Does Acceleration Affect Time Dilation?

[Chris Miller]

I understand that time slows with increased velocity and gravity, but even after reading this (article http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html) I'm a little confused as to whether or not acceleration alone, as in independent of (i.e., in addition to) velocity slows time.

E.g., consider a clock in a centrifuge out in space passed by another non-accelerating clock whose relative velocity is momentarily 0 (i.e., equal to the centrifuge clock's tangential velocity). Are they running at the same rate?

E.g., During the instant I smash into a brick wall (go from say 150 mph to 0), is my car's clock running faster or slower than before the collision?

 

[phinds]

I understand that time slows with increased velocity and gravity, but even after reading this (article http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html) I'm a little confused as to whether or not acceleration alone, as in independent of (i.e., in addition to) velocity slows time.

No, it does not. It APPEARS to an observer to run slower but locally it does not slow. Consider this; right now as you read this you are MASSIVELY time dilated and traveling at near light speed relative to a particle in the CERN accelerator. You are also traveling fast and mildly time dilated according to a passing asteroid. You are also stationary and not time dilated at all relative to the chair you are sitting in. Does your clock reflect any of this? Of course not.

 

[Bartolomeo]

E.g., consider a clock in a centrifuge out in space passed by another non-accelerating clock whose relative velocity is momentarily 0 (i.e., equal to the centrifuge clock's tangential velocity). Are they running at the same rate?

The both (rotating and tangential) clocks will measure, that clock in the center of the circumference is ticking $\gamma$ times faster. The clock in the center will measure, that the both rotating and tangential clock dilate $\gamma$ times.

 

[jambaugh]

Your questions are a bit ill posed due to some implicit assumptions. Like "have you stopped beating your wife?" it is hard to answer directly. But here's a go.

Firstly when you say "time slows" there is the begged question "relative to what?" It is well to pay attention to Einstein's oft quoted definition of time. "Time is what a clock reads". So time as measured by a given clock never slows relative to that clock. To formulate your question better let's then just talk about moving clocks and accelerating clocks and what they will read as predicted by Einstein's theories. Let's also be very very careful not to take a "God's eye view" and imagine we can peek at all the clocks "at the same time" because that is not well defined without a.) consulting another clock and b.) explaining how to communicate time information between distant points.

I will mention that by the equivalence principle: gravity = acceleration so both should have equivalent effects (including formations of stationary event horizons).
To compare two moving clocks we need to bring them into very close proximity so that a signal between them takes effectively no time (as seen by all observers)
or to bring them into a co-moving inertial frame with 0 relative velocity so that they can pass singles back and forth enough to synchronize.

So let me propose an explanatory thought experiment which, I hope, will clarify the issue for you. Imagine two clocks traveling on circular tracks at the same tangential velocity (one circle inside the other but touching at a point.)

Let the clocks start clockwise around these circles with the same speed as measured by you who are free floating and see the circles as stationary.
The inner circling clock will experience a stronger acceleration than the outer one but both are traveling at the same speed in your frame and according to each other when they happen to come together at the tangent point. Let's also make the the big circle's circumference a rational multiple of smaller's so that after the inner clock makes n turns and the outer clock makes m turns they will both be at the tangent point at the same time (though each may measure that time as a different value). The question then is what will the two clocks read at this point? Different values or the same value?

I think they will agree on times (and be slower than your clock if you also are standing at the tangent point.) But I need to sit down with pencil and paper and work out the details. (It's been awhile since I played with this.) I will do this and post my figures. In the mean time, does this scenario address your question?

 

[jambaugh]

I should have recalled but yes they will agree. In this scenario the differential time rates will only depend on their speeds relative to the stationary observer and since they travel at the same speed for the same amount of time they will have the same proper times.

$$d\tau^2 = dt^2 - dx^2 \to d \tau/dt = \sqrt{1-v^2}$$
in c=1 units.

Then you may ask "why do clocks slow when inside a gravity well?" That situation is more akin to comparing the clocks going around the circle to the clock of the stationary observer at the tangent point. Do note that again it is tricky to speak about comparing two clocks when they are separated in space and you can't have different gravity for different objects at the same location. But this is a more classic question and I refer you to the texts on the matter.

 

[Ibix]

You have to be a bit careful with your definitions here. In an inertial reference frame, the speed of the clock as measured in that frame is the only thing that you need to determine its tick rate. The only way acceleration enters into it is that it can affect speed.

What the tick rate is at the instant of a car crash isn't a well-posed question. It's assuming a discontinuity in the speed of the car, a time where speed isn't actually defined, and asking "what is the value of something that depends on the speed that isn't defined". In practice a car cannot be perfectly rigid and there will be a velocity profile to its deceleration; the velocity profile in some frame defines the clock rate in that frame.

Regarding the centrifuge question, at the instant that the centrifuge clock is stationary with respect to the inertial clock they tick at the same rate (at least, there's a well-defined limit where that's true).

 

[Chris Miller]

I should have recalled but yes they will agree. In this scenario the differential time rates will only depend on their speeds relative to the stationary observer and since they travel at the same speed for the same amount of time they will have the same proper times.

$$d\tau^2 = dt^2 - dx^2 \to d \tau/dt = \sqrt{1-v^2}$$
in c=1 units.

Then you may ask "why do clocks slow when inside a gravity well?" That situation is more akin to comparing the clocks going around the circle to the clock of the stationary observer at the tangent point. Do note that again it is tricky to speak about comparing two clocks when they are separated in space and you can't have different gravity for different objects at the same location. But this is a more classic question and I refer you to the texts on the matter.

Thanks for sorting through my poor phrasing to understand my question, and for the more exact thought experiment. So, because both clocks have the same velocity relative to some observer, she will observe them both to run at the same speed. That one clock is under greater acceleration than the other has no impact. (Ibix also seems to confirm this for me, thanks).

I don't so much ask why does gravity affect time as why doesn't acceleration? If I understand Einstein's elevator thought experiment, then it seems like it'd be easy to tell, from inside, whether you were on the surface of Earth or accelerating at 1g through space. Only if on Earth would the clock on the ceiling tick more slowly than the one on the floor?

 

[Chris Miller]

No, it does not. It APPEARS to an observer to run slower but locally it does not slow. Consider this; right now as you read this you are MASSIVELY time dilated and traveling at near light speed relative to a particle in the CERN accelerator. You are also traveling fast and mildly time dilated according to a passing asteroid. You are also stationary and not time dilated at all relative to the chair you are sitting in. Does your clock reflect any of this? Of course not.

I understand that my clock only appears slower to an observer on the asteroid or particle, just as theirs does to me. Sorry for the muddled phrasing.

 

[Chris Miller]

The both (rotating and tangential) clocks will measure, that clock in the center of the circumference is ticking $\gamma$ times faster. The clock in the center will measure, that the both rotating and tangential clock dilate $\gamma$ times.

Little confused by this. If only relative velocity, and not acceleration, is affecting time, then both the rotating and stationary clock should measure the other as running slower?

EDIT: Is it their different world lines?

 

[Bartolomeo]

Little confused by this. If only relative velocity, and not acceleration, is affecting time, then both the rotating and stationary clock should measure the other as running slower?

EDIT: Is it their different world lines?

Not quite. Everything is very simple.
First: Look for Mossbauer rotor time dilation test - ONLY blueshift of frequency for rotating absorber, redshift of frequency if source rotates and absorber is in the center.
https://en.wikipedia.org/wiki/Ives–Stilwell_experiment
Second: look for transverse Doppler Effect - blueshift of frequency is observer moves in the reference frame of the source - "light received at closest approach in the receiver frame will be blueshifted relative to its source frequency"
https://en.wikipedia.org/wiki/Relativistic_Doppler_effect
Third: Look for Champeney and Moon time dilation test, when absorber and source placed on opposite sides of the rim - no shift of frequency, the both clock measure the same ticking rate.
http://iopscience.iop.org/article/10.1088/0370-1328/77/2/318/meta
Fourth:
https://www.researchgate.net/publication/304781760_Specific_Features_of_Time_Dilation_During_Circular_Movement

Also the last chapter
http://www.mathpages.com/home/kmath587/kmath587.htm

 

[Hiero]

I don't so much ask why does gravity affect time as why doesn't acceleration? If I understand Einstein's elevator thought experiment, then it seems like it'd be easy to tell, from inside, whether you were on the surface of Earth or accelerating at 1g through space. Only if on Earth would the clock on the ceiling tick more slowly than the one on the floor?

Viewing things on the surface of the Earth is supposed to be (on a small scale) like viewing things in an accelerating room. If the floor clock and ceiling clock have no relative motion in an accelerating room, will they be moving relative to each other in the non-accelerating frame?

 

[Chris Miller]

Viewing things on the surface of the Earth is supposed to be (on a small scale) like viewing things in an accelerating room. If the floor clock and ceiling clock have no relative motion in an accelerating room, will they be moving relative to each other in the non-accelerating frame?

The clocks have a relative velocity of 0 in both scenarios. But on earth, the lower clock is deeper in the gravity well and should run slower. Actually... even if acceleration did affect time the way gravity does, all elevator clocks would still tick at the same rate.

 

[phinds]

The clocks have a relative velocity of 0 in both scenarios.

yes

But on earth, the lower clock is deeper in the gravity well and should run slower

yes

. Actually... even if acceleration did affect time the way gravity does, all elevator clocks would still tick at the same rate.

Well, sort of. Clocks in a gravity well (and you are positing for the moment that acceleration acts the same) tick LOCALLY at the same rate as all local clocks, but they are on a different world line than a clock higher up in the gravity well and if you were to bring them together the one lower in the gravity well would have ticked fewer times.

 

[Hiero]

The clocks have a relative velocity of 0 in both scenarios.

yes

Wait, what? If I see a spaceship accelerating relative to me, and two clocks (say on the floor and ceiling of the spaceship) have no relative motion within the accelerating frame, will I not see the two clocks as moving towards each other? (Because the proper distance between the clocks is being contracted more and more as the speed between the frames increases...?)

 

[phinds]

Wait, what? If I see a spaceship accelerating relative to me, and two clocks (say on the floor and ceiling of the spaceship) have no relative motion within the accelerating frame, will I not see the two clocks as moving towards each other? (Because the proper distance between the clocks is being contracted more and more as the speed between the frames increases...?)

Yes, but I took his statement to be about the local relationship of the clocks not what we would perceive. As you just said, within the accelerating frame they have no relative motion. I see now that his response wasn't really a direct answer to the post he quoted and led me astray.

 

[Chris Miller]

Yes, but I took his statement to be about the local relationship of the clocks not what we would perceive. As you just said, within the accelerating frame they have no relative motion. I see now that his response wasn't really a direct answer to the post he quoted and led me astray.

No, it was about the local relationship of the clocks, and being able to determine from inside the 1 g accelerating "elevator" that you were not on earth. No external observer.

 

[Chris Miller]

Wait, what? If I see a spaceship accelerating relative to me, and two clocks (say on the floor and ceiling of the spaceship) have no relative motion within the accelerating frame, will I not see the two clocks as moving towards each other? (Because the proper distance between the clocks is being contracted more and more as the speed between the frames increases...?)

Different topic/question. And no, I don't think you'd see the clocks moving toward each other as much as the scale changing. Else if a light year long train were to (somehow) accelerate to .85c in a few days, you'd measure the relative velocity of the engine and caboose to be way faster than c.

 

[Chris Miller]

yes

yes

Well, sort of. Clocks in a gravity well (and you are positing for the moment that acceleration acts the same) tick LOCALLY at the same rate as all local clocks, but they are on a different world line than a clock higher up in the gravity well and if you were to bring them together the one lower in the gravity well would have ticked fewer times.

But wouldn't their world lines in an accelerating elevator (unlike in a gravity well) be parallel?

 

[Buzz Bloom]

In an inertial reference frame, the speed of the clock as measured in that frame is the only thing that you need to determine its tick rate.

Hi Ibix:

Your statement above raises the following questions with respect to the two circle example in post #4.
When moving at a uniform speed in a circle, is the rotating reference frame inertial?
When moving at a uniform speed in a circle, is the non-rotating reference frame inertial?

Regards,
Buzz

 

[Ibix]

Your statement above raises the following questions with respect to the two circle example in post #4.
When moving at a uniform speed in a circle, is the rotating reference frame inertial?
When moving at a uniform speed in a circle, is the non-rotating reference frame inertial?

When you are at rest in a frame, do you weigh anything? If yes, it is not an inertial frame. Whether or not the frame you are working in is inertial has no bearing on whether or not any other frame is inertial.

Someone at rest on a train circling either of those tracks will be plastered against the outside wall of the train.

 

[Hiero]

I don't think you'd see the clocks moving toward each other as much as the scale changing.

You mean the scale of the accelerated frame is changing right? In other words, I (non-accelerating) see rulers of theirs (accelerating) becoming shorter in time, right? How is this any different than saying the clocks move toward each other? (The clocks are a fixed number of shrinking rulers apart.)

Else if a light year long train were to (somehow) accelerate to .85c in a few days, you'd measure the relative velocity of the engine and caboose to be way faster than c.

There's no problem with that. I meant "relative velocity " in the sense of the speed at which two points come together, as viewed by a third point. That's why I spoke of two different relative motions; the 'proper relative motion' and the 'relative motion viewed by someone else.'

Different topic/question.

You said that in Einstein's elevator we could tell we are in gravity by seeing if the lower clock ticks at a slower rate. I believe your thinking was that two accelerating points will have the same time dilation factor if they accelerate identically... but in 'Einstein's elevator' (the accelerating frame) those points would appear to be moving apart. If we want the two points to maintain the same displacement in Einstein's elevator then we must accelerate the "lower" point faster (to compensate the scale change) and so the lower clock should read a smaller time than the higher clock.

I'm actually having trouble (and lacking time) in turning this qualitative view into a quantitative expression for (uniform) gravitational time dilation (I am still learning as well) so I can't say forsure that this perspective is the entire story until the math works out. I still believe it captures the essence of your misunderstanding though.

 

[Buzz Bloom]

Someone at rest on a train circling either of those tracks will be plastered against the outside wall of the train.

Hi Ibix:

Sorry to be so dense.

From what you just said, neither of the travelers on the two trains on circular tracks are in an inertial frame. Therefore based only on your earlier quote, it is not necessarily true that the only thing those travelers need to do determine the speed of their respective clocks is to count tick rates.

So, do they or do they not have to do anything more than count tick rates?

Regards,
Buzz

 

[Dale]

That one clock is under greater acceleration than the other has no impact.

This is also known as the clock hypothesis and has been tested with muons up to something like 10^18 g if I recall correctly (experiment by Bailey et al)

 

[Ibix]

So, do they or do they not have to do anything more than count tick rates?

Their own clock ticks at one second per second, as always. To determine the rate of the other guy's clock they can count ticks, yes. You can always count ticks of someone's clock (at least in SR...).

The complication in every inertial frame except the one where the circles are at rest is that the speed of the circling trains is not constant, so calculating the elapsed time for the circling observers isn't necessarily trivial. In non-inertial frames the relationship between velocity and tick rate also depends (in general) on the location and time that you're thinking about, which is yet more complexity.

Not sure I'm answering your question.

 

[Raymond Potvin]

Hi everybody,

At Wiki, the clock hypothesis states that the rate of a clock does not depend on its acceleration but only on its instantaneous velocity, so it means that, for two clocks at rest one beside the other, if a clock stays at rest while the other accelerates away, the one that is accelerating begins to slow down at the beginning of the acceleration, and goes on slowing down during the acceleration, which means that we can predict which clock is going to slow down during the acceleration, whereas relativity tells us that we cannot predict anymore which clock is going to slow down after the acceleration has stopped. Of course that in this case, we know which clock will me moving away from the other after the acceleration has stopped, and it is incidentally also the case for the Twins because we know which one is accelerating to make the roundtrip. So when we know which clock has accelerated, why not simply use it to solve SR problems?

 

[Z3r0]

Well I know for a fact that my clock at work ticks much slower than the clock at the bar.

Cheers :D

 

[Raymond Potvin]

Unbelievable! Show us the data! :0)

 

[phinds]

Hi everybody,

At Wiki, the clock hypothesis states that the rate of a clock does not depend on its acceleration but only on its instantaneous velocity,

No, it does not depend on either. All clocks tick at one second per second. You are likely referring to what it APPEARS to do to an observer.

so it means that, for two clocks at rest one beside the other, if a clock stays at rest while the other accelerates away, the one that is accelerating begins to slow down at the beginning of the acceleration, and goes on slowing down during the acceleration

No, it does neither, it just keeps ticking at one second per second. Again, you are referring to what an observer sees, not what the clock is doing.

, which means that we can predict which clock is going to slow down during the acceleration

We know which clock will appear to be going slower, yes

whereas relativity tells us that we cannot predict anymore which clock is going to slow down after the acceleration has stopped.

This is not correct. The clocks are symmetrical after acceleration stops. Each appears to the other to be running slow but both are ticking at one second per second.

 

[Dale]

which means that we can predict which clock is going to slow down during the acceleration

Think this through a little more. What if the clocks start out moving and one of them accelerates to rest?

 

[pervect]

Thanks for sorting through my poor phrasing to understand my question, and for the more exact thought experiment. So, because both clocks have the same velocity relative to some observer, she will observe them both to run at the same speed. That one clock is under greater acceleration than the other has no impact. (Ibix also seems to confirm this for me, thanks).

I don't so much ask why does gravity affect time as why doesn't acceleration? If I understand Einstein's elevator thought experiment, then it seems like it'd be easy to tell, from inside, whether you were on the surface of Earth or accelerating at 1g through space. Only if on Earth would the clock on the ceiling tick more slowly than the one on the floor?

I could be wrong, but my impression is that when you talk about "gravity affecting time", you presume some universal notion of time, absolute time, that exists for gravity to effect. This is not the case in special - or general - relativity. Unfortunately, it seems a bit of a digression to go off and try to explain this point, but it also seems like an obstacle to prevent any further progress without explaiing it.

I shall try another approach, that might be of some help. This is to propose a different experiment, basically a variant of something that has actually been carried out, the "Harvard clock tower" experiment, aka the Pound-Rebka experiment, and compare it to the experiment you suggest.

Basically , a signal emitter is placed at a high altitude, and a receiver is placed at a low altitude. And one looks for a doppler shift of the transmitted signal. One predicts from conservation of energy arguments, and measures experimentally, that the doppler shift exists.

Perhaps it is not at first obvious what this has to do with 'time'. The emitting source can be regarded as being some sort of clock in its own right, and it can be compared locally to some standard clock (currently the standard is a cesium atomic clock), and it can be found that the emitting source keeps the same sort of time as the standard clock.

An identical emitting source can be placed at the receiver's position. This lower emitting source, too, can be syncronized to a standard atomic clock at the same lower altitude location.

But the doppler-shifted signal from the upper clock will not, due to the doppler shift, will not and can not have the same frequency as the non-doppler shifted signal from lower emission source.

So we at least start to glimpse the issue here. The experiment you proposed doesn't show any difference in clock rates. But the Pound-Rebka experiment does. "Clock rates" is an ambiguous term, here, we have several clocks, and just as importantly, we need the details of how we compare these clocks. So we need to be more precise in our language.
So, we need to have the right words to talk concisely about the difference between the two experiments. The one set of experiments shows no difference in 'time', but the other set of experiments does. The right words here turn out to be 'proper time' and 'coordinate time'. There are some other active (and long) threads on this already, it would be off topic to go into all of the details here, I think. The point I want to make is that one can concisely describe the results of the two experiments by saying that in an inertial frame of reference, the ratio of proper time to coordinate time does not depend on acceleration, only on velocity. In a non-inertial frame of reference, though, the ratio of proper time to coordinate time depends on acceleration and position within the frame.

This later observation about non-inertial frames is true both in the case of the non-inertial frame of an accelerating elevator, or in the non-inertial frame that's due to gravity.

 

[Adel Makram]

I think they will agree on times (and be slower than your clock if you also are standing at the tangent point.) But I need to sit down with pencil and paper and work out the details. (It's been awhile since I played with this.) I will do this and post my figures. In the mean time, does this scenario address your question?

How do they agree in their time despite that both of them are non-inertial frames with different acceleration?

 

[Ibix]

How do they agree in their time despite that both of them are non-inertial frames with different acceleration?

Because their paths through spacetime are the same "length".

 

[Adel Makram]

Because their paths through spacetime are the same "length".

But I think their metric are different in their respective accelerating frames. And the metric is a function of the acceleration not as simple as in Minkowski metric.

 

[Adel Makram]

But I think their metric are different in their respective accelerating frames. And the metric is a function of the acceleration not as simple as in Minkowski metric.

But this confuses me again because this would mean the proper time is not invariant. I mean in accelerating frame with no gravity, will the metric in front of $dt$ be a function of the acceleration? if yes, then how the proper time attached to the clock at rest in this frame is invariant?

 

[Dale]

But this confuses me again because this would mean the proper time is not invariant.

Proper time is invariant.

if yes, then how the proper time attached to the clock at rest in this frame is invariant?

In the case of the accelerated frame the proper acceleration is in the metric. In the case of an inertial frame the proper acceleration is in the expression of the worldline. Either way the proper time is affected by the proper acceleration.