal of the issue I brought up is unjustifed (and why this is so revealing about the way understanding of the new physics developed).

If Einstein had come up with a thought experiment that showed that something he believed to be true broke the law of conservation of energy, far from considering conservation of energy to be "irrelevant" he would surely have considered this very significant. He would have had to conclude that if the thought experiment was valid either his belief was incorrect or the law of conservation of energy was genuinely broken, which could be as radical a discovery as special relativity.

In March of Einstein's "miracle year" of 1905 Einstein had published a paper showing that light was composed of particles, despite exhibiting wave properties. Obviously, since quantum mechanics was still undiscovered, this was something only partially understood. My question is, was his improved (but still partial) understanding of the nature of light enough in June of the same year (when he published his first paper on special relativity) to have given him the chance to conclude that the energy in a light signal transmitted between two points at different time dilations would change according to the time dilation?

My considered view is he probably could have if he had combined his two pieces of work. He knew that the packets of light had energy equal to h nu. With this frequency clearly being measured relative to a clock, the energy shift is clear, as long as one rejects the unacceptable idea that the number of photons changes depending on which frame one counts them in.

Pretty straightforward with a century of hindsight, but not at all in the year when special relativity and the quantisation of light were born. Given that, two years later in 1907, Einstein was the first person to realize that gravitational time dilation occurred, perhaps he can be forgiven for having not inferred this with a simple thought experiment in 1905!

 

[PAllen]

I will explain why PAllen's curt dismissal of the issue I brought up is unjustifed (and why this is so revealing about the way understanding of the new physics developed).

If Einstein had come up with a thought experiment that showed that something he believed to be true broke the law of conservation of energy, far from considering conservation of energy to be "irrelevant" he would surely have considered this very significant. He would have had to conclude that if the thought experiment was valid either his belief was incorrect or the law of conservation of energy was genuinely broken, which could be as radical a discovery as special relativity.

In March of Einstein's "miracle year" of 1905 Einstein had published a paper showing that light was composed of particles, despite exhibiting wave properties. Obviously, since quantum mechanics was still undiscovered, this was something only partially understood. My question is, was his improved (but still partial) understanding of the nature of light enough in June of the same year (when he published his first paper on special relativity) to have given him the chance to conclude that the energy in a light signal transmitted between two points at different time dilations would change according to the time dilation?

My considered view is he probably could have if he had combined his two pieces of work. He knew that the packets of light had energy equal to h nu. With this frequency clearly being measured relative to a clock, the energy shift is clear, as long as one rejects the unacceptable idea that the number of photons changes depending on which frame one counts them in.

Pretty straightforward with a century of hindsight, but not at all in the year when special relativity and the quantisation of light were born. Given that, two years later in 1907, Einstein was the first person to realize that gravitational time dilation occurred, perhaps he can be forgiven for having not inferred this with a simple thought experiment in 1905!

Does doppler violate conservation of energy? Does time dilation between inertial frames? It 'appears to', and there is much nonsense written about this.

Einstein would see that, if his prediction was true, energy was conserved in the polar frame for the process of emission; or in the equator frame for the process of emission. Then the rest is just relativistic doppler, which he already understood (and certainly did not consider to violate conservation).

 

[ghwellsjr]

Einstein would see that, if his prediction was true, energy was conserved in the polar frame for the process of emission; or in the equator frame for the process of emission. Then the rest is just relativistic doppler, which he already understood (and certainly did not consider to violate conservation).

Einstein considered only one frame in his original prediction of the Twin Paradox at the end of section 4 of his famous 1905 paper introducing Special Relativity. The observer/object/clock on the equator is not inertial. The observer/object/clock at one of the poles is inertial. The frame under consideration is one in which the observer/object/clock at one of the poles is at rest. Since the observer/object/clock on the equator is in constant motion in the frame under consideration and the observer/object/clock at one of the poles remains at rest in the same frame, the time dilation applies only for the observer/object/clock on the equator and the effect of time dilation is not symmetrical or reciprocal. Einstein did not attempt to explain the Twin Paradox from a frame in which the observer/object/clock on the equator was at rest in his 1905 paper. Neither did he employ Relativistic Doppler in his explanation.

 

[PAllen]

Einstein considered only one frame in his original prediction of the Twin Paradox at the end of section 4 of his famous 1905 paper introducing Special Relativity. The observer/object/clock on the equator is not inertial. The observer/object/clock at one of the poles is inertial. The frame under consideration is one in which the observer/object/clock at one of the poles is at rest. Since the observer/object/clock on the equator is in constant motion in the frame under consideration and the observer/object/clock at one of the poles remains at rest in the same frame, the time dilation applies only for the observer/object/clock on the equator and the effect of time dilation is not symmetrical or reciprocal. Einstein did not attempt to explain the Twin Paradox from a frame in which the observer/object/clock on the equator was at rest in his 1905 paper. Neither did he employ Relativistic Doppler in his explanation.

Right, but Elroch has modified Einstein's original proposal to one involving redshift. I am then left to guess how Einstein would respond to this in 1905. My guess is that he would think as I outlined, and deny there is any issue with conservation of energy. He would see it as simply an instance of transverse doppler, and shrug, what's the problem? Energy is conserved in the emitter frame, the rest is doppler.

 

[Elroch]

PAllen, with all due respect, you appear to be missing a crucial point. The comparison of energies takes place at the same place at the same velocity, so is nothing to do with the Doppler effect (which is about comparing something in two different frames, whether they be Galilean or Lorentzian). You can't make energy in the same frame fail to add up using the Doppler effect.

In the thought experiment, you have two paths some stuff can take from A to B, with no external transfer of energy, and if you believe there is a difference in clocks, you have an energy imbalance. It is trivial to modify the thought experiment to create a perpetual motion machine. This is why the conclusion could not have been brushed off as something irrelevant.

The only way to make the energy add up is if whether you annhilate the particles at A and send to radiation to B or send the particles to B and then annihilate them, you get exactly the same radiation at B. If the only effect was transverse Doppler, the radiation would be a different color. The energy argument indicates that at sea level something compensates exactly for this transverse Doppler effect. This something is gravitational time dilation, of course.

[As for the frames, the exact nature of them doesn't really affect the argument. In a nutshell, all that matters is that if a mass can move between them doing zero work, radiation can move between them without being shifted in frequency]

[If there is any doubt about the link between frequency and the clocks, consider measuring the length of a day by counting the cycles of the radiation]

 

[PAllen]

PAllen, with all due respect, you appear to be missing a crucial point. The comparison of energies takes place at the same place at the same velocity, so is nothing to do with the Doppler effect (which is about comparing something in two different frames, whether they be Galilean or Lorentzian). You can't make energy in the same frame fail to add up using the Doppler effect.

In the thought experiment, you have two paths some stuff can take from A to B, with no external transfer of energy, and if you believe there is a difference in time dilation, you have an energy imbalance. It is trivial to modify the thought experiment to create a perpetual motion machine. This is why the conclusion could not have been brushed off as something irrelevant.

In your description in #27 and #30, I see doppler. You compare doing a light emittting reaction at the pole, measuring frequency; with doing the same reaction at the equator and transmitting the radiation to the pole. That transmission entails transverse doppler due to the equator's motion. It is a change of reference frame. Maybe you mean something different, but that is how I read your posts.

 

[ghwellsjr]

Right, but Elroch has modified Einstein's original proposal to one involving redshift. I am then left to guess how Einstein would respond to this in 1905. My guess is that he would think as I outlined, and deny there is any issue with conservation of energy. He would see it as simply an instance of transverse doppler, and shrug, what's the problem? Energy is conserved in the emitter frame, the rest is doppler.

If the Earth were transparent them maybe Relativistic Doppler might have been a consideration, unless somebody ran a long fiber optic cable from the pole to the equator.

 

[PAllen]

If the Earth were transparent them maybe Relativistic Doppler might have been a consideration, unless somebody ran a long fiber optic cable from the pole to the equator.

That's Elroch's problem not mine. He just said 'imagine transmitting the light'. So I tried to imagine it.

 

[Elroch]

Glad to hear you're using your imagination, PAllen. :)

Transverse Doppler is relevant to the physics, but entirely irrelevant to the logic of the argument. Similarly the precise nature of the frames is irrelevant to the argument. If you do not realize this yet, please think a bit.

The top level logic is as follows:

If the clocks at the equator and at the pole run at different speeds, you can make a perpetual motion machine, therefore they don't.

To put in a bit of detail:

If the clocks run at different speeds, radiation transmitted between the two points must be shifted in frequency (because the cycles can be used as a clock), which changes the amount of energy in it. Since mass can be transferred between the points without a change in energy, conservation of energy is broken.

Note 1: This implicitly uses the fact that the distance between the two points is constant (you could use the fact that the two points repeatedly return to the same relative position instead)
Note 2: The part about mass being transferred between the two frames without input of energy relies on the special property of (idealised) sea level, of course.

I suspect anyone following this will agree with me that all the components of this argument were established in 1905, but no-one happened to put them together. Does anyone disagree? If so, please specify which exact part of what I wrote above was not available at that time.

 

[PAllen]

If the clocks run at different speeds, radiation transmitted between the two points must be shifted in frequency (because the cycles can be used as a clock), which changes the amount of energy in it. Since mass can be transferred between the points without a change in energy, conservation of energy is broken.

.
This is transverse doppler. Transverse doppler is due precisely to time dilation different between inertially moving frames (these are not quite, but Einstein was ignoring this, and so shall we ...).

Note 1: This implicitly uses the fact that the distance between the two points is constant (you could use the fact that the two points repeatedly return to the same relative position instead)
Note 2: The part about mass being transferred between the two frames without input of energy relies on the special property of (idealised) sea level, of course.

If a body goes from the pole to the equator, a polar observer sees it gaining speed - accelerating. It changes frame, and time dilates. It is being acted upon - if the surface were frictionless, it would be clear that force would need to be applied on the way from the pole to the equator and back, to achieve the change in frames.

I suspect anyone following this will agree with me that all the components of this argument were established in 1905, but no-one happened to put them together. Does anyone disagree? If so, please specify which exact part of what I wrote above was not available at that time.

I continue to see that Einstein would view this as an unsurprising application of transverse doppler.

 

[ThomasT]

That is absurd. Basic thought experiments make it plainly obvious why time slows for a moving body.

I agree that there's a perfectly acceptable and formally correct explanation in terms of different spacetime paths via the standard geometric interpretation of SR.

As Naty1 pointed out, and with which I agree, " PAllen's post # 9 gives as precise an explanation as is currently known".

I was just expressing my personal opinion that the spacetime path explanation is not as deep an explanation as might satisfy one, based on (also just my personal opinion) the notion that a deeper explanation (either a testable formalism or at least a somewhat well developed conceptualization) in terms of wave mechanics or particle interactions might be possible.

I don't think that that's an absurd opinion to have, but it might be considered as such by those who think that the spacetime path explanation is as deep an explanation and understanding of relativistic differential aging as can or will ever be developed.

 

[PAllen]

In SR, if you start with clocks together, separate them, and bring them together, they will generally differ (the path with greater deviation form an inertial path will accumulate less time). The exact same thing is true in GR.

In SR, mutual observation of inertial clocks is symmetric - each sees the other slower. In GR, this is still true for 'nearby clocks' with different inertial motion. Quite generally, SR global behavior becomes GR local behavior.

Finally, for comparing inertial and non-inertial clocks (or two non-inertial clocks) without bringing them together, you have asymmetric effects in both SR and GR. For example, in SR, clocks will run slower at the front of a long uniformly accelerating rocket[edit: compared to the rear of the rocket], and this effect is not symmetric.

Since this has been referenced a few times as a description of what 'is' (how deep is not of particular interest to me), I need to correct a typo. In a long, uniformly accelerating rocket in empty space, the clocks at the rear runs slower than the clocks in front. I was thinking 'faster' and wrote 'slower'.

 

[PAllen]

Elroch,

Let me try to explain why I don't see the whole issue of idealized sea level as relevant.

Consider two cases: (A) without gravity; (B) with gravity.

A) We remove the Earth altogether. We have a long axle with an orthogonal bar protruding. The bar is spinning about the axle. Assume this framework is of negligible mass. You have a clock at the top of the axle and at the tip of the bar. Einstein predicts, completely correctly, that a clock on the bar would run slow compared to one on the axle, and light from it would be red shifted as observed from the axle top observer. All of this is correct and and involves no violations of energy conservation. Einstein believed, in 1905, the the Earth would be equivalent to this (there was, as yet, no basis for him or anyone else to believe otherwise). Since this is a completely correct prediction, why would your arguments about sea level and gravitational equipotential lead to a change in analysis?

B). Same set up, but where the bar joins the axle, we have superdense mass concentration. Now, it happens that, along the bar, there will be clocks running slower than the 'pole clock', then (a bit closer to the axle) the same speed, then (a bit closer yet) faster than the polar (axle top) clock. Now in this set up, describe precisely, your conservation of energy argument, without any confusion now about friciton and gaining energy from the angular momentum of the Earth (which your original setup entailed).

 

[Elroch]

[EDIT: this was a response to PAllen's penultimate post]

PAllen, I believe that you are intelligent enough to be able to understand the logic of my post rather than ignore what I say and repeat points that are irrelevant to the logic. Why don't you?

As well as that, you make an error in believing that it takes energy to move a mass from sea level at the pole to sea level at the equator (ignoring irreversible losses due to friction). If it did, the sea could find a lower energy state by adding some water at one point and subtracting it at the other (indirectly, of course). The fact that the sea is moving with the Earth does not affect this principle (though of course it has a big influence on the sea level at different latitudes in order to make the principle true. Think of it this way: there is no place in the (idealised) oceans where you can waterski without a boat (I am assuming an idealised smooth sea which rotates perfectly in step with the Earth, or this becomes very messy and moves from physics to some other applied subject). This local fact implies moving masses about on the sea takes no energy (in principle).

 

[PAllen]

PAllen, I believe that you are intelligent enough to be able to understand the logic of my post rather than ignore what I say and repeat points that are irrelevant to the logic. Why don't you?

As well as that, you make an error in believing that it takes energy to move a mass from sea level at the pole to sea level at the equator (ignoring irreversible losses). If it did, the sea could find a lower energy state by adding some water at one point and subtracting it at the other (indirectly, of course). The fact that the sea is moving with the Earth does not affect this principle (though of course it has a big influence on the sea level at different latitudes in order to make the principle true. Think of it this way: there is no place in the (idealised) oceans where you can waterski without a boat (I am assuming an idealised smooth sea which rotates perfectly in step with the Earth, or this becomes very messy and moves from physics to some other applied subject).

I genuinely don't understand your point or its relevance, and don't find you being at all helpful in clarifying it, if it is correct.

I also don't see you engaging my arguments at all with logic, only with ad hominem comments.

For example, your comments here about the ocean seem completely irrelevant to my arguments. I'm saying (and did, in an earlier post) suppose at the surface of the ocean, making it smooth, we placed a frictionless surface. Then launch (slowly) a clock from the pole (still subject to gravity). It would reach the equator in a state of motion a thousand mph different than a clock floating on the ocean. Without taking gravitiational time dilation into account, why wouldn't Einstein (sitting at the pole) think the clock at the equator, that looked stationary, skimming on the frictionless surface, runs the same as his. While the clock carried by the ocean at 1000 mph runs slower.

 

[Elroch]

[EDIT: Another cross post! Sorry if I haven't appeared helpful. I got the wrong impression you understood but chose to not address the points directly.]

One preparatory note. When I talk about a point at sea level, I mean a point moving at the rotational speed. This is crucial to the energy arguments.

Elroch,

Let me try to explain why I don't see the whole issue of idealized sea level as relevant.

With all due respect, the reason you don't see it as relevant is because you don't yet understand it. The crucial point is that you can move a mass from one point at the surface of an ocean in equilibrium to another without using energy. The reason this is so is that you can't reduce the energy state by moving a little bit of the surface water from one point to another. You see?

Consider two cases: (A) without gravity; (B) with gravity.

A) We remove the Earth altogether. We have a long axle with an orthogonal bar protruding. The bar is spinning about the axle. Assume this framework is of negligible mass. You have a clock at the top of the axle and at the tip of the bar.

In your system it takes energy to move a mass from one point to the other and you can extract energy by moving it back. So crucially different.

Einstein predicts, completely correctly, that a clock on the bar would run slow compared to one on the axle, and light from it would be red shifted as observed from the axle top observer. All of this is correct and and involves no violations of energy conservation.

Yes, if the point at the end of the bar is at a lower potential, clocks will go slower.

Einstein believed, in 1905, the the Earth would be equivalent to this (there was, as yet, no basis for him or anyone else to believe otherwise).

Einstein could have seen the relevance of being at sea level, which roughly means being at equal potential. Admitedly, it would appear that he probably didn't consider this factor and merely thought of the equator moving when he wrote his paper on special relativity. But he did have enough information to do so.

I hope that makes the crucial difference clear.

 

[PAllen]

With all due respect, the reason you don't see it as relevant is because you don't yet understand it. The crucial point is that you can move a mass from one point at the surface of an ocean in equilibrium to another without using energy. The reason this is so is that you can't reduce the energy state by moving a little bit of the surface water from one place to another. You see?

Let's focus on this. Again, I agree with this but don't see the relevance. I hope you also see, that despite this, the ocean is doing work on a boat moving from the pole to the equator (in any inertial frame). Consider introducing a huge number of giant boats at the pole (one at a time) and moving each to the equator with minimal energy. The angular momentum of the Earth (minus the boats) is now lower, with the boats having picked up kinetic energy and angular momentum in an inertial frame at rest with respect to the poles.

Thus, despite your equipotential argument (and again, assuming we can make no recourse to gravitational time dilation), I see a simple case of boats being accelerated relative to inertial motion; at the equator, they are (moment to moment) nearly inertial observers at 1000 mph relative to inertial polar observer.

Perhaps it would help if you did something which I haven't see yet: describe precisely a sequence of steps in which a violation of conservation of energy is seen in a single inertial (non-rotating) frame of reference. I have understood all of your conservation arguments to involve mixing of frames, which is not valid.

 

[Elroch]

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.

As for the single frame violation of conservation of energy, simply do one of the paths backwards to make a loop. Not practical but makes the point.

Eg start at an inertial frame at the North pole, annihilate an electron and a positron, send the gamma rays to the equator. If time dilation there will be some energy left over if you recreate an electron and positron, send the spare energy and the two particles back to the pole. perpetuum mobile

 

[PAllen]

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

Now who is ignoring who? I literally have nothing to respond to unless you describe a violation of conservation of energy in a single inertial frame. [Note, the boats have KE and angular momentum in any inertial frame].

Note, that if you try treat a non-inertial frame as if it is inertial, you trivially have violation of conservation of energy.

 

[Elroch]

:-) 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.

Thank you for discussing this in a way that will be productive. This is refreshing.

 

[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.

You can't ignore friction, else you can't get from the pole to the equator moving along with the water.

 

[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.