Regarding centrifugal effects on the time dilation of a moving clock

[tade]

So in the model of Special Relativity, an object which is in motion is time-dilated. And let's say that we're trying to measure the kinematic time dilation of a moving clock.
So an atomic clock is placed on a dolly on a track and its sped along the track.

And due to the curvature of the Earth's surface, the clock would at least be subject to some centrifugal accelerations as it speeds along the track.
Are these centrifugal accelerations expected to have an effect on the rate at which the clock ticks, and if so, how significant is the effect?

 

[Dale]

For time dilation there is no additional effect of acceleration beyond the velocity. This is known as the clock hypothesis and has been experimentally validated up to about $10^{18} \ g$

 

[Nugatory]

Are these centrifugal accelerations expected to have an effect on the rate at which the clock ticks,

Centripetal not centrifugal, but much more important is “the rate at which the clocks ticks” compared to what other clock? It makes no sense to talk about a clock running fast or slow except as compared to some other clock.

 

[tade]

Centripetal not centrifugal, but much more important is “the rate at which the clocks ticks” compared to what other clock? It makes no sense to talk about a clock running fast or slow except as compared to some other clock.

Compared to a synchronized pair of "stationary" clocks that are placed beside the track

 

[Vanadium 50]

Please, I hope we are past the stage of "The One True Stationary Clock".

Like the title, these threads run in circles.

 

[Nugatory]

Compared to a synchronized pair of "stationary" clocks that are placed beside the track

Stationary relative to what? The rotating earth’s surface? The sun or fixed stars? Someone on Mars watching the proceedings through a telescope?

 

[tade]

Stationary relative to what? The rotating earth’s surface? The sun or fixed stars?

oh sorry, i meant that they're planted on the ground beside the track, so relative to the earth's surface

 

[tade]

Please, I hope we are past the stage of "The One True Stationary Clock".

Like the title, these threads run in circles.

nah no worries i'm not thinking about the concept of a "one true" anything at all

 

[tade]

For time dilation there is no additional effect of acceleration beyond the velocity. This is known as the clock hypothesis and has been experimentally validated up to about $10^{18} \ g$

thanks I see. and by the way the Twin Paradox is an important example of the clock hypothesis, and also, for the perspective of the travelling twin, is it that, when his spaceship is making the turnaround, the ticking rate of the earth-staying twin becomes faster than that of the travelling twin?

 

[Ibix]

for the perspective of the travelling twin, is it that, when his spaceship is making the turnaround, the ticking rate of the earth-staying twin becomes faster than that of the travelling twin?

No. Directly looking through a telescope at an Earth clock, the travelling twin will see the Earth clock switch from slow ticking to fast ticking as he turns around. This is due to the change in sense of the Doppler effect from red shift to blue shift. However, there is no unique prescription for describing how a non-inertial observer like the travelling twin should interpret these observations to determine what the clock is doing "now".

The underlying reason here is that simultaneity is relative, and you have quite a lot of freedom to define what you mean by the word "now" in a question like "what time is shown on clocks on Mars now?" In the case of inertial observers almost any sensible procedure to determine "now" will reduce to Einstein clock synchronisation. But for a non-inertial observer you can get many different interpretations.

One choice of "now" definition is called radar time. In this case you do get fast ticking of the Earth clock "during" the acceleration. But the length of time you call "during" the acceleration depends on distance and you get slow ticking of clocks on the other side of the ship, again with duration dependent on distance. On the other hand (and by far the simplest solution mathematically), the traveller can simply interpret everything using the Earth's inertial frame, and regard his own clocks as ticking slow the whole time.

 

[PeroK]

Compared to a synchronized pair of "stationary" clocks that are placed beside the track

Imagine constructing a large circle of inertial clocks, all at rest relative to each other. Then having another clock move round the circle at some constant speed $v$. This "moving" clock will run slow relative to the circle of clocks by precisely the expected gamma factor:$$\gamma = \frac 1 {\sqrt{1 - v^2/c^2}}$$This is not so much time dilation as differential ageing, as the "moving" clock measures each clock in the circle to have recorded more time during each completed circle. Differential ageing is, therefore, an invariant, physically meaningful quantity.

Time dilation itself is a coordinate effect. In this case, in the usual inertial coordinates, the time dilation equals the differential ageing, as measured by the circle of clocks. The situation for the moving clock is complicated by the fact that it is moving non-inertially and there is no standard way to define space and time as it (centripetally) accelerates round the circle. In this more general scenario time dilation is not really a thing. Instead, the invariant differential ageing is the physically meaningful concept.

 

[Nugatory]

thanks I see. and by the way the Twin Paradox is an important example of the clock hypothesis, and also, for the perspective of the travelling twin, is it that, when his spaceship is making the turnaround, the ticking rate of the earth-staying twin becomes faster than that of the travelling twin?

From the perspective of the traveling twin, their clock is at rest and the earth clock is moving relative to them throughout both the outbound and return legs. Thus according to the time dilation formula the earth clock is always ticking more slowly. There’s never a time when the earth clock is ticking faster. But somehow the earth twin is older at the reunion - that’s what makes the twin paradox a paradox. The resolution of the paradox is to either not think in terms of clock rates or to properly allow for the relativity of simultaneity when you do.

Followup discussion of the twin paradox belongs in its own thread, and after reading the stickied post at the top of this subforum.

 

[Sagittarius A-Star]

There’s never a time when the earth clock is ticking faster.

That's only the case for an inertial rest-frame of the traveler (outbound and return legs). If the traveler i.e. has a constant proper acceleration towards the earth clock, then a Rindler frame may be defined, in which the traveler is at rest (at an x-coordinate, where his proper time rate equals the coordinate time rate) and the Earth clock is ticking faster than the coordinate time (turnaround).

when his spaceship is making the turnaround, the ticking rate of the earth-staying twin becomes faster than that of the travelling twin?

But somehow the earth twin is older at the reunion - that’s what makes the twin paradox a paradox.

I think that's misleading, because the "twin paradox" is no paradox.

 

[tade]

Followup discussion of the twin paradox belongs in its own thread, and after reading the stickied post at the top of this subforum.

so i'm still on my OP, and as Dale brought up the clock hypothesis, I'd like to clarify some things about the Twin Paradox as I think that they might have relevance to my OP
such as whether my OP scenario shares some of the intricacies of what makes the Twin Paradox so complex

though yeah i guess that the discussion might become quite complex, can i remain with this thread or should i create a new one

 

[PeroK]

so i'm still on my OP, and as Dale brought up the clock hypothesis, I'd like to clarify some things about the Twin Paradox as I think that they might have relevance to my OP
such as whether my OP scenario shares some of the intricacies of what makes the Twin Paradox so complex

though yeah i guess that the discussion might become quite complex, can i remain with this thread or should i create a new one

It's pointless moving from one question to another until you have understand the first postulate of SR, which implies that:

All inertial motion is relative. There is no such thing as a "moving" clock; only a clock moving in some system of coordinates. No inertial clock is ever ticking slowly or quickly - its tick rate is measured to be slow or fast in a system of coordinates in which the clock is moving.

You should never talk about a moving clock. Only a clock moving relative to something.

Moreover, velocity-based time dilation is relative and reciprocal. If you measure my clock running slow, then I measure your clock running slow by the same amount.

You need to digest completely these consequences of the first postulate - before you go any further.

 

[tade]

It's pointless moving from one question to another until you have understand the first postulate of SR, which implies that:

All inertial motion is relative. There is no such thing as a "moving" clock; only a clock moving in some system of coordinates. No inertial clock is ever ticking slowly or quickly - its tick rate is measured to be slow or fast in a system of coordinates in which the clock is moving.

You should never talk about a moving clock. Only a clock moving relative to something.

Moreover, velocity-based time dilation is relative and reciprocal. If you measure my clock running slow, then I measure your clock running slow by the same amount.

You need to digest completely these consequences of the first postulate - before you go any further.

yeah that's true, i believe that i have digested the first postulate

 

[Nugatory]

I'd like to clarify some things about the Twin Paradox as I think that they might have relevance to my OP such as whether my OP scenario shares some of the intricacies of what makes the Twin Paradox so complex

You have taken the time to read the sticky thread I mentioned? If not, do so now.

The essential aspect of the twin paradox is that two clocks at the same place are compared; they separate and go their own way for a while; then reunite and are compared again. Are you considering the situation in which the track goes all the way around the earth, so we compare the two clocks before and after one full circuit? If so, that variant of the twin paradox has been covered in many other threads here.

 

[tade]

You have taken the time to read the sticky thread I mentioned? If not, do so now.

The essential aspect of the twin paradox is that two clocks at the same place are compared; they separate and go their own way for a while; then reunite and are compared again. Are you considering the situation in which the track goes all the way around the earth, so we compare the two clocks before and after one full circuit? If so, that variant of the twin paradox has been covered in many other threads here.

Yeah I have read it, and I've also read the articles by Michael Weiss. I'd like to ask about some of the topics in Weiss's article on the Equivalence Principle Analysis.

 

[Nugatory]

Yeah I have read it, and I've also read the articles by Michael Weiss. I'd like to ask about some of the topics in Weiss's article on the Equivalence Principle Analysis.

New questions, new threads.

 

[Dale]

Yeah I have read it, and I've also read the articles by Michael Weiss. I'd like to ask about some of the topics in Weiss's article on the Equivalence Principle Analysis.

I am not familiar with those articles. I have no strong opinions on whether it goes here or in another thread, but I would need a link

 

[Dale]

thanks I see. and by the way the Twin Paradox is an important example of the clock hypothesis, and also, for the perspective of the travelling twin, is it that, when his spaceship is making the turnaround, the ticking rate of the earth-staying twin becomes faster than that of the travelling twin?

The traveling twin is non inertial, so their perspective is a non inertial reference frame. Things are considerably more complicated in such frames, including the fact that there are no universally accepted definitions of such frames

Here is my favorite approach, with an explicit application to the twins scenario.

https://arxiv.org/abs/gr-qc/0104077

Figure 9 is particularly surprising and informative. It answers the question you just asked

 

[cianfa72]

Imagine constructing a large circle of inertial clocks, all at rest relative to each other. Then having another clock move round the circle at some constant speed $v$. This "moving" clock will run slow relative to the circle of clocks by precisely the expected gamma factor:$$\gamma = \frac 1 {\sqrt{1 - v^2/c^2}}$$This is not so much time dilation as differential ageing, as the "moving" clock measures each clock in the circle to have recorded more time during each completed circle. Differential ageing is, therefore, an invariant, physically meaningful quantity.

Just to add that, considering any of the "stationary" clock in the circle, the worldline of the "moving" clock around the circle intersects the "stationary" clock worldline at both the starting and end events respectively. Therefore the differential ageing is actually an invariant since one compares two different spacetime timelike paths between the two given points/events.

 

[tade]

The traveling twin is non inertial, so their perspective is a non inertial reference frame. Things are considerably more complicated in such frames, including the fact that there are no universally accepted definitions of such frames

Here is my favorite approach, with an explicit application to the twins scenario.

https://arxiv.org/abs/gr-qc/0104077

Figure 9 is particularly surprising and informative. It answers the question you just asked

oh yeah, I remember Figure 9, its really intriguing due to, from Barbara's perspective, the distance to Alan appearing to be constant for an extended duration, quite a strange effect. Is there a PF thread which explains this strange effect in more detail?

 

[Dale]

Is there a PF thread which explains this strange effect in more detail?

I am not sure about that. There have been many threads discussing that paper, so probably at least one contains a detailed explanation.

I think the paper itself does a good job. It is simply a direct result of extending the second postulate to the traveler’s non-inertial frame.

 

[Ibix]

Is there a PF thread which explains this strange effect in more detail?

It's fairly straightforward to see why it happens. Add a second traveller who does exactly the same as the first except mirrored in the Earth's x direction. If both travellers send out a radar pulse at the same time they will both arrive at Earth at the same time and therefore the echoes will arrive back at their original ships at the same time as the pulse from the other ship arrives.

If one ship sends out a series of evenly spaced radar pulses before turnaround that arrive at the other ship after turnaround must arrive evenly spaced, just because everything is moving inertially. So evenly spaced radar pulses echoing off Earth must return evenly spaced.

So the measured distance change can only be linear as a function of time. But the distance to Earth time $T$ before return must be the same as at time $T$ after departure, from symmetry. The only way you can do that with a linear distance change is if the distance change is zero.

 

[Histspec]

Yeah I have read it, and I've also read the articles by Michael Weiss. I'd like to ask about some of the topics in Weiss's article on the Equivalence Principle Analysis.

I am not familiar with those articles. I have no strong opinions on whether it goes here or in another thread, but I would need a link

He apparently refers to the series of twin paradox articles in the Physics FAQ written by Michael Weiss:
https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

 

[A.T.]

thanks I see. and by the way the Twin Paradox is an important example of the clock hypothesis, and also, for the perspective of the travelling twin, is it that, when his spaceship is making the turnaround, the ticking rate of the earth-staying twin becomes faster than that of the travelling twin?

Don't confuse the proper acceleration of a clock (which has no effect on its tick rate per Clock Hypothesis), with analyses done in non-inertial coordinates (where, depending on the chosen simultaneity convention, clocks at different locations might tick at different rates).

In the later case, the tick rates of two co-located clocks with different proper accelerations will still be the same, in agreement with the Clock Hypothesis.

 

[PeterDonis]

the tick rates of two co-located clocks with different proper accelerations but the same coordinate velocities will still be the same

I added a qualifier in bold that is necessary.