Understanding Time Dilation in Einstein's Special Theory of Relativity

[Al68]

Note to Al68: keep in mind that cos seems to be trying to lead people into saying that the equatorial clock is ticking slower at every instant, not just over the course of an entire orbit, and not just visually when an observer next to one of the clocks looks at the other. I'm sure you'd agree that at any given moment, we can pick an inertial frame where the equatorial clock's instantaneous velocity is smaller than the polar clock's, and that it would therefore be the polar clock that is ticking slower at this moment in this frame (which is just as good as any other inertial frame).

Hi JesseM,

Sure, but if I understand cos correctly, he wants to analyze things from the accelerated rest frame of the equatorial clock, ignoring gravity. Basically the one reference frame in which both clocks are stationary.

 

[cos]

The complication in considering rotating frames is that this is an advanced topic in special relativity, halfway towards the mathematics of general relativity.

I'm going to modify your scenario slightly and consider a wheel rolling along a road. As always we ignore gravity.

Clock A is fixed to the wheel rim.
Clock B is fixed to the wheel centre.
Clock C is fixed to the road.

If the wheel is stationary, all 3 clocks agree that they are ticking at the same rate as each other. Now let the wheel roll.

We consider what is happening just at the moment that the wheel rolls over clock C in such a way that clocks A and C are momentarily at the same place.

From the point of view of inertial clock B, A and C are both ticking at the same rate as each other, both slower than B.

From the point of view of inertial clock C, A is ticking at the same rate as C, B is ticking slower than A and C.

From the point of view of accelerating clock A, C is ticking at the same rate as A, B is ticking faster than A and C.

Now how do you explain those 3 points of view in terms of "physically ticking", whatever that means?

In all 3 of those points of view the word "physically" can be placed in front of every "ticking".

On the basis that the wheel is initially at rest then starts rolling along the road your clock C is then Einstein's section 4 STR clock B (to differentiate let's call his clock B') which remains at rest whilst your clock B (Einstein's A') ticks over at a slower rate than it did before it accelerated ergo your clock C (Einstein's B') ticks over at a faster rate than your clock B (Einstein's A') not at a slower rate as you present above.

I won't bother dissecting the rest of your depictions; it's tiresome and uneccessary but perhaps I could save both of us a lot of work -

A person is located at the center of a stationary (neither moving nor spinning) hypothetically zero-mass disc in an imaginary otherwise empty universe. There is, at the rim of this disc, a large clock (A) identical to his own clock (B).

The disc starts spinning; according to Einstein's section 4 depiction - clock A (now moving in a closed curve around the stationary clock B) will tick over at a slower rate than B.

The person located at the center of the wheel will see clock A continuously ticking over at a slower rate than his own clock and on the basis that observation (determination) creates reality he is fully entitled to be of the opinion that clock A is ticking over at a slower rate than his own clock.

He has every right to anticipate that if he then moves to the rim of the disc his clock will be subjected to the same 'law' of physics that caused clock A to tick over at the slower rate than his own clock when A started moving thus that when he moves to the rim his clock will also be ticking over at a slower rate than a clock at the center of the disc.

Being located on the rim of the disc he would observe that the central clock is ticking over at a faster rate than his own clock however for him to assume that his clock has not been subjected to the same 'law' of physics as was clock A but that the central clock's rate of operation increased he, presumably being a scientist, should ask himself what indeterminable force - what phenomenon - caused the central clock to physically undergo an increase it's rate of operation?

It is my understanding that the idea of the central clock's rate of operation increasing (i.e. time contraction) was, for Einstein, an anathema.

Your depiction of a wheel rolling along a road complies with sections 1 through 3 of STR.

My depiction of a disc rotating in space complies with section 4 of STR.

 

[atyy]

"Thus, while the static observers in the cylindrical chart admits a unique family of orthogonal hyperslices T = T0, the Langevin observers admit no such hyperslices." http://en.wikipedia.org/wiki/Born_coordinates

Is quoted statement correct, and if so, is it applicable to this discussion?

 

[Al68]

A person is located at the center of a stationary (neither moving nor spinning) hypothetically zero-mass disc in an imaginary otherwise empty universe. There is, at the rim of this disc, a large clock (A) identical to his own clock (B).

The disc starts spinning; according to Einstein's section 4 depiction - clock A (now moving in a closed curve around the stationary clock B) will tick over at a slower rate than B.

This is correct, but doesn't depend on whether or not the wheel was previously spinning or not.

The person located at the center of the wheel will see clock A continuously ticking over at a slower rate than his own clock and on the basis that observation (determination) creates reality he is fully entitled to be of the opinion that clock A is ticking over at a slower rate than his own clock.

Also correct, but it's not just his opinion, it is objectively true in his rest frame.

He has every right to anticipate that if he then moves to the rim of the disc his clock will be subjected to the same 'law' of physics that caused clock A to tick over at the slower rate than his own clock when A started moving thus that when he moves to the rim his clock will also be ticking over at a slower rate than a clock at the center of the disc.

Also correct, if he also accelerates continuously to stay stationary with clock A (and clock B) after he gets to the rim. (if he instead just moves to the rim and becomes inertial, the central clock will run slow relative to his, as he would be moving in a straight line tangent to the rim, in inertial motion, and would soon be far away.)

Being located on the rim of the disc he would observe that the central clock is ticking over at a faster rate than his own clock however for him to assume that his clock has not been subjected to the same 'law' of physics as was clock A but that the central clock's rate of operation increased he, presumably being a scientist, should ask himself what indeterminable force - what phenomenon - caused the central clock to physically undergo an increase it's rate of operation?

The fact that he is now accelerating toward the central clock (just like clock A is) caused the central clock's rate to increase relative to his, the central clock's rate doesn't change in its own frame, or in any absolute sense. If this observer's clock is running slower than the central clock then the central clock is running faster than his. The "increased" rate of the central clock is only a relative increase-relative to the observer's clock. The observer never sees his own clock change its rate of ticking. The central clock doesn't tick faster than it used to in any sense whatsoever, except relative to the observer's clock.

And again, no clock ever changes its own rate of operation in its own frame. Any clock that does is "broken" in SR.

And it's still important to note that in the rotating rest frame in which clock A is stationary, clock B is also stationary, and the relative velocity between them is zero. Obviously we can't attribute the time dilation to velocity in this frame, because there is none. But attributing the time dilation to gravitational time dilation in the accelerated frame is mathematically equivalent to attributing it to relative velocity in the inertial frame in which the central clock is stationary and the rim clock is in relative motion.

 

[cos]

Having made the comment "Right" can I take it that you agree that an equatorial clock is ticking over at a slower rate than a polar clock ("...under otherwise identical conditions.")?

Yes, in the rest frame of the polar clock, and in the accelerated frame of the equatorial clock, if we ignore gravity. Note again that this is not standard time dilation between inertial frames. Both clocks are stationary in the "rotating" reference frame. Relative velocity between the clocks is zero in this frame.

You do not need to continuously refer to the fact that we ignore gravity nor do you need to point out, 'again', that this is not standard time dilation between reference frames.

Relative velocity is not zero in the picture referred to. The equatorial clock is moving in a closed curve around the polar clock as is Einstein's analogous clock that is made to move in a closed curve around an at rest clock.

An observer located at the center of the wheel sees clock A ticking over at a slower rate than his own clock (and, obviously, at a slower rate than it was before the wheel started spinning).

Yes, in the center observers rest frame. But not for an observer stationary at clock A. In his non-inertial frame, clock A runs at the same rate it always did, and the center clock B started running at a faster rate when the wheel started spinning.

An observer located alongside clock A is not moving relative to clock A but is orbiting clock B.

When the wheel starts spinning that person is then subjected to a g force thus knows that his is no longer an inertial reference frame.

The central clock is not 'running at a faster rate' when the wheel starts spinning.

It is, from that person's point of view, ticking over at a faster rate than his own clock but to suggest that the central clock is ticking over at a faster rate than it was before the wheel started turning requires 'something' - some force or phenomenon - that has physically created this faster rate of operation however that observer has no reason whatsoever to be of the opinion that the central clock (other than appearing to be spinning on its axis which could be eliminated by it's being mounted on a free-spinning base) is moving.

There is no discernible force that has engendered a suitable equal and opposite reaction.

Having read and accepted Einstein's section 4 STR depiction of a clock that is made to move in a closed curve relative to another clock (and which will, as a result, incur time dilation relative to the 'at rest' clock) observer B realizes that clock A is ticking over at a slower rate than it was before the wheel started spinning due to the fact that it is now moving whilst he has remained at rest.

Yes, if you're referring to the inertial frame in which clock B is spinning, and clock A is in relative motion. If you're referring to the rotating reference frame in which both clocks are stationary after the wheel starts spinning, then the difference in clock rates is not due to relative motion...

I am of the opinion that it should be blatantly obvious that I am referring to the frame wherein clock A is in relative motion not to the rotating...(etc.)

I am of the opinion that observer B could be aware of the fact that if he were to move to the rim of this wheel his own clock would then also be ticking over at a slower rate than it is whilst he remains at the center of the wheel; that the 'law' of physics that caused A to start ticking over at a faster rate than it was before the wheel started spinning will also apply to his clock.

No single clock changes its own rate in its own rest frame ever. If it does, then it doesn't qualify as a good time keeper in SR. That being said, if clock B were moved to the rim, it would run slow relative to a third clock "C" that was at the center and remained there.

That's precisely what I said!

I am of the opinion that he does not "observe the center clock to run faster than his own, [because] he is accelerating toward the center clock continuously." but he observes the central clock to be running faster than his own clock for the simple reason that it, having remained at rest, is physically ticking over at a faster rate than his own clock.

In the frame of clock A, both clocks are at rest. Both clocks are stationary in the accelerated frame of clock A.

The central observer sees clock A orbiting around him; he is obviously of the opinion that clock A is moving; he moves to A's location and, for some reason, decides that he and clock A are no longer moving?

My depiction is not in relation to what clock A determines but what the previously centrally located observer (having moved to A's location) determines.

The observer I depicted moves to the rim thus then sees the clock at the center of the wheel ticking over at a faster rate than his own clock however on the basis that he can find no reason whatsoever as to why that clock would now be ticking over at a faster rate than it was before he moved to the rim of the wheel he can only(sensibly) conclude that his clock has slowed down in the same way as did clock A when the wheel started spinning.

He has plenty of basis. He is no longer in an inertial frame.

How does the fact that he is no longer in an inertial frame provide a reason for the central clock ticking over at a faster rate than it did before he moved to the rim?

He can't use the standard time dilation equations in his frame.

How would using the standard time dilation equations provide him with an identification of an indeterminable force?

Light doesn't even travel at c in his accelerated reference frame.

Light emanating from the central light clock does reach him at c in precisely the same way that light from the sun reaches us at c.

The standard lorentz transformations do not apply in this frame.

I'm not sure to which frame you are referring however Einstein pointed out that the slower rate of operation of the clock on the rim is in accordance with the equation .5tv^2/c^2.

He can prove to himself that he is not at rest in an inertial frame by the simple fact that if he releases his clock into freefall it will not stay near him. The clock must be accelerated to stay in the frame.

Having moved from the centre of the wheel to it's rim he already knows that he is not at rest in an inertial reference frame.

Again, any clock that changes its own rate in its own frame does not qualify as a valid clock in SR.

In section 4 Einstein pointed out that a clock that moves to another clock's location does change it's own rate. It, according to Einstein, 'goes more slowly' than it did before it started moving.

In sections 1 through 3 of SR "...any clock that changes its own rate in its own frame does not qualify as a valid clock in SR." however in section 4 Einstein introduced a clock that does 'change it's own rate' - by moving!

He is unable to carry out any experiment to confirm that his clock is ticking over at a slower rate than it was prior to his relocation - his heart-beat has also slowed down as have his mental processes however his intelligence has (presumably; hopefully) not been impaired.

Well, his heartrate may very well be different under different circumstances.

And your reason for making that comment was...? I really don't believe that it contributed anything.

But if he has a watch that chimes every hour according to his clock A before the wheel started spinning, it will chime every hour according to clock A afterward as well.

Is it not feasible that because his seconds are shorter than they were before he started moving that his hours are also shorter?

 

[Dale]

I believe that to be a correct assumption. My interpretation of what is 'real' or 'physical' or 'actual' or 'normal' is what takes place in an observer's reference frame not what appears to be taking place from the point of view of a person who is moving relative to that reference frame i.e. a point of view that is frame dependent.

So, do I understand this correctly? If A and B are in relative motion you would describe A in terms of A's rest frame and B in terms of B's rest frame and call that "real" etc. Any description of A from B's rest frame or B from A's rest frame would be "illusion" etc. Is that correct?

If my understanding is correct then I think this entire thread is pretty easy to resolve.

What you describe as "real" is what is usually called "proper" (e.g. an observer's proper time is the time displayed on a clock carried by the observer which is thus at rest in the observer's frame). As you mention, proper quantities (proper time, proper length, proper acceleration, proper mass, etc.) are not frame dependent. This is one good reason for classifying "proper" quantities as "real".

Now, looking at the scenarios of interest here, we know that between when A and B start and when they meet A accumulates less proper time than B. This is a frame-invariant fact and involves only descriptions of each clock in its own rest frame. All frames agree on this, and this is what you would call "real".

However, as soon as you begin comparing the rate of one clock to another clock then you are talking about "what appears to be taking place from the point of view of a person who is moving relative to that reference frame". This is not "real" according to your definition above, and therefore it should not be surprising that different reference frames disagree on the details since they are all just "illusions" anyway. You simply cannot make any "real" statements about the relative rates of A and B.

 

[Al68]

Relative velocity is not zero in the picture referred to. The equatorial clock is moving in a closed curve around the polar clock as is Einstein's analogous clock that is made to move in a closed curve around an at rest clock.

Relative velocity between the clocks is zero in the rotating reference frame, not any inertial frame.

The central clock is not 'running at a faster rate' when the wheel starts spinning.

Not in an absolute sense, or relative to any inertial frame. It does run fast relative to clock A in the accelerated frame of clock A.

The central observer sees clock A orbiting around him; he is obviously of the opinion that clock A is moving; he moves to A's location and, for some reason, decides that he and clock A are no longer moving?

He decides he is no longer moving relative to clock A. The relative velocity of both clocks is now zero relative to him.

How does the fact that he is no longer in an inertial frame provide a reason for the central clock ticking over at a faster rate than it did before he moved to the rim?

The central clock didn't change its rate, it always ticked at a faster rate than a clock accelerating toward it in the accelerating clock's frame.

Light emanating from the central light clock does reach him at c in precisely the same way that light from the sun reaches us at c.

Light from the sun doesn't reach us at precisely c. Light only travels at c relative to inertial reference frames.

I'm not sure to which frame you are referring however Einstein pointed out that the slower rate of operation of the clock on the rim is in accordance with the equation .5tv^2/c^2.

Right, in the inertial frame of the center clock. But this equation can't be used in accelerated frames. If it was used, it would say that clock B ticked at the same rate as clock A in A's frame, which is clearly wrong.

In section 4 Einstein pointed out that a clock that moves to another clock's location does change it's own rate. It, according to Einstein, 'goes more slowly' than it did before it started moving.

It 'goes more slowly' than the stationary clock, not necessarily more slowly that it did before. We could easily say that the "moving" clock is "going faster than it did before" relative to a third clock at rest with it after it starts moving. How does a single clock slow down and speed up at the same time? Because nothing happened to the clock itself, its rate is frame dependent.

In sections 1 through 3 of SR "...any clock that changes its own rate in its own frame does not qualify as a valid clock in SR." however in section 4 Einstein introduced a clock that does 'change it's own rate' - by moving!

The clock never changed its own rate, it always had different rates in different reference frames because its rate was always frame dependent.

Anytime Einstein speaks of a clock running slow relative to another, his assumption is that nothing physical is different about, or happening to the clocks. They are simply both keeping proper time. It is time itself that passes at different rates for observers in relative motion, and their clocks are just tools that record this, assuming that each clock works identically regardless of its own state of motion.

If we define two events, less proper time will elapse for the rim clock than for the center clock between those two events. The clocks just measure this effect, they don't cause the effect by "changing their rate of operation". The effect isn't caused by anything happening to the clocks.

 

[phyti]

Anytime Einstein speaks of a clock running slow relative to another, his assumption is that nothing physical is different about, or happening to the clocks. They are simply both keeping proper time. It is time itself that passes at different rates for observers in relative motion, and their clocks are just tools that record this, assuming that each clock works identically regardless of its own state of motion.

Time is the tick rate of the clock, which is a function of the ratio of its speed to light speed. It's also the rate of activity for all material objects composed of basic particles, because light is the mediator of energy transitions. Proper time occurs when the clock and observer have no relative motion, thus the observer cannot detect the change. The clock function is a real, physical effect. Clocks don't measure time since time is a relationship of events. The clock only provides a standard event to measure the rate of activity. An observer with a clock that ticks at half the rate of a 2nd clock merely records twice as many (external to their frame) events in an interval as the 2nd observer. The number of external events remains constant.

 

[cos]

Being located on the rim of the disc he would observe that the central clock is ticking over at a faster rate than his own clock however for him to assume that his clock has not been subjected to the same 'law' of physics as was clock A but that the central clock's rate of operation increased he, presumably being a scientist, should ask himself what indeterminable force - what phenomenon - caused the central clock to physically undergo an increase it's rate of operation?

The fact that he is now accelerating toward the central clock (just like clock A is) caused the central clock's rate to increase relative to his, the central clock's rate doesn't change in its own frame, or in any absolute sense. If this observer's clock is running slower than the central clock then the central clock is running faster than his. The "increased" rate of the central clock is only a relative increase-relative to the observer's clock. The observer never sees his own clock change its rate of ticking. The central clock doesn't tick faster than it used to in any sense whatsoever, except relative to the observer's clock. And again, no clock ever changes its own rate of operation in its own frame. Any clock that does is "broken" in SR

In section 4 STR Einstein implied that a clock on the rim of the wheel (i.e. a clock that is moving in a closed curve around another clock) will 'go more slowly' (i.e. will tick over at a slower rate) than the 'at rest' clock by a factor of .5tv^2/c^2.

The v in that equation is, of course, the speed at which the moving clock is orbiting the stationary clock.

The traveler would 'see' the stationary clock ticking over at a faster rate than his clock in accordance with that equation yet there is nothing in that equation which refers to his centripetal acceleration toward the other clock!

The traveler 'sees' the central clock 'ticking over at a faster rate than it was before he started moving' thus assumes that it has changed it's own rate of operation ergo that clock is "broken".

No action taken by the traveler - accelerating or decelerating; moving at any uniform velocity toward or away from the stationary clock - has any physical effect whatsoever on that clock's rate of operation! For him to be of the opinion that it is ticking over at a faster rate than it was before he moved to the rim he must be of the opinion that it is his having moved to the rim that has caused the central tick to tick over at a faster rate than it was when he was at that location!

In section 4 Einstein effectively, analogously, wrote that the 'going more slowly' (i.e. time dilation) of the rim observer is dependent upon his rate of travel around the stationary clock. You are, in my opinion, insinuating that from the point of view of your, now, rim observer Einstein was wrong - that the moving clock does not 'go more slowly' (i.e. ticks over at a slower rate) than the stationary clock but that the stationary clock 'goes more quickly' (i.e. ticks over at a faster rate - time contraction) than the accelerated clock.

It is my understanding that the idea of time contraction was, for Einstein, an anathema.

The traveler maintaining an identical distance from the central clock (i.e. being anchored to the rim of the wheel) is analogous to an astronaut whose ship is up against an invisible immovable barrier in space some distance from a clock. He fires his rockets yet cannot move toward that clock. The fact that he fires his rocket does not increase the rate of operation of that clock!

He can pour as much power as he likes into his rocket (analogous to the rim observer increasing his rate of travel around the central clock whilst maintaining a constant distance from same i.e. the wheel is turning faster) but this will have no affect whatsoever on that clock's rate of operation yet in the case of the rim observer, the central clock's rate of operation 'does' increase (in reality, his clock 'goes [even] more slowly' than it did before the rate of spin of the wheel increased).

You wrote "If this observer's clock is running slower than the central clock then the central clock is running faster than his." and I have consistently agreed with that comment however for the traveler to be of the opinion that the central clock has undergone a change in it's rate of operation and that it is now ticking over at a faster rate than it was when he was at that location indicates to me not only a challenge to Einstein's section 4 depiction but also an indication of his gross ignorance and stupidity!

And it's still important to note that in the rotating rest frame in which clock A is stationary, clock B is also stationary, and the relative velocity between them is zero. Obviously we can't attribute the time dilation to velocity in this frame, because there is none. But attributing the time dilation to gravitational time dilation in the accelerated frame is mathematically equivalent to attributing it to relative velocity in the inertial frame in which the central clock is stationary and the rim clock is in relative motion.

Whilst it well may be "...important to note that in the rotating rest frame in which clock A is stationary, clock B is also stationary, and the relative velocity between them is zero." for the observer at the center of the wheel to be of the opinion that when he moves to it's rim he will be stationary is, in my opinion, asinine!

The wheel is spinning at perhaps several hundred thousand Ks a second. Having moved to the rim he feels a tremendous 'force' attempting to pull him away from the wheel - a force to which he was not being subjected at the center of the wheel.

Assuming that he is of the opinion that he is 'stationary' is he not likely to ask himself what is creating this 'pull'?

Is he incapable of realizing either before, during or after his relocation that being on the rim of the wheel he will be moving at the same velocity as was a clock at the rim before he moved?

 

[cos]

Anytime Einstein speaks of a clock running slow relative to another, his assumption is that nothing physical is different about, or happening to the clocks.

On the basis that one clock is ticking over at a slower rate than another clock something physical is different about those clocks! They are ticking over at different rates!


If a clock at sea-level is ticking over at a slower rate than a clock on top of a mountain something physical is creating this variation. One of them is in a stronger gravitational tidal area than the other one!

When Hafele and Keating carried out the first leg of their experiment something physical did happen to the clocks in the aircraft! They were 'going more slowly' (i.e. ticking over at a slower rate) than the laboratory clocks. The laboratory clocks remained unchanged!

Clifford M Will pointed out in Was Einstein Right? that the clocks in the aircraft should more correctly have been compared with a (relatively 'stationary') master clock at the center of the planet which (gravitational effects being allowed for) is ticking over at the same rate as Einstein's polar clock.

During that flight Hafele and Keating were analogous to a person moving from a point part-way across the rotating wheel (A') to the rim of that wheel who (erroneously) assumes that his clock is not ticking over at a slower rate than it was when he was at A's location but that A' has started ticking over at a faster rate than it was when he was at that location.

Hafele and Keating could also have been (but presumably were not) of the opinion that their clocks did not slow down when they 'moved to that more distant location on the spinning wheel' but that the laboratory clocks (clock A') ticked over at a faster rate than they did before the flight commenced ergo that the laboratory cloks changed their rate of operation!

They are simply both keeping proper time. It is time itself that passes at different rates for observers in relative motion, and their clocks are just tools that record this, assuming that each clock works identically regardless of its own state of motion.

According to Einstein - a clock that has been accelerated 'goes more slowly' (i.e.ticks over at a slower rate) than a clock that has remained at rest!

If we define two events, less proper time will elapse for the rim clock than for the center clock between those two events. The clocks just measure this effect, they don't cause the effect by "changing their rate of operation". The effect isn't caused by anything happening to the clocks.

Less proper time elapses for the rim clock than for the center clock because the rim clock is 'going more slowly' (i.e. ticking over at a slower rate) than the center clock!

I have made no suggestion to the effect that the clocks will cause the times between when those events occur to vary.

 

[JesseM]

In section 4 Einstein effectively, analogously, wrote that the 'going more slowly' (i.e. time dilation) of the rim observer is dependent upon his rate of travel around the stationary clock. You are, in my opinion, insinuating that from the point of view of your, now, rim observer Einstein was wrong - that the moving clock does not 'go more slowly' (i.e. ticks over at a slower rate) than the stationary clock but that the stationary clock 'goes more quickly' (i.e. ticks over at a faster rate - time contraction) than the accelerated clock.

It is my understanding that the idea of time contraction was, for Einstein, an anathema.

If you are just comparing the rates of the two clocks to one another in given frame, then the contrast you're trying to draw here doesn't make sense--if clock X is ticking slower than Y, then of course Y is ticking faster than X, these are just two ways of saying the same thing (just like there's no difference between the statements 'Dave is taller than Stan' and 'Stan is shorter than Dave'). On the other hand, if you are talking about the rate a clock is ticking relative to coordinate time in a given frame, then it's true that in inertial frames there is no "time contraction"--a clock at rest in an inertial frame ticks at the same rate as coordinate time, while all moving clocks tick slower than coordinate time. But in a non-inertial frame like the rotating frame where the clock at the equator is at rest and ticking at the same rate as coordinate time, it is perfectly possible for a clock to tick faster than coordinate time, so in this sense "time contraction" can occur in non-inertial frames.

The wheel is spinning at perhaps several hundred thousand Ks a second. Having moved to the rim he feels a tremendous 'force' attempting to pull him away from the wheel - a force to which he was not being subjected at the center of the wheel.

Assuming that he is of the opinion that he is 'stationary' is he not likely to ask himself what is creating this 'pull'?

In non-inertial frames, G-forces which an inertial observer would attribute to acceleration are instead explained as a consequence of a "pseudo-gravitational field"--see the equivalence principle analysis from the twin paradox page.

 

[atyy]

In section 4 STR Einstein used that equation in relation to several accelerated frames!

That's interesting if he did so.

 

[JesseM]

In section 4 STR Einstein used that equation in relation to several accelerated frames!

That's interesting if he did so.

You can read section 4 here, nowhere does Einstein refer (explicitly or implicitly) to any non-inertial frames, although he does analyze the time on accelerating clocks from the perspective of inertial frames.

 

[atyy]

You can read section 4 here, nowhere does Einstein refer (explicitly or implicitly) to any non-inertial frames, although he does analyze the time on accelerating clocks from the perspective of inertial frames.

Thanks! Very standard stuff (for our times) then.

 

[cos]

In section 4 STR Einstein used that equation in relation to several accelerated frames!

That's interesting if he did so.

In all but his reference to a clock at the equator he points out that each clock A starts off at rest then moves to another location and although he does not specifically refer to acceleration per se I believe that a relocation of clock A requires acceleration.

 

[Al68]

In section 4 STR Einstein implied that a clock on the rim of the wheel (i.e. a clock that is moving in a closed curve around another clock) will 'go more slowly' (i.e. will tick over at a slower rate) than the 'at rest' clock by a factor of .5tv^2/c^2.

Yes, if the equation is used in the inertial frame in which the center clock is at rest.

The v in that equation is, of course, the speed at which the moving clock is orbiting the stationary clock.

Yes, again in the inertial frame in which the center clock is at rest.

The traveler would 'see' the stationary clock ticking over at a faster rate than his clock in accordance with that equation yet there is nothing in that equation which refers to his centripetal acceleration toward the other clock!

That's because the equation isn't used in the "traveler's" rest frame.

The traveler 'sees' the central clock 'ticking over at a faster rate than it was before he started moving' thus assumes that it has changed it's own rate of operation ergo that clock is "broken".

Not if he understands SR.

In section 4 Einstein effectively, analogously, wrote that the 'going more slowly' (i.e. time dilation) of the rim observer is dependent upon his rate of travel around the stationary clock. You are, in my opinion, insinuating that from the point of view of your, now, rim observer Einstein was wrong - that the moving clock does not 'go more slowly' (i.e. ticks over at a slower rate) than the stationary clock but that the stationary clock 'goes more quickly' (i.e. ticks over at a faster rate - time contraction) than the accelerated clock.

If A>B then B<A.

You wrote "If this observer's clock is running slower than the central clock then the central clock is running faster than his." and I have consistently agreed with that comment however for the traveler to be of the opinion that the central clock has undergone a change in it's rate of operation and that it is now ticking over at a faster rate than it was when he was at that location indicates to me not only a challenge to Einstein's section 4 depiction but also an indication of his gross ignorance and stupidity!

The faster rate is relative, not absolute.

Whilst it well may be "...important to note that in the rotating rest frame in which clock A is stationary, clock B is also stationary, and the relative velocity between them is zero." for the observer at the center of the wheel to be of the opinion that when he moves to it's rim he will be stationary is, in my opinion, asinine!

Stationary with respect to the rim clock, not in any other sense.

The wheel is spinning at perhaps several hundred thousand Ks a second. Having moved to the rim he feels a tremendous 'force' attempting to pull him away from the wheel - a force to which he was not being subjected at the center of the wheel.

Assuming that he is of the opinion that he is 'stationary' is he not likely to ask himself what is creating this 'pull'?

Is he incapable of realizing either before, during or after his relocation that being on the rim of the wheel he will be moving at the same velocity as was a clock at the rim before he moved?

If he's moving at the same velocity as the rim clock, in the center clock's frame, then he is stationary with respect to the rim clock. The "pull" he feels is evidence of proper acceleration, not velocity.

Having moved to the rim and seeing the central clock ticking over at a faster rate than his own clock (i.e seemingly at a faster rate than it was before he moved) the observer can only assume that the central clock has changed it's rate of operation.

That would be like saying that the car driving in front of me "sped up" because its speed relative to me increased when I hit my brakes.

In section 4 STR Einstein used that equation in relation to several accelerated frames!

He certainly did not. He used the equation for clocks that have accelerated, not for accelerated reference frames.

Here's a question: Let's call the clock that moves from the center to join the rim clock clock "C". The statement that clock C runs slower at the rim than it did at the center is simply not true in every reference frame. For example, let's say I'm in inertial motion at the rim, local to and momentarily co-moving with the rim clock. In my frame, clock C is running faster than it did at the center. What caused clock C to "speed up its rate of operation"?

 

[JesseM]

In all but his reference to a clock at the equator he points out that each clock A starts off at rest then moves to another location and although he does not specifically refer to acceleration per se I believe that a relocation of clock A requires acceleration.

But again, he's analyzing things from the perspective of an inertial frame, not from the perspective of A's non-inertial rest frame. Do you understand the difference between 1) analyzing an accelerating object from the perspective of an inertial frame, and 2) using a non-inertial frame?

 

[Al68]

Time is the tick rate of the clock, which is a function of the ratio of its speed to light speed.

That ratio is always zero for the inertial rest frame of the clock.

 

[cos]

In section 4 STR Einstein implied that a clock on the rim of the wheel (i.e. a clock that is moving in a closed curve around another clock) will 'go more slowly' (i.e. will tick over at a slower rate) than the 'at rest' clock by a factor of .5tv^2/c^2.

Yes, if the equation is used in the inertial frame in which the center clock is at rest.

So the observer is located at the center of the wheel; he determines that the rim clock (A) is moving around him at v and, applying Einstein's section 4 STR equation (i.e. "...the equation is used in the inertial frame in which the center clock is at rest."), he calculates the slower rate at which the rim clock is ticking compared to his own clock's rate of operation (i.e. clock B).

Still located at the center of the wheel is he not entitled to be of the opinion that if he moved to A's location that his clock would, then, also be ticking over at the same slower rate than a central clock?

Is he not entitled to be of the opinion that the same 'law' of physics that causes clock A to tick over at a slower rate than B would equally affect his clock?

If he sends another clock, that is synchronous with his clock, out to A's location would he not, then, see that clock ticking over at a slower rate than it was before it moved?

I am of the opinion that this is the very crux of the discussion so at this stage will delay responding to the rest of your post until I receive a response to this message.

I hope that doesn't sound pretentious - it was not intended to be; it is merely an attempt to save both of us some time.

 

[JesseM]

Still located at the center of the wheel is he not entitled to be of the opinion that if he moved to A's location that his clock would, then, also be ticking over at the same slower rate than a central clock?

There isn't really room for differing "opinions" in SR, there are just statements of fact about what is true in a given frame, no one ever disagrees about what's true in a specific frame. It's certainly true that in the inertial frame where the center of the wheel is at rest, an observer's clock will tick slower if he moves from the center to the edge of the wheel. But without the context of a particular frame, it's meaningless to offer "opinions" about which clock is ticking slower at a given moment.

 

[Al68]

So the observer is located at the center of the wheel; he determines that the rim clock (A) is moving around him at v and, applying Einstein's section 4 STR equation (i.e. "...the equation is used in the inertial frame in which the center clock is at rest."), he calculates the slower rate at which the rim clock is ticking compared to his own clock's rate of operation (i.e. clock B).

Still located at the center of the wheel is he not entitled to be of the opinion that if he moved to A's location that his clock would, then, also be ticking over at the same slower rate than a central clock?

Yes, but it's not just his opinion, it's objective fact in his frame.

Is he not entitled to be of the opinion that the same 'law' of physics that causes clock A to tick over at a slower rate than B would equally affect his clock?

Yes, he is.

If he sends another clock, that is synchronous with his clock, out to A's location would he not, then, see that clock ticking over at a slower rate than it was before it moved?

Yes, he would. But not everyone would. Some hypothetical observers would see that same clock tick at a faster rate than it did before it moved. Because in some reference frames, it "runs slower" and in some frames it "runs faster" than it did when it was at the center.

 

[phyti]

Jesse;

To avoid the complications of too many words, here is a drawing.
Here A and B are initially at rest in the F-frame. When clock F reads 0, A and B clocks read 0, and A and B accelerate (instantly) to speed v. Per synch convention, clock B concludes clock A is ahead by d, for the duration t as indicated on A and B clocks. At the end of t, both decelerate (instantly) to zero in the F-frame. Clock A reads t + d, clock B reads t.
Do you think this scenario is correct, specifically the clock readings?

 

[JesseM]

Jesse;

To avoid the complications of too many words, here is a drawing.
Here A and B are initially at rest in the F-frame. When clock F reads 0, A and B clocks read 0, and A and B accelerate (instantly) to speed v. Per synch convention, clock B concludes clock A is ahead by d, for the duration t as indicated on A and B clocks. At the end of t, both decelerate (instantly) to zero in the F-frame. Clock A reads t + d, clock B reads t.
Do you think this scenario is correct, specifically the clock readings?

No. If two clocks are initially in sync in the F frame, and then simultaneously in the F frame they both accelerate to velocity v, and later come to rest simultaneously in the F frame, then they will naturally remain synchronized in the F frame because their velocities are the same at every moment in this frame and thus their rate of ticking (=the rate they are accumulating proper time) is also the same at every moment in this frame.

If you want to look at the frame F' in which they are at rest during the phase where they were moving at velocity v in the F frame, then if we look at the prior phase where both were at rest in the F frame, in the F' frame both clocks were moving at velocity v during this phase and the time on B's clock was ahead of the time on A's clock by some constant amount vx/c^2 (where x is the distance between them in the F frame). In frame F' B comes to rest before A comes to rest (another consequence of the relativity of simultaneity), so B is then ticking faster than A and the difference between their readings increases, then A comes to rest too and the difference between their readings remains constant for a bit, and then a little later B accelerates away from rest again before A accelerates away from rest, so during this period A is ticking faster than B and the difference between their readings is decreasing, with the net result that once A accelerates and they are both moving at constant velocity in frame F' again, the time difference between their readings in frame F' will once again have returned to vx/c^2.

 

[DrGreg]

An illustration, in the F frame (left) and the F' frame (right)

 

[JesseM]

Thanks for the illustration DrGreg! Did you use any special graphing program to put those together or just make it in a drawing program? I'd like to find some simple program to put together spacetime diagrams quickly, they'd come in handy on a lot of these threads...

 

[DrGreg]

Thanks for the illustration DrGreg! Did you use any special graphing program to put those together or just make it in a drawing program? I'd like to find some simple program to put together spacetime diagrams quickly, they'd come in handy on a lot of these threads...

No, I just used Microsoft Powerpoint as a drawing tool (the 2007 version conveniently is able to export as PNG which I can then crop to size before uploading, otherwise I could have done a screen dump).

In the past I've used the specialist software MATLAB to draw accurate graphs, but that was using someone else's computer. I think you could use Microsoft Excel, or other graph-plotting software, in a similar way. But in this case I just drew some lines and circles and arranged them by eye.

 

[cos]

So the observer is located at the center of the wheel; he determines that the rim clock (A) is moving around him at v and, applying Einstein's section 4 STR equation (i.e. "...the equation is used in the inertial frame in which the center clock is at rest."), he calculates the slower rate at which the rim clock is ticking compared to his own clock's rate of operation (i.e. clock B).

Still located at the center of the wheel is he not entitled to be of the opinion that if he moved to A's location that his clock would, then, also be ticking over at the same slower rate than a central clock?

Yes, but it's not just his opinion, it's objective fact in his frame.

Thank you.

Is he not entitled to be of the opinion that the same 'law' of physics that causes clock A to tick over at a slower rate than B would equally affect his clock?

Yes, he is.

Ditto.

If he sends another clock, that is synchronous with his clock, out to A's location would he not, then, see that clock ticking over at a slower rate than it was before it moved?

Yes, he would. But not everyone would. Some hypothetical observers would see that same clock tick at a faster rate than it did before it moved. Because in some reference frames, it "runs slower" and in some frames it "runs faster" than it did when it was at the center.

So he sends another clock to A's location and determines the objective fact that that clock is ticking over at a slower rate than it was before it moved (ergo slower than his own clock) however he then takes into account that from the point of view of some hypothetical observers that clock "runs slower" or "runs faster" than it did when it was at the center of the wheel.

Does that determination arrived at by some hypothetical observer affect his opinion, determination, prediction, calculation, ******* (insert your choice of word) of the clock's slower rate? Does it alter the objective fact that that clock is ticking over at a slower rate than the central clock?

Are any of the numerous hypothetical observers entitled to realize that whilst that relocated clock is, from their point of view (in accordance with their mathematical calculations, determinations, predictions,*******) ticking over at a slower rate (or faster rate) than it was when it was at the center of the wheel that in the original observer's frame the relocated clock is ticking over at a slower rate than it was before it moved?

Does anything that those hypothetical observers determine have any affect whatsoever on that clock?

I suggest that they do not!

For all intents and purposes, as far as the real observer, is concerned those hypothetical observers are just that!

For all intents and purposes, as far as the real observer is concerned, those hypothetical observers do not exist!

Having determined the objective fact that a clock at the rim is ticking over at a slower rate than his own clock and having sent a clock to A's location and determined the objective fact that it is then ticking over at a slower rate than his own clock he moves to A's location.

Before he moves from the center of the wheel - you agree with me that he can determine that when he moves to the rim his clock will then be subjected to the same 'law' of physics that caused the rim clock to be ticking over at a slower rate than his centrally located clock - that his clock will be ticking over at a slower rate than it is whilst he is still at the center of the wheel.

He moves to A's location; is he then not entitled to be of the opinion that the same 'law' of physics that caused the rim clock to be ticking over at a slower rate than the central clock no longer applies to him and his clock?

Prior to moving from the center of the wheel he notices that a clock at that location is stationary alongside him. It is (relative to him) not spinning on an axis and he is not being subjected to any g force (he is actually spinning very slowly - it is a wheel of enormous diameter - but cannot feel that he is on the basis that it is an otherwise empty universe).

He moves to the wheel rim which he knows is spinning around it's hub on the basis that, having arrived at that location, he is now looking at a central clock that is spinning on it's axis but, more importantly, he is then being subjected to an enormous g force attempting to move him further away from the center of the wheel ergo he knows that his is not an inertial frame.

He knows (having previously determined the speed at which a clock on the rim is spinning around the center of the wheel) that his clock is moving at that same speed ergo applies Einstein's equation and determines the, then, slower rate of operation of his clock.

It is ticking over at it's 'proper' time but he knows that it is (as is Einstein's section 4 clock traveling in a closed curve around an at rest clock) moving and that in accordance with section 4 (as well as his determination of an objective fact whilst he was at the center of the wheel) it is 'going more slowly' than it was before he moved to the rim.

I'm not suggesting that he must or that he should arrive at this conclusion but that he could!

 

[atyy]

Does that determination arrived at by some hypothetical observer affect his opinion, determination, prediction, calculation, ******* (insert your choice of word) of the clock's slower rate? Does it alter the objective fact that that clock is ticking over at a slower rate than the central clock?

Yes.

Does anything that those hypothetical observers determine have any affect whatsoever on that clock?

No.

I'm not suggesting that he must or that he should arrive at this conclusion but that he could!

I can arrive at any conclusion I want any time.

Proposed answers only. If they don't make sense, read what JesseM has to say!

 

[Al68]

So he sends another clock to A's location and determines the objective fact that that clock is ticking over at a slower rate than it was before it moved (ergo slower than his own clock) however he then takes into account that from the point of view of some hypothetical observers that clock "runs slower" or "runs faster" than it did when it was at the center of the wheel.

Does that determination arrived at by some hypothetical observer affect his opinion, determination, prediction, calculation, ******* (insert your choice of word) of the clock's slower rate? Does it alter the objective fact that that clock is ticking over at a slower rate than the central clock?

Well, like I pointed out, that's only an objective fact in his frame. Those other hypothetical observers agree with the objective fact that clock "C" slowed down in the frame of your observer.

The same laws of physics that caused the clock to "slow down" in your observer's frame also caused the same clock to "speed up" in other frames, whether actual observers are present or not. After all, your observer is hypothetical as well.

Whether or not the clock slowed down or sped up is frame dependent, not observer dependent or a matter of opinion.

 

[JesseM]

So he sends another clock to A's location and determines the objective fact that that clock is ticking over at a slower rate than it was before it moved (ergo slower than his own clock) however he then takes into account that from the point of view of some hypothetical observers that clock "runs slower" or "runs faster" than it did when it was at the center of the wheel.

Does that determination arrived at by some hypothetical observer affect his opinion, determination, prediction, calculation, ******* (insert your choice of word) of the clock's slower rate? Does it alter the objective fact that that clock is ticking over at a slower rate than the central clock?

Again, it is meaningless to talk about clock rates without specifying a choice of frame. He believes that the clock at the edge of the wheel is ticking slower in the frame where he is at rest, and so does the hypothetical observer at rest in a different frame. Likewise, the hypothetical observer B at rest in the other frame believes that the clock at the center is ticking slower than the clock at the edge (at some specific moment) in the frame where that observer B is at rest, and the observer at the center of the wheel agrees. Either observer can make determinations/predictions/calculations in any frame they choose, and as long as they specify which frame a particular statement about clock rates is meant to apply in, their statements will be objective facts that all observers should agree on. On the other hand, if you talk about clock rates without specifying a choice of frame, your statements are too ill-defined to be judged true or false.

He knows (having previously determined the speed at which a clock on the rim is spinning around the center of the wheel) that his clock is moving at that same speed ergo applies Einstein's equation and determines the, then, slower rate of operation of his clock.

It is ticking over at it's 'proper' time but he knows that it is (as is Einstein's section 4 clock traveling in a closed curve around an at rest clock) moving and that in accordance with section 4 (as well as his determination of an objective fact whilst he was at the center of the wheel) it is 'going more slowly' than it was before he moved to the rim.

I'm not suggesting that he must or that he should arrive at this conclusion but that he could!

He can certainly conclude that his clock is now going more slowly in the inertial rest frame where the center of the wheel is at rest, and every possible observer would agree with that statement. But if you don't add that qualifier about which frame you mean your statements to apply to, but just say something like "my opinion is that his clock is moving more slowly once he's at the rim of the wheel", then you aren't making well-defined statements about physics.