As part of a project for my degree I am writing code to simulate N-body gravitational interactions, however I have to then use this code to investigate something. Struggling to think of ideas I wondered whether black hole binary systems would be possible? I would only be investigating something to do with their formation from two dwarf galaxies containing single black holes, so could cut it off when they got too close but would need to run it while they were fairly close.

At what stage would I need to introduce corrections to simple Newtonian gravity, and how complicated/feasible is it to do so? Any advice would be great (or if you have any ideas for anything else to investigate, that would be amazing!)

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    $\begingroup$ en.wikipedia.org/wiki/Parameterized_post-Newtonian_formalism $\endgroup$ Apr 4, 2012 at 18:20
  • $\begingroup$ I saw that earlier but was hoping (probably futilely) that someone would be able to explain or link to somewhere explaining how best to implement it. $\endgroup$
    – Mithra
    Apr 4, 2012 at 20:05
  • $\begingroup$ It's a matter of how much detail you get. PPN gets pretty complicated even by the time you get enough detail that you get non-zero gravitational radiation (2.5th order). If you want to see a treatment of it, it's developed pretty well in Weinberg's Cosmology textbook and in MTW. If you want to see past numerical simulations using PPN, there is probably something that the LIGO consortium has put together on the topic, but I don't know where for sure. This is definitely what the numerical relativity community uses for far-orbit black holes, though. $\endgroup$ Apr 4, 2012 at 20:55

1 Answer 1


The corrections to Newtonian gravity are called post-Newtonian (not to confuse with PPN or parametrized post-Newtonian expressions used to compare modified theories of gravity).

The corrections are usually done by expanding Einsteins equation into series in $\dfrac{G}{c^2}$ or $\dfrac{v}{c}$, the latter being more common, as the expnasion parameter is dimensionless ($v$ stands for the velocities of the bodies). Such an expansion effectively produces a set of corrections to the newtonian forces. The corrections proportional to $(\dfrac{v}{c})^n$ and called $nPN$ corrections. To my knowledge, one can find in the literature the expressions for the forces for up to $3.5PN$ order.

Concerning feasibility, the higher the order $n$, the more complicated are the expressions. While $1PN$ corrections can be implemented almost "for free", the following orders may be described by the equations couple of pages long. These guys talk about more or less the same problem you are doing http://labs.adsabs.harvard.edu/ui/abs/2009ApJ...695..455B, and use up to $2.5PN$ corrections.

There might be more proficient triggers to turn on/off the correction terms, but as a good one may compare directly the expansion factors $(\dfrac{v}{c})^n$ to some threshold values, where $v$ here is the relative velocity of the black holes.

There is a huge body of relevant research to your project. If you are more interested in dynamics, there are many other factors, such as collisions with stars/interstellar medium(if relevant in dwarf galaxies)/dynamic friction/etc, which probably are much more important than PN corrections during most of the galaxy merger. However, if you want to have accurate descriprion of the black hole late inspiral, say in order to extract gravitational waveforms, including PN corrections is really important.

Practically, I would trace the factor $\dfrac{v}{c}$ during everyday simulations, check its values and then decide, whether PN corrections are needed. Alternatively, search the literature on ads.


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