Gradient in the rate of time vs acceleration

[synch]

(context) I can remember reading about an atomic clock that could show time running slightly differently rates at different heights, due to the differences in gravitation.
Is it realistic to think of it the other way round, ie gravity as an effect of miniscule time rate difference ?
If an object finds itself in a rate-of-time gradient, does that automatically instigate acceleration ?
(my apologies, if this shows exasperating ignorance on my part, it is a genuine question though)

 

[Nugatory]

Is it realistic to think of it the other way round, ie gravity as an effect of miniscule time rate difference ?

No, because we have a theory (General Relativity) that explains gravity without reference to the time rate difference - and then finds the time rate difference as a further consequence of gravity. To go the other way, we'd need a theory that explains the time dilation without reference to gravity and then derives gravity from there. That's not what we have.

If an object finds itself in a rate-of-time gradient, does that automatically instigate acceleration ?

That rate-of-time gradient can occur in a curved spacetime (gravity is present, no acceleration) or when accelerating in a flat spacetime (gravity is not present, acceleration is).

You might want to come at the question from a different direction... fhow curvature can produce the same effects as an attractive force (search this forum for videos done by member A.T.) and how the "equivalence principle" (google is your friend) explains gravitational time dilation.

 

[A.T.]

I can remember reading about an atomic clock that could show time running slightly differently rates at different heights, due to the differences in gravitation.

Even without differences in gravitation (uniform field) you would have different clock rates at different heights.

Is it realistic to think of it the other way round, ie gravity as an effect of miniscule time rate difference ?

Both effects are a consequence of a non-inertial frame of reference.

If an object finds itself in a rate-of-time gradient, does that automatically instigate acceleration ?

Yes, a gradient of resting clocks' rates implies coordinate acceleration for free falling objects, along that gradient (towards slower clocks).

 

[andrewkirk]

Is it realistic to think of it the other way round, ie gravity as an effect of miniscule time rate difference ?

For this one, as explained above, it is not realistic. I'll just add another reason to those given above. That is that the mathematical object that determines what we call gravity at a point in spacetime is called the Einstein Tensor, and it needs ten numbers to fully define it. By contrast, a time dilation can be defined by a single number. Hence the gravity measure contains more information than the time dilation measure, so while it is conceivable that time-dilation could be inferred from gravity (and indeed it can be), it is not even conceivable that the full measurement of gravity could be inferred from time dilation. It's like how one can infer somebody's BMI (body mass index) from their height and weight but we cannot infer height or weight from just a BMI.

If an object finds itself in a rate-of-time gradient, does that automatically instigate acceleration ?

For this one though, it's conceivable that one could interpret the question in such a way that the answer is 'yes'. But we'd need to be a bit careful about exactly what we meant by a 'rate-of-time gradient' and 'instigate acceleration', as neither of the terms has a standard, accepted meaning in the context of General Relativity.

 

[synch]

Thank you, I will read up along those lines, it is fascinating. I recently started to think of gravity as an acceleration instead of a force .which is a good sign I think !
I will keep reading, much appreciated.

 

[A.T.]

I will keep reading, much appreciated.

The relation between gravity and gravitation time dilation is visualized here:

http://demoweb.physics.ucla.edu/content/10-curved-spacetime

and here:

http://www.adamtoons.de/physics/gravitation.swf

 

[synch]

By the way I will post the thought chain/reason the question came up, for comments/interest maybe. The idea was the situation with two atomic clocks -say A and B. One is in stronger gravity than the other, so it's time is proceeding at a slightly slower rate. So, there appears to be a rate-of-time gradient between the two clocks. If an object is on a line between the two clocks, the time the object experiences towards clock A will be proceeding at a different rate than the time towards clock B...so the object's existence will be happening more towards one clock than the other. Which looks very like an acceleration (?) (still thinking / reading more)

 

[pervect]

(context) I can remember reading about an atomic clock that could show time running slightly differently rates at different heights, due to the differences in gravitation.
Is it realistic to think of it the other way round, ie gravity as an effect of miniscule time rate difference ?
If an object finds itself in a rate-of-time gradient, does that automatically instigate acceleration ?
(my apologies, if this shows exasperating ignorance on my part, it is a genuine question though)

I would suspect from the nature of the question that you are envisioning 'the rate of time" as some distortion of clocks compared to some sort of underlying "absolute time". The basic problem is that there isn't any such thing as an absolute time :(. Of course, this raises the issue - explaining what we mean by "absolute time", so it's not just word soup. And also (and not incidentally) explaining why it doesn't work.

There are two different things that might help. The first is for you to explain how you think "time rate differences" are measured. What is the "time rate" different from? I would assume you think it's different from something. My guess is that something is what I call "absolute time". But perhaps I'm wrong, and you mean something else. IF so you can explain. Even if you can't explain, thinking about the question might be helpful.

The other thing that can potentially help is to look at "Einstein's train" thought experiment. If you do believe in "absolute time", this thought experiment will illustrate the difficulties with this belief.

It's possible that "Einstein's train" isn't familiar to you, so I'll give a link to Einstein's original description of the thought experiment. http://www.bartleby.com/173/9.html. You can also find more discussions under the keywords "Einstein's train" and "Relativity of Simultaneity". There's quite a lot written about the topic, but it's a very common stumbling block.

 

[synch]

No, I am not thinking of absolute time., or a distortion. The rate difference would show simply by taking two synchronised clocks, and moving one to a high gravity environment for say a year, then moving it back and comparing the two. I think a similar experiment was done for the effect of velocity, by flying one clock in a jet for an extended period and then observing the slight difference.
Overall it becomes a philosophical question. I can see though, even if there is a rate difference, the fact that events depend on the time would automatically compensate exactly so there could be no known effect. Certainly not enough for my original question in any case, though it was an interesting track to follow.

 

[synch]

>> ".. atomic clock that could show time running slightly differently rates at different heights, due to the differences in gravitation"

I am assuming, that if two such clocks were running close by, one on top of a stack of lead such that it was in a stronger gravity, that they would exhibit different times with the difference increasing at a steady rate. Maybe that is incorrect ?

 

[PeterDonis]

The rate difference would show simply by taking two synchronised clocks, and moving one to a high gravity environment for say a year, then moving it back and comparing the two.

In other words: you take two clocks that start out at one event in spacetime; they follow different paths through spacetime, but end up back together again at some other event. Their elapsed times are different. Yes, this has been experimentally confirmed (the first time with atomic clocks was the Hafele-Keating experiment).

I am assuming, that if two such clocks were running close by, one on top of a stack of lead such that it was in a stronger gravity, that they would exhibit different times with the difference increasing at a steady rate.

If the clocks are "close by", they are experiencing the same gravity. In order to get a difference, they have to separate and then come back together, as described above. They can't do that and remain "close by".

If the clocks separate and then move back together, you can measure their time difference at the end, when they come back together, but there is no way to assign a "rate of increase" to their time difference while they are separated.

 

[pervect]

No, I am not thinking of absolute time., or a distortion. The rate difference would show simply by taking two synchronised clocks, and moving one to a high gravity environment for say a year, then moving it back and comparing the two. I think a similar experiment was done for the effect of velocity, by flying one clock in a jet for an extended period and then observing the slight difference.
Overall it becomes a philosophical question. I can see though, even if there is a rate difference, the fact that events depend on the time would automatically compensate exactly so there could be no known effect. Certainly not enough for my original question in any case, though it was an interesting track to follow.

OK, I think I can work with this. The key assumption here is that you have some way of synchronizing clocks. And I believe it's also implied in your thinking that you are eliminating the notion of time dilation caused by velocity by spending much more time "at rest" than "moving". Which means that you are assuming we have a concept of "not moving" or "at rest". This is also something that needs to be defined.

To my way of thinking, what you are essentially doing is sufficient to define most of a coordinate system, where the time coordinate is given by a number assigned to all synchronized clocks, and the space-coordinate is constant for objects "at rest". Some details are missing, though, for instance how exactly to assign a specific number to a set of synchronized clocks, the number that will be the "time coordinate". It's usual to pick one particular clock (wordline) in the set and use that particular clocks proper time to assign the specific coordinate number for time, but one needs to single out that particular worldline (usually the origin of the coordinate system) for a complete definition.

Given this framework, then, I would identify your notion of "time rate dilation" with one component of the metric tensor, usually call $g_{00}$ and/or $g_{tt}$. If we further assume that "gravity" is the proper acceleration of an object "at rest", I believe we can say the answer is "yes" under some circumstances. But I'm not quite sure what circumstances are sufficient and necessary offhand. The relation that holds sometimes is that the proper acceleration is proportional to the spatial gradient of $g_{00}$, i.e. the x-component of the acceleration would be $\partial g_{tt} / \partial x$. Sorry I can't be more definite at this point, perhaps someone else can be. I was working on being more definite, but it got too involved for a short post.

A meta-point here though is that this view is breaking up the mathematically unified space-time into separate "space" and "time". This is inherently an observer-dependent process, so we can't escape a lot of definitions of how we perform this breakup in attempting this process. You are also focussing in fine on one particular component of the metric tensor by this approach - and ignoring all the other components, which are necessary for a complete understanding.

Another related issue is that the framework of "forces on test particles" isn't sufficient to describe everything that's going on according to GR. Effects on spatial geometry aren't covered conceptually by this framework, nor are some of the gravitomagnetic effects (such as differences of trajectory of spinning vs non-spinning test particles, for instance).

 

[A.T.]

One is in stronger gravity than the other, so it's time is proceeding at a slightly slower rate.

The clock rates don't depend on the local strength of gravity, but their relative positions along gravity. Even when both clocks experience the same gravity strength, the lower will tick slower.

If an object is on a line between the two clocks, the time the object experiences towards clock A will be proceeding at a different rate than the time towards clock B...so the object's existence will be happening more towards one clock than the other. Which looks very like an acceleration (?) (still thinking / reading more)

To me, the simplest way to understand the geometric relationship is given in the link in post #6.

 

[pervect]

The clock rates don't depend on the local strength of gravity, but their relative positions along gravity. Even when both clocks experience the same gravity strength, the lower will tick slower.

To me, the simplest way to understand the geometric relationship is given in the link in post #6.

I like those diagrams as well. My observation though is they're only helpful to those interested readers who can draw and understand a space-time diagram. Unfortunately, I see a lot of resistance in getting readers to draw and understanding space-time diagrams, and it seems to be difficult to write useful words about it, words that are simpler than the diagrams. I do help feeling that there should be some words that are needed/helpful, because I'[ve seen my share of diagrams that I scratch my head looking at wondering what they are diagram of, exactly. But I'm still not sure what words are helpful in getting across the concept needed for a space-time diagram. It's not terribly hard, or shouldn't be. It's just a graph of position versus time, really. But it seems to be a stumbling block, and I am not sure if there is a way around it.

It's also not quite answering the OP's question directly, though it's definitely worth pointing out that there is a "better approach". One can hope the reader will appreciate that the approach is better, but sometimes they don't, or are interested in exploring their own ideas and want to see how they play out. Sometimes - almost always - it's useful to have multiple ways of understanding something.

So - It's not totally clear what the direct answer to the question is, but I'm leaning towards the following. If one has a static / stationary space-time (one whose geometry is "the same" as time progresses), and it's not rotating, I think the explanation works reasonably well for what it does. As we've remarked, it only focuses on those aspects of gravity that are modeled as a force. It's worth pointing out that this is not all of gravity.

Note that I don't have a mathematical proof of this answer, nor a textbook reference - so take it with a grain or two of salt, please. My basic thinking though is that if you have a stationary space-time, one can define a potential, and the gradient of that potential is the force, and in this case I believe it's related to "time dilation". Some specific examples that meet this criterion - Rindler space-time (which is flat space-times of "Einstein's elevator") and the Schwarzschild space-time.

I can see some definite issues in rotating space-times. It is assumed (at least in my interpretation of answering the question) that one can synchronize clocks that are "at rest", but this turns out not to be possible in rotating space-times if one uses the Einstein definition of synchronization. Perhaps the OP is using some different definition of synchronization, in which case we'd need a whole set of posts to straighten out just what we're both talking about so we're all talking about the same thing. A practical example of a "rotating space-time" - the Kerr space-time of a rotating black hole.

As far as the argument about space time being stationary (unchanging). The operational procedure given for determining "time dilation" relies on the geometry of space-time not changing. If the space-time geometry is evolving, it's a lot less clear how one can even define "time dilation" in the sense that the OP wants to define it. Basically one has to ask "what is the time dilation NOW". But we run again into the issue about a shared understanding of clock synchronization.