Does acceleration cause time dilation?

[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

$$\lim_{B\rightarrow A}\frac{x^0(B')-x^0(A')}{x^0(B)-x^0(A)}$$​

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

$$\lim_{B\rightarrow A}\frac{x^0(B')-x^0(A')}{x^0(B)-x^0(A)}$$​

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.