Can an object fall into a black hole at faster that the speed of light?

[Herbascious J]

TL;DR

I am wondering if an object can fall into a blackhole at faster that the speed of light. I have heard that the expansion of the universe can make distant galaxies appear to recede faster than the speed of light. Can a black hole also cause geodesics to appear faster than c?

I am wondering if an object can fall into a black hole at faster that the speed of light. I have heard that the expansion of the universe can make distant galaxies appear to recede from one another at velocities faster than the speed of light.



Intuitively, this makes sense to me. I am assuming that according to GR the expansion of space time is a metric expansion of space itself and therefore the galaxies are simply receding due to expansion and not due to local velocities (in a sense it’s as if they are standing still as space grows). 



But here is my confusion. If GR allows for the expansion of space to behave like this, is it not also true for any object freely following a geodesic in space-time. Aren’t even orbiting objects kind of ‘standing still’ as well and it’s just the curvature of space-time that makes them appear to gain velocity as they ‘fall’ through a gravitational field.

If GR allows this on the large scale, it seems that if a black hole were massive enough, it might curve space enough to bring objects past light speed as well. I understand that a local velocity is very different from a distant velocity, but I can’t intuitively imagine at what point things change and how. Thank you for any clarification.

 

[PAllen]

Every time you say velocity, you have to say in relation to what. Further, in GR as opposed to SR, relative velocity is only unambiguous for two nearby bodies. With this in mind, NO, there is no possible nearby test body relative to which something falling into a BH can move faster than light.

As for distant galaxies, recession rate is a completely different quantity than relative velocity. Further, it is a coordinate dependent quantity, and if you set up cosmological type coordinates in flat spacetime you end up with arbitrarily large recession rates. In contrast, though relative velocity at a distance is ambiguous in GR, it is actually never greater than c.

On this last point I’ll get a little technical: relative velocity involves comparing vectors; if the vectors are distant, you must bring them together to compare. The process of bringing them together without rotating them is called parallel transport. In flat spacetime, the result of this operation is independent of the path over which they are transported, so distant vector comparison is unambiguous in flat spacetime. In curved spacetime, the result depends on the path used. This is the precise reason for the ambiguity. However, no matter what path is used, you never end up with greater than c relative velocity.

 

[Herbascious J]

So, to clarify, is it correct to say that local objects never have a relative velocity from one another above c, and I also imagine a distant region, which also has local objects confined to c, but that these two distant regions appear to be receding faster than c. So, is it then impossible for me to 'measure' the velocity of a very distant object that would appear to be traveling close to my velocity. I hope this last question is clear. I'm imagining that this distant object is traveling so fast toward me as to be outpacing the expansion of space, which I assume would violate GR and the speed of light speed limit. Thank you.

 

[PAllen]

Well it all depends what you mean by measuring a distant velocity. The whole point is that in GR, this cannot have any unambiguous meaning. However I’ll answer a different question that is perfectly well defined because it only deals with measured quantities. In any galaxy, however far away, it is possible for there to be body in it, moving less than c relative to that galaxy (towards you) such that you observe that body to have neither red shift nor blueshift. This much is unambiguous physics.

Now you can choose to define that this absence of observed frequency shift means that the velocity of this body e.g. a billion years ago, relative to you now, is zero. However this is just an arbitrary definition.

 

[Herbascious J]

If I may press a little further, if I am a telescope observer with the right equipment, I observer a galaxy that is very far, far enough to have a redshit that implies a recession velocity that is significantly higher than c (in step with normal cosmological expansion). Perhaps this redshift implies a velocity more than 2x c (or not. I don't how important that would be). I observe a star 'passing through' that galaxy and it has no shift in it's light. From a Newtonian point of view, I think to myself that the galaxy is moving more quickly than c, but that the star moving toward me (or through the galaxy really) is in fact standing still relative to me. I would conclude the object is also passing through the galaxy at faster than the speed of light (in fact it's velocity to the galaxy would be same as my velocity to the galaxy because the star seems still to me). How do I interpret in GR an extreme red-shift discrepancy between two objects that appear to be local to one another, but far from me. The light from them has traveled quite far, but I assume that the light from each object left at the same time and has arrived at the same time, each with their own red-shift. Even though I think the velocity difference is huge between them, they do not measure a velocity above c?

 

[PAllen]

I think you didn’t get something from an earlier post of mine. Recession rate is a different quantity than relative velocity. Distant galaxies have superluminal recession rates, but (ambiguously) subluminal relative velocity relative to us.

How is this possible? Recession rate is defined as growth in distance (with a coordinate dependent definition) divided by proper time of a comoving observer. Let us consider a simple analog in special relativity. A rocket is traveling at .999c velocity relative to the Earth on the way to alpha Centauri. If we divide the distance traveled in our coordinates by the proper time of travel for the rocket, we will get a rate of over 22 times c. This is analogous to recession rate. Meanwhile, the relative velocity of the rocket to Earth is .999c.

 

[Herbascious J]

Thank you for the answers and discussion! I will post my follow up question as a separate thread, as I think it belongs in a new post.

 

[stevebd1]

I am wondering if an object can fall into a black hole at faster that the speed of light. I have heard that the expansion of the universe can make distant galaxies appear to recede from one another at velocities faster than the speed of light.



Intuitively, this makes sense to me. I am assuming that according to GR the expansion of space time is a metric expansion of space itself and therefore the galaxies are simply receding due to expansion and not due to local velocities (in a sense it’s as if they are standing still as space grows). 



But here is my confusion. If GR allows for the expansion of space to behave like this, is it not also true for any object freely following a geodesic in space-time. Aren’t even orbiting objects kind of ‘standing still’ as well and it’s just the curvature of space-time that makes them appear to gain velocity as they ‘fall’ through a gravitational field.

If GR allows this on the large scale, it seems that if a black hole were massive enough, it might curve space enough to bring objects past light speed as well. I understand that a local velocity is very different from a distant velocity, but I can’t intuitively imagine at what point things change and how. Thank you for any clarification.

You might find this old blog post regarding various types of radial infalling plungers into a BH in various reference frames of interest (velocity is towards the bottom) -

https://www.physicsforums.com/threads/properties-of-in-falling-radial-plungers.768338/