Are gravitational force and gravitational time dilation proportional?

Question

Particles in gravitational fields are subject to gravitational time dilation. The closer a particle is near a gravitational source, the slower is running its clock. I would like to know more about the relation between gravity and gravitational time dilation.

In order to get a rough impression I used Newton's gravity equation (which may be used for weak fields, and I found that gravity and time dilation are (approximately) proportional: Can this result be confirmed on the base of Einstein's field equation (maybe even for stronger fields)?

dτ = proper time of a particle in the gravitation field of Earth, dt = proper time of an observer in infinity, rs = Schwarzschild radius of Earth, r = distance particle - center of Earth

Gravitational time dilation:

$\frac{dτ}{dt}=\sqrt{1- \frac{r_s}{r}} ≈ 1- \frac{r_s}{2r}$

Time dilation (difference):

$1-\frac{dτ}{dt} ≈ \frac{r_s}{2r}= \frac{GM}{c^2 r}$

Gravitational force (Newton's equation):

$F=G \frac{mM}{r^2}$

\begin{equation} \frac{Gravitational\:force}{Time\:dilation\:(difference)} ≈ \frac{G\frac{mM}{r^2}}{\frac{GM}{c^2 r}}=\frac{mc^2}{r}=\frac{rest\:energy\:(of\:the\:particle\:subject\:to\:time\:dilation)}{distance\:(of\:the\:particle)} \end{equation}

(As a result, time dilation would be approximately gravitaty, divided by the rest energy of the particle, multiplied by its distance.)

Answer 1

If you have a look at my answer to Deriving a Schwarzschild radius using relativistic mass I discuss how the weak field approximation gives us an approximate metric for the Newtonian gravitational potential $\phi$:

$$ ds^2 \approx -\left( 1 + \frac{2\phi}{c^2}\right) c^2dt^2 + \frac{1}{1 + 2\phi/c^2}\left(dx^2 + dy^2 + dz^2\right) $$

To extract time dilation from this we take a stationary object, so $dx = dy = dz = 0$ and use the relationship between the line element and the proper time $ds^2 = -c^2d\tau^2$ to get:

$$ \frac{d\tau}{dt} \approx \sqrt{ 1 + \frac{2\phi}{c^2}} $$

To clarify this, take two observers $A$ and $B$ with gravitational potential energies $\phi_A$ and $\phi_B$, then the equation tells us that the elapsed times recorded by $A$ and $B$ are related by:

$$ \frac{dt_A}{dt_B} \approx \sqrt{ 1 + \frac{2(\phi_A - \phi_B)}{c^2}} $$

This equation is only valid when $2\Delta\phi/c^2 \ll 1$, in which case we can use the binomial expansion:

$$ \frac{dt_A}{dt_B} \approx 1 + \frac{\Delta\phi}{c^2} + \text{higher terms} $$

and dropping the higher terms and rearranging:

$$ \frac{dt_A - dt_B}{dt_B} \approx \frac{\Delta\phi}{c^2} $$

And this is sort of what you describe. Remember that the potential energy $\phi$ is the potential energy per unit mass, so if we multiply the top and bottom of the right side by the mass to get the total potential energy $\Phi$ we get:

$$ \frac{dt_A - dt_B}{dt_B} \approx \frac{\Delta\Phi}{mc^2} $$

which is indeed the gravitational potential divided by the rest energy.

But this is an approximation that works (reliably) only in the weak field limit. As it happens the weak field expression works for any values of $r$ in the Schwarzschild metric, but as discussed in the linked question this is an accidental coincidence and can't be relied on.

Answer 2

They don't have to be related.

For instance if you have a hollow spherical shell of matter then the inside of the sphere is a flat spacetime region and it has the same time dilation as the shell.

But since the inside is flat, there is no gravitational force inside the shell. Yet there is time dilation.