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Author Topic: What is the time dilation at the gravitational center of a planet or star?  (Read 7686 times)

Offline jeroboambramblejam

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Since there is no gravitational gradient ( acceleration), one might expect time dilation (or the Gravitational Red Shift for that matter) to be similar to what it is at some great distance from the planet or star;  but there is actually equal and opposite force acting on (actually, surrounding) the centered mass, so it could be said to be in a more intense gravitational 'field', however nullified.


 

Offline JP

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Check out this thread on the subject: http://www.thenakedscientists.com/forum/index.php?topic=29156.0

In summary, clocks at the center of a planet/star and deep space won't agree because even though there is no net gravitational force, the space-time is "squashed" there compared with in deep space due to the mass of the planet/star.
 

Offline graham.d

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JP, your post here is in disagreement with your last post in the thread you cite. Which do you think is right?
 

Offline graham.d

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I think your previous post (in the cited thread) was right because the gravitational potential at the centre of a spherically symmetric massive object would be the same as that in free space, but I wouldn't bet my life on it.
« Last Edit: 25/04/2010 11:45:59 by graham.d »
 

Offline JP

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I think you're looking at the last post I made on the first page of that thread.  I was later corrected (and I agree with that correction.)   I'm by no means an expert on GR, but it sounds plausible to me (based on a basic understanding of GR) that time runs differently at the center of a massive object than it does in deep space.

Basically, gravitational "force" has to do with the curvature of space-time and time dilation can happen if space-time is squashed in one region compared to another, even if it's not curved.  At the center of your planet/star, you would expect no curvature, but a squashed space-time.
 

Offline graham.d

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Oh yes. It was a long thread and I didn't spot that. I will take a look again. At the moment I can't see how the centre of a planet/star would be different from other parts of the universe that have flat space-time. I thought the time dilation only depended on the gravitational potential; the fact that the only way to link the two spaces would involve travel through a region of curvature would not, at first sight, seem to affect that. Any way I will look through the thread and see if I get convinced. I can't believe this is something that has not been worked out convincingly somewhere.

Thinking about it, I can imagine some sort of hollow sphere and that the curvature occurs outside but then the curvature lessens inside the shell and then flattens when completely inside. Could this leave the potential different inside from outside? I think I would like to see the maths.
« Last Edit: 25/04/2010 15:00:22 by graham.d »
 

Offline graham.d

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Hmm, I read the thread (rather quickly I confess) but I'm not convinced. What was the compelling evidence that changed your mind, JP?
 

Offline graham.d

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After further thought I now think I agree that there is time dilation at the centre of a massive sphere, even though the space is locally flat. A simple way to look at this is that if you shone a light (out of a deep bore hole for example) from the centre, when viewed from distance it would be redshifted. There is no compensating blueshift as you come through the bulk of the sphere and the centre of the sphere is at the lowest gravitational potential.
 

Offline graham.d

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Sorry about this. I seem to be talking to myself :-)
 

Offline JP

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You answered your own question before I woke up this morning.  :)

Basically, I went through the same thought process as you when convincing myself.  Also, while curvature is important in GR as it tells things how to move, it's only one component of the structure of space-time itself.  Time dilation is tied to the "squashed-ness" of space-time as far as I can tell, rather than the curvature.  (You can see this in the equations of GR since they depend on the geometry of space-time, not just the curvature.)
 

Offline JP

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Of course, again, this is outside my area of expertise, so if an expert came along and corrected me, I wouldn't be incredibly surprised.  But this explanation makes the most sense to me based on what I do know about GR.
 

Offline jeroboambramblejam

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Well, thanks for the cogitatin' fellows (?)... I posed this Q to another site, but with unsatisfying results as of yet.  Sorry I missed your prior discussion under the 'boiled egg' heading... but the idea of  'squashed' space-time is new to me.   By the way, I was intrigued by an article describing a device for creating a flattened volume of space-time in Earth orbit.
 

Offline JP

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I think having you ask the question again makes it much clearer.  The boiled egg thread took many twists and turns before we all seemed to agree on the answer.  If the other site you posted on has something to say about the topic, I'd be interested in hearing it.
 

Offline jeroboambramblejam

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Nothing yet... thanks again for your interest;  There's a hint that Feynman wrote something on the topic (from the lecture title), but I couldn't get back to it (404'd):
newbielink:http://jsomers.net/papers/feynman-lectures-on-physics/ [nonactive]
 

Offline graham.d

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This subject has been discussed again in other threads on this website. I am sure that clocks will run slower at the centre of the earth (say) because they are at a lower gravitational potential, though it took a bit of reasoning to get there. However I note that there are some who disagree. I think the reason for this is that, whilst it is often stated that gravitational time dilation is due to gravitational potential, the equation that people use is one that is derived for clocks external to a massive sphere. This equation relates to the gravitational field rather than potential and is not valid inside the sphere. Of course whilst this aspect is realised by most they continue to (incorrectly) relate the field inside the sphere to the time dilation rather than potential. Outside the sphere the potential increases as the field decreases (moving away from the body) but inside the sphere (moving towards the centre) the potential decreases and the field decreases. The potential should be derived by integration of the field with radius.

Perhaps surprisingly, I have not found the equation for gravitational potential inside a massive sphere anywhere on the web. I think I will work it out myself when time allows. Even the person who wrote most of the Wiki page on the subject has a note to say that he will do this at some time.

I could not access your reference to Feynman Lectures, jeroboam, as it seems to need a password etc. There is nothing relevent that I could find in the 3 book set "Feynman Lectures on Physics" which I have had since my undergraduate days.
« Last Edit: 31/05/2010 09:50:13 by graham.d »
 

Offline Farsight

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It's all pretty simple really. If you took a flat slice through the middle of the earth, you can then plot gravitational potential like the one on the wikipedia page: http://en.wikipedia.org/wiki/Gravitational_potential

The slope indicates the local strength of the gravitational field, the depth indicates gravitational potential. Where the latter is deepest, the gravitational time dilation is at a maximum. That's at the centre of the earth. But the slope there is "flat", so there's no discernible gravitational field. If you were in a void at the centre of the earth, you'd float around weightless. But that's where the clocks run slowest.
« Last Edit: 31/05/2010 17:40:06 by Farsight »
 

Offline amrit

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"time dilatation" means that clock run slower.....there is no time behind run of clocks
 

Offline graham.d

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"there is no time behind run of clocks" - Amrit

What does that mean, Amrit? The sentence does not make any sense.
 

Offline Geezer

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Graham,

I probably have it wrong, but I thought that (according to shell theorem) the gravitational field inside a sphere is zero anywhere within the sphere.
 

Offline graham.d

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Hi Geezer. You are right. The field at the centre would therefore be zero, but the time dilation is a function of gravitational potential. The field is the differential of the potential. Inside the sphere, as you travel towards the centre the field gets less until it becomes zero, but you are still moving to a lower potential; at the centre the potential is at a minimum. You have to use energy to climb out of a hole and so does light. It gets redshifted. 
 

Offline imatfaal

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Graham,

I probably have it wrong, but I thought that (according to shell theorem) the gravitational field inside a sphere is zero anywhere within the sphere.

Shell theorem says that within a HOLLOW sphere that no gravitational force is felt by any body within the shell.  In the case of a position within a SOLID sphere (help I don't like confined spaces) the force perceived varies linearly with the distance to the centre.  Matthew
 

Offline Geezer

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Graham,

I probably have it wrong, but I thought that (according to shell theorem) the gravitational field inside a sphere is zero anywhere within the sphere.

Shell theorem says that within a HOLLOW sphere that no gravitational force is felt by any body within the shell.  In the case of a position within a SOLID sphere (help I don't like confined spaces) the force perceived varies linearly with the distance to the centre.  Matthew

Well, right. But a body can only exist within a hollow sphere. If the sphere is solid, there can be no body in it. (Unless the mob were somehow involved.)
 

Offline imatfaal

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It could be at the bottom of a very deep and very hot mine shaft.  or as you say mob-related.  i would think that any calculations regarding speed of solar nuclear interactions have to take into account time dilation due to gravitational potential within the solar mass.

and to be honest i was still thinking along the lines of Amrit's south african mine and photon clock.     
 

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