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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.)

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  • $\begingroup$ Hi Moonraker. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. $\endgroup$ – Qmechanic Mar 31 '16 at 10:27
  • $\begingroup$ Hi @Qmechanic, thank you for your information, and I did read the page you indicated. - I don't know why you think that my question falls under this category. My question is clearly indicated in the title, and there is no other question: Is there some relation of proportionality between gravitational force and gravitational time dilation. I could even reformulate my question without the formulas I provided (if you wish so). I just need the information about Einstein field equations, nothing more. $\endgroup$ – Moonraker Mar 31 '16 at 11:56
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    $\begingroup$ Also, for future reference, please don't delete and repost questions. Instead, if they're not well received, you should edit them to improve them. $\endgroup$ – David Z Mar 31 '16 at 12:36
  • $\begingroup$ @David Z♦ : OK, I undeleted the former question. Thank you for this formal comment, I hope I fixed it. I would be keen on knowing your comment on the topic of my question? Sincerely $\endgroup$ – Moonraker Mar 31 '16 at 13:23
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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.

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  • $\begingroup$ Nice, interesting explanations, but you are far from answering my question! You confirmed that my rough calculation with Newton's equation yields approximately correct results for weak fields. I know this, even if I could not explain it with Schwarzschild metrics as you did. But my calculations (and yours also) induce the idea that gravitational force and gravitational time dilation are proportional, with a factor of proportionality equal to the rest energy of the particle divided by the distance. - Are they proportional or not? $\endgroup$ – Moonraker Mar 31 '16 at 6:48
  • $\begingroup$ @Moonraker: I thought that was obvious from my final equation. In the weak field limit the relative time dilation is proportional to the gravitational potential. However this is only true in the weak field limit so it works for calculating the time dilation of geostationary satellites but not for calculating the time dilation near a black hole. $\endgroup$ – John Rennie Mar 31 '16 at 7:08
  • $\begingroup$ My question is not for purposes of technological applications, but for exploring the nature of gravitational time dilation and gravity (even in stronger fields). Your final approximate equation does not include more insight than my final approximate equation, Newton's equation is appropriate for weak fields. $\endgroup$ – Moonraker Mar 31 '16 at 8:09
  • $\begingroup$ I appreciate that you showed the Schwarzschild approximation, but there might be other insights deriving from Einstein's field equation which are putting some light on this supposed proportionality. Mass is producing gravity, and mass is producing gravitational time dilation - may be there is a direct relation between both of them?? $\endgroup$ – Moonraker Mar 31 '16 at 8:09
  • $\begingroup$ @Moonraker: I have to confess that I don't understand what you are getting at. The linear proportionality is a weak field phenomenon and is an inevitable consequence of the fact that in the weak field limit GR must reproduce Newtonian gravity. Outside the weak field limit the time dilation is obviously related to the mass (more precisely the stress-energy tensor) but remember you can choose any coordinate system you want and the time dilation will depend on the coordinates you choose, so the relationship is a complicated one. $\endgroup$ – John Rennie Mar 31 '16 at 8:22
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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.

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