<p>If you're asking whether it's sufficient to use a retarded (time-delayed) positions to calculate gravitational forces, then no, that would be much worse than Newtonian gravity. For example, that would predict that the Earth should spiral into the Sun on the order of about 400 years. See also answers to </p>

<ul>
<li><a href="https://space.stackexchange.com/q/21467/12102">Besides retarded gravitation, anything else to worry about when calculating MU69's orbit from scratch?</a></li>
<li><a href="https://astronomy.stackexchange.com/q/5869/7982">Is the influence of gravity instantaneous?</a></li>
</ul>

<hr>

<blockquote>
  <p>Most small-scale N-body simulations (e.g., planetary systems, solar system, stellar clusters, ...) use classical Newtonian gravity.</p>
</blockquote>

<p>This is true, but some simulations use relativistic corrections.</p>

<p>The post-Newtonian expansion works in order <span class="math-container">$\epsilon\sim v^2\sim U$</span>, where <span class="math-container">$U$</span> is (the negative of) Newtonian potential, and including the first-order post-Newtonian correction to the N-body equations of motion forms the <a href="https://en.wikipedia.org/wiki/Einstein%E2%80%93Infeld%E2%80%93Hoffmann_equations" rel="noreferrer">Einstein–Infeld–Hoffmann equations</a>. Because they contain <span class="math-container">$3$</span>-body terms, the simulation would now be <span class="math-container">$\mathcal{O}(N^3)$</span> instead of the Newtonian simulation <span class="math-container">$\mathcal{O}(N^2)$</span>.</p>

<p>The post-Newtonian three-body equations are known to 2PN, while the two-body problem to 3.5PN. Effects beyond the first order can be important in e.g. binary simulations because gravitational radiation losses are seen at 2.5PN order. However, they are not translatable to an N-body problem.</p>

<p>A technique of making the EIH equations more tractable by making a second expansion that depends on the problem at hand can be found in Will (2014), which is applicable when most the relativistic dynamics are dominated by a few contribution (e.g., a supermassive central black hole in galaxy simulations).</p>

<blockquote>
  <p>Most large-scale N-body simulations (e.g., galaxy clusters, ...) use general relativity.</p>
</blockquote>

<p>As <strong>Walter</strong> says, most of them use Newtonian gravity on a expanding background. Not necessarily directly computer particle-particle Newtonian force, but perhaps modeling Poisson's equation through mesh methods, or perhaps some combination thereof (e.g., direct force calculations for close particles, mesh methods for larger scales).</p>

<blockquote>
  <p>My question: if I were to make a <strong>large-scale</strong> N-body simulation using <strong>Newtonian</strong> gravity, would it be enough to "correct" that to first-order by modifying the equations of motion to take into account the fact that gravitational fields propagate at the speed of light?</p>
</blockquote>

<p>If you discount cosmological expansion, it will simply be wrong. However, first-order corrections to the Newtonian gravity part of simulations do exist in the <em>post-Friedmann formalism</em>. I'm not really familiar with PFF, but according to Bruni <em>et al.</em> (2014), the leading-order correction to Newtonian gravity introduces a vector potential in the <span class="math-container">$g_{0i}$</span> term in the metric that can produce weak lensing effects but <em>doesn't</em> affect matter dynamics. Which is entirely sensible because that kind of gravitomagnetic effects should be suppressed by a further <span class="math-container">$v/c$</span> term for matter, just like magnetism.</p>

<p>For references on how low order GR corrections to Newtonian gravity simulations are often made, see answers to <a href="https://space.stackexchange.com/q/23408/12102">How to calculate the planets and moons beyond Newtons's gravitational force?</a></p>

<hr>

<p>References:</p>

<ol>
<li>Will, C. M. "Incorporating post-Newtonian effects in N-body dynamics". Phys. Rev. D <strong>89</strong>, 044043 (2014) [arXiv:<a href="http://arxiv.org/pdf/1312.1289v3" rel="noreferrer">1312.1289</a>]</li>
<li>Bruni, M., Thomas, D. B., Wands, D. "Computing general-relativistic effects from Newtonian N-body simulations: Frame dragging in the post-Friedmann approach." Phys. Rev. D <strong>89</strong>, 044010 (2014) [arXiv:<a href="http://arxiv.org/abs/1306.1562" rel="noreferrer">1306.1562</a>]</li>
</ol>

CC BY-SA 4.0
2015-02-06T19:02:19.023Z
2019-07-13T00:47:48.440Z
2019-07-13T00:47:48.440Z
Stan Liou
<p>Most small-scale N-body simulations (e.g., planetary systems, solar system, stellar clusters, ...) use  classical Newtonian gravity. </p>

<p>Most large-scale N-body simulations (e.g., galaxy clusters, ...) use general relativity.</p>

<p>My question: if I were to make a <strong>large-scale</strong> N-body simulation using <strong>Newtonian</strong> gravity, would it be enough to "correct" that to first-order by modifying the equations of motion to take into account the fact that gravitational fields propagate at the speed of light?</p>

CC BY-SA 3.0
2015-02-06T09:36:23.033Z
*Most large-scale N-body simulations (e.g., stellar clusters, galaxies, galaxy clusters, ...) use general relativity* **Wrong** For up to galaxy scales, one safely uses Newtonian gravity. For larger scales (so-called cosmological $N$-body simulations), one uses GR only to describe the (assumed) smooth background (universal expansion), but Newtonian physics otherwise.
Walter
CC BY-SA 3.0
2015-02-06T10:56:42.713Z
@Walter true, edited. As I understand it, this is justified for galaxies because the time scales involved in galactic evolution dwarfs the travel time of gravity through the galaxy, so any effect is negligible. Still! My question stands...Perhaps a good follow-up question is then: how to quantify the error between the models; plain Newtonian, corrected Newtonian, and GR?
Rody Oldenhuis
<general-relativity><n-body-simulations><newtonian-gravity>
Does one need to take into account finite gravity speed in N-body simulations?
Rody Oldenhuis
# Does one need to take into account finite gravity speed in N-body simulations?

(Asked by: username_0 on 2015-02-06)

Most small-scale N-body simulations (e.g., planetary systems, solar system, stellar clusters, ...) use classical Newtonian gravity.

Most large-scale N-body simulations (e.g., galaxy clusters, ...) use general relativity.

My question: if I were to make a **large-scale** N-body simulation using **Newtonian** gravity, would it be enough to "correct" that to first-order by modifying the equations of motion to take into account the fact that gravitational fields propagate at the speed of light?

**Tags:** general-relativity, n-body-simulations, newtonian-gravity
> **Comments:**
> **username_1**: *Most large-scale N-body simulations (e.g., stellar clusters, galaxies, galaxy clusters, ...) use general relativity* **Wrong** For up to galaxy scales, one safely uses Newtonian gravity. For larger scales (so-called cosmological $N$-body simulations), one uses GR only to describe the (assumed) smooth background (universal expansion), but Newtonian physics otherwise.  
> **username_0**: @Walter true, edited. As I understand it, this is justified for galaxies because the time scales involved in galactic evolution dwarfs the travel time of gravity through the galaxy, so any effect is negligible. Still! My question stands...Perhaps a good follow-up question is then: how to quantify the error between the models; plain Newtonian, corrected Newtonian, and GR?  

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## 1 answers

### Answer by username_2 on 2015-02-06. Score: 9 | Accepted answer

If you're asking whether it's sufficient to use a retarded (time-delayed) positions to calculate gravitational forces, then no, that would be much worse than Newtonian gravity. For example, that would predict that the Earth should spiral into the Sun on the order of about 400 years. See also answers to

* [Besides retarded gravitation, anything else to worry about when calculating MU69's orbit from scratch?](https://space.stackexchange.com/q/21467/12102)
* [Is the influence of gravity instantaneous?](https://astronomy.stackexchange.com/q/5869/7982)

---

> Most small-scale N-body simulations (e.g., planetary systems, solar system, stellar clusters, ...) use classical Newtonian gravity.

This is true, but some simulations use relativistic corrections.

The post-Newtonian expansion works in order $\epsilon\sim v^2\sim U$, where $U$ is (the negative of) Newtonian potential, and including the first-order post-Newtonian correction to the N-body equations of motion forms the [Einstein–Infeld–Hoffmann equations](https://en.wikipedia.org/wiki/Einstein%E2%80%93Infeld%E2%80%93Hoffmann_equations). Because they contain $3$-body terms, the simulation would now be $\mathcal{O}(N^3)$ instead of the Newtonian simulation $\mathcal{O}(N^2)$.

The post-Newtonian three-body equations are known to 2PN, while the two-body problem to 3.5PN. Effects beyond the first order can be important in e.g. binary simulations because gravitational radiation losses are seen at 2.5PN order. However, they are not translatable to an N-body problem.

A technique of making the EIH equations more tractable by making a second expansion that depends on the problem at hand can be found in Will (2014), which is applicable when most the relativistic dynamics are dominated by a few contribution (e.g., a supermassive central black hole in galaxy simulations).

> Most large-scale N-body simulations (e.g., galaxy clusters, ...) use general relativity.

As **Walter** says, most of them use Newtonian gravity on a expanding background. Not necessarily directly computer particle-particle Newtonian force, but perhaps modeling Poisson's equation through mesh methods, or perhaps some combination thereof (e.g., direct force calculations for close particles, mesh methods for larger scales).

> My question: if I were to make a **large-scale** N-body simulation using **Newtonian** gravity, would it be enough to "correct" that to first-order by modifying the equations of motion to take into account the fact that gravitational fields propagate at the speed of light?

If you discount cosmological expansion, it will simply be wrong. However, first-order corrections to the Newtonian gravity part of simulations do exist in the *post-Friedmann formalism*. I'm not really familiar with PFF, but according to Bruni *et al.* (2014), the leading-order correction to Newtonian gravity introduces a vector potential in the $g\_{0i}$ term in the metric that can produce weak lensing effects but *doesn't* affect matter dynamics. Which is entirely sensible because that kind of gravitomagnetic effects should be suppressed by a further $v/c$ term for matter, just like magnetism.

For references on how low order GR corrections to Newtonian gravity simulations are often made, see answers to [How to calculate the planets and moons beyond Newtons's gravitational force?](https://space.stackexchange.com/q/23408/12102)

---

References:

1. Will, C. M. "Incorporating post-Newtonian effects in N-body dynamics". Phys. Rev. D **89**, 044043 (2014) [arXiv:[1312.1289](http://arxiv.org/pdf/1312.1289v3)]
2. Bruni, M., Thomas, D. B., Wands, D. "Computing general-relativistic effects from Newtonian N-body simulations: Frame dragging in the post-Friedmann approach." Phys. Rev. D **89**, 044010 (2014) [arXiv:[1306.1562](http://arxiv.org/abs/1306.1562)]


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