# Would studying massive stars that are experiencing time dilation due to proximity to a black hole be useful?

I saw this question (Could a star closely orbit a black hole long enough for the star to have lost 0.5B+ years to time dilation?) and came up with an interesting thought. Massive stars live for less than 100 million years, but if it is traveling at relativistic speeds ($$\sim 0.1c$$) by orbiting an SMBH, we could observe it for slightly longer. Would this be useful, or would it have to be the other way around?

NB: By "other way around," I mean an observer moving at $$\sim 0.5c$$ or so watching stars age and die.

## 1 Answer

I don't think this is feasible, because of two competing effects.

To have significant time dilation, the velocity needs to be large. The velocity in a Keplerian orbit of radius $$r$$ scales as $$r^{-1/2}$$, so the star would need to be close to the black hole.

Tidal forces from the black hole (the difference in gravitational force on the near side vs. far side of the star) scale as $$r^{-3}$$, so they get much stronger for orbits closer to the black hole. So there would be strong tidal forces trying to pull the star apart. Massive stars have outer layers that are only loosely bound in the first place (they are close to the Eddington luminosity) and thus have significant mass loss due to radiation pressure pushing on gas in their outer layers. Tidal forces would exacerbate this, so the star would quickly start shedding mass.

The mass loss would have (at least) two effects. First, it would generate a lot of luminosity as it heats up and spirals into the black hole; these accretion disks are in fact one of the ways that black holes are observed. So the star itself would probably be hard to observe separately. Second, the star would feel a drag from the gas in the accretion disk, and would spiral in toward the black hole, which would make the tidal forces stronger. It might not seem like the star should feel drag from the gas if both are orbiting at the same distance, and thus should have the same orbital velocity. But the gas disk is hot, so its internal pressure (or more fundamentally, its pressure gradient with $$r$$) partially supports it, allowing the gas to have a stable orbit with a slightly sub-Keplerian velocity. The star, as a macroscopic object, has to orbit with the regular Keplerian velocity, so it feels a "headwind" from the gas.

I would guess that complete tidal disruption of the star would happen before it inspirals all the way to the event horizon, but either way, I don't think it would last long.

EDIT: As @ProfRob points out in the comments, this analysis doesn’t work for supermassive black holes. They have very low densities inside the event horizon, and thus relatively weak tidal forces there.

• Excellent answer! Apr 22, 2021 at 2:23
• Surface gravity for a massive star is about ten times that of Earth. You can get arbitrarily close to the EH of a supermassive black hole (as specified in the Q) without serious issues due to tidal forces, providing the SMBH is of order $10^8$ solar masses (or more). Apr 22, 2021 at 5:53
• Ok, good point. Edited. Apr 22, 2021 at 12:09