Does acceleration cause time dilation?

[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

\Phi = -(g_{00} - 1)c^2/2[/tex]<br /> <br /> 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[/color] 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

 

[Dale]

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?

probably. In addition, there was some extra confusion from two separate threads getting mixed together. The first quote of mine in your previous message was in response to a purely SR question.

Also, I am afraid that you misinterpreted my use of the word "gravity" where I was specifically referring to the acceleration measured by an accelerometer in a gravitational field (g). I think you interpreted my comments more generally than I intended. I was only saying that the thing which is measured by an accelerometer in does not by itself cause time dilation whether we are talking about SR (a) or GR (g)

 

[DrGreg]

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.

In GR, two possible methods occur to me.

1. Equate gravitational time dilation with gravitational red shift. If two observers remain a constant distance apart, they can exchange light signals and measure doppler shifts. As their separation is constant, any doppler shift must be attributed to a difference in clock rates. If the two observers make consistent measurements of each other (i.e. the red shift measured by one is equivalent to the blue shift measured by the other) we could consider that shift to determine relative clock rates.

2. Suppose we have a coordinate system already established, with a known metric d\tau^2 = g_{\mu\nu}\, dx^{\mu} \,dx^{\nu}, a timelike coordinate x^0 = t and three spacelike coordinates, such that our two observers A and B each lie at constant space coordinates. Then we can calculate d\tau_A / dt and d\tau_B / dt along the worldlines of A and B, and consider the relative clock rate between A and B to be the ratio of these numbers. This equates to the ratio of the two values of \sqrt{g_{00}} at A and B.

The question is, are these two techniques compatible with each other or with Fredrik's method (which, in my view, undoubtably makes sense in SR)?

I am still very much a beginner in GR, but a book I possess* seems to indicate that the above techniques are compatible in the case of what is called a "stationary spacetime", which, roughly speaking, means that the metric is constant over time, i.e. there are no "moving gravitational sources", and where the observers are "stationary relative to the source(s)".

*Rindler, W. (2006 2nd ed), Relativity: Special, General and Cosmological, Oxford University Press, Oxford, ISBN 978-0-19-856732-5.

 

[Mentz114]

Hi DrGreg,
from my understanding,

1) frequency shifts can be attributed to clock rates of sender and receiver. This is true in SR and GR.

2) Restricting the observers to have stationary spatial coordinates is tricky, because they may not be freely falling. If they are under external force ( like rockets) this may affect the clock rate.

I think (2) is true for freely falling worldlines where the observers are momentarily at rest wrt to each other.

M

 

[Fredrik]

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).

I actually made a mistake in my definition of the relative ticking rate, but maybe you looked past it and interpreted what I said as what I should have said.

My mistake was to use two different co-moving coordinate systems instead of just one. The simultaneity lines I used at events A and B aren't parallel and that means that t' as I defined it will be negative if the clocks are far apart. As B goes to A, the lines will become parallel, but if t' is negative, then t'/t will be negative even in the limit when B goes to A.

This is what I should have said in #64:

I'm going to define the relative ticking rate of clock 2 from the point of view of clock 1 at an event on the world-line of clock 1. Let x be an inertial frame that's co-moving with clock 1 at event A and let B be a later event on the world-line of clock 1. Let's call the events on the world line of clock 2 that are simultaneous (in x) with these events A' and B' respectively. Now define the relative ticking rate as

\lim_{B\rightarrow A}\frac{x^0(B&#039;)-x^0(A&#039;)}{x^0(B)-x^0(A)}​

I think this will work in SR no matter what the two world lines look like.

 

[Mentz114]

I'm going to define the relative ticking rate of clock 2 from the point of view of clock 1 at an event on the world-line of clock 1. Let x be an inertial frame that's co-moving with clock 1 at event A and let B be a later event on the world-line of clock 1. Let's call the events on the world line of clock 2 that are simultaneous (in x) with these events A' and B' respectively.

Fredrik,
I'm trying to draw a space-time diagram of this and I find A and A' are the same event. Any chance of a diagram ?

[Edit] OK, I got it. Your definition looks like \frac{d\tau}{dt} where \tau is the proper time of the accelerating frame.

M

 

[DrGreg]

I actually made a mistake in my definition of the relative ticking rate, but maybe you looked past it and interpreted what I said as what I should have said.

My mistake was to use two different co-moving coordinate systems instead of just one. The simultaneity lines I used at events A and B aren't parallel and that means that t' as I defined it will be negative if the clocks are far apart. As B goes to A, the lines will become parallel, but if t' is negative, then t'/t will be negative even in the limit when B goes to A.

This is what I should have said in #64:

I'm going to define the relative ticking rate of clock 2 from the point of view of clock 1 at an event on the world-line of clock 1. Let x be an inertial frame that's co-moving with clock 1 at event A and let B be a later event on the world-line of clock 1. Let's call the events on the world line of clock 2 that are simultaneous (in x) with these events A' and B' respectively. Now define the relative ticking rate as

\lim_{B\rightarrow A}\frac{x^0(B&#039;)-x^0(A&#039;)}{x^0(B)-x^0(A)}​

I think this will work in SR no matter what the two world lines look like.

I still think the definition you gave in post #64 is a sensible definition, and it is that definition that is used in Rindler coordinates in posts #66, #68 and #69. The fact that the dilation factor can be zero or even negative is just the way things are.

Your new definition in post #74 is another way of looking at the problem but gives a different answer to post #64. Actually the way you've written it, with the same coordinate x⁰ in both numerator and denominator, makes little sense, as the answer would always be 1 (exactly, even before taking the limit). I assume that you really intended to put the proper time measured by clock 2 in the numerator and the proper time measured by clock 1 in the denominator. The rest of this post is on that assumption.

Method #74 (i.e. my corrected version) ignores the acceleration of clock 1. (And, I have just realized, I think both methods ignore the acceleration of clock 2, which is an asymmetry in the original method #64). Any clock moving through event A with the same momentary velocity as clock 1 would calculate the same dilation as clock 1 by method #74 -- in fact it would (in SR) be just the standard Lorentz factor \gamma.

I haven't fully grasped the details of "parallel transport" in GR, but I think method #64 seems to use parallel transport and method #74 doesn't. Or to put it another way, does #64 use covariant differentiation and #74 use coordinate differentiation? (Or the other way round??)

Which leaves me even more confused than ever, as now I'm not sure which of your two methods (if any) is equivalent to either of the two methods I suggested in post #72.

 

[Fredrik]

Actually the way you've written it, with the same coordinate x⁰ in both numerator and denominator, makes little sense, as the answer would always be 1 (exactly, even before taking the limit). I assume that you really intended to put the proper time measured by clock 2 in the numerator and the proper time measured by clock 1 in the denominator.

I actually meant it as I wrote it, but you're right. That makes no sense. Maybe I got it right in #64, but it still bothers me that the result can be negative. (I'm not sure why though. I don't have a mathematical reason). Maybe the definition only makes sense as long as the result is positive. It would be a bit like how the Rindler coordinates only makes sense for x>0. (I'm just speculating now because I don't have much time to think this through today).

 

[pmb_phy]

probably.

For future reference you should take note of the fact that the term gravitational time dilation does not have the meaning that you appear to think it has. Its definition is given here

http://en.wikipedia.org/wiki/Gravitational_time_dilation

Gravitational time dilation refers to the fact that time passes at different rates in regions of different gravitational potential. I.e. it is identical to gravitational redshift. It does not refer to the rate at which one clock runs. In fact time dilation has never meant that in any context. Time dilation is always about comparing the rates at which otherwise identical clocks tick.

Pete

 

[DrGreg]

I actually meant it as I wrote it, but you're right. That makes no sense. Maybe I got it right in #64, but it still bothers me that the result can be negative. (I'm not sure why though. I don't have a mathematical reason). Maybe the definition only makes sense as long as the result is positive. It would be a bit like how the Rindler coordinates only makes sense for x>0. (I'm just speculating now because I don't have much time to think this through today).

In the case of Rindler coords, two observers, one with constant positive Rindler x-coord, the other constant negative x-coord, (or one of them at x = 0), it is impossible for either to send light signals to the other -- they are separated by an "event horizon". This should be pretty obvious from a spacetime diagram. So a negative or zero dilation according to definition #64 means, in this case, an event horizon, so there is no doppler shift to measure!

(Rindler coordinates do still make sense when x < 0, it's just that the Rindler t coordinate runs backwards relative to the proper time of a Rindler-stationary observer. Remember the t coordinate is synchronised to an observer at x = c²/a. There is, of course, a singularity at x = 0, where nothing makes sense in Rindler coords -- but nothing unusual there in inertial coords.)

For what it's worth, I did a back-of-envelope calculation last night and managed to persuade myself that methods #64, #72(1) and #72(2) all give the same answer for a pair of Rindler observers each at a constant positive Rindler x-coord, namely the ratio of their x-coords. (Method #74, reinterpreted by me #76, would give 1.)

 

[rcgldr]

I think the OP just wanted a simple yes or no, still it's an interesting thread.

As a more general question, two there are two identical clocks, both resting on scales. Clock #1 experiences 1.0g as an unlown combination of gravity and acceleration. Clock#2 experiences 0.0 g as an unlown combination of gravity and acceleration. The velocity of the clocks have identical magnitude and direction (they're moving in parallel at the same speed at all time, even if the speed is varying, don't worry about trying to figure out an actual example, I'm sure it's complicated but possible, with circular paths). It is observed that clock #1 has a slower rate than clock #2.

Does it make any difference in the relative rates of each clock what the ratio of gravity versus acceleration is? For example, if clock #1's 1g is due to all gravity or due to all acceleration, while clock #2 is experiencing neither. A simple yes or no is good enough here, but feel free to continue with all the math stuff.

 

[Fredrik]

...two identical clocks, both resting on scales. Clock #1 experiences 1.0g as an unlown combination of gravity and acceleration. Clock#2 experiences 0.0 g as an unlown combination of gravity and acceleration. The velocity of the clocks have identical magnitude and direction (they're moving in parallel at the same speed at all time, even if the speed is varying, don't worry about trying to figure out an actual example, I'm sure it's complicated but possible, with circular paths).

I would be very surprised if that turns out to be possible, and even more surprised if it's possible with circular paths. How do you make gravity pull you away from the center at every point on circle? Also, I don't know if it can ever make sense to describe the two world lines as "parallel". In general, you can't say that curve 1 at event A is parallel to curve 2 at event A' because there's no path-independent way to compare the tangent vectors of the two curves. (You want to take the tangent vector of 1 at A and compare it to the tangent vector of 2 at A', but those vectors are members of different vector spaces, so you need a way to identify the two vector spaces before you can compare the vectors. The metric suggests a way to do that, but the identification is different for different paths from A to A').

Does it make any difference in the relative rates of each clock what the ratio of gravity versus acceleration is?

I don't think there's a way to make sense of the question in the (probably impossible) scenario you described, but if we go back to the two clocks attached to the floor and ceiling of a small box with no windows, their relative ticking rates will not depend on what mix of gravity and acceleration the box is in. It only depends on the "force" felt inside the box.

I haven't done any calculations to verify that this is what GR predicts, but it shouldn't be necessary since we know that GR was constructed with the explicit goal to make sure that this is true.

 

[DrGreg]

Does it make any difference in the relative rates of each clock what the ratio of gravity versus acceleration is? For example, if clock #1's 1g is due to all gravity or due to all acceleration, while clock #2 is experiencing neither. A simple yes or no is good enough here, but feel free to continue with all the math stuff.

The answer is definitely no. No maths are necessary. It's the principle of equivalence on which the whole of GR is based. "Proper acceleration" and "gravity" are postulated to be indistinguishable.

The particular scenario is possible in GR: put #1 on the surface of the Earth at the north pole and #2 at the centre (extremely difficult in practice but conceptually possible).

Another example that has definitely been achieved: put #1 on the surface of the Earth on the equator and put #2 in synchronous equatorial orbit around the Earth (i.e. one revolution per 24 hours).

(When I say "gravity" above, I use the word loosely to refer to the Newtonian concept. In GR gravity means the same thing as the proper acceleration of a frame.)

The velocity of the clocks have identical magnitude and direction (they're moving in parallel at the same speed at all time, even if the speed is varying)

Note that the precise meaning of this depends on which frame you measure with. My answer above assumes they are each stationary relative to some common frame of reference (which could be an accelerating frame). If, instead, a single inertial observer is measuring the velocities to be equal at all times you could be in the Bell's paradox scenario, so there will be additional time dilation due to relative motion.

 

[rcgldr]

How do you make gravity pull you away from the center at every point on circle?

It would be simpler to have gravity pull you towards the center at every point on a circle. I didn't state what direction the acceleration or pull of gravity was, just that the total effect would be equivalent to 1 g.

 

[Dale]

Gravitational time dilation refers to the fact that time passes at different rates in regions of different gravitational potential. I.e. it is identical to gravitational redshift.

(emphasis added)

That is all I was saying. It is different gravitational potential that is important for gravitational time dilation, not different gravity (g).

 

[yuiop]

Does it make any difference in the relative rates of each clock what the ratio of gravity versus acceleration is? For example, if clock #1's 1g is due to all gravity or due to all acceleration, while clock #2 is experiencing neither. A simple yes or no is good enough here, but feel free to continue with all the math stuff.

It does make a difference and the difference is measurable. The question "Does acceleration cause time dilation" is usually asked in the context of the twin's paradox and the answer is no. To suggest that gravitational time dilation is caused by gravitational acceleration rather than gravitational potential would cause you to get the wrong answer when trying to calculate what happens to clock rates in some situations. For example if you assumed time dilation is caused by acceleration, then knowing that there is no acceleration due to gravity inside a hollow massive sphere would cause you to calculate that there is no time dilation at the centre of a massive sphere and you would be wrong. The potential inside a hollow sphere is non zero and in fact the time dilation is greater inside the sphere than at the surface. Another example is that assuming acceleration causes time dilation would cause you to calculate the wrong value for the time dilation of a clock on the perimter of a rotating turntable. It is important to make the distinction and to be clear about the distinction if you want to able to do correct calculations. The distinction is not merely philosophical.

So the answer to the OP is a resounding "NO".

Dalespam and myself are clear about that.

 

[DrGreg]

Another example is that assuming acceleration causes time dilation would cause you to calculate the wrong value for the time dilation of a clock on the perimter of a rotating turntable.

This is one of those "it depends what you mean by..." questions and this example's a good illustration.

Inertial observer A at the centre of the turntable sees observer B moving round the circumference of the turntable and attributes the dilation between them due to the velocity of B relative to A. B's acceleration is irrelevant to A.

B sees A as being stationary relative to B's rotating frame of reference and attributes dilation to B's proper acceleration.

In the context of SR you can say A is "right" and B is "wrong". In the context of GR you can't say that: they are both equally correct.