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

[Nugatory]

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]

that also leads to the contradiction with white holes

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...

 

[PeterDonis]

Maybe isotropic coordinates?

Those won't help, they don't cover the region at or below the horizon.

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

Try Painleve coordinates:

https://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates

 

[PAllen]

But a Kerr BH is just a vacuum solution...

The exterior of a realistic collapse becomes a vacuum Kerr solution to any chosen precision in a very short time. As I noted, the interior, which is non vacuum (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]

The exterior of a realistic collapse becomes a vacuum 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]

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.

 

[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.

 

[PAllen]

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)

Yes, any chart that spans the horizon may be used, and all will make identical physical predictions. Peter has suggested a convenient coordinate chart to use and provided a link (for the idealized non rotating BH).

Another less commonly used chart that I happen to like is the Lemaitre chart:

https://en.wikipedia.org/wiki/Lemaître_coordinates

These have the feature of maintaining 1 timelike and 3 spacelike coordinates throughout the exterior and interior (Gullestrand-Panlieve coordinates are all spacelike inside the horizon). Kruskal coordinates also maintain 1 timelike and 3 spacelike coordinates everywhere, but I find them harder for many computations. In Lemaitre coordinates, free fall trajectories from infinity have a very simple representation, and the time coordinate gives proper time along such trajectories.

 

[jerromyjon]

In Lemaitre coordinates, free fall trajectories from infinity have a very simple representation, and the time coordinate gives proper time along such trajectories.

How would I hook that to a white hole?

 

[PAllen]

How would I hook that to a white hole?

Lemaitre coordinates don't include the white hole portion of the full Kruskal geometry. They include two of its 4 quadrants. This is not a bad thing because there are good reasons to believe the the other two quadrants don't exist in our universe. This is because there is no evolution from a prior state not in including them, that can result in their existence. A BH formed by collapse includes only geometry of the type covered by Lemaitre coordinates.

 

[jerromyjon]

This is because there is no evolution from a prior state not in including them, that can result in their existence.

Why not?

 

[jerromyjon]

I seem to have a system in my head where the entire universe (a white hole for argument sake) pushing in on every system in the universe (especially black holes) what can't work mathematically...

 

[PAllen]

Why not?

Because it is a mathematical theorem? Not sure what you are looking for, but a white hole can only exist as an eternal object. If anything like FLRW cosmology is true, white holes are impossible because the initial state doesn't include them.

 

[Dale]

I seem to have a system in my head where the entire universe (a white hole for argument sake) pushing in on every system in the universe (especially black holes) what can't work mathematically...

Well, nothing can work mathematically without doing the math. Please see the professional literature for that, we don't accept personal speculation here.