Mass & Weight Impact on Spiraling Into a Black Hole

[Hornbein]

TL;DR Summary

Does the mass make any difference as to the rate of inspiraling?

Suppose you have two small masses spiraling into a black hole in similar trajectories. Does the mass of these small masses make any difference to the rate of inspiraling? What if one is a thousand times more massive than the other...

 

[PeroK]

TL;DR Summary: Does the mass make any difference as to the rate of inspiraling?

Suppose you have two small masses spiraling into a black hole in similar trajectories. Does the mass of these small masses make any difference to the rate of inspiraling? What if one is a thousand times more massive than the other...

The masses would need to be significant compared to the mass of the black hole. Otherwise, they would be considered "test" masses and too small to significantly affect the local spacetime geometry.

 

[anuttarasammyak]

In ordinary treatment geodesics which BH decides is considered for motion of falling objects. The effect of falling objects mass is neglected.

 

[Ibix]

It's worth noting that an orbiting body will not in general spiral in unless it is sufficiently massive and fast moving to be emitting significant gravitational radiation. In that case the mass does matter because it affects the emitted power.

Test bodies' orbits are not affected by their mass. This is an approximation analogous to the Newtonian calculation where we assume the Sun (or whatever primary we are considering) is stationary, rather than orbiting a barycenter.

 

[PeroK]

It's worth noting that an orbiting body will not in general spiral in unless it is sufficiently massive and fast moving to be emitting significant gravitational radiation. In that case the mass does matter because it affects the emitted power.

I assume you mean spiralling in from an otherwise stable orbit?

 

[Ibix]

I assume you mean spiralling in from an otherwise stable orbit?

There are orbits that terminate inside the black hole, yes, but I'm not sure if they really qualify as 'spiralling in' - more like 'plunging in' I think. I know you can orbit a couple of times just above the photon sphere and escape on a freefall path. I don't know how many orbits you can get if you dip below it. Maybe if you're doing nearly light speed in the same direction as the rotation of a Kerr black hole and you cross the photon sphere nearly tangentially you can orbit a few times? I don't know.

 

[pervect]

TL;DR Summary: Does the mass make any difference as to the rate of inspiraling?

Suppose you have two small masses spiraling into a black hole in similar trajectories. Does the mass of these small masses make any difference to the rate of inspiraling? What if one is a thousand times more massive than the other...

My recollection is that it depends on the details. If one of the masses is a significant fraction of the mass of the black hole, the system loses energy and there can be an inspiral for this reason. And in this case the mass ratios matter. It sounds like in your case the masses are small and there would be negligible difference.

Objects with negligible masses can also spiral into a black hole if they don't have enough angular momentum. https://www.fourmilab.ch/gravitation/orbits/ has the equations and a graphic illustration, but it might be a bit tricky to use. A very short hilight - to use the application, set the angular momentum value and other parameters, then click on the yellow "effective potential" diagram to set the starting position and start the simulation.

The effective potential method is a method that reduces the 2d problem to an equivalent 1d problem, as per the wiki https://en.wikipedia.org/wiki/Effective_potential. It's basically the differential equation of motion for the 1 dimesnaionl problem, but rather than writing down the diffential equation, one computes and/or displays a potential that would have the same result. One imagines a ball rolling down the hill of the effective potential diagram - or more literally, a frictionless ball sliding along the hill illustrated by the diagram. The effective potential method originated with the Newtonian problem, but there is a GR version applicable to the GR equations of motion for a test mass as well. Note that the method assumes that there is no energy loss due to gravitational radiation. Thus the effective potential method isn't good for the inspiral problem where the inspiral occurs because of energy radiated away by gravitational radiation - rather, it is applicable to an unstable orbit with insufficient angular momentum.

Also note that there is no such minimum angular momentum in the Newtonian orbital case, the minimum angular momentum is GR specific.

Finally, note that the GR equation and diagram uses slightly different parameters to describe the physics that the Newtonian version. The radial coordiante "r" in the GR case is the Schwarzschild r coordinate, which is different than distance. It is still true though that as the r coordiante approaches the Schwarzschild radius, the infalling object approaches the event horizon, but there are confusing singularities in the Schwarzschild r coordinate near the event horizon. The time coordinate for the GR problem is the proper time tau, which is the time that would be read on an idealized clock or idealized wristwatch carried on the infalling mass, rather than a coordinate time.

 

[PAllen]

Of course test particles are an abstraction. But not by much for e.g. the Earth sun orbit. The Earth sun orbit emits around 200 watts of GW per simple computation. If all other enormously larger effects were excluded, this would mean the Earth would eventually spiral into the sun in about a trillion trillion years i.e. a hundred trillion times the age of the universe

 

[Nugatory]

Spiraling in will be seen in the accretion disks around stellar black holes. Orbiting matter in the disk loses energy by electromagnetic radiation and just plain ordinary friction.