# B What (if anything) limits the speed of something falling into a black hole?

#### jerromyjon

I became uncertain of my understanding of the physics after the plane starts to descend.

What I imagine happens is that your forward velocity would remain constant and you would be accelerated towards the Earth at about 9.8m/s2. The part I am most uncertain about is that since you are isolated from the air resistance in the atmosphere you would continue to increase velocity without bound until you pull up or crash.

Assuming I have that correct my next question is what would limit the velocity of a mass approaching a black hole in a vacuum? I know matter cannot be accelerated to c, so what in physics describes the "terminal velocity" of mass? Is it simply the curvature of spacetime which limits the speed?

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#### mfb

Mentor
The part I am most uncertain about is that since you are isolated from the air resistance in the atmosphere you would continue to increase velocity without bound until you pull up or crash.
Without relevant bound, as long as relativistic effects are negligible.
Assuming I have that correct my next question is what would limit the velocity of a mass approaching a black hole in a vacuum?
It will approach c towards the event horizon (in suitable coordinate systems). That is the definition of the event horizon - the place where the escape velocity reaches c.
Inside, radial speed is not a very useful concept.

#### Dale

Mentor
Is it simply the curvature of spacetime which limits the speed?
No, the limit of c occurs in flat spacetime also. It is not due to curvature

#### jerromyjon

No, the limit of c occurs in flat spacetime also. It is not due to curvature
Yes, I understand that no mass can be accelerated to c, even in the absence of gravity. I was specifically curious about how GR relates to the limit, which by what mfb says the event horizon would be the boundary where the curvature meets the velocity limit, if I understand that correctly.

#### sophiecentaur

Gold Member
what would limit the velocity of a mass approaching a black hole in a vacuum?
I think you have taken a step too far here by trying to extend a simple classical model. The reason for reaching terminal velocity in a normal atmosphere in non-relativistic conditions is that the molecular thermal motion of the air molecules 'beats' the gravitational attraction. A large, heavy object accelerates until the resistance force balances its weight. Why not just assume that, at some stage on the way down towards a black hole, there will be an equivalent 'atmosphere' in which molecules are kept aloft due to thermal effects and would provide some resistance to a large falling body?
You have a choice - you either take a classical model and increase the numbers to get a meaningless answer out of the process or you do the full analysis. I find many of these 'What if?" type questions provide very little actual enlightenment about advanced subjects. They are false friends because they devalue the subject.

#### Dale

Mentor
I was specifically curious about how GR relates to the limit,
In GR the limit is a local limit. It is only valid in local inertial frames, which are by definition both free falling and small enough that curvature is negligible.

the event horizon would be the boundary where the curvature meets the velocity limit,
The curvature can be arbitrarily small at the event horizon if the black hole is arbitrarily large. Curvature is essentially tidal gravity, and a very large BH will have an event horizon with small tidal forces

#### jerromyjon

The curvature can be arbitrarily small at the event horizon if the black hole is arbitrarily large. Curvature is essentially tidal gravity, and a very large BH will have an event horizon with small tidal forces
So you are saying matter could be pulled into a black hole at v<<c?

#### jartsa

Assuming I have that correct my next question is what would limit the velocity of a mass approaching a black hole in a vacuum? I know matter cannot be accelerated to c, so what in physics describes the "terminal velocity" of mass? Is it simply the curvature of spacetime which limits the speed?
When radiation falls into a black hole it blue shifts, which means its energy increases. At the event horizon the energy has increased to infinity. The speed of the radiation is c, regardless of the energy of the radiation.

When matter falls into a black hole its energy increases. At the event horizon the energy has increased to infinity. The speed of the matter is c when the energy is infinite.

#### Dale

Mentor
So you are saying matter could be pulled into a black hole at v<<c?
That is not what I was saying, but it is true.

#### jerromyjon

That is not what I was saying, but it is true.
I've been reading and I think I was just using the wrong term... curvature as you say is just a small part of GR, the tensors are the main "attractive force", is that right?

#### Nugatory

Mentor
So you are saying matter could be pulled into a black hole at v<<c?
Yes, although this would be a good time to ask the most basic sanity-check question of all: What is v relative to?

#### Nugatory

Mentor
To the black hole of course.
I'm not sure whether the smiley is because you think that answer is obvious, or because you think it is obvious why that answer is meaningless.

(Saying that a velocity is relative to something is essentially stating the velocity in coordinates in which the spatial coordinates of all events on the timelike worldline of that something are constant. Doing that when the something is "the black hole" is tricky).

#### jerromyjon

because you think it is obvious why that answer is meaningless.
I'm not trying to compare manifolds or reference frames or anything that serious, I just want to have a basic conceptual understanding of "things that occur in nature"... I'm certainly not anywhere near the level to care how the velocity has to relate to something computable. Able to be calculated. Is there any other reason it matters?

#### Dale

Mentor
curvature as you say is just a small part of GR, the tensors are the main "attractive force", is that right?
Hmm, this is a very jumbled question. Tensors are just a general class of mathematical objects, conceptually they are a generalization of vectors. So I wouldn't say that tensors are an attractive force, but all of the physically important things in GR are mathematical represented as tensors.

At this point, you may be better served by reading a coherent presentation of the material, like Carroll's lecture notes on GR.

#### Nugatory

Mentor
I'm not trying to compare manifolds or reference frames or anything that serious, I just want to have a basic conceptual understanding of "things that occur in nature"...
Of course you aren't... but you still have to say what the velocity is relative to and to do that you have to suggest some stationary object.

I'm thinking that when you you said "relative to the black hole" you actually meant "relative to some observer hovering at some distance from the black hole".

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#### jerromyjon

So I wouldn't say that tensors are an attractive force
I didn't mean that.
but all of the physically important things in GR are mathematical represented as tensors.
Yeah, I got that now, thanks!

At this point, you may be better served by reading a coherent presentation of the material, like Carroll's lecture notes on GR.
Yeah I've tried numerous approaches and I always get bogged down trying to comprehend the math. My mind doesn't work that way, so I'm going my own way about it trying to start with "how gravity functions", then how GR models it, then the math...
"relative to some observer hovering at some distance from the black hole".
Yeah, if that works, I'll volunteer!

#### jerromyjon

Yeah, if that works, I'll volunteer!
* Writes make-believe paper * "On the "trippyness" of frame dragging"

#### jerromyjon

When matter falls into a black hole its energy increases. At the event horizon the energy has increased to infinity. The speed of the matter is c when the energy is infinite.
Matter couldn't possibly have an infinite quantity of energy, could it? I was thinking more along the lines of approaching infinity, but I can't really believe it's possible, just like it can't actually attain lightspeed, right?

#### Dale

Mentor
Yeah I've tried numerous approaches and I always get bogged down trying to comprehend the math.
Have you tried Sean Carroll's lecture notes? The first two chapters should be enough.

This is a reasonable approach, but it may be very difficult for other people to provide this type of information to you. How gravity functions is so well and precisely described by the math, and English simply is not built to describe it precisely.

#### jartsa

Matter couldn't possibly have an infinite quantity of energy, could it? I was thinking more along the lines of approaching infinity, but I can't really believe it's possible, just like it can't actually attain lightspeed, right?

Bob is falling into a black hole. At the event horizon Bob passes a thing that is hovering at the event horizon. What is Bob's speed relative to the hovering thing?

Only thing that can hover at the event horizon is a light pulse heading straight up. So the "hovering thing" must be that thing. And now we know how fast Bob passes the event horizon.

#### jerromyjon

Sean Carroll's lecture notes
Yeah the preface sounded encouraging but for someone who has never gotten past basic algebra in school, I get lost quickly. (I was programming machine code out of my head before then)

What would work really nicely would be a simple "theory of operation" explaining in laymen's terms how it works. I like equation 1.1 s2 = (∆x)2 + (∆y)2 . Pythagorean theorem, I get that much. I know how sine and cosine work. I'm getting into tangents. Everything makes more sense the further I get.

#### jerromyjon

Only thing that can hover at the event horizon is a light pulse heading straight up.
How could it possibly get in that position? Maybe we could just say "Bob observes some Hawking radiation at some uncertain time at twice the expected energy as he passes the event horizon and quickly calculates that he must be going c as he's spaghettified..." or something along those lines I don't know exactly how the radiation's energy would be calculated or how far it would be blue shifted or when he'd be shredded I think that happens before the EH...

#### jartsa

How could it possibly get in that position? Maybe we could just say "Bob observes some Hawking radiation at some uncertain time at twice the expected energy as he passes the event horizon and quickly calculates that he must be going c as he's spaghettified..." or something along those lines I don't know exactly how the radiation's energy would be calculated or how far it would be blue shifted or when he'd be shredded I think that happens before the EH...

Or Bob falls feet first and wears sneakers with LED lights.

#### jartsa

When radiation falls into a black hole it blue shifts, which means its energy increases. At the event horizon the energy has increased to infinity. The speed of the radiation is c, regardless of the energy of the radiation.

When matter falls into a black hole its energy increases. At the event horizon the energy has increased to infinity. The speed of the matter is c when the energy is infinite.

Is there perhaps some problem with the infinite energy? Like where does it come from.

Well kinetic energy of a falling object comes from the gravity field. The gravity field may not have an infinite amount of energy. Well then the energy of the gravity field just becomes very negative when it gives away an infinite amount of energy.

"What (if anything) limits the speed of something falling into a black hole?"

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