# Why does time slow down the faster you go?

PAllen
2019 Award
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
2019 Award
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).

[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).

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PAllen
2019 Award
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.

[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.

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PAllen
2019 Award
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.

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

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PAllen
2019 Award
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.

:-) 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
2019 Award
:-) 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
2019 Award
:-) 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
2019 Award
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.]

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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.

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 realise 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.

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PAllen
2019 Award
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
Mentor
2019 Award
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.

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.

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PAllen
2019 Award
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
2019 Award
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.
]

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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.

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PAllen
2019 Award
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
2019 Award
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.

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.

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