Time Dilation: Does Gravitational Field Strength Matter?

[sawer]

Hi
I have 2 questions.

There are 2 planets and one clock on each of them. One of them has a bigger gravitational field strength. And two clock have same distance from the core.

1-) Does time dilation occur between two? Which clock ticks slower?
2-) If time dilation occurs, which formula calculates this phenomenon? (I think Schwarzschild factor isn't related this phenomenon)

 

[Orodruin]

One of them has a bigger gravitational field strength.

This is a very ambiguous statement. A gravitational field strength cannot be assigned to a planet as the strength of the gravitational field depends on the distance. Are you meaning to say that one of the planets is more massive than the other, thus creating a larger field strength at the same distance from the corresponding centres?

1-) Does time dilation occur between two? Which clock ticks slower?
2-) If time dilation occurs, which formula calculates this phenomenon? (I think Schwarzschild factor isn't related this phenomenon)

1) Given that the answer to the above is "Yes, one mass is greater than the other." then there will be a relative time dilation. The clock on the surface of the more massive planet will tick slower. However, it does not directly relate to gravitational field strength, but instead to gravitational potential. 2) The Schwarzschild factor is crucial to the problem. The relative ticking of the clocks to a clock at infinity is given by the time component of the metric and therefore the relative ticking of the clocks relative to each other is given by the ratio of those metric components.

 

[Ibix]

I assume that the planets are isolated. There are no pesky stars with their gravity to complicate things, and the planets are far enough from each other that their mutual gravitation is negligible. And also, as Orodruin says, that you have two planets of equal size but different mass.

Then you can just add a notional third clock at infinity. What you appear to be calling the Schwarzschild factor gives you the tick rates of the two planetary clocks compared to the notional clock at infinity. Consistency then requires that the rates of the planetary clocks be the ratio of the two factors.

 

[sawer]

Thank you Orodruin and Ibix
I got it.

However, it does not directly relate to gravitational field strength, but instead to gravitational potential.

If gravitational field strength relates to mass and mass is a parameter of Schwarzschild factor, then why do say "it does not directly relate to gravitational field strength"? Mass itself, also slows time. The bigger the mass means more time dilation. Right? Height in a gravittional field and strength of gravitational field makes opposite effect. Right?
$$r_S=\frac{2MG}{c^2}.$$

 

[Orodruin]

If gravitational field strength relates to mass and mass is a parameter of Schwarzschild factor, then why do say "it does not directly relate to gravitational field strength"?

Because it doesn't. You cannot draw that conclusion from looking at a single solution. Consider the gravitational field at the centre of a body - due to symmetry it will be zero - but a clock at the centre would be still be time dilated. The important quantity is the gravitational potential.

 

[Dale]

If gravitational field strength relates to mass and mass is a parameter of Schwarzschild factor, then why do say "it does not directly relate to gravitational field strength"?

This argument doesn’t make any sense to me.

You can show that it depends on the potential instead of the field by using the equivalence principle. You calculate the Doppler shift between the top and the bottom of an elevator accelerating uniformly far from gravity. By the equivalence principle that is time dilation in a uniform gravitational field. It depends on the potential, gh, and not just on the field strength, g.

 

[Janus]

Thank you Orodruin and Ibix
I got it.

If gravitational field strength relates to mass and mass is a parameter of Schwarzschild factor, then why do say "it does not directly relate to gravitational field strength"? Mass itself, also slows time. The bigger the mass means more time dilation. Right? Height in a gravittional field and strength of gravitational field makes opposite effect. Right?
$$r_S=\frac{2MG}{c^2}.$$

Gravitational field strength is a measure of force per unit mass exerted at a certain point.
So let's say you have two planets, A and B. Planet B has both 4 times the mass and 2 times the radius of planet A. As a result, the gravitational field strength at the surface of each planet is the same.
However, time dilation in determined by the relationship of
$$ t = \frac{t`}{\sqrt{1- \frac{2GM}{rc^2}}} $$

Which gives the result that the time dilation at the surface of planet B will be greater than that at the surface of planet A.