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

[jerromyjon]

When replying to this thread: https://www.physicsforums.com/threads/the-nasa-zero-gravity-flight.927136/
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?

 

[mfb]

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]

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]

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]

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]

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]

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?

 

[jerromyjon]

What is v relative to?

To the black hole of course.

 

[Nugatory]

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]

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]

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".

 

[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]

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.

trying to start with "how gravity functions"

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.

 

[Nugatory]

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?

Energy is frame-dependent, even in classical physics and flat spacetime (for example the total energy of the book in my lap is zero using a frame in which the airliner I'm on is at rest, but not zero using a frame in which the ground 10,000 meters below the airliner is at rest).

Thus, you can make the energy come out to be pretty much whatever you want it to be by choosing your coordinates to produce that particular value (of course energy is still conserved, although the value of the conserved energy is different in different frames). Especially in a curved spacetime, we can find coordinates in which sensible physical quantities such as the energy of an object or the time between ticks of a clock become infinite somewhere - usually this just means that we've made a poor choice of coordinates.

Jartsa's suggestion about speed reaching $c$ and infinite energy at the horizon is an example. The coordinates that work well for observers hovering above the even horizon are Schwarzschild coordinates and just abut everything you'll hear about black holes outside of a serious GR textbook is based on calculations using these coordinates. However, Schwarzschild coordinates have a coordinate singularity at the event horizon, so the infinite energy at the event horizon should not be taken seriously.

It's worth taking the time to study and understand a Kruskal diagram, which draws the spacetime around a black hole using coordinate that do not have a singularty at the event horizon. This allows you to visualize what's really going on as an object falls to and through the horizon.

 

[Nugatory]

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.

You are making the mistake (although it is better concealed here than in most examples) of trying to assign a rest frame to a flash of light. Although you can "calculate" a coordinate velocity of zero in Schwarzschild coordinates for an outgoing flash of light at the horizon, that doesn't mean that the flash of light is hovering, it means that your coordinates are singular there.

 

[PeterDonis]

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.

No, we don't. We know that the light at the horizon is moving at $c$ relative to Bob when Bob passes it. But that's because light always moves at $c$. It's not because Bob is moving at $c$.

 

[PeterDonis]

what would limit the velocity of a mass approaching a black hole in a vacuum?

"Velocity" in curved spacetime is a local concept in the general case. So, strictly speaking, the concept of the "velocity" of a mass that is free-falling very near a black hole's horizon, relative to an observer far away, makes no sense.

The fundamental reason why the above is true is that, in a general curved spacetime, there is no way to identify "points in space" the way you intuitively think of them. However, there is a special class of spacetimes, called "stationary spacetimes", in which you can do this, in a limited sense. A stationary spacetime is a spacetime in which there is a family of timelike worldlines, along each of which the spacetime curvature stays the same. You can think of these worldlines as the worldlines of a family of observers, each of whom is "at rest" in a sense, and each of whom marks out a well-defined "point in space" that stays the same over time.

Spacetime outside the horizon of a black hole (an idealized one whose mass is constant) is stationary, so we can imagine a family of observers, each one "hovering" at a constant altitude above the hole; these observers meet the requirement I just described--spacetime curvature is constant along each of their worldlines. So we can treat the local velocity of an infalling object, relative to whichever one of these family of observers is nearby, as the "velocity" of the object in the sense you are using the term. And if we do this, the answer to your question is that the velocity of the object will always be less than $c$, but it will approach $c$ as the object approaches the hole's horizon.

But as soon as the object reaches the horizon, this no longer works, because at the horizon and beneath it, spacetime is no longer stationary, and all the stuff I described above doesn't work any more. So at and beneath the horizon, the only concept of "velocity" that works is the local one (within a local inertial frame). The concept of the "velocity" of an object beneath the horizon, relative to an observer outside it, doesn't make sense.

 

[PAllen]

Well, there is a sense of velocity of a body inside the horizon relative to one outside the horizon that does make sense. Consider two free fallers separated by some modest distance heading for the BH. In either one's local inertial frame, while one is inside the horizon and the other outside, their relative velocity can be whatever you want depending on how you set up the scenario. Perhaps 1 meter per second, or .1 c, whatever you want. The horizon will just be a lightlike surface on the way from one to the other that has nothing to do with their relative velocity.

Another observation is that locally, for an observer just inside the horizon, a stationary observer just outside is equivalent to a Rindler observer relative to an inertial observer that has just crossed the Rindler horizon. Per the inertial observer, the Rindler stationary observer always has relative velocity less than c, but due to its acceleration, a light signal corresponding to the Rindler horizon never quite catches it.