Gravitational time dilation in the center of a mass

[Xilor]

Hi, I understand that according to general relativity the time dilation experienced due to gravity gets more significant the more you go down in a gravitational well, so the maximum should be at the center of the mass. But I can't really rhyme that idea with the idea that the time dilation has to do with being present in a non-inertial reference frame. In the center of the earth, gravity should not cause any acceleration because of equal mass-distributions in all directions. Without acceleration, is there really an accelerated reference frame? If not, why does gravitational time dilation still happen there then?

And if gravitational time dilation is caused more by a combination of the space-curvatures of all the mass in the universe, shouldn't the dilation be much higher then? If you stand on the surface then gravitational 'forces' of all the mass pulling you to to your left should cancel out with the force of all the mass pulling you to the right. But the time dilation wouldn't cancel out, so dilation should change differently than the force experienced, right?

What am I missing here?

 

[A.T.]

Without acceleration, is there really an accelerated reference frame?

No, the rest frame of the Earth's center is free falling and locally inertial.

If not, why does gravitational time dilation still happen there then?

The acceleration is connected to the 1st derivative of the gravitational time dilation. At the center the gravitational time dilation has an extremum, so the 1st derivate is zero.

 

[Xilor]

Aha, the derivative. Well that makes a lot more sense already. So, does that mean that time dilation increases exponentially when going from infinity to surface (or upon entering the atmosphere), but starts to transform into a logarithmic function from there till the center?

 

[Bill_K]

So, does that mean that time dilation increases exponentially when going from infinity to surface (or upon entering the atmosphere), but starts to transform into a logarithmic function from there till the center?

The gravitational time dilation is proportional to the Newtonian gravitational potential.

But I can't really rhyme that idea with the idea that the time dilation has to do with being present in a non-inertial reference frame.

The special relativistic time dilation has to do with relative velocity, not acceleration. E.g. the time correction for a GPS satellite needs to take into account the orbital speed.

 

[A.T.]

Aha, the derivative. Well that makes a lot more sense already. So, does that mean that time dilation increases exponentially when going from infinity to surface (or upon entering the atmosphere), but starts to transform into a logarithmic function from there till the center?

It's not really exponential or logarithmic, but I guess you mean the general look. In this applet you see the gravitational time dilation as inflation of the space-propertime cylinder for a radial line through the center. The gravitational time dilation is maximal at the center. The green lines indicate the surface.

 

[Xilor]

It's not really exponential or logarithmic, but I guess you mean the general look. In this applet you see the gravitational time dilation as inflation of the space-propertime cylinder for a radial line through the center. The gravitational time dilation is maximal at the center. The green lines indicate the surface.

Yes, that's what I meant and the applet was very helpful for visualization. Thank you very much.