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Author Topic: How will the clock tick in the middle of the Earth  (Read 4958 times)

Offline yor_on

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Imagine us digging to the absolute center of the earth. Securing our chamber with what we have, mostly chewing gum, and we all know how sticky and strong that can be, we place a trusty atomic clock down there, keeping its exact twin on the surface.

Now inside that chamber there should be no gravity, right?

So how will the clocks compare? Will they 'tick' the same?
Will the one in the chamber go faster?
Or will it go slower.

And Why?
The 'why' is the 'biggie-thingie' as we say in Sweden.

 W H Y do you say that?:)

says I.
« Last Edit: 22/12/2010 00:23:10 by yor_on »


 

Offline JP

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How will the clock tick in the middle of the Earth
« Reply #1 on: 22/12/2010 00:55:49 »
This was discussed a while back, but I can't find the link.  The consensus was that there is gravity at the earth's center, in that space-time is squashed relative to space-time in deep space, far from gravitating matter.  There's no gravitational force, since all the forces cancel there, but there is gravity. 

The net effect of this is that your clock will run slow compared to one in deep space, because of the difference in space-time between the two points.
 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #2 on: 22/12/2010 01:16:30 »
"Because of the difference in space-time between the two points."

:)

would you mind expanding on that answer JP?
In form of 'gravitational fields'?

Like we say that Earth is a 'gravitational field' or a 'gravity well'.
Here we have no gravity but a slowing clock.
 

Offline CPT ArkAngel

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How will the clock tick in the middle of the Earth
« Reply #3 on: 22/12/2010 02:55:54 »
At a point of no gravity in the middle of the earth, the timerate is higher. It is like the earth was not there at all. You have to think of it in 3D not 2D.
 

Offline JP

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How will the clock tick in the middle of the Earth
« Reply #4 on: 22/12/2010 03:08:56 »
The best way to visualize this is with the rubber-sheet analogy.  Think of space-time as a rubber sheet with a bunch of perpendicular lines drawn on it, like graph paper.  To compare clocks at two points in space, you need to compare the graph lines at those two points.  If space is curved at one point and not at the other, the lines have different structure, which indicates that clocks won't agree when compared.  This analogy does have some issues, since it's obviously a 2D shape in space, not a 4D space-time shape, but it does get the basic point across about the geometry of space-time.

Now, let's say you want to model the earth's gravitational field.  You press a metal ball into the sheet so that it curves the sheet.  Other objects placed on the sheet will roll towards it.  If you somehow remove the earth, but keep the sheet bent, you will see that an object placed at the center of the earth (at the center of the bend) won't roll anywhere, since it's at a local minimum.  Similarly, a ball placed really far away from the earth will be on a flat section of the sheet and won't roll.  However, at the earth's center the sheet is still bent, while really far away it isn't.  If you compare the shape of the graph lines at these two points, they won't agree--they'll be stretched out at the earth' center compared to the flat section.  Also, any direction you move away from the earth's center curves upwards, while the sheet is flat in any direction from the point far away from the earth.  This is why you can have a point at the earth's center that doesn't experience gravitational force, but whose clocks run slower than those far away form the earth: the geometry of the sheet is different at those two points, and in GR, the geometry of space-time at different positions will tell you about time dilation due to gravity.
 

Offline Soul Surfer

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How will the clock tick in the middle of the Earth
« Reply #5 on: 22/12/2010 08:51:51 »
The most important thing about ALL these clock ticking questions is that if you are standing by the clock it ALWAYS ticks at a "normal" rate and of you carry a clock with you anywhere at any speed it ALWAYS appears to you to tick normally.  It is only when you look at the clock FROM A DISTANCE in a gravitational field or moving quickly there is a difference in the ticking rate, or compare a clock with another identical clock after one has gone on a round trip journey there is a difference in elapsed time.
 

Offline imatfaal

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How will the clock tick in the middle of the Earth
« Reply #6 on: 22/12/2010 12:14:34 »
Can you not think of it in terms of gravitational potential? Clocks run slower in positions of lower gravitational potential compared to higher.  The surface of the earth is a higher absolute potential than the centre and thus clocks in the centre run slower than at the surface.  I would say JP is right and CPT is incorrect. 

The Surfer is, of course, correct that these are all relative measurements
 

Offline Foolosophy

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How will the clock tick in the middle of the Earth
« Reply #7 on: 22/12/2010 13:53:24 »
The dfference in the passage of "time" for 2 atomic clocks - one situated on the surface of the earth and the other orbiting the earth at some altitude is due to 2 effects.

Firstly, there is the relativistic effect of a clock moving at some velocity and the other effect is due to the change in the gravitational force.

The atomic clocks located in orbiting satellites that are used for GPS systems must correct for both of these effects.

 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #8 on: 22/12/2010 15:24:50 »
Very nice points from you all :)

Helps me see it, I've seen some speak of it in form of 'gravitational potential'? Would you say that this is an appropriate expression, or should one discuss it as a 'twisted' metric?
 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #9 on: 22/12/2010 15:30:11 »
You could also use this example as a argument for what a 'free fall' is couldn't you? You would be stationary in that chamber relative the Earth surrounding you, but you would still be in a 'free fall'.

Am I right, am I, am I :)
 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #10 on: 23/12/2010 01:10:05 »
The interesting thing here to me is that they don't 'meet'. the 'weightlessness' and the clock. Assuming that a slower 'timerate' is a time dilation relative Earth. No this isn't right. What does it say? That clocks will follow not gravity in itself, but the 'gravitational potential' as Imatafaal express it, and more people that him, if I may add, use those exact words. Maybe it's the correct expression?

So what the he* is a gravitational potential? in our 2-D scheme I thought of it as those geodesics getting 'bent/stretched', expressed as 2-D lines as I understands it? Or am I just jumping to conclusion here? Looking at it, it seems a Newtonian concept, treating gravity as a 'force'?

But it's easy to see why it might become expressed as a 'force' as our example with the clocks creates a 'time dilation' where we also expect a 'weightlessness' and if I'm right, you also can look at it as the clock is following a geodesic (aka free falling) even though encapsulated by the Earth as it does it.
===

And while I remember it, the weirdest effect is that the clock slows down as that implies that it is 'accelerating' relative the the clock on the surface, as a slower time relative the observer at the surface should be equivalent to an acceleration, if I'm thinking correctly?

But maybe I'm looking at it wrong?
« Last Edit: 23/12/2010 01:19:25 by yor_on »
 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #11 on: 23/12/2010 01:29:48 »
Looking at this explanation. Gravitational potential. I get a distinct feeling of a 'force', which I don't particularly like myself. :) as i look at gravity as a property of SpaceTime.

Here is another way of describing it.

"In the case of motion-induced Fitzgerald contraction, the effect is, as
I said, arguably a "trick of the light" (but it seems pretty real to me,
at least in some cases).  Space itself is notably not affected; only the
measurements seem to change.

In the case of a gravity well, space itself is unambiguously distorted.
 The metric tensor changes, and to deal with this you need to dive into
general relativity and pseudo Riemannian geometry.  To deal with the
gravitational field around a planet, you can look at the Schwarzschild
metric (Wiki has a page on it, for example), which will give you some
feel for how the distortion "looks".

But there is a very important point here, which is that there are
several kinds of distortion, and they can happen together or separately.

 -- The "curvature" is nonzero in the presence of a gravitational field
which has tidal effects.  This results when there are nonzero second
derivatives of the components of the metric tensor.  Nearly all real
gravitational fields have tidal effects.  The Schwarzschild metric, for
instance, is curved.

-- The "connection" is nonzero when there's a gravitational gradient --
i.e., when there's a direction things fall in.  This is a result of
nonzero values for the the first derivatives of the components of the
metric tensor.  This is associated with distortion along one axis, I
think, and is somewhat analogous to Fitzgerald contraction.  However,
note well this this has very little to do with time dilation!

-- Finally, there's gravitational time dilation, which is correlated
with the gravitational *potential*, *not* with the local gravitational
field strength!

For example, if we dig a spherical chamber in the center of a planet,
there will be *no* gravitational "field" within that chamber caused by
the mass of the planet.  However, the gravitational potential is lower
in that chamber than it is on the surface, and clocks in the chamber
will run SLOWER than clocks on the surface.

As another example, if we dig a spherical chamber OFF CENTER inside a
*uniformly* *dense* planet, the gravitational field within that chamber
caused by the mass of the planet will be almost exactly uniform.  In
other words, there will be no tidal effects, and it will be impossible
for an observer within the chamber to determine whether he's being
subjected to a gravitational field, or is simply accelerating at a
uniform rate.  Clocks will runs slower at the "bottom" of the chamber
than at the "top". "

But what does it say?
that gravity is a 'force'?

If gravity's potential is called a 'force' then I have a hard time excluding that from 'gravity'?

 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #12 on: 23/12/2010 01:45:14 »
You might argue that there is no acceleration involved in there and they are at 'rest' versus each other no clocks 'ticking' accelerating relative the other. But for this 'time dilation', as I call it, to happen we must have introduced a moment of 'acceleration' or 'slowing down' of that clock inside the Earths middle. Do you agree?

Where did that happen?
As we lowered the clock?
 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #13 on: 23/12/2010 02:01:17 »
Let's put it this way. How do we split 'gravity' into two concepts?

It's 'potential' that then will create its own coordinate system, defining a 'highest and lowest' gravitational impact on, for example, our trusty 'clock'. But then also define gravity as something acting another way, having a different coordinate system?

What brings those coordinate systems together?
==

Could I see it as Weightlessness versus 'Masslessness'?

We have the invariant mass still there, but 'weightlesness' is a description of where gravity takes out itself? What happens with those geodesics at a Lagrange point? Do SpaceTime 'curl up' there?

And maybe a Lagrange point is the correct description of that middle too? Or can I see a object in a Lagrange point as following a geodesic too? Think about 'uniform motion', you observing that object at its Lagrange point (unbeknownst to you), or is that too far-fetched?
« Last Edit: 23/12/2010 02:21:14 by yor_on »
 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #14 on: 23/12/2010 02:29:21 »
And another thing, as soon as we have two equivalent 'systems', as our two clocks, and then find one of them go slower than the other. Am I right in assuming that this is equivalent to the introduction of an acceleration somewhere in its 'path'. And this leads to another question that I just lost.. Da*n it was a good one, I could have asked it directly but then you wouldn't have seen how I came to it. My memory is a sieve. Da*n again, anyone more recognizing this. Forgetting your question while you write it, or is it only me :)
==

I better leave this be for a while ::))
Total recall, now that was a good movie.
« Last Edit: 23/12/2010 02:31:13 by yor_on »
 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #15 on: 23/12/2010 03:23:13 »
Maybe it was this one, I know it was a good one, but this one isn't that bad, maybe? If I look at that clock that 'slowed down' then it had to be that the 'gravitational potential' there was stronger than on Earths surface right?

So, where do we find a 'masslessness', as I somewhat flippantly will insist on calling the concept, until someone hit me with that big hammer whereupon i might change my mind, or brain? Does that concept exist? And, what does it say about a Lagrange point, Will our clock slow down there too?

It won't, right, and that takes care of that idea. The middle of the Earth is no mystical Lagrange point, or is it? After all, they're not defined by gravitational potential are they? They are defined by gravity acting upon itself? And suddenly it feels as if I'm discussing 'forces' again. Da*n again :)
===

thinking of it. No, all Lagrange points must be dynamic 'systems' uniformly moving. They're only 'still' relative us. and that only as long as no other mass is introduced.
« Last Edit: 23/12/2010 03:29:42 by yor_on »
 

Offline JP

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How will the clock tick in the middle of the Earth
« Reply #16 on: 23/12/2010 06:52:58 »
I still think the most general description is to look at the local structure of space-time.  This tells you how fast clocks run when placed at different locations in a gravitational field.  In certain cases you can tell how fast clocks will run by using the gravitational potential instead of computing the structure of space-time (the Schwarzschild metric, which is a good approximation to the earth's gravitational field, for example). 

I think your post a few up hit the nail on the head.  It's the metric tensor (which describes the structure of space-time) that tells you about how gravity effects things, and provides all the information about time dilation.  The metric tensor can be different at two points, yielding disagreement between clocks, even though the connection is zero at each point (i.e. things don't fall at either point).  The connection is just one property of the metric tensor, and it isn't enough to fully determine the structure of space-time.

Or in less technical terms, if you tell me an object isn't falling at a point, that isn't enough for me to tell you about the structure of space-time there.  It's just one factor that goes into describing the structure of space-time.
 

Offline yor_on

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How will the clock tick in the middle of the Earth
« Reply #17 on: 23/12/2010 12:41:55 »
But it's still a very good argument for space being something else than just 'empty' or 'not there'. Isn't it :)
 

Offline jartza

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How will the clock tick in the middle of the Earth
« Reply #18 on: 24/12/2010 01:10:53 »
There is a gravitational potential of 220000 J/kg between earth's equator and north pole.
(it means objects at equator contain 220000 J/kg extra potential energy compared to objects at north pole)


Objects at equator contain 100000 J/kg extra kinetic energy compared to objects at north pole.

If the potential energy difference and the kinetic energy difference happened to be the same, then clocks would run at the same rate at the poles as at the equator.

 

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How will the clock tick in the middle of the Earth
« Reply #18 on: 24/12/2010 01:10:53 »

 

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