gravity - Degree of Time Dilation At a Distance From the Sun where acceleration = g?

At higher altitudes above a body, clocks tick more slowly, and gravitational field is weaker. But what is the relationship? It is tempting for a GR newbie such as myself to think that anywhere that the gravity is equal to that of the earth's surface that time-dilation would be the same, i.e. dilation as a simple function of local gravitational acceleration. I have been assured that this is not the case. I wonder if someone can calculate the following situation, which strikes me as the simplest 2nd case to compare with earth's surface: At the surface of the Sun, gravity is roughly 28 times that at the Earth's surface. If we move out to a suitable distance, we can find a place where the gravitational acceleration is the same as the surface of the Earth - 1g - and we can hold a clock there with a rocket. What is the time dilation there with regard to a "Far away" observer? I understand to the dilation on the Earth's surface to be 0.0219 seconds per year compared with the distant observer.

Answer

For a Schwarzschild observer the gravitational time dilation means the local time at a distance $r$ from a mass $M$, appears to be running slow by a factor of:

$$ \sqrt {1 - \frac{2GM}{r c^2}} $$

You get this straight from the Schwarzschild metric:

$$ ds^2 = -\left(1-\frac{2GM}{r c^2}\right)dt^2 + \left(1-\frac{2GM}{r c^2}\right)^{-1}dr^2 + r^2 d\Omega^2 $$

because hovering above the Sun, both $dr$ and $d\Omega$ are zero.

Anyhow, I reckon the acceleration due to the Sun's gravity is 9.81m/s$^2$ at a distance of about $3.68 \times 10^9$m (using the Newtonian expression for $a$), and at this distance I work out the time dilation to be about 12.6 seconds per year. So the time dilation is not simply proportional to the gravitational acceleration.

You can get an approximate, but very good, relationship between the acceleration and the time dilation by using the Newtonian expression for the acceleration to get $r$ and substituting in the expression for the time dilation. Doing this gives:

$$ \sqrt{1 - \frac{2\sqrt{aGM}}{c^2}} $$

which I don't think is terribly illuminating!