Does acceleration cause time dilation?

[pmb_phy]

Einstein's first derivation of gravitational time dilation is found in Jahrbuch der Radioaktivität und Elektronik (1907). Note that Einstein stated the clock hypothesis in the section labeled Space and time in a uniformly accelerated reference system. After stating the equivalence principle and just prioer to his derivation of the gravitational time dilation effect he wrote

First of all, we have to bear in mind that a specific effect of acceleration on the rate of the clocks in $\Sigma$ need not be taken into account, since they would have to be of order $\gamma^2$.

I would hazard to guess that Einstein later accepted that there is no effect at all on the acceleration of an ideal clock. With this assumption in mind Einstein derived the gravitational time dilation relation. So its erroneous to hold that the clock hypothesis implies that gravitational time dilation doesn't happen. It is a misuse of that hypothesis since this dilation effect has to do, not with the rate a single clock runs, but at the rates between two different clocks which are separated in an accelerating frame.

Pete

 

[rcgldr]

Ok, one thing that is confusing me. Why would altitude in an accelerating rocket in open space (assume the ideal case that there is zero gravitational effects in the region the rocket is traveling through) cause any difference in time dilation effects.

It's my belief that once in a stable accelerating state, every molecule of the rocket and all attached components are moving at the exact same velocity, and experience the exact same acceleration (otherwise there would be continuous compression or expansion) (ignoring the spent fuel and the flow of unspent fuel here).

Given this, then how could the positioning of two clocks within such a rocket experience any difference in time dilation?

 

[pmb_phy]

Ok, one thing that is confusing me. Why would altitude in an accelerating rocket in open space (assume the ideal case that there is zero gravitational effects in the region the rocket is traveling through) cause any difference in time dilation effects.

I explained the reason for the time dilation above. In essece the farther up in the rocket the longer it takes the light, as observed from an observer in an inertial frame of reference, to go from tale of the rocket to the nose of the rocket. The greater the speed the greater the redshift in the light due to the greater speed of the nose.

It's my belief that once in a stable accelerating state, every molecule of the rocket and all attached components are moving at the exact same velocity, and experience the exact same acceleration (otherwise there would be continuous compression or expansion) (ignoring the spent fuel and the flow of unspent fuel here).

That is correct.

Given this, then how could the positioning of two clocks within such a rocket experience any difference in time dilation?

Did you read the explanation of the derivation that I provided above? Was it unclear>

 

[Fredrik]

Why would altitude in an accelerating rocket in open space [...] cause any difference in time dilation effects.
...
how could the positioning of two clocks within such a rocket experience any difference in time dilation?

I didn't read pmb_phy's explanation so I can't comment on that, but one way to understand this is to just draw the world lines of the front and rear of the rocket in a space-time diagram corresponding to an inertial frame (or just imagine doing it). The two world lines will not be identical in the inertial frame. If they were, the rocket would remain the same length in the inertial frame even though it's being Lorentz contracted more and more as the speed increases. So the world line of the rear must be curved more than world line of the front, making it look like the rear is "catching up" with the front. In the inertial frame, the rear is always moving faster than the front.

Now consider the fact that what a clock really measures is the integral of $\sqrt{dt^2-dx^2}$ along its world line. The contribution from the dx displacements make the total smaller, and the dx displacements are bigger on the world line of the rear, and that means less proper time. So the clock in the rear runs slower than the clock in the front.

 

[rcgldr]

accelerating rocked two clocks

My intended observer would also be inside the rocket accelerating and moving at the same speed as the rocket. It this observer going to see a difference between the two clock rates?

To elminate the ever changing velocity issue, then go back to the rotating space station, clock #1 at the perimeter on one side of the space station, clock #2 on the perimeter on the opposite side of the space station. The observer is at the center of the space station. Does the observer see both clocks running at the same rate? If the observer is next to clock #1, does clock #2 "above" appear to be running at a different rate?

 

[yuiop]

Ok, one thing that is confusing me. Why would altitude in an accelerating rocket in open space (assume the ideal case that there is zero gravitational effects in the region the rocket is traveling through) cause any difference in time dilation effects.

It's my belief that once in a stable accelerating state, every molecule of the rocket and all attached components are moving at the exact same velocity, and experience the exact same acceleration (otherwise there would be continuous compression or expansion) (ignoring the spent fuel and the flow of unspent fuel here).

Given this, then how could the positioning of two clocks within such a rocket experience any difference in time dilation?

A diagram illustrating the point made by Fredrik, that a continuously accelerating rocket undergoes continuous length contraction is shown here. http://www.mathpages.com/home/kmath422/kmath422.htm

 

[Fredrik]

Does the observer see both clocks running at the same rate?

Yes. They have the same speed, so he sees the same time dilation effect on both of them.

If the observer is next to clock #1, does clock #2 "above" appear to be running at a different rate?

Yes. If the speed relative to the center of the space station is v, clock #2 is moving with speed 2v/(1+v²) relative to clock #1. So he sees a time dilation corresponding to that speed.

 

[yuiop]

...
Yes. If the speed relative to the center of the space station is v, clock #2 is moving with speed 2v/(1+v²) relative to clock #1. So he sees a time dilation corresponding to that speed.

I don't think is quite right. Both clocks are effectively at the same altitude and moving at the same speed so they will be running at the same rate and see each other to be running at the same rate.

 

[Dale]

The ratio of the the two clocks are a function of $\gamma$ where, for the accelerating frame of refernce, of acceleration a, that $\gamma^2 = (1+ az/c^2)$. For the gravitational field $\gamma^2 = (1+ gz/c^2)$. The equivalence principle is evident here in the relation a = g.

don't you see that we are in agreement here? As you have derived and as I have stated several times: the gravitational time dilation is a function of the gravitational potential, gz, rather than just a function of the gravity, g. "What we have here is failure to communicate"

Why are you using that metric?

because jeff's question was specifically set up as an experiment with a sea level clock and a space clock, so swarzschild is the appropriate 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.

yes, I know. And I pointed that out to Jeff too

 

[Fredrik]

I don't think is quite right. Both clocks are effectively at the same altitude and moving at the same speed so they will be running at the same rate and see each other to be running at the same rate.

I disagree. This is a simple SR problem. The effect that makes two clocks on a rocket tick at different rates isn't present here (in this case the two world lines look identical in the inertial frame, so the rear isn't catching up with the front, or anything like that), but there is still plenty of time dilation due to the velocity difference.

I also think that even though the equivalence principle can be useful sometimes, it's usually just a cause of confusion. It was invented as a criterion that candidate theories of gravity must satisfy to be taken seriously. For example if you find a theory of gravity that doesn't predict that the ceiling ages faster than the floor, as in SR (on an accelerating rocket), the theory would be dismissed. There may be times when the equivalence principle can help you quickly find the correct result in SR from a known result in GR, but I think those times are rare. It's probably more useful when we want to push an SR calculation over to GR.

It's important to note that there are no situations when the equivalence principle must be used. The alternative theories have already been dismissed, and we're left with a theory that does predict, all by itself, that the ceiling ages faster than the floor. So we should only use it when it significantly simplifies the calculations.

 

[Fredrik]

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.

These are some very good comments, obviously, but I didn't realize when I read them the first time that they suggest an answer to a question that (I believe) was left unresolved in the discussions in recent threads: What is a homogeneous gravitational field in GR?

A "homogeneous gravitational field" must be a metric that somehow causes this effect.

That statement is rather vague and needs to be made more formal. Unfortunately I haven't figured out how to do that yet.

 

[yuiop]

I don't think is quite right. Both clocks are effectively at the same altitude and moving at the same speed so they will be running at the same rate and see each other to be running at the same rate.

I disagree. This is a simple SR problem. The effect that makes two clocks on a rocket tick at different rates isn't present here (in this case the two world lines look identical in the inertial frame, so the rear isn't catching up with the front, or anything like that), but there is still plenty of time dilation due to the velocity difference.

I also think that even though the equivalence principle can be useful sometimes, it's usually just a cause of confusion. It was invented as a criterion that candidate theories of gravity must satisfy to be taken seriously. For example if you find a theory of gravity that doesn't predict that the ceiling ages faster than the floor, as in SR (on an accelerating rocket), the theory would be dismissed. There may be times when the equivalence principle can help you quickly find the correct result in SR from a known result in GR, but I think those times are rare. It's probably more useful when we want to push an SR calculation over to GR.

It's important to note that there are no situations when the equivalence principle must be used. The alternative theories have already been dismissed, and we're left with a theory that does predict, all by itself, that the ceiling ages faster than the floor. So we should only use it when it significantly simplifies the calculations.

I do not disagree with anything you say in this post, but that is not the situation I was talking about. I agree that the ceiling ages faster than the floor. When asked if clocks A and B would both run at the same rate as seen from the centre (C) we both said yes they would. In this case the perimeter where A and B are located represents the floor, while the centre represents the ceiling. Clocks A and B both run slower than clock C.

When asked if if an observer located by clock A on the perimeter, would see clock A as running at a different rate to clock B (also located on the perimeter), the answer should also be yes, because both clocks A and B are on the floor and not moving relative to each other.

 

[pmb_phy]

My intended observer would also be inside the rocket accelerating and moving at the same speed as the rocket. It this observer going to see a difference between the two clock rates?

As I mentioned above ...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, .. So we have already addressed this above. The answer is that even though the clocks are at rest with respect to each other there is a change in the light as it moves between them.

To elminate the ever changing velocity issue, then go back to the rotating space station, clock #1 at the perimeter on one side of the space station, clock #2 on the perimeter on the opposite side of the space station. The observer is at the center of the space station. Does the observer see both clocks running at the same rate? If the observer is next to clock #1, does clock #2 "above" appear to be running at a different rate?

No. Since these clocks are at the same gravitational potential they will run at the same rate.

Pete

 

[pmb_phy]

don't you see that we are in agreement here?

No. In fact I can't seem to figure out what it is you're saying. What you wrote here

As you have derived and as I have stated several times: the gravitational time dilation is a function of the gravitational potential, gz, rather than just a function of the gravity, g.

is inconsistent with what you wrote above, i.e.

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

What do you mean in this last statement when you use the term time dilation. Do you mean something different than gravitational time dilation? If so then I see the communication problem.

Pete

 

[yuiop]

I think all Dalespam is trying to say is that gravitational time dilation is caused by gravitational potential and not gravitational acceleration.

An example I showed earlier showed that an observer in small hollow spherical cavity at the centre of the Earth, experiences no gravitational acceleration, yet is subject to gravitational time dilation due to the non zero gravitational potential there.

 

[Dale]

So its erroneous to hold that the clock hypothesis implies that gravitational time dilation doesn't happen.

certainly that is erroneous, and I have never made that claim.

In your expressions above what english names would you give to the terms g and gz? I call g "gravity" and I call gz "gravitational potential", but I would be glad to use whatever terms you prefer just to make it clear to you that we have no disagreement.

My claim is and has always been that gravitational time dilation is a function of "gravitational potential" (gz), not "gravity" (g). This holds in the uniformly accelerating flat spacetime, the Swarzschild spacetime, and I would assume any other stationary spacetime.

 

[Dale]

I think all Dalespam is trying to say is that gravitational time dilation is caused by gravitational potential and not gravitational acceleration.

An example I showed earlier showed that an observer in small hollow spherical cavity at the centre of the Earth, experiences no gravitational acceleration, yet is subject to gravitational time dilation due to the non zero gravitational potential there.

Yes, exactly!

 

[pmb_phy]

certainly that is erroneous, and I have never made that claim.

In your expressions above what english names would you give to the terms g and gz? I call g "gravity" and I call gz "gravitational potential", but I would be glad to use whatever terms you prefer just to make it clear to you that we have no disagreement.

My claim is and has always been that gravitational time dilation is a function of "gravitational potential" (gz), not "gravity" (g). This holds in the uniformly accelerating flat spacetime, the Swarzschild spacetime, and I would assume any other stationary spacetime.

A function f(x, y, z) = xyz is a function not merely of "xyz" but is a function of x, y, z. Stating otherwise is a misuse of the phrase "is a function of"

I see that we agree that that a variation of g_00 can occur in both a non-inertial frame as well as a gravitational field and thus a difference in the rate at which clocks tick in such a spacetime also occurs. Is that correct?

 

[pmb_phy]

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.

In Newtonian mechanics the gravitational potential was a scalar quantity. In GR its a tensor quantity. There are ten indpendant components of the metric tensor. Eintein (and many, if not most, others) define these components as gravitational potentials. The Newtonian potential is related to g₀₀ in the weak field limit as

[itex]\Phi = -(g_{00} - 1)c^2/2[/tex]

Pete

 

[MeJennifer]

In GR its a tensor quantity. There are ten indpendant components of the metric tensor. Einstein (and many, if not most, others) define these components as gravitational potentials.

Sure that is all correct. The metric tensor describes a complete spacetime with a unique curvature since under General relativity the connection is considered torsion free. But I do not see any readily identifiable potentials in that tensor.

Obviously it is easy to define a gravitational potential in a static spacetime but a static spacetime is very special. To make the claim that time dilation is caused by a difference between two or more gravitational potentials one first has to define gravitational potential in a coordinate free way.

One can easily find the amount of local curvature at two given spacetime points but that does not fully determine the time dilation factor except for the simplest spacetimes.

 

[pmb_phy]

Sure that is all correct. The metric tensor describes a complete spacetime with a unique curvature since under General relativity the connection is considered torsion free. But I do not see any readily identifiable potentials in that tensor.

On the contrary, by definition the $g_{\mu\nu}$ are the gravitational potentials. That's a definition of the term gravitational potential.

Obviously it is easy to define a gravitational potential in a static spacetime but a static spacetime is very special.

The term does not depend on a particular spacetime since it is defined indepentant of the spacetime. It holds in all possible cases since its a definition.

To make the claim that time dilation is caused by a difference between two or more gravitational potentials one first has to define gravitational potential in a coordinate free way.

I disagree. Gravitational time dilation is a coordinate dependant phenomena as is the presence of the gravitational field. Gravitational time dilation mnight exist in one coordinate system and not in another. When you are discussing gravitational time dilation you are in fact talking about a coordinate dependant phenomena. And if you're talking about a a coordinate system in which the components of the metric tensor (i.e. the gravitational potentials) are an explicit function of time then there is no reason to assume that gravitational time dilation will be present at all times since the rate of the clocks will be a function of time in general.

In any case the metric tensor g is the geometric quantity that is referred to as the gravitational potential tensor[/b]. Think of this as the analogy of the magnetic field potential Ain EM. This potential is not a scalar but a vector.

One can easily find the amount of local curvature at two given spacetime points but that does not fully determine the time dilation factor except for the simplest spacetimes.

Curvature and gravitational time dilation are independant concepts.

Pete

 

[MeJennifer]

Pete, I did not write that gravitational time dilation is a coordinate independent phenomenon.

As to the discussion on gravitational potential in general relativity, so not in terms of Newtonian gravitation or weak field limits, but in general cases it seems we just have to agree to disagree. :)

 

[pmb_phy]

Pete, I did not write that gravitational time dilation is a coordinate independent phenomenon.

As to the discussion on gravitational potential in general relativity, so not in terms of Newtonian gravitation or weak field limits, but in general cases it seems we just have to agree to disagree. :)

Okey-dokey.

Best wishes

Pete

 

[Fredrik]

In this case the perimeter where A and B are located represents the floor, while the centre represents the ceiling. Clocks A and B both run slower than clock C.

When asked if if an observer located by clock A on the perimeter, would see clock A as running at a different rate to clock B (also located on the perimeter), the answer should also be yes, because both clocks A and B are on the floor and not moving relative to each other.

But they are moving relative to each other, in this problem. That's all you need to know.

You came to a different conclusion by using the equivalence principle to justify why the solution of a difficult problem should also be the solution of an easy problem. We know that the solution you obtained this way is wrong because it disagrees with the time dilation formula of SR, so we can be sure that you didn't use the equivalence principle correctly.

 

[Dale]

A function f(x, y, z) = xyz is a function not merely of "xyz" but is a function of x, y, z. Stating otherwise is a misuse of the phrase "is a function of"

I get your math point here, but it is a rather useless point in physics since in physics you often make new variables out of old ones. Following your example if we have a derived quantity k=xyz then it is no misuse to say f is a function of k only. After all, if something were a function of work we would not usually say that it was a function of force and distance.

As it applies to this thread it would be correct to say "gravitational time dilation is a function of gravity (g) and height (z) only" or to say "gravitational time dilation is a function of gravitational potential (gz) only" but it would not be correct to say "gravitational time dilation is a function of gravity (g) only"

I see that we agree that that a variation of g_00 can occur in both a non-inertial frame as well as a gravitational field and thus a difference in the rate at which clocks tick in such a spacetime also occurs. Is that correct?

I don't know enough GR math to know. If g_00 reduces to gz for the uniform field/acceleration case and GM/R in the Swarzschild case, then that is probably exactly what I am looking for. I have been calling these things gravitational potentials because that is what they are in classical terms, I don't know if the GR potential tensor reduces to the Newtonian concept in the appropriate limit.

 

[Fredrik]

Yes. If the speed relative to the center of the space station is v, clock #2 is moving with speed 2v/(1+v²) relative to clock #1. So he sees a time dilation corresponding to that speed.

I started thinking about this again, and I realize now that my answer is wrong.

My mistake is actually the same mistake that people usually make in the twin paradox problem. If you only consider the time dilation and ignore that the hypersurface of simultaneity gets tilted in a different direction as we go from one inertial frame to another, you get the wrong answer. I am 100% aware of this, but I still managed to make that mistake here.

This is the correct solution: In an inertial frame that's co-moving with clock #1, the event where clock #1 shows time t is simultaneous with the event where clock #2 shows t. This is true for every t, and it's easy to see if we imagine a space-time diagram. (I can't draw it very well because the world lines are spirals around the surface of the cylinder that represents the walls).

It is however, also correct to say as I did, that in an inertial frame that's co-moving with clock #1, clock #2 is ticking at a different rate because of the velocity difference. This is not wrong, but it's just half the story, just like the common mistake in the twin paradox problem. The hypersurfaces of simultaneity are rotating as #1 moves on its spiral path through space-time, so when #1 moves ahead by a small amount from time t to t+dt, the event where clock #2 shows t now has a lower time coordinate than the event where clock #1 shows t! This effect exactly cancels the time dilation!

 

[ThomasT]

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

OK. I'm just trying to untangle the semantics.

I had written in a previous post that differential aging is a consequence of different acceleration histories.

If, during a certain interval, object B is accelerated and object A isn't, then during that interval B's clock will have accumulated less time than A's -- and that difference will be directly proportional to the acceleration parameters.

Of course, if no relative velocity is observed, then no time dilation and no differential aging will be observed either.

Different relative velocities are a function of different acceleration histories.

If objects A and B have undergone exactly the same accelerations while moving away from and toward each other, then no difference in their accumulated times will be recorded even though time dilation will be observed.

If time dilation is a function of relative velocity, then time dilation with respect to objects A and B is a symmetric artifact of objects A and B moving either toward or away from each other. I think it would be correct to say that time dilation is caused by acceleration, because any instantaneous velocity is a product of some acceleration history.

To reiterate, time dilation and differential aging refer to two different things. Time dilation is a necessary, but not a sufficient, condition to produce differential aging. Differential aging requires different acceleration histories.

This is my current understanding -- any criticism is welcome.

 

[George Jones]

Suppose that clock 1 and clock 2 both experience events A and B, and that the two clocks don't necessarily traverse the same worldlines between A and B. Then, readings for the elapsed time between A and B can be compared directly.

I think it would be useful if people gave their operational definitions of the meaning of "Clock 1 ticks more slowing than clock 2." for clocks that don't share pairs of coincidence events.

I haven't waded through all the posts in this thread, so maybe this has already been done.

 

[Fredrik]

Suppose that clock 1 and clock 2 both experience events A and B, and that the two clocks don't necessarily traverse the same worldlines between A and B. Then, readings for the elapsed time between A and B can be compared directly.

I think it would be useful if people gave their operational definitions of the meaning of "Clock 1 ticks more slowly than clock 2." for clocks that don't share pairs of coincidence events.

I haven't waded through all the posts in this thread, so maybe this has already been done.

I second that request. I've been thinking the same thing, and I don't think anyone has attempted a definition in this thread. I'll start by explaining what I mean when I say that the two clocks on opposite walls of the rotating space station tick at the same rate. Pick any event on the world line of either clock. That event is simultaneous in a co-moving inertial frame with the event where the other clock shows the same time. (A space-time diagram makes this obvious). I would interpret that as "they tick at the same rate".

Suppose now that the world lines of the clocks are different. Pick an event (A) on the world line of clock 1, and find out which event (A') on the world line of clock 2 is simultaneous with A (in an inertial frame that's co-moving with 1 at A). Suppose that the clocks are set to 0 at these events. Now consider the event (B) where clock 1 shows t, and find out which event (B') on the world line of clock 2 is simultaneous with B (in an inertial frame that's co-moving with 1 at B). If clock 2 shows t' at that event, then maybe we can define relative ticking rate at A as the limit of t'/t as B goes to A. I'm not sure that this makes sense. Maybe that limit is always 1, I haven't really thought it through.

Even if this definition makes sense in SR, it clearly doesn't in GR, since there's (in general) no natural way to extend a local inertial frame to a region large enough to include a simultaneous event on the world line of the other clock. I don't think there's a way to define the relative ticking rate that makes sense in general. We seem to need a preferred coordinate system to define simultaneity. The idea of "gravitational time dilation" probably only makes sense because there is such a preferred coordinate system on a Schwarzschild space-time.

 

[yuiop]

I second that request. I've been thinking the same thing, and I don't think anyone has attempted a definition in this thread. I'll start by explaining what I mean when I say that the two clocks on opposite walls of the rotating space station tick at the same rate. Pick any event on the world line of either clock. That event is simultaneous in a co-moving inertial frame with the event where the other clock shows the same time. (A space-time diagram makes this obvious). I would interpret that as "they tick at the same rate".

Suppose now that the world lines of the clocks are different. Pick an event (A) on the world line of clock 1, and find out which event (A') on the world line of clock 2 is simultaneous with A (in an inertial frame that's co-moving with 1 at A). Suppose that the clocks are set to 0 at these events. Now consider the event (B) where clock 1 shows t, and find out which event (B') on the world line of clock 2 is simultaneous with B (in an inertial frame that's co-moving with 1 at B). If clock 2 shows t' at that event, then maybe we can define relative ticking rate at A as the limit of t'/t as B goes to A. I'm not sure that this makes sense. Maybe that limit is always 1, I haven't really thought it through.

Even if this definition makes sense in SR, it clearly doesn't in GR, since there's (in general) no natural way to extend a local inertial frame to a region large enough to include a simultaneous event on the world line of the other clock. I don't think there's a way to define the relative ticking rate that makes sense in general. We seem to need a preferred coordinate system to define simultaneity. The idea of "gravitational time dilation" probably only makes sense because there is such a preferred coordinate system on a Schwarzschild space-time.

I have an interesting thought. In Special Relativity, if one inertial observer at rest with 2 spatially separated clocks considers the two clocks to be running at the same rate then any other inertial observer with motion relative the the first observer, will also measure the same two clocks to be running at the same rate. Obviously doppler effects and light travel times have to be taken into account. It seems that even a non inertial will reeach the same conclusion even though I have not rigorously proved this. I will call this this the "clock rate conjecture" and see what other PF members make of it :) Please note that I am referring specifically to clock rates and not whether different observers consider the clocks to be synchronised, as obviously that is not true. Note also that if an observer considers one clock to be running running slower by a factor of x then an observer not at rest with the original observer will not consider the two clocks to running slower by a factor of x unless x is exactly unity.

On this conjectured basis, if the observer at the centre of the spacestation considers clocks A and B to be running at running at the same rate, then an observer located at A or B will consider clocks A and B to be running at the same rate.

 

[Fredrik]

It seems that even a non inertial will reeach the same conclusion even though I have not rigorously proved this. I will call this this the "clock rate conjecture" and see what other PF members make of it :)

The problem isn't in the proof, it's in the definition. How do you define the ticking rate of an inertial clock in an accelerating frame? (The real problem is "how do you define the accelerating frame?"). Obviously it's the number of ticks per unit of time, but in order to say that "this tick happened at time t", you have to decide which space-like hypersurfaces are simultaneous with the event at time t on the accelerating observer's world line.

There is in general no natural way to associate a global coordinate system with an accelerating observer. If the acceleration is constant, Rindler coordinates can be considered the natural choice. Look at the space-time diagram, and note the slope of the simultaneity lines. If the world line of our accelerating observer is one of the time-like curves drawn in that diagram, then clocks that are stationary in the inertial frame are ticking faster the further to the right they are. A clock at t=0 isn't ticking at all, and a clock at t<0 is actually ticking backwards.

I think the definition of "ticking rate" that I suggested agrees with the result obtained by using Rindler coordinates.

 

[yuiop]

The problem isn't in the proof, it's in the definition. How do you define the ticking rate of an inertial clock in an accelerating frame? (The real problem is "how do you define the accelerating frame?"). Obviously it's the number of ticks per unit of time, but in order to say that "this tick happened at time t", you have to decide which space-like hypersurfaces are simultaneous with the event at time t on the accelerating observer's world line.

There is in general no natural way to associate a global coordinate system with an accelerating observer. If the acceleration is constant, Rindler coordinates can be considered the natural choice. Look at the space-time diagram, and note the slope of the simultaneity lines. If the world line of our accelerating observer is one of the time-like curves drawn in that diagram, then clocks that are stationary in the inertial frame are ticking faster the further to the right they are. A clock at t=0 isn't ticking at all, and a clock at t<0 is actually ticking backwards.

I think the definition of "ticking rate" that I suggested agrees with the result obtained by using Rindler coordinates.

Certainly the clocks that are stationary in the inertial frame appear to be running at different rates to the acclerating observers upon initial examination while running at the same rate according to the the inertial observers. The reason I said "upon initial inspection" is because the clocks in the inertial frame appear to be free falling according to accelerating observers that may consider themselves to be at rest in a gravitational field. As they free fall they have motion and I think that the apparent initial difference in clock rates can be reconciled with Newtonian doppler shift due to their falling motion and apparent acceleration in the gravitational field. After taking non-relativistic doppler shift and light travel times into consideration (I think) they will consider the free falling clocks all to be running at the same rate. It will take a much better mathematician than me to prove that ;)

It is also worth noting that the accelerated observers can not actually see any light signals from the objects to the left of the origin as they are effectively behind an event horizon.

 

[Fredrik]

Certainly the clocks that are stationary in the inertial frame appear to be running at different rates to the acclerating observers upon initial examination while running at the same rate according to the the inertial observers.

If you consider that a certainty, then you must have a definition of what this statement means. Is it the definition imposed on us by the Rindler coordinates, or something else entirely?

It's immediately obvious from the space-time diagram that in a Rindler coordinate system, the two clocks tick at different rates. Just imagine the world lines of the clocks as two vertical lines, with a dot at each ticking event (e.g. every event where the second hand moves forward one discrete step). Now look at the simultaneity lines. They make it clear that at any time t>0 (t=0 is when the observer is co-moving with the clocks) on the time axis of the Rindler coordinates, the clock to the left hasn't ticked as many times as the one on the right. (I imagine the accelerating observer to be to the right of both of them).

I noticed one more interesting thing while looking at the simultaneity lines. It is also immediately obvious that the distance between the two clocks in the Rindler frame is increasing! (I think you had a different opinion in the thread about Bell's spaceship paradox).

 

[DrGreg]

I have to agree with almost everything Fredrik has said in the last few posts.

If we take the coordinates as

$$T = \frac {x}{c} \sinh \frac {at}{c}$$ ... (1)
$$X = x \cosh \frac {at}{c}$$ ... (2)​

(t, x) are the Rindler coordinates of an accelerating observer A located at $x = c^2 / a$ with proper time t and constant proper acceleration a. (T, X) are the inertial coordinates of the co-moving inertial frame I when t = 0. All events with same t coordinate are simultaneous according to the accelerating observer's co-moving frame at t.

This is compatible with Fredrik's definition of clock rate comparison in post #64 (which, by the way, I think should work for any accelerating observer in SR, not just uniform acceleration). The clock rate of I relative to A is simply $dT/dt$ subject to X being constant. (Or, if you like, $1 / (\partial t / \partial T)$.)

Calculate this by dividing (1) by (2) to obtain

$$\frac{T}{X} = \frac{1}{c} \tanh \frac {at}{c}$$ ... (3)​

Holding X constant, we obtain

$$\frac{1}{X} \frac{dT}{dt} = \frac{a}{c^2} \, sech^2 \, \frac {at}{c}$$ ... (4)​

Putting t = 0 to consider when the two clocks are relatively stationary

$$\frac{dT}{dt} = \frac{aX}{c^2}$$ ... (5)​

This is 1 when $x = X = c^2 / a$ (at t = 0), i.e. at A. This is zero when X = 0 and negative when X < 0.

 

[pmb_phy]

Dalespam - I'm trying to understand where we had a miscommunication. Please bear with me.

In your first post (post #10 in this thread) you wrote

Acceleration does not cause time dilation. This is known as the clock hypothesis and has been experimentally verified up to about 10^18 g.

Later on in post #25 you wrote

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

It appears from these comments that you were not talking about the phenomena of gravitational time dilation. Is that correct? If not then what were you referring to? Why did you bring up the clock hypothesis? Did you interpret the original question to be about something other than the phenomena of gravitational time dilation? If so then what?

I think that the mix up had to do with our different ideas of what the op was talking about. Is that your take on this?

Pete