# Could a star closely orbit a black hole long enough for the star to have lost 0.5B+ years to time dilation?

I was wondering how stable a close star-black hole system could plausibly be, and thus how much time a star could plausibly miss out on (from an outside observer's perspective) due to being in an orbit at relativistic speeds around a black hole.

E.g. could there plausibly be a scenario where a star has only aged less than 13 billion years, but is in fact one of the universes's earliest Population II stars, because it has been in a fairly stable, close, rapid orbit around a black hole for the past few billion years*, and thus was subject to time dilation during a few billion years of orbits?

(*Or was orbiting a BH at some point, and escaped due to a chance interaction with a newcomer third body that tossed it out.)

Perhaps this would require a larger black hole, so that the tidal forces would be lower, which would presumably causing less gas stripping for an orbit that would result in the same time dilation? I realize that even well outside the photon sphere of a black hole, there can be decay of orbits due to gravity waves, but evidently there can be a counteracting separating phenomenon well, due to stars being stripped of some mass (e.g. as described in this report of a star orbiting a black hole at a velocity of about %1 of c).

So to put a number on this to make it more answerable, could a star plausibly orbit a black hole long enough and closely enough without spiralling in (or outward due to gas stripping) to make it half a billion years older in origin (according to the CMB rest frame) than it has experienced?

## 2 Answers

In the comments of the other answer, the question came up whether decay of the orbit would limit the time amount of time dilation. The answer is of course yes. But by how much? This question can be answered using some modern results for the modelling extreme mass -ratio binaries in the limit of extremal spins.

The answer will depend crucially on ratio of the masses of the star and the supermassive black hole. So suppose, we take a supermassive black hole of $$10^9$$ solar masses with spin parameter $$a=0.999$$, and a solar mass star orbiting it in a quasicircular inspiral. The rate of decay of such an inspiral has been tabulated on the Black Hole Perturation Toolkit repository (See arXiv:1603.01221).

Using this information and the time dilation for circular orbits (also available form the BHPT), we can obtain the evolution of coordinate time as a function of proper time. The plot below plots the difference of the two, assuming the binary is currently at the last stable orbit. We see that the difference between the proper time experienced by the star and coordinate time experience by an asymptotic observer is about 1.2 Gyr over 13 Gyr (i.e. roughly the age of the universe). In particular, we see (somewhat to my surprise) that the 0.5 billions requested by the OP is entirely within the realm of possibility. (The 10+ billions suggest by the other answer however seems unreachable).

Yes, but not very likely.

The closest orbit that does not require constant expenditure of energy to maintain it is the prograde equatorial ISCO. For a Kerr black hole the time dilation factor on this orbit is $$\frac{dt}{d\tau}\approx \frac{2^{4/3}}{\sqrt{3}(1-a/M)^{1/3}},$$ which at the astrophysically likely Thorne limit $$a = 0.998M$$ gives a dilation of 11.5470. For a Schwarzschild black hole ($$a=0$$) it is merely 1.22. (Hamilton 2018, p. 654)

So a 13 billion year black hole could have a star orbiting that had just experienced 13/11.5470=1.12583 billion years!

Now, the gravitational wave emission from orbiting like this would no doubt sink the star. After all, stellar remnants starting at least part of an AU apart are merging today, and the emission scales up fast as one gets closer. Ergosphere processes might happen, but they also reduce $$a$$ and hence the time dilation.

• Please see this answer for an explanation of the spin parameter a, and a plot showing the spins of 19 supermassive black holes. – PM 2Ring Nov 2 '19 at 9:02
• GW emission would severely limit the length of time that can be spent in this regime. Even with the largest supermassive black hole at the lightest brown dwarf, GW emission would keep the the time spent in the regime where the redshift is significant well under a million years (probably by orders of magnitude). – mmeent Nov 2 '19 at 12:13
• Note: I accepted the other answer which discussed decay further, but this one seemed to provide a suitable answer, too. Unfortunately, I don't have the level of understanding necessary to determine with certainty which answer most deserved the acceptance; I clicked Accept on that one but upvoted both. If it had been possible, I would have split acceptance between the two. – Jacob C. Nov 19 '19 at 22:35