Clock in isotropic gravitational field

[Calimero]

1. How would clock in isotropic gravitational field, for example at centre of earth, tick compared to the clock at surface of earth?

2. How would clock in the center of Earth tick compared to the clock at center of sun?

 

[bcrowell]

1. How would clock in isotropic gravitational field, for example at centre of earth, tick compared to the clock at surface of earth?

It would run more slowly. This is an example of a gravitational redshift, as in the celebrated Pound-Rebka experiment. To remember the direction of the effect, think of light getting red-shifted as it emerges from just outside the event horizon of a black hole. To a distant observer, it seems like the oscillator that emitted the light must have been vibrating more slowly.

Some gratuitous nitpicking:

-In Newtonian mechanics, there isn't really any such thing as an isotropic gravitational field. The field at the center of the Earth is simply zero.

-In GR, unlike Newtonian mechanics, the gravitational field isn't even frame-independent. For example, a free-falling observer near the surface of the Earth says there's zero field. Since the gravitational field isn't a meaningful concept, there's no way that time dilation can depend on it; it actually depends on the gravitational potential.

2. How would clock in the center of Earth tick compared to the clock at center of sun?

The one at the center of the sun is at an even lower gravitational potential than the on at the center of the earth, so it runs even slower.

 

[Calimero]

Thanks BC.

Now, one more question. Clocks in the past, when universe was denser, were running slower than now. True?

 

[Dale]

Clocks in the past, when universe was denser, were running slower than now. True?

How would you intend to compare them? I mean, assuming that you had an ancient clock that you could have set up however you wanted, what physical experiment would you do to compare it to the rate of a modern clock.

 

[Calimero]

Well I can't. But if we know that clocks in the lower gravitational potential run slower, we can conclude that in the past they were running slower (presuming ideal homogeneity of universe). Or we can't?


Edit: Or maybe I can. Assume ancient clock in far away place. If we know scale factor of the universe (by some other means then the redshift) at the time light ventured towards us, we could easily see if it is running slower.

 

[Dale]

if we know that clocks in the lower gravitational potential run slower, we can conclude that in the past they were running slower (presuming ideal homogeneity of universe). Or we can't?

No, you can only make a potential in a static spacetime, not in a general spacetime like an expanding FLRW metric.

Edit: Or maybe I can. Assume ancient clock in far away place. If we know scale factor of the universe (by some other means then the redshift) at the time light ventured towards us, we could easily see if it is running slower.

AFAIK, essentially all we have is the redshift. Now, when a clock is at a lower gravitational potential we say that it is running slow and we see it as redshifted, so they are related effects in a static spacetime. If you are willing to accept a measurement of gravitational redshifting as a measurement of ancient clocks being slow, then there is plenty of such evidence. But if you are specifically excluding that then I don't believe there is any sense in which you can say that ancient clocks ran slow.

 

[Calimero]

What I am wondering about is that we take redshift and interpret it straight away as consequence of expansion. Cmb, for example is at z=1090, so we say that since then scale factor grew 1091 times. There is no doubt that universe was much, much denser at the time, but we just don't count that into the redshift. Why is that so?

 

[Dale]

I don't understand your comment.

 

[Calimero]

Ok, simply we don't count for possibility that ancient clocks ran slower.

 

[Dale]

If by "ran slower" you mean that signals from ancient clocks would be redshifted then we do count for that possibility. If you mean something else then I am not sure what you mean observationally.

 

[Passionflower]

-In GR, unlike Newtonian mechanics, the gravitational field isn't even frame-independent. For example, a free-falling observer near the surface of the Earth says there's zero field.

That all hinges on how one defines things. What is a gravitational field? Some define it as a non zero Riemann curvature tensor while others define it differently. One is not more right than the other it simply depends on how you define it in English.