<p>In the static weak-field approximation, the metric is approximated in terms of the the Newtonian gravitational potential $\Phi$ as:
$$\mathrm{d}s^2 = -\left(1+2\Phi\right)\mathrm{d}t^2 + (1-2\Phi)\mathrm{d}S^2\text{,}$$
where $\mathrm{d}S^2$ is the metric for Euclidean $3$-space. A more general situation requires one to solve a multi-body problem in general relativity, although the <a href="https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism" rel="nofollow">parametrized post-Newtonian formalism</a> provides a canonical approximation scheme. So the completely general answer to your question is "it's very difficult problem of numerical general relativity that doesn't have a neat analytic solution."</p>

<p>However, at large distances from slowly-moving objects (i.e., large compared to their Schwarzschild radii, we can approximate a multi-body time dilation relative to a stationary observer at infinity as just given by the sum of their gravitational potentials:
$$\frac{\mathrm{d}\tau}{\mathrm{d}t} = 1 - \frac{1}{c^2}\sum_k\frac{GM_k}{r_k} \text{.}$$
Note that this is consistent with the Schwarzschild formula for large radial coordinates, since by the Taylor-MacLaurin expansion,
$$\frac{\mathrm{d}\tau}{\mathrm{d}t} = \sqrt{1-\frac{2GM}{rc^2}} = 1 - \frac{GM}{rc^2} + \mathcal{O}\left(\frac{R^2}{r^2}\right)\text{,}$$
where $R = 2GM/c^2$ is the Schwarzschild radius. This also means that simply multiplying the factors is a pretty good approximation as long as those factors are close to $1$.</p>

CC BY-SA 3.0
2016-03-30T19:11:22.087Z
2016-03-30T19:11:22.087Z
Stan Liou
<p>I wondered how time would be dilated for an object in the middle of 2 black holes, however 
<a href="https://en.wikipedia.org/wiki/Gravitational_time_dilation" rel="nofollow">https://en.wikipedia.org/wiki/Gravitational_time_dilation</a> only provides a formula for one source of gravity</p>

<p>i tried approximating it by multiplying the time dilation factors, however when i used 1 single source with twice the mass the results differed from 2 objects with half the mass quite drastically</p>

<p>the examples:<br>
mass $1,00E+036$kg distance $3000000000$m<br>
time:$0,7105905766$</p>

<p>mass $2,00E+036$kg distance $3000000000$m<br>
time:$0,099387802$</p>

<p>approximation: $0,7105905766*0,7105905766=0,5049389676
$
which is off by a factor of 5 and would be even more off the closer the distance gets to the schwarzschild radius</p>

<p>so how can the time dilation of multiple objects be approximated ?</p>

<gravity><black-hole><time-dilation>
gravitational time dilation multiple sources
asdf
# gravitational time dilation multiple sources

(Asked by: username_0 on 2016-03-30)

I wondered how time would be dilated for an object in the middle of 2 black holes, however
<https://en.wikipedia.org/wiki/Gravitational_time_dilation> only provides a formula for one source of gravity

i tried approximating it by multiplying the time dilation factors, however when i used 1 single source with twice the mass the results differed from 2 objects with half the mass quite drastically

the examples:  
mass $1,00E+036$kg distance $3000000000$m  
time:$0,7105905766$

mass $2,00E+036$kg distance $3000000000$m  
time:$0,099387802$

approximation: $0,7105905766\*0,7105905766=0,5049389676
$
which is off by a factor of 5 and would be even more off the closer the distance gets to the schwarzschild radius

so how can the time dilation of multiple objects be approximated ?

**Tags:** gravity, black-hole, time-dilation


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## 1 answers

### Answer by username_1 on 2016-03-30. Score: 2 | Accepted answer

In the static weak-field approximation, the metric is approximated in terms of the the Newtonian gravitational potential $\Phi$ as:
$$\mathrm{d}s^2 = -\left(1+2\Phi\right)\mathrm{d}t^2 + (1-2\Phi)\mathrm{d}S^2\text{,}$$
where $\mathrm{d}S^2$ is the metric for Euclidean $3$-space. A more general situation requires one to solve a multi-body problem in general relativity, although the [parametrized post-Newtonian formalism](https://en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism) provides a canonical approximation scheme. So the completely general answer to your question is "it's very difficult problem of numerical general relativity that doesn't have a neat analytic solution."

However, at large distances from slowly-moving objects (i.e., large compared to their Schwarzschild radii, we can approximate a multi-body time dilation relative to a stationary observer at infinity as just given by the sum of their gravitational potentials:
$$\frac{\mathrm{d}\tau}{\mathrm{d}t} = 1 - \frac{1}{c^2}\sum\_k\frac{GM\_k}{r\_k} \text{.}$$
Note that this is consistent with the Schwarzschild formula for large radial coordinates, since by the Taylor-MacLaurin expansion,
$$\frac{\mathrm{d}\tau}{\mathrm{d}t} = \sqrt{1-\frac{2GM}{rc^2}} = 1 - \frac{GM}{rc^2} + \mathcal{O}\left(\frac{R^2}{r^2}\right)\text{,}$$
where $R = 2GM/c^2$ is the Schwarzschild radius. This also means that simply multiplying the factors is a pretty good approximation as long as those factors are close to $1$.


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