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

#### Nugatory

Mentor
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, Schwarzchild 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.

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#### Nugatory

Mentor
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 Schwarzchild 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

Mentor
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 $c$.

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#### PeterDonis

Mentor
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.

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#### PeterDonis

Mentor
there is a sense of velocity of a body inside the horizon relative to one outside the horizon that does make sense
Within the confines of a single local inertial frame that straddles the horizon, yes. I mentioned in my post that this local concept of relative velocity still works in a general curved spacetime.

#### pervect

Staff Emeritus
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?
Unfortunately, if you want even a basic understanding, you need to realize that velocity has to be measured relative to something. Furthermore, we need to check that your understanding of velocity is what we are talking about, otherwise we can't possibly communicate.

To measure a velocity, we need to define an observer who has zero velocity. Presumably this observer is hovering above a black hole by means of a powerful rocket. This is not a very complete description, but hopefully it's complete enough. If that's not what you had in mind, then the answer I'm going to give will perhaps not be right, but you'd need to explain what you mean by velocity in order for us to communicate.

I find that people in general find it somewhere between hard and impossible to explain what they mean, so what I try to do is to offer an explanation of what I mean, in hopes that they can compare it to what they mean. This sometimes works (and sometimes doesn't, if they don't follow the explanation of what I mean). Unfortunately, if they don't follow the explanation of what I mean, I see no way to proceed with the discussion.

How do we measure velocity? Well, we have two observers, and one of them (preverably both of them) carry around clocks and rods with them. Then one observer uses his clocks to measure the time interval it takes the other observer to pass by the length of his rod.

We find that velocity is reciprocal - it doesn't matter which one of the observers has the clocks and rods , we get the same answer either way.

So, lets get into the details. An observer "at rest" exists only outside the event horizon of a black hole. Said observer "at rest" needs to have a proper acceleration away from the black hole, or else they'll fall in. Something needs to hold them in place.

At the event horizon, the required proper acceleration is infinite - no material observer can "hold station" at the event horizon of a black hole. If we want the observer to measure the velocity, it needs to be a material observer (one that is not moving at light speed). There is a FAQ that explains why there is no reference frame of an observer moving at "c", the explanation is not terribly complex but some people get hung up on this point anyway. For the moment I'll just refer to the FAQ on this point rather than digress and break the thread of what I want to explain.

So, the observer "at rest" is presumed to be a material observer, who carries along some clocks and measuring rods, and he uses these clocks and measuring rods to measure the velocity of the infalling observer.

With this approach, the velocity of the infalling observer at the horizon can only be computed as a limit, by taking observers closer and closer to the event horizon. To make a long story short, that limit is "c", the speed of light. So when we speak informally, we say that an infalling observer crosses the event horizon at "c", though what we actually mean is this limiting process.

There is perhaps an easier way to do the same thought experiment that provides more insight. We still need two obserers, one moving, one "at rest", to measure the relative velocity between. However, we put the clocks and rods on the falling observer, and have the falling observer measure the relative velocity of the stationary observer, instead of the other way around. As we remarked earlier, this gives us the same answer. The two numbers turn out to be the same, measuring the velocity is a reciprocal process.

Doing things this way though clarifies what happens at the event horizon. There is no "observer" with clocks and rods located exactly at the event horizon, but we can imagine a light pulse moving out from the black hole that's "stuck" at the horizon. This doesn't qualify as an observer, because a light pulse can't have clocks (or measuring rods). To see why, you'd have to expand your interest to look at things you say you are not interested in, but turn out to be important to answer your question even though you think you're not interested.

With this approach, we can see that the relative velocity between the light pulse (located at the event horizon), and the infalling observer (who is a material observer with clocks and rods) must be equal to c, because the velocity of a light pulse relative to a material object is always "c". The trick here is the event horizon is not some sort of normal place. It can't be. Assumign that it is, and that there is some sort of "reference frame" that exists there leads to problems, the problems which you are probably encountered. The solution is reasonably simple - don't do that, don't assume there is a "reference frame" at the event horizon.

The math tells us the same thing, except we instead talk about the poor behavior of the Schwarzschild coordinates at the event horizon.

#### PeterDonis

Mentor
don't assume there is a "reference frame" at the event horizon.
Just to clarify: the thing not to assume is that there is a local inertial frame in which the horizon is "at rest". There can't be, because the horizon is a lightlike surface.

#### jerromyjon

I'm not sure if I'm more or less confused than I was before, but at least I'm learning new things about black holes.
I just wish there was a way to draw a picture which shows how gravity functions according to GR, because all this complicated math is doing is confusing me.

#### PeterDonis

Mentor
I just wish there was a way to draw a picture which shows how gravity functions according to GR
That would require drawing a 4-dimensional picture, which is unfortunately beyond PF's current technical capabilities.

#### Nugatory

Mentor
That would require drawing a 4-dimensional picture, which is unfortunately beyond PF's current technical capabilities.
Although if you're willing to limit yourself to observers all one a single radial line (no orbits, no rotation, no sideways motion, no tangential velocities, just falling in or firing your rockets straight up and down, ....) a Kruskal diagram has a lot to offer in terms of intuitive understanding.

#### jerromyjon

a Kruskal diagram has a lot to offer in terms of intuitive understanding.
Yeah, that was the best tidbit I found. Thank you very much. But that also leads to the contradiction with white holes, unless our entire universe is the result of a white hole, and the black hole is taken as the other extreme, then it might make some sense cancelling out the singularities...
That would require drawing a 4-dimensional picture, which is unfortunately beyond PF's current technical capabilities.
Tesseract as a sphere using polar coordinates? I seem to want to think about it as wrapping the circumference in a manifold of "Kruskal diagrams", does that make any sense?

#### PeterDonis

Mentor
You don't need to believe that the entire Kruskal diagram represents something real, in order to use the portion of it that describes the exterior and interior of the black hole (usually labeled as regions I and II) to help with your understanding. It's a tool, that's all.

Also, there are Kruskal-style diagrams for, e.g., the Oppenheimer-Snyder model of a spherically symmetric star collapsing to a black hole, which only includes physically reasonable regions. See, for example, here:

https://www.physicsforums.com/insights/schwarzschild-geometry-part-4/

Tesseract as a sphere using polar coordinates? I seem to want to think about it as wrapping the circumference in a manifold of "Kruskal diagrams", does that make any sense?
Not really, no. It is true that each point in the Kruskal diagram represents a 2-sphere, but you can't "invert" that the way you describe, because the radius of the 2-spheres is different for different parts of the Kruskal diagram.

#### jerromyjon

Tesseract as a sphere using polar coordinates?
Maybe isotropic coordinates? I feel like I'm grasping at straws here but all I really need to do is get the short straw. I am trying to imagine how I could simulate a test particle falling into a generic black hole in some type of metric that makes sense but I keep running off on tangents that I can't get the gist of. Since everything I've found about black hole simulations seems to indicate it hasn't or perhaps can't be done, I have more ambition to try, and I'm not the type to give up. This isn't just some passing fancy, I've been very interested in and working towards computer simulations most of my getting long life. So, anyway, any help would be graciously appreciated and I'm back off searching...

#### PAllen

Maybe isotropic coordinates? I feel like I'm grasping at straws here but all I really need to do is get the short straw. I am trying to imagine how I could simulate a test particle falling into a generic black hole in some type of metric that makes sense but I keep running off on tangents that I can't get the gist of. Since everything I've found about black hole simulations seems to indicate it hasn't or perhaps can't be done, I have more ambition to try, and I'm not the type to give up. This isn't just some passing fancy, I've been very interested in and working towards computer simulations most of my getting long life. So, anyway, any help would be graciously appreciated and I'm back off searching...
I'm confused. Test particles falling into ideal BH is a basic exercise in GR courses, and is straightforward in all major BH coordinate systems in use except Schwarzschild coordinates. Unless, perhaps, by generic BH you mean the result of a realistic collapse. In this case, it is established that the exterior settles to a Kerr BH, while the the generic interior state is, indeed, unknown. Not just in terms of coordinates, but in basic physics even classically. It is presumed to be chaotic, but the general features are unknown.

#### jerromyjon

describes the exterior and interior of the black hole (usually labeled as regions I and II[in the Kruskal diagram])
The big "a-ha" moment for me was thinking in term of the entire universe, and then a "black hole/white hole unitary viewpoint" then of course to create "objects" in that universe...

#### jerromyjon

I'm confused.
Welcome to the club. Sorry to do that to you.
Test particles falling into ideal BH is a basic exercise in GR courses
Sorry I missed mine.
[it] is straightforward in all major BH coordinate systems in use except Schwarzschild coordinates.
So which one is easiest to model?
By easiest I meant which way might be the least intensive to calculate... I have a hunch quarternions might help...

#### jerromyjon

Unless, perhaps, by generic BH you mean the result of a realistic collapse. In this case, it is established that the exterior settles to a Kerr BH, while the the generic interior state is, indeed, unknown.
But a Kerr BH is just a vacuum solution...

Mentor

#### PAllen

But a Kerr BH is just a vacuum solution...
The exterior of a realistic collapse becomes a vaccuum Kerr solution to any chosen precision in a very short time. As I noted, the interior, which is non vaccuum (in part), is a currently an open question, even classically, more so with quantum considerations. The idealized classical interiors are fun for exercises, but no one has a clue how much corresponds to what an infalller would experience in a real BH. Fortunately, for observations we can make, e.g. LIGO or the horizon imaging projects, only the exterior matters.

#### Nugatory

Mentor
The exterior of a realistic collapse becomes a vaccuum Kerr solution to any chosen precision in a very short time.
And if OP is willing to accept one simplifying unrealistic assumption....
He can assume that there is no angular momentum involved because the collapsing matter started out not rotating. Now he can use the Schwarzschild spacetime in the vacuum on both sides of the horizon.

But @jerromyjon, what exactly are you looking for here? You say you are "trying to imagine how you would simulate a test particle falling into a generic black hole" but I don't understand exactly what you mean by that. This thread started with a question about what limits the speed of an infalling object, so it sounds as if perhaps you are trying to understand the trajectory of an infalling object relative to various observers?

#### jerromyjon

so it sounds as if perhaps you are trying to understand the trajectory of an infalling object relative to various observers?
The point of view you choose shouldn't affect the reality of the physics, should it? I mean even if different observers have different perspectives there should be some calculable "reality" as to what happens which all should agree on when spacetime distortions are considered. Can't there be one chart in some sense that can be translated into any reference frame or is that physically unrealistic? (neglecting quantum effects)

#### jerromyjon

Fortunately, for observations we can make, e.g. LIGO or the horizon imaging projects, only the exterior matters.
I remember the "ring-down" from LIGO during the first detected event which corresponded to the BH merger and it emanated from beyond the EH though, isn't that at least "some type of observation" via indirect means? I'm not trying to be annoying or argumentative, simply thorough.

#### Nugatory

Mentor
The point of view you choose shouldn't affect the reality of the physics, should it? I mean even if different observers have different perspectives there should be some calculable "reality" as to what happens which all should agree on when spacetime distortions are considered.
What you are calling the "some calculable reality" are the things that are invariant, that are frame-independent, that have the same values in all frames (that was three different ways of saying the same thing). These are indeed the "actual physics". The worldline of an object falling into a black hole is such a thing; it's the set of points in spacetime the object passes through, and for any given point in spacetime it is a simple fact that either the object was there or it wasn't.
However, in the first post of this thread you asked about the speed of the infalling object. That's not a frame-independent invariant, and different observers will have different perspectives on what it is.
Can't there be one chart in some sense that can be translated into any reference frame? (neglecting quantum effects)
"Chart" has a specific technical meaning, so the question as asked is ill-formed. However, if we substitute "description" for "chart" the answer is yes - the worldline of the infalling object is what you're looking for. We can use it to calculate the coordinate speed of the infalling object in any frame you please.

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#### PAllen

I remember the "ring-down" from LIGO during the first detected event which corresponded to the BH merger and it emanated from beyond the EH though, isn't that at least "some type of observation" via indirect means? I'm not trying to be annoying or argumentative, simply thorough.
No, the ring down does not emanate from the BH interiors.

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

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