Does acceleration cause time dilation?

[kahoomann]

Simple question (derived from some unanswered posts from various posters)...
Does acceleration cause time dilation? Can someone shed light on this one?

There seems two conflicting claims:

acceleration cause time dilation

For example, in the most 'natural' way to attach a comoving coordinate chart to an accelerating observer, you have the effect that when he accelerates towards a clock, it is observed to run faster. (The increase is proportional to coordinate distance, and probably also to the magnitude of the acceleration) And similarly, when accelerating away from a clock, it is observed to run slower, or even backwards..

acceleration does not cause time dilation

My recommendation is to forget the acceleration. It only serves to break the symmetry and does not cause time dilation by itself. Instead, use the spacetime interval which allows you to extend the analysis to arbitrarily accelerating twins. .

 

[granpa]

acceleration causes a change in what the object being accelerated considers to be simultaneous. that is what Hurkyl was referring to. it does not contradict what DaleSpam said.

 

[kahoomann]

acceleration causes a change in what the object being accelerated considers to be simultaneous. that is what Hurkyl was referring to. it does not contradict what DaleSpam said.

>> when accelerating away from a clock, it is observed to run slower, or even backwards..

This sounds pretty much like time dilation to me

 

[granpa]

nope. look again.

 

[kahoomann]

Yep, look again

 

[shalayka]

The question "Does acceleration cause time dilation?" is similar in context to "Does HIV kill you?". The answer is "yes, but not directly". Acceleration causes velocity, which causes time dilation. HIV causes AIDS, which causes death. It's not meant to be a perfect analogy, since one can undergo HIV without developing AIDS, but one can never undergo acceleration without developing velocity.

Just because one experiences time dilation while undergoing acceleration does not mean that acceleration is the direct cause. If that were the case, then one would experience a constant rate of time dilation under constant acceleration. However, we know better: the rate of time dilation only stays constant with constant velocity. Therefore, it's directly related to velocity.

The real question is: "What is time dilation constant with respect to?"

The real answer is: "Velocity."

 

[granpa]

>> when accelerating away from a clock, it is observed to run slower, or even backwards..

This sounds pretty much like time dilation to me

how would you change what is considered to be simultaneous without some clocks seeming to move forward and some clocks seeming to move backward?

 

[ehj]

So accumulated proper time during acceleration can be found using a simple integral? I thought SR couldn't handle acceleration?

 

[Fredrik]

So accumulated proper time during acceleration can be found using a simple integral? I thought SR couldn't handle acceleration?

That's just a misunderstanding that's extremely common because of how SR is taught in introductory texts.

As I pointed out in the "Einstein simultaneity" thread, Einstein's postulates are ill-defined and can't be the axioms of a mathematical theory. They're just there to help us guess the real axioms of the theory. It turns out we need only one: Space-time can be represented mathematically by Minkowski space.

The properties of the minkowski metric imply that inertial frames exist, but it would be absolutely preposterous to pretend that those are the only coordinate systems we're allowed to consider, since a coordinate system is just a function that assigns numbers to events. Some authors claim that we're doing GR when we're considering other coordinate systems on Minkowski space. In my opinion that's just an obsolete way of thinking about SR that should have been abandoned decades ago.

If the space-time we're considering is Minkowski space, then we're doing special relativity. If the manifold is curved, we're doing general relativity.

 

[Dale]

Acceleration does not cause time dilation. This is known as the http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Clock_Hypothesis" and has been experimentally verified up to about 10^18 g. Consider also muons created from cosmic rays in the upper atmosphere. They do not accelerate but instead are created at their high relative velocity. They are a textbook example of time dilation without acceleration.

You can either say that velocity causes time dilation or that time dilation is just what happens when a clock takes a shorter path through spacetime. I prefer the second approach, which is the spacetime geometric explanation.

 

[ehj]

So what they mean when saying "SR can't handle acceleration" would for instance be the traveling twins frame during the turnaround?

 

[yuiop]

So what they mean when saying "SR can't handle acceleration" would for instance be the traveling twins frame during the turnaround?

I tend to agree with Fredrik thoughts in post #9 that suggest these type of statements are a common misconception.

Special Relativity is perfectly capable of handling twins type paradoxes by plotting paths (even paths that change direction) on Minkowski type diagrams with distance on one axis and time on the other axis. Saying General Relativity is required to explain the twins paradox is a gross exageration. The power of General Relativity is only really required when things get really complicated such as when tidal effects have to be taken into account. (IMHO)

 

[DemTings]

The question "Does acceleration cause time dilation?" is similar in context to "Does HIV kill you?". The answer is "yes, but not directly". Acceleration causes velocity, which causes time dilation. HIV causes AIDS, which causes death. It's not meant to be a perfect analogy, since one can undergo HIV without developing AIDS, but one can never undergo acceleration without developing velocity.

Just because one experiences time dilation while undergoing acceleration does not mean that acceleration is the direct cause. If that were the case, then one would experience a constant rate of time dilation under constant acceleration. However, we know better: the rate of time dilation only stays constant with constant velocity. Therefore, it's directly related to velocity.

The real question is: "What is time dilation constant with respect to?"

The real answer is: "Velocity."

It's science, but it's real. Nothing but real science here...

 

[shalayka]

It's science, but it's real. Nothing but real science here...

Let me guess... you believe that a static axisymmetric body rotating on its axis of symmetry emits gravitational waves? If so, try spinning a disc under water and see how well the disc's edge pushes the water out of the way. Doesn't work quite as well as a spinning stick, does it? PhD? My God, they're giving them out in Cracker Jack boxes these days.

Hopefully one of us is making sense.

Either way, if you have an actual point, make it. Otherwise stop trolling.

 

[ThomasT]

Does acceleration cause time dilation?

Time dilation is a symmetric artifact of relative motion.

Sometimes when people use the term time dilation they actually mean differential aging.

Differential or asymmetric aging is a consequence of different acceleration histories.

 

[paw]

I think this shows pretty conclusively that acceleration by itself isn't the cause of differential aging (time dilation in the OP). Have a look at the spacetime diagram attached. It's a rework of the problem posed in an earlier thread.

A is an asteroid moving left to right at 0.3c according to M (the midpoint observer). B is a spaceship moving right to left at 0.3c according to M. M sends a light pulse (event 'flash') and A and B set their clocks to zero when they receive the flash (A0 and B0).

A and B collide at C. Since mass A >> mass B the spacetime path of A is unaffected by the collision.

Moments before the collision A's clock and B's clock have recorded the same number of ticks. This should be clear from the symmetry and the calculated proper times (A0 to C and B0 to C) are the same. Moments after the collision, B is co-moving with A so from this point on both clocks should accumulate time at the same rate.

Sometime after the collision (A1_B1), astronauts recover B's clock which is still ticking (it's a Timex). Both clocks will read the same time even though B underwent an (somewhat substantial) acceleration. Which, while it lacks the rigor of a proof, makes it pretty clear acceleration doesn't directly cause time dilation.

Special thanks to Mentz114 for his 'Spactime plotter' software. Very nice and useful Mentz114.

 

[yuiop]

Time dilation is a symmetric artifact of relative motion.

Sometimes when people use the term time dilation they actually mean differential aging.

Differential or asymmetric aging is a consequence of different acceleration histories.

There is a very simple proof that acceleration does not cause differential ageing.

Take 2 turntables. One has a radius that is 10 times bigger than the other. Place observer A on the perimeter of the small turntable and observer B on the perimeter of the large turntable. Spin up both turntables so that they reach the same perimeter velocity relative to inertial observer C who is not on a turntable. A and B experience different proper accelerations but when the turntables have stopped observer A and B note that the proper times recorded on their own clocks are the same despite experiencing different proper accelerations. They also note that the elapsed proper time on their clocks is less than that of observer C and can be accounted for by their equal perimiter velocities.

Repeat the experiment, but this time spin the turntables so that both A and B experience the same proper centripetal accelerations. This time their perimeter velocities are different and they record different proper elapsed times.

How is this a proof when I have shown no maths? Easy, an analogue of the experiment was actully carried out in a lab with a centrifuge where the equipment experienced centripetal acceleration millions of times greater than that of the Earth. The time dilation was proportional to the instantaneous linear velocity on the perimeter and independent of the proper acceleration. Case closed.

 

[pmb_phy]

I'd like to point out an example of a gravitational field which produces the exact same metric as the one used for a rotating frame of reference. If one takes a shell which has a finite thickness and a uniform mass distribution and sets it rotating about its axis of symetry while the observer inside remains in a coordinate system which remains non-rotating then the rotating shell will create a a gravitational field incuding frame-dragging effect. Even though the spacetime inside the shell remains flat there will still be a gravitational field. The field manifests itself by causing different clocks at different distances from the center of rotation run at different lengths. Observers who were initially at rest inside will experience a gravitational force directed radially away from the center of rotation.

Pete

 

[ThomasT]

There is a very simple proof that acceleration does not cause differential ageing.

Take 2 turntables. One has a radius that is 10 times bigger than the other. Place observer A on the perimeter of the small turntable and observer B on the perimeter of the large turntable. Spin up both turntables so that they reach the same perimeter velocity relative to inertial observer C who is not on a turntable. A and B experience different proper accelerations but when the turntables have stopped observer A and B note that the proper times recorded on their own clocks are the same despite experiencing different proper accelerations. They also note that the elapsed proper time on their clocks is less than that of observer C and can be accounted for by their equal perimiter velocities.

Repeat the experiment, but this time spin the turntables so that both A and B experience the same proper centripetal accelerations. This time their perimeter velocities are different and they record different proper elapsed times.

How is this a proof when I have shown no maths? Easy, an analogue of the experiment was actully carried out in a lab with a centrifuge where the equipment experienced centripetal acceleration millions of times greater than that of the Earth. The time dilation was proportional to the instantaneous linear velocity on the perimeter and independent of the proper acceleration. Case closed.

Given two identical oscillators, sitting side by side, keeping time at exactly the same rate, then if neither clock undergoes an acceleration, then there will be no difference between their accumulated times.

On the other hand, if one of the clocks is accelerated for a certain interval, then brought back to rest beside the unaccelerated clock, then there will be a difference between the accumulated times.

No acceleration, no differential aging. Isn't this correct, or am I missing something?

 

[rcgldr]

These posts are confliciting with the thread I created, where the "conclusion" was that the "equivlance" of gravity and acceleration held in the case of time dilation and have the same time dilation effect in GR. 2 objects, one experiencing 1g of gravity, the other 1g of acceleration experience the same time dilation component effect (their clocks would run slower than clocks experiencing 0g assuming the clocks at 0 g are at the same velocity).

Going back to my analogy:

Clock #1 on the equator of the Earth at sea level, velocity is 465.09 m/s (relative axis of earth), and the clock experiences a gravitational force equivalent to 9.78033 m / s² of acceleration.

Clock #2 on the perimeter of a rotating space station in open space, with a radius of 22116.71 meters rotating at 0.0210289 radians / s, with a speed of 465.09 m / s, and experiencing centripetal acceleration of 9.78033 / s². The space station is traveling at the same velocity as the earth, using rocket engines to duplicate the Earth's orbital speed around the sun, as well as the sun's orbital speed around the galaxy.

Clock #3 at the center of the same space station as clock #2, experiencing zero acceleration.

Assuming that "equivalency" holds for time dilation in GR with respect to gravity and acceleration, then clock #1 and clock #2 should run at the same rate, but slower than clock #3.

 

[yuiop]

These posts are confliciting with the thread I created, where the "conclusion" was that the "equivlance" of gravity and acceleration held in the case of time dilation and have the same time dilation effect in GR. 2 objects, one experiencing 1g of gravity, the other 1g of acceleration experience the same time dilation component effect (their clocks would run slower than clocks experiencing 0g assuming the clocks at 0 g are at the same velocity).

Going back to my analogy:

Clock #1 on the equator of the Earth at sea level, velocity is 465.09 m/s (relative axis of earth), and the clock experiences a gravitational force equivalent to 9.78033 m / s² of acceleration.

Clock #2 on the perimeter of a rotating space station in open space, with a radius of 22116.71 meters rotating at 0.0210289 radians / s, with a speed of 465.09 m / s, and experiencing centripetal acceleration of 9.78033 / s². The space station is traveling at the same velocity as the earth, using rocket engines to duplicate the Earth's orbital speed around the sun, as well as the sun's orbital speed around the galaxy.

Clock #3 at the center of the same space station as clock #2, experiencing zero acceleration.

Assuming that "equivalency" holds for time dilation in GR with respect to gravity and acceleration, then clock #1 and clock #2 should run at the same rate, but slower than clock #3.

Clock #1 and #2 run at the same rate because they have the same velocity. Change the radius of the of the space station and adjust the angular velocity so that the instantaneous linear velocity of the clock on the perimeter is still 465.09m/s and their clocks will still run at the same rate despite their proper accelerations being different.

Another thing to note is that gravitational time dilation is more precisely a function of gravitational potential and not of gravitational acceleration although they are closely related. FOr example if you descend down a very deep mine shaft on the Earth the gravitational time dilation continues to increase even though the gravitational acceleration is decreasing (assuming a non rotating Earth).

Second of all, the figure you give of 465.09m/s is the instaneous velocity of clock #1 on the surface of the Earth's equator due to rotation of the Earth. You are treating the time dilation at the surface of the Earth as due to simple rotation. Clock #1 will time dilate due to the instantaneous linear velocity due to rotation AND addititional gravitational time dilation proportional to the gravitational potential at the surface of the Earth which equates to an additional kinetic time dilation factor proportional to the escape velocity at the surface of the Earth.

Together with the arguments presented in my previous post and those presented here I hope you are convinced that time dilation is due to acceleration. Whoever told you that the analogy you gave in your post is correct has been misleading or confusing you.

There is another demonstration. Two twins start at the same point in space near a marker. An additional witness stays at the marker at all times. Twin 1 accelerates off into spaceand then cruises at constant velocity. After a period of one year twin 2 accelerates of into space at the same rate as twin 1 did and at the same time in the witness frame twin 2 decelerates to a stop and accelerates back towards twin 1. When they meet twin 2 turns around and comes back with twin 1 and finally they both decelerate to a stop at the marker where the witness is. Assume they have been careful to execute identical acceleration profiles at each manouver in their journeys, then twin 2 will have aged less than twin 1 despite both twins having experienced identical acceleration. Both twins will have aged less than the witness, but twin 1 more so. Differential time dilation is due to different lengths of the paths through spacetime rather than acceleration per se.

 

[yuiop]

Given two identical oscillators, sitting side by side, keeping time at exactly the same rate, then if neither clock undergoes an acceleration, then there will be no difference between their accumulated times.

On the other hand, if one of the clocks is accelerated for a certain interval, then brought back to rest beside the unaccelerated clock, then there will be a difference between the accumulated times.

No acceleration, no differential aging. Isn't this correct, or am I missing something?

Well, its more a case of no relative velocity, no differential ageing. See my previous post for more detail ;)

 

[rcgldr]

Clock #1 and #2 run at the same rate because they have the same velocity.

As in my previous thread, I'm trying to eliminate the velocity aspect and only focus on gravity versus accelertion and their effect on time dilation.

Change the radius of the of the space station and adjust the angular velocity so that the instantaneous linear velocity of the clock on the perimeter is still 465.09m/s and their clocks will still run at the same rate despite their proper accelerations being different.

This would mean that the GR "equivalency principle" for gravity and acceleration doesn't apply to time dilation?

You are treating the time dilation at the surface of the Earth as due to simple rotation.

That wasn't my intention, I was only interested in the near 1 g of gravitational field strength at the equator. The speed was only mentioned so I could show another example of a clock moving at the same speed and experiencing near 1 g of acceleration.

Clock #1 will time dilate due to the instantaneous linear velocity due to rotation and addititional gravitational time dilation proportional to the gravitational potential at the surface of the Earth which equates to an additional kinetic time dilation factor proportional to the escape velocity at the surface of the Earth.

OK, but doesn't acceleration on clock #2 at the perimeter of the space station have the same component of time dilation due to acceleration as clock #1 does due to gravitational potential?

 

[Hans de Vries]

Many posters here need to seriously rethink their claims...


The Equivalence principle says:

-The ceiling in a house on Earth ages faster as the floor,
-The ceiling in a linear accelerating rocket also ages faster as the floor

They do so by the same amount if both g and h(eight) are the same.
If the windows are blindfolded then you can't tell the difference. You
can't tell in which room you are by comparing the clocks on the floor
and at the ceiling.



The time dilation is differential: between the ceiling and the
floor. One should not try to relate the time dilations on Earth with
those in the rocket. The clocks in the rocket will go slower and slower
compared with those on Earth but that is not the point. The point is
that you can not tell in which room you are. The one in a gravitational
field or the one in an accelerating rocket.

The elementary particles in the ceiling in a house age faster because
they have a higher potential energy. Their de Broglie frequency is higher.

The effect in the the rocket is explained with SR and non simultaneity.
The x'-axis of the Minkovski diagram continuous to rotate closer to the
45 degrees axis because the rocket continuous to go faster and faster.

The ceiling (which the astronaut considers to be in the same reference
frame as his floor) lays further and further ahead in time as the floor in
the rest frame. This means that the ceiling has been traveling longer as
the floor and for that reason has aged more, and it continues to age
more (faster) as long as the rocket keeps accelerating.



Regards, Hans

 

[Dale]

These posts are confliciting with the thread I created, where the "conclusion" was that the "equivlance" of gravity and acceleration held in the case of time dilation and have the same time dilation effect in GR.

Did you not read my posts in that other thread? They are equivalent: neither acceleration nor gravity cause time dilation.

Your scenarios have nothing to do with the equivalence principle, as I already pointed out in the other thread.

 

[pmb_phy]

They are equivalent: neither acceleration nor gravity cause time dilation.

You can't be serious can you? Its a well known fact of general relativity that gravitational time dilation occurs in both accelerated frames and in gravitational fields. Einstein himself proved this and it was observed back as far as the early 60's. The GPS system depends on it.

Pete

 

[yuiop]

As in my previous thread, I'm trying to eliminate the velocity aspect and only focus on gravity versus accelertion and their effect on time dilation.

When analysing a complex issue you should start with the simplest possible thought experiment and make it more general when you have a better handle on the issue.

Comparing a model of a planet with gravitation and rotation to a turntable is not the simplest model. It is better to compare a model of a planet with gravitation and no rotation with with a rotaing turntable with gravitation. That way, you can isolate and compare gravitaion effects with rotation effects.

This would mean that the GR "equivalency principle" for gravity and acceleration doesn't apply to time dilation?

The equivalence principle does apply to time dilation, but the issues are subtle.

That wasn't my intention, I was only interested in the near 1 g of gravitational field strength at the equator. The speed was only mentioned so I could show another example of a clock moving at the same speed and experiencing near 1 g of acceleration.

The answer is that the clock #1 on the Earth experiences additional time dilation due to gravitational potential (not acceleration) relative to clock #2 on the spacestation.

OK, but doesn't acceleration on clock #2 at the perimeter of the space station have the same component of time dilation due to acceleration as clock #1 does due to gravitational potential?

The acceleration on clock #2 at the perimeter of the space station contibutes no component of time dilation to clock #2. The proof, is in this experiment in the FAQ of this forum (Experimental basis of Special Relativity) titled "The Clock Postulate" http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html#Clock_Hypothesis

This is where all the hand waveing stops. We cannot argue with what is actually measured in experiments. The FAQ item states:

" The clock hypothesis states that the tick rate of a clock when measured in an inertial frame depends only upon its velocity relative to that frame, and is independent of its acceleration or higher derivatives. The experiment of Bailey et al. referenced above stored muons in a magnetic storage ring and measured their lifetime. While being stored in the ring they were subject to a proper acceleration of approximately $$10^{18}g$$ where (1 g = 9.8 m/s2). The observed agreement between the lifetime of the stored muons with that of muons with the same energy moving inertially confirms the clock hypothesis for accelerations of that magnitude."

In other words, the time dilation of muons experiencing 10,000,000,000,000,000,000 times the surface gravitational acceleration of the Earth, did not experience any additional time dilation above the time dilation accounted for by the instantaneous linear velocity.

However illogical or unreasonable the result of the experiment seems, that is the facts of the case.

 

[Dale]

You can't be serious can you? Its a well known fact of general relativity that gravitational time dilation occurs in both accelerated frames and in gravitational fields. Einstein himself proved this and it was observed back as far as the early 60's. The GPS system depends on it.

I am 100% serious. Speed differences cause time dilation in SR. In GR gravitational potential differences also cause time dilation. In neither case is the cause of time dilation the acceleration.

The GPS satellites are at a higher gravitational potential than the ground, similaraly with the Pound-Rebka experiment you refer to. Similarly in an accelerating reference frame.

Look at the expression for gravitational time dilation in the Swartzschild metric: $$\frac{t_g}{t_f} = \sqrt{1 - \frac{GM}{Rc^2}}$$ which is clearly only a function of the gravitational potential, $$\frac{GM}{R}$$, and not only a function of the gravitational acceleration, $$\frac{GM}{R^2}$$. You find a similar conclusion in a uniform field or accelerating rocket.

 

[Fredrik]

A clock in free fall at ground level ticks at the same rate as a clock on the satellite, right? Doesn't that contradict your claim that gravitational potential causes time dilation?

 

[MeJennifer]

I am 100% serious. Speed differences cause time dilation in SR. In GR gravitational potential differences also cause time dilation. In neither case is the cause of time dilation the acceleration.

I like this line of thinking however I prefer to say that velocity (not necessarily speed) differences and non Euclidean spacetimes ('gravitational potential' is nice but how would you define 'gravitational potential' in GR?) cause time dilation. And needless to say that time dilation is always a comparison between two or more clocks, as there is no absolute notion of time.

 

[yuiop]

A clock in free fall at ground level ticks at the same rate as a clock on the satellite, right? Doesn't that contradict your claim that gravitational potential causes time dilation?

Your statement is a little vague. A clock dropped dropped vertically from a couple of meters above the ground is briefly if freefall and so is a clock that is orbiting a couple of meters above the ground around some planet with no atmosphere.

Assuming you meant the latter and assuming a non rotating planet and assuming a non rotating satellite that is orbiting much higher up, then the clock rates will not necessarily be the same. The lower orbiting clock is at a lower gravitational potential (higher gravitational time dilation) and a higher orbital velocity and so it also has greater kinetic time dilation.

A object in free fall "feels no gravity" but is still subject to gravitational time dilation. The "feels no gravity" part just means that an accelerometer attached to the object would not measure any acceleration and in a free falling closed lab for example, it would appear as if there is no gravity as far as the occupents are concerned (if we ignore some tidal effects that are hard to detect in a small volume).

 

[pmb_phy]

I like this line of thinking however I prefer to say that velocity (not necessarily speed) differences and non Euclidean spacetimes ..

I believe that non-Euclidean spacetimes are neccesary but not sufficient for gravitational time dilation. For instance Minkowski spacetime will not produce gravitational time dilation but is a non-Euclidean spacetime.

...'gravitational potential' is nice but how would you define 'gravitational potential' in GR..

The term gravitatational potential is more precisely defined in the plural, i.e.gravitatational potentials in general relativity is defined as the components of the metric tensor. They are referred to as a set of ten independant gravitational potentials. The metric tensor can then be thought of as a tensor potential (or potential tensor).

Pete

 

[Fredrik]

Your statement is a little vague. A clock dropped dropped vertically from a couple of meters above the ground is briefly if freefall and so is a clock that is orbiting a couple of meters above the ground around some planet with no atmosphere.

Assuming you meant the latter and assuming a non rotating planet and assuming a non rotating satellite that is orbiting much higher up, then the clock rates will not necessarily be the same. The lower orbiting clock is at a lower gravitational potential (higher gravitational time dilation) and a higher orbital velocity and so it also has greater kinetic time dilation.

A object in free fall "feels no gravity" but is still subject to gravitational time dilation. The "feels no gravity" part just means that an accelerometer attached to the object would not measure any acceleration and in a free falling closed lab for example, it would appear as if there is no gravity as far as the occupents are concerned (if we ignore some tidal effects that are hard to detect in a small volume).

I meant that any clock in free fall should tick at the same rate as any other, but I have to admit that I don't fully understand gravitational dime dilation and I also haven't really thought this through, so maybe I'm way off. Doesn't any clock in free fall define a "default ticking rate" that we compare everything else to? (I guess you have already answered that with a no). I'm going to have to start thinking about this now.

 

[pmb_phy]

I am 100% serious. Speed differences cause time dilation in SR. In GR gravitational potential differences also cause time dilation. In neither case is the cause of time dilation the acceleration.

That is incorrect. No offense my frien but you keep confusing the clock hypothesis with the phenomena of gravitational time dilation. The clock hypothesis refers to the notion that the rate at which a clock runs as measured locally depends only on the speed of the clock as measured locallyand not the acceleration of the clock. It does not refer to the relative rates at which two different clocks run which are not locally compared but which are separated by a finite distance, either in a higher gravitational potential or higher up in the acclerated frame. I can then be shown, by utilizing the clock hypothesis, that the clock at z = 0 will run at a different rate than an identical clock at z = h, even though they will run at identical rates when compared locally. Each clock having, for all practical purposes, the same acceleration (even though the acceleration makes no difference). The reason? Consider an observer at rest in frame S' which is a frame of reference which is accelerating with respect to an inertial frame S. Let the origin of S and S' be coincident at t = 0. Let light be emitted from z = 0 where the coordinate clock is located, in the +z direction which is the direction in which the frame is accelerating. The light will arrive at z = h where there is an identical clock of identical construction. The light takes a finite amount of time to reach the clock as observed from both S and S'. Observers at rest in S will determine that the observer at rest at z = h in S' will be moving with respect to S with a speed v as measured in S. Since the observer at h has a clock whose rate does not depend on acceleration but merely on speed then that observer will detect a redshift in the frequency of light. Now look at this from observers at rest in S'. The clock located at z = 0 is not moving relative to the observer at z = h. Since the equivalence principle tells us that this same thing will happen in a uniform gravitational field, or in a region of a gravitational field in which the field is uniform for all practical purposes. The later has been measured in the lab with positive results. The ratio of the the two clocks are a function of $\gamma$ where, for the accelerating frame of refernce, of acceleration a, that $\gamma = (1+ az/c^2)$. For the gravitational field $\gamma = (1+ gz/c^2)$. The equivalence principle is evident here in the relation a = g.

Funny thing! Nobody has explained they physics up until now!

The GPS satellites are at a higher gravitational potential than the ground, similaraly with the Pound-Rebka experiment you refer to. Similarly in an accelerating reference frame.

I['m confused. Elsewhere you told me that Therefore gravity does not cause time dilation.. Please clarify for me.

Look at the expression for gravitational time dilation in the Swartzschild metric: $$\frac{t_g}{t_f} = \sqrt{1 - \frac{GM}{Rc^2}}$$ which is clearly only a function of the gravitational potential, $$\frac{GM}{R}$$, and not only a function of the gravitational acceleration, $$\frac{GM}{R^2}$$. You find a similar conclusion in a uniform field or accelerating rocket.

Why are you using that metric? The week form of the equivalence principle states that a uniformly accelerating frame of reference is equivalent to a uniform gravitational field. It doesn't say that any accelerating frame of reference is equivalent to the Earth's gravitational field.

Best wishes

Pete

 

[yuiop]

I meant that any clock in free fall should tick at the same rate as any other, but I have to admit that I don't fully understand gravitational dime dilation and I also haven't really thought this through, so maybe I'm way off. Doesn't any clock in free fall define a "default ticking rate" that we compare everything else to? (I guess you have already answered that with a no). I'm going to have to start thinking about this now.

It is a complex and subtle issue and I am not 100% convinced I am right and there seems to quite a lot of difference in opinion in this forum. The confusion comes about I think because the equivalence principle is usually stated in terms of acceleration but I believe the equal acceleration is coincidental because as I have shown in my other posts we can construct thought experiments where the acceleration is equal but the time dilation is different and equally we show situations where the acceleration is different but the time dilation is equal.

Another example is to consider what happens when a clock is lowered down a long mine shaft to the centre of a non rotating earth. (Assume it does not melt). If we have a small hollow spherical cavity at the center the acceleration of gravity at the centre of the planet is zero according to both Newton and Einstein. Now if you assume zero acceleration = zero time dilation you would get the wrong answer. The clock at the centre runs slower than a clock at the surface of the gravitational body. So why is the clock at the centre running slower than the clcok at the surface, when the clock at the centre has zero acceleration and zero velocity? The answer is that the gravitational potential at the centre of the Earth is not zero and is less than the potential at the surface which is why objects fall towards the centre. For a given gravitational potential there is an equivalent escape velocity. The escape velocity at the center of the Earth is not zero. The escape velocity is the "effective velocity" and you can think of the time dilation as kinetic time dilation proportional to that that "effective velocity". It is a bit like an object sitting on a table experiences acceleration even though it is not actually moving anywhere. Just by sitting there it has an effective veocity which is numerically the escape velocity which is a function of gravitational potential.

Just for info, the Newtonian acceleration inside a body of even density is not proportional to GM/R^2 but:

$$\frac{GMR_{inside}}{R_{surface}^3}$$

where the variable R(inside)<=R(surface) and R(surface) is constant.
At least I think that is right, but I could not find a handy reference to it. At the surface R(inside) = R(surface and the expression becomes equal to the regular GM/R^2.