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If we perform a simple experiment by orbiting identical spacecraft around a non-rotating black hole and a non-rotating neutron star of equal mass at a "safe" distance, and ignore differences other than gravitation (e.g. magnetic fields and charge) would there be any differences in their orbits predicted by general relativity?

"Safe" distance means far from the event horizon of the black hole and where tidal forces don't disrupt the spacecraft.

After writing this answer I am now wondering if it's 100% correct.

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Birkhoff's theorem is very useful: in general relativity, if you are in vacuum and there is a spherically symmetric gravitational field, then it will be the Schwarzschild solution. This solution only depends on the mass, not on the size of the object. So the neutron star and the black hole will give rise to exactly the same orbits.

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    $\begingroup$ A question. If the mass of the black hole sufficed to made it a black hole, can a neutron star with the same mass exist? If yes, I suspect it depends on the mechanism of formation. $\endgroup$
    – Alchimista
    May 10 at 9:05
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    $\begingroup$ @Alchimista Good question. See en.wikipedia.org/wiki/… It's possible that there are black holes with mass under the TOV limit, given sufficient pressure. $\endgroup$
    – PM 2Ring
    May 10 at 9:29
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    $\begingroup$ And of course, you could have a primordial black hole that has ended up with the same mass as a neutron star. $\endgroup$ May 10 at 11:38
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    $\begingroup$ @AndersSandberg yes, I had to go through all of that before pushing the "post" button and I decided primordials were going to be my explanation if anybody asked, or in this case Gedankenprimordials ;-) $\endgroup$
    – uhoh
    May 10 at 12:50
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    $\begingroup$ @uhoh I put primordial BHs in the "given sufficient pressure" category. ;) $\endgroup$
    – PM 2Ring
    May 10 at 17:27
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If they were spinning they would be distinguishable (in principle), otherwise not.

Astrophysical black holes and neutron stars are expected to spin. In the case of a neutron star that automatically means that the mass/energy distribution is not spherically symmetric and therefore that the detail of the potential outside the surface depends on the detail of how the mass is arranged (e.g. the mass quadrupole and the spin octopole), which in turn depends on the equation of state of the neutron star, not just on its mass and angular momentum - see for example Pappas & Apostolatos (2013); Pappas (2017); Frutos-Alfaro (2018) for calculations of the metric outside spinning neutron stars.

For a spinning black hole, the potential is determined by the Kerr metric and simply depends on the total mass and angular momentum with no additional details required.

i.e. The spacetime metrics outside a spinning black hole and a spinning neutron star with the same $J$ and $M$ are different.

The difference in the metrics has observable consequences (in principle) for say the innermost stable circular orbit (ISCO), the orbital frequency at the ISCO and for orbital precession. For example, see the plot below (from Luk & Lin 2018), which shows there are big differences in the ISCO radius if the value of $J$ is large (in practice, I think this means rotation frequencies of $\sim 1$ kHz are required).

RISCO vs J

For non-spinning objects, Birkhoff's theorem applies, as described by Anders Sandberg, and there would be no distinction.

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Technically, there would be minute differences due to the gravitational field generated by the mass of the spacecraft tidally deforming the neutron star leading a small response in the gravitational field of the neutron star, which in turn effects the orbit of the spacecraft.

This tidal response is governed by the so called tidal Love numbers of the Neutron star. (The Love number for a a non-spinning black hole is zero).

However, this effect would be absolutely tiny for any reasonable spacecraft. Not only does it only appear at 5PN order, but it is also suppressed by the ratio of the mass of the spacecraft to the mass of the neutron star.

For more massive orbiting objects it does become relevant however. For example, it is the leading correction to the inspiral of a binary neutron star compared to a binary black hole of the same mass.

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    $\begingroup$ @mmeent and while I specified ""Safe" distance means... where tidal forces don't disrupt the spacecraft" I certainly left an opening for this answer :-) $\endgroup$
    – uhoh
    May 10 at 12:52
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    $\begingroup$ Yes please do, thank you! $\endgroup$
    – uhoh
    May 10 at 13:02

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