Time dialation at Earth's center

[DrMatrix]

I understand that a clock on the surface of the Earth is observed to run more slowly than a clock in orbit or a clock "at infinity". What about a clock situated at the center of the earth. Would it be slowed because of the concentration of mass? Or would it run normal speed because there is no net effect of gravity? Is there a slow down simply because it is in a gravity well?

How does a clock situated at the cener of the Earth run compared to a clock on the surface and compared to a clock at "infinity"? And (briefly) why?

Thanks.

 

[pervect]

Time dilation is largest at the center of the Earth. The reason for this is that it is proportional to the Newtonian potential U or the negative of the Newtonian potential \Phi which is at a maximum (for U) or minimum (for Phi) there. Gravity is zero at the center of the Earth, but that's proportional to the gradient of the potential, i.e. how fast it changes. The potential itself is nonzero at the center. Sign conventions are that U is positive and Phi is negative at the center of the Earth - usually, one sees Phi used, but for gravitational time dilation its handy to use U instead.

 

[DaveC426913]

So that means you can have a strong time dilation factor while being in a zero net force gravity field.

Pardon my wild speculation, but does that mean our universe could be enclosed in a sphere such that the time dilation is a factor, yet we would not experience a force of gravity?

 

[Garth]

So that means you can have a strong time dilation factor while being in a zero net force gravity field.

Pardon my wild speculation, but does that mean our universe could be enclosed in a sphere such that the time dilation is a factor, yet we would not experience a force of gravity?

For time dilation you need to compare one clock against another, in your scenario where would the other clock be situated?

Garth

 

[Chris Hillman]

"Time dilation" etc requires specifying pair of observers!

Hi, DrMatrix,

How does a clock situated at the cener of the Earth run compared to a clock on the surface and compared to a clock at "infinity"? And (briefly) why?

Garth already alluded to this, but to clarify a bit: to discuss "gravitational red shift" (aka "gravitational time dilation") you need to specify a pair of observers, an emitter and a receiver. You specified the emitter but didn't completely specify the receiver, although we can make an obvious guess: that you had in mind a receiver who is a "very distant static observer". This is a terribly important point which I seem to find myself repeating endlessly (I've done so several times in the past week alone, I think).

A second issue is that you are tacitly asking about a "stellar model", i.e. a model of an isolated star, presumably modeled, in the context of gtr, as a perfect fluid ball. You can obtain such a model by matching an exterior vacuum solution across the world sheet of a spherical surface to a perfect fluid solution, such that the surface of the "star" has zero pressure. Some of the simplest models of this kind are static spherically symmetric stellar models, the first of which was constructed by Schwarzschild himself just before his death in 1916. Look for "Schwarzschild fluid" or "Schwarzschild interior solution" in the arXiv or in textbooks like Schutz. In particular, look for recent surveys coauthored by Matt Visser, since there has recently been some remarkable progress in this area!

Coming to the point of the second issue: a qualitative answer to your second question will depend to some extent upon fluid model you adopt, assuming that you naively assume that a lightlike signal is propagated from the center of the star along a radial outgoing null geodesic to the surface and then out to the distant observer as in the usual analysis for two static observers in the Schwarzschild vacuum (see any gtr text for that!).

Third issue: the point I just mentioned, deep inside even an ordinary star, a real signal would not be able to propagate freely. Roughly speaking, photons don't get very far inside the Sun before they run into something.

Fourth issue: when I see questions like this, I have learned to be wary of a possible "ambush" by a creationist; so-called "Young Earth Creationists" have been known to make some hilariously incorrect misstatements claiming (quite incorrectly) that when one takes account of gravitational time dilation, the six billion year old Earth is seen to be six thousand years old, as per Bishop Ussher. Strange but true. So if you've been "debating" with creationists, it can be useful to mention this context.

Attempts to hijack science (or rather, profound misconceptions about science) for ideological purposes sometimes makes for strange bedfellows. Years ago I found a marxist website (long since vanished) which made essentially the same mistake but was (needless to say) arguing in favor of very different conclusions from the creationists!

What about a clock situated at the center of the earth. Would it be slowed because of the concentration of mass? Or would it run normal speed because there is no net effect of gravity? Is there a slow down simply because it is in a gravity well?

That's the idea. What matters is that (details depending upon what model you choose), the density will usually be maximal at the center; it is true that (by symmetry) the magnitude of acceleration of a static bit of fluid will be zero at the center (just as in Newtonian theory), but this is not what causes the redshift of outgoing signals. Namely: the curvature of spacetime (which as you know from the EFE is directly related to the stress-energy tensor) causes initially parallel outgoing null geodesics to diverge, which means that when they are received by our distant static observer, the signal will be redshifted.

 

[Chris Hillman]

So that means you can have a strong time dilation factor while being in a zero net force gravity field.

I got to go, but very briefly: you are confused the magnitude of acceleration of a static observer with the tidal forces on an observer (more or less identified with curvature and with "gravitational field" in gtr). The former might vanish at the center of a star, but certainly not the latter.

Now consider a hollow thin massive nonrotating spherical shell. Just as in Newtonian gravitation, the field (curvature) vanishes inside the shell, and just as for our stellar models, the exterior is matched to a Schwarzschild vacuum (with mass parameter set to "the mass" of the shell). Ignoring issue three in my previous post, can you figure out what would happen to radially outgoing signals in this case? See any gtr textbook and modify the usual computation appropriately.

 

[MeJennifer]

Namely: the curvature of spacetime (which as you know from the EFE is directly related to the stress-energy tensor) causes initially parallel outgoing null geodesics to diverge, which means that when they are received by our distant static observer, the signal will be redshifted.

Would a hypothetical clock the size of a local Lorentz frame in the center of a very large star run slower than one far removed from the star?

 

[pervect]

I got to go, but very briefly: you are confused the magnitude of acceleration of a static observer with the tidal forces on an observer (more or less identified with curvature and with "gravitational field" in gtr). The former might vanish at the center of a star, but certainly not the latter.

Now consider a hollow thin massive nonrotating spherical shell. Just as in Newtonian gravitation, the field (curvature) vanishes inside the shell, and just as for our stellar models, the exterior is matched to a Schwarzschild vacuum (with mass parameter set to "the mass" of the shell). Ignoring issue three in my previous post, can you figure out what would happen to radially outgoing signals in this case? See any gtr textbook and modify the usual computation appropriately.

I would say that Dave is simply thinking of the gravitational field as the proper accelration of a static observer, something that most people do. This is closely related to the idea of the gravitational field as a connection coefficient, but I think the above description is closer to what layman (i.e. anybody but someone who has spent a lot of time studying GR) actually means when they say "gravitational field"

Assuming this defintion of "gravitational field", he's correct in his observation.

 

[Chris Hillman]

"Clock run slow?" DMS as stated

Would a hypothetical clock the size of a local Lorentz frame in the center of a very large star run slower than one far removed from the star?

DMS = doesn't make sense.

Same reason as in my reply to the first post in this thread: statements about "frequency shift", "time dilation", "ideal clocks running at different rates" only make sense in reference to a specific pair of observers in a specific spacetime model, assuming that the comparison is carried out using "initially parallel" null geodesics from the world line of one observer to the world line of the other observer.

To rephrase your question properly: "Would an ideal clock carried by a bit of matter at the center of the "star" in some model, in gtr, of a spherically symmetric static isolated star run more slowly than an ideal clock carried by a very distant static observer, as compared using null geodesics running from the first to the second observer?"

Generally speaking, the expected answer would be yes. (If I had more energy right now, I'd type in some examples; in lieu of that, see MTW or Schutz's gtr textbook for fine discussions of Schwarzschild's static spherically symmetric fluid model from 1916.)

 

[MeJennifer]

Same reason as in my reply to the first post in this thread: statements about "frequency shift", "time dilation", "ideal clocks running at different rates" only make sense in reference to a specific pair of observers in a specific spacetime model, assuming that the comparison is carried out using "initially parallel" null geodesics from the world line of one observer to the world line of the other observer.

I know what you mean but didn't I specify that in my question?

To rephrase your question properly: "Would an ideal clock carried by a bit of matter at the center of the "star" in some model, in gtr, of a spherically symmetric static isolated star run more slowly than an ideal clock carried by a very distant static observer, as compared using null geodesics running from the first to the second observer?"

Actually the question is particular to an ideal and hypothetical clock the size of a local Lorentz frame, in other words the clock runs in flat space-time.

 

[Hans de Vries]

I understand that a clock on the surface of the Earth is observed to run more slowly than a clock in orbit or a clock "at infinity". What about a clock situated at the center of the earth. Would it be slowed because of the concentration of mass? Or would it run normal speed because there is no net effect of gravity? Is there a slow down simply because it is in a gravity well?

How does a clock situated at the cener of the Earth run compared to a clock on the surface and compared to a clock at "infinity"? And (briefly) why?

Just to fill in some actual numbers:The potential in the center of the Earth is lower and therefor clocks run
slower. If one assumes uniform mass density then the potential is 50%
lower at the center compared to the surface of the earth.

A clock at the surface of the Earth runs 1 second per 45.6 year slower as
a clock at infinity. while a clock at the center of the Earth runs 1 second
per 30.4 year slower as a clock at infinity, assuming a uniform mass density.

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html#c5
http://www.phys.uri.edu/~gerhard/PHY204/tsl94.pdf

The last link shows how to calculate the potential for a solid charged sphere
which you might use to calculate the gravitational (Newtonian) potential.
(Also for more realistic mass distributions)Regards, Hans

 

[Chris Hillman]

Specifying states of motion

I know what you mean but didn't I specify that in my question?

No, because you failed to say that you were asking about lightlike signal sent by a static observer at the center of an isolated star radially outward to a very distant static observer.

Do you remember the thread in which I was trying to get you and someone else to work some exercises on redshifts observed by various observers in Schwarzschild spacetime? One of the cases were discussed involved inertial (hence non-static) "emitting" observers, although in each case we discussed in detail, the "receiving" observers were distant static observers. As you remember, this makes a big difference!

 

[MeJennifer]

No, because you failed to say that you were asking about lightlike signal sent by a static observer at the center of an isolated star radially outward to a very distant static observer.

Ok, I understand what you are saying.

But what, per se, does a light signal between them have to do with dilation?

For instance consider the following experiment:

I have two synchronized ideal clocks, A and B, in flat space-time. These clocks are programmed to stop after running for exactly 1 million years of proper time. I transport clock A to the center of a very massive object.
After 1 million years on B's clock I verify that clock B indeed stopped and I travel to A to verify A's state. The initial transport of clock A and the travel to verify A's state afterwards was done with an identical proper time interval.
Would I witness the stopping right at the moment I reach it or would it still be running for a while?

In this scenario there is no light signaling.

 

[Chris Hillman]

Hi, Jennifer,

Ok, I understand what you are saying.

But what, per se, does a light signal between them have to do with dilation?

I got to go, but you can look back for the thread in which you, I and Jorrie were discussing frequency shifts? That should answer your question.

 

[yogi]

Jenniferme - Is the problem the same as setting two clocks at opposite ends of an accelerating rocket to run for a fixed same duration. Say the A clock is at the foot and B is at the front of the rocket at launch and each clock is programmed to start and run for 10 seconds at the time the rocket begins to accelerate at a constant rate. Assume also that A clock sends a pulse each usec and the pulses are received by the B clock every two usec. The B clock will receive less (approx by half) the number of pulses sent by A during its 10 second transmitting period than those sent by the A clock - so the A clock would be considered as running slower than B based upon signals received from B - but when the rocket stops accelerating after 15 seconds or so - the stopped A and B clocks are now in the same frame again and both would have recorded 10 seconds when they are compared...so in reality they do not really run at different rates?