How Does Size Affect Time in Thought Experiments?

[akshay848]

hi people!

please can anyone explain me the conclusions of a thought experiment given below ?

consider yourself standing straight with both nyour hands wide apart horizontal to the floor on the surface of the Earth wearing a stop watch on your wrist, now bring your hands together start the stop watch and go back to theninitial position and back together and stop then stop watch. then time recorded on your stop watch is say 2 seconds. say the diastance covered by your hands in this motion is 4 meterrs now imagine you expanding and growing bigger and bigger, imagine you being bigger than our galaxy. now repeat the experiment, it will again take 2 seconds to perform the same action but now the distance covered is a few hundred light years. does nthat mean you started moving faster than light ? or the time changes for different sizes ?

 

[PeterDonis]

imagine you being bigger than our galaxy. now repeat the experiment, it will again take 2 seconds to perform the same action

No, it won't; if your arms are each 10,000 light years long, then it will take at least 10,000 years for the information that you have started to move your arms to propagate to the ends of your arms. Internal forces in objects, which is what causes the ends of your arms to move when you move your arms, can't propagate faster than light.

(I say "at least 10,000 years" because most internal forces in objects propagate a lot *slower* than light. In your body they probably propagate at a speed something like the speed of sound in a solid, which is a few thousand meters per second, so if your arm is a meter long it takes less than a millisecond for internal forces to propagate from one end of your arm to the other. That explains why, even though this propagation speed is finite, you don't perceive it that way: it takes tens of milliseconds for nerve impulses to travel within your body, and a hundred milliseconds or so for you to become conscious of changed input, so the movement of your arms appears to you to happen instantly. That wouldn't be the case if your arms were 10,000 light years long. )

 

[Vanadium 50]

If your arms as as big as a galaxy, you can't clap your hands in 2 seconds.

 

[KGH]

Regarding

imagine you expanding and growing bigger and bigger

Imagine that you have a companion who does not grow with you. Imagine also that you are wearing a wristwatch that counts time based on how many times a pulse of light bounces back and forth between two mirrors. As you and your watch get bigger, the space between the mirrors widens. Your watch slows down. So do all the other physical and mental processes associated with "you" as "you" grow bigger. So, according to your watch (and indeed by any instrument you use to measure time), the time it takes to clap your hands may not have changed, but your companion will find that you are clapping slower and slower as you grow.

This is more or less what is going on in gravitational time dilation, as far as I understand it. You and your companion start off far away from a source of gravity, and you then move toward it, perhaps coming to rest on the surface. The "space" part of spacetime is stretched so that everything you (and your watch) do takes longer. You measure time to run at a different rate for you on the surface than for your companion far away.

 

[PeterDonis]

Imagine that you have a companion who does not grow with you. Imagine also that you are wearing a wristwatch that counts time based on how many times a pulse of light bounces back and forth between two mirrors. As you and your watch get bigger, the space between the mirrors widens. Your watch slows down. So do all the other physical and mental processes associated with "you" as "you" grow bigger. So, according to your watch (and indeed by any instrument you use to measure time), the time it takes to clap your hands may not have changed, but your companion will find that you are clapping slower and slower as you grow.

This is all fine as far as it goes (except that I'm a little hazy on what laws of physics your body is obeying in order for your expansion to slow down *all* of your physical processes in exact sync with your light clock, but it's just a thought experiment so I'm willing to let that pass). But note that it means that if your companion has his own light clock, bouncing light back and forth between his own pair of mirrors, then it's easy to tell that you're growing and he's not by measuring the distance between the mirrors. The distance between your pair of mirrors grows; the distance between his pair does not. This is important when you compare this scenario with gravitational time dilation; see below.

This is more or less what is going on in gravitational time dilation, as far as I understand it. You and your companion start off far away from a source of gravity, and you then move toward it, perhaps coming to rest on the surface. The "space" part of spacetime is stretched so that everything you (and your watch) do takes longer.

No, this is not correct. The way you can tell it's not correct is this: if you both have light clocks, bouncing light between pairs of mirrors, and you both measure the distance between your respective pairs of mirrors, you get the same answer--unlike the above "growing" scenario, where you don't. So whatever is going on in gravitational time dilation, it can't be explained by a simple "stretching of space" this way.

You may ask, but if your light clocks both have the same distance between their respective pairs of mirrors, how can they tick at different rates? What does that even mean? The answer is this: suppose you and your companion exchange light signals between each other (not the same ones that are bouncing around inside each of your light clocks), and you each count how many ticks of your respective clocks elapse during one round trip of a light signal between you. You will find that your clock registers *fewer* ticks elapsed during each round trip than your companion's clock. That's what "gravitational time dilation" means.

 

[KGH]

I acknowledge my transgression of the forum rules in having included a link to an external document of mine in my original post (now redacted). I did so in the interest of not including its full text here. What I had also said is that the question, "Is time size dependent?" has been very useful to me in forming a framework by which I can more easily understand currently-accepted theory. I am not espousing an alternate theory here, just a different way of talking about what is shown mathematically by Einstein, Schwarzschild, Wheeler, et al.

No, this is not correct. The way you can tell it's not correct is this: if you both have light clocks, bouncing light between pairs of mirrors, and you both measure the distance between your respective pairs of mirrors, you get the same answer--unlike the above "growing" scenario, where you don't. So whatever is going on in gravitational time dilation, it can't be explained by a simple "stretching of space" this way.

My objection to this objection is that in order for one observer to make direct measurements of the separation of the other's mirrors, that observer must be co-located with the mirrors he is measuring. This would make the measurement meaningless if the point of the measurement is to determine whether the size of the clock changes due to the gravitational field.

However, after being co-located and verifying that the two clocks are of identical size, either observer may use a presumably constant speed of light between the other's mirrors (indirectly measurable in the rate of the other's clock) to make an inference about the distance between them. As the observers separate and the stationary observer notices the rate of the other's clock changing, it would not seem incorrect to conclude that the distance between the mirrors is changing accordingly.

To me this seems precisely what the Schwarzschild metric is saying regarding the stretching of space, though I do welcome further criticism of this interpretation.

If permissible, I would like to mention that I was quite excited this year to find a perspective very similar to mine in Lee Smolin's Time Reborn; Smolin writes that the slowing of time and the stretching of space are functionally equivalent. I find the latter idea more easy to get a grip on, and if it is not incorrect, I'd like to develop it further for use in teaching. Of course, if it is incorrect, I would like to understand how and why.

 

[PeterDonis]

My objection to this objection is that in order for one observer to make direct measurements of the separation of the other's mirrors, that observer must be co-located with the mirrors he is measuring.

There are lots of ways around this. For example: set up a long ruler with detectors on it that get triggered when a mirror passes them. Then set up a communications network that collects the data from the detectors. The "growing" observer in your first scenario will find different detectors being triggered as he grows, because his mirrors are moving along the ruler. The other observer will not. But in the gravitational case, both observers see the same detectors being triggered for all time--neither set of mirrors is moving relative to the ruler and detectors set up by the observer who is using that set of mirrors as a clock.

However, after being co-located and verifying that the two clocks are of identical size, either observer may use a presumably constant speed of light between the other's mirrors (indirectly measurable in the rate of the other's clock) to make an inference about the distance between them.

No, this won't work either, because in curved spacetime you can't assume that the speed of light is the same everywhere. More precisely, you can't assume that the speed of light that is spatially distant from you is the same, measured in your coordinates, as the speed of light that is at your spatial location.

As the observers separate and the stationary observer notices the rate of the other's clock changing, it would not seem incorrect to conclude that the distance between the mirrors is changing accordingly.

Another objection to this is that in the gravitational case, this would give a different answer than a local measurement, such as the ruler-and-detector setup I mentioned above.

Perhaps it's worth expanding on this in some detail. Call the distant observer (the one far away from the gravitating body) O and the observer who is deep in the gravity well D. Both O and D have ruler-detector setups arranged to measure the separation between their mirrors; essentially this measurement outputs the distance between the mirrors, as marked off on the ruler. Then O and D can verify the following:

(1) The distance between both of their mirrors is the same, as verified by their respective ruler-detector setups;

(2) If O and D exchange round-trip light signals, D's clock ticks off fewer ticks between two successive signals than O's does.

In other words, D's clock is running slower even though it is *the same size* as O's clock, as verified by the mirror-detector setups. So interpreting D's clock running slower as his mirrors being further apart is not consistent with the results of the mirror-detector measurements.

Smolin writes that the slowing of time and the stretching of space are functionally equivalent.

I haven't read the book, so I can't comment on this. If you can find a presentation of his argument somewhere online, by all means link to it.

 

[KGH]

Thanks, this is is a good discussion for me because I am learning what I trust is the more common viewpoint while the need to explain my own is helping me clarify it in my own mind.

There are lots of ways around this. For example: set up a long ruler with detectors on it that get triggered when a mirror passes them.

This presumes that a one-meter interval at one end of the ruler is equal to a one-foot interval on the other end when the ends are at different radii from the source of gravitation. In this way, it is similar to the problem I described earlier of knowing whether the light clocks are "really" the same size once they are moved to different locations. You might say that they are the same size but that the underlying metric of spacetime has a different value, causing light to move at a different speed between the mirrors. I might say that is true, but that it could just as truly be said that light moves at the same speed in both locales but that the sizes of the two clocks are different. The underlying truth in both of these viewpoints would be the varying value of the metric (for example, at different radii from a spherically symmetrical large mass).

To reiterate: Supposing that objects of identical manufacture remain the same size when separated, and saying that

you can't assume that the speed of light is the same everywhere. More precisely, you can't assume that the speed of light that is spatially distant from you is the same, measured in your coordinates, as the speed of light that is at your spatial location.

describes a situation which I understand to be semantically different from but mathematically the same as to say that the speed of light is everywhere constant but that "you can't assume that the size of identically manufactured objects are the same everywhere. More precisely, you can't assume that the size of such an object which is spatially distant from you is the same, measured in your coordinates, as such an object that is at your spatial location." This, including the light clock example, is almost exactly how Smolin describes something called "shape dynamics" in his book and in the resources I describe below. I must confess that he describes "shape dynamics" as an alternative theory to GR rather than an alternate expression of it, and if that is true I may be guilty of again violating forum rules. I am not sure I agree with him in that regard, however.

I haven't read the book, so I can't comment on this. If you can find a presentation of his argument somewhere online, by all means link to it.

A Google search for "Lee Smolin shape dynamics" will return a couple of articles which review the book. The LA Review of Books review contains an excerpt having the key phrase "shape dynamics" which is most relevant to our discussion; and Peter Woit's blog review of Smolin's book has several comments added by Smolin himself. The back-and-forth comments between Smolin and Chris Kennedy are on this topic; again search for that same key phrase. I hesitate to include the actual links because as I look more closely at Smolin's exposition, it doesn't seem to describe ideas as widely accepted as I thought.

 

[PeterDonis]

This presumes that a one-meter interval at one end of the ruler is equal to a one-foot interval on the other end when the ends are at different radii from the source of gravitation.

I don't understand; why do you think this? If both rulers are constructed identically, a one-meter interval on one is equal to a one-meter interval on the other.

In this way, it is similar to the problem I described earlier of knowing whether the light clocks are "really" the same size once they are moved to different locations.

This question is not well-defined as it stands; you need to define what you mean by "really", i.e., you need to specify exactly what measurements you would make to tell whether they are "really" the same size.

You might say that they are the same size but that the underlying metric of spacetime has a different value

This makes no sense; the metric of spacetime determines the "size" of things. More precisely, the metric of spacetime determines the physical interval corresponding to a given coordinate interval. I suspect that you are (perhaps without realizing it) putting a physical interpretation on coordinate intervals directly, instead of using the metric to determine the physical meaning of coordinate intervals.

I might say that is true, but that it could just as truly be said that light moves at the same speed in both locales but that the sizes of the two clocks are different. The underlying truth in both of these viewpoints would be the varying value of the metric (for example, at different radii from a spherically symmetrical large mass).

But what, exactly, is "varying"? Remember that specific metric coefficients are coordinate-dependent; you can change coordinate charts and that will change the metric coefficients. For example, in the Painleve coordinate chart, the coefficient of $dr^2$ is 1, indicating that there is no radial variation in the metric in this chart.

Supposing that objects of identical manufacture remain the same size when separated

More precisely, suppose that if I take two objects of identical manufacture, which are identical when placed next to each other, if we take one and move it somewhere else, it will still be the same size as the other, provided we do the movement slowly enough (so that the structure of the object remains intact).

you can't assume that the size of such an object which is spatially distant from you is the same, measured in your coordinates, as such an object that is at your spatial location.

This is true, but as I noted above, what counts is the size of the object as measured by the metric, not by coordinates. For example, if two rulers are identical when they are far out in empty space, they will both occupy the same coordinate length $\Delta r$ if placed radially. Since both rulers are far out in empty space, their physical length will be the same as their coordinate length, i.e., $\Delta r$.

If I then move one ruler deep into a gravity well, it will occupy a *different* coordinate length $\Delta R < \Delta r$; but its *physical* length will be $\Delta R / (1 - 2m / r) = \Delta r$, i.e., the *same* physical length.

I must confess that he describes "shape dynamics" as an alternative theory to GR rather than an alternate expression of it

Based on what you've described here so far, I think this must be the case, because in standard GR, the physical interpretation of coordinate lengths is as I described it above.

A Google search for "Lee Smolin shape dynamics" will return a couple of articles which review the book. The LA Review of Books review contains an excerpt having the key phrase "shape dynamics" which is most relevant to our discussion; and Peter Woit's blog review of Smolin's book has several comments added by Smolin himself. The back-and-forth comments between Smolin and Chris Kennedy are on this topic; again search for that same key phrase.

I'll look these up when I get a chance.