Simulation of a neutron star's impact on a supermassive black hole?

[The Bill]

Please forgive the awkward title. "Supermassive black hole" uses up a lot of the title character limit.

Has anyone made a simulation of what would happen if a neutron star impacted a simplified (Schwarzschild) supermassive black hole? I've seen simulations of a neutron star colliding with a stellar mass black hole, and of two stellar mass black holes colliding, but my Google searches haven't found me this yet.

Specifically, would the event horizon temporarily change shape near the impact?

I'm asking this here because someone I know asked a related question out of curiosity. Out of politeness, I'd like to be able to get an answer to my question here back to them within a month or two.

 

[PeterDonis]

Has anyone made a simulation of what would happen if a neutron star impacted a simplified (Schwarzschild) supermassive black hole?

I don't know, but I don't think you need one to answer the specific question you're asking. See below.

would the event horizon temporarily change shape near the impact?

The question is not well-formed. The event horizon is a surface formed of radially outgoing light rays. It doesn't act anything like the surface of an ordinary object; it's not even a "place" with a "shape" in the ordinary sense of the word. The neutron star would just fall into the hole; there would be no "impact" at all, since the hole is vacuum--empty space.

 

[The Bill]

The question is not well-formed.

How precise do I have to be?

Is it precise enough if I say: "The shape of the event horizon of the idealized black hole we are considering is "spherically" symmetrical in the Schwarzschild metric. That is, if we rotate the solution in any way using the angular coordinates of the Schwarzschild metric, the shape of the event horizon is unchanged. Now, if we model a neutron star approaching along a radial direction(that is, it has no angular momentum relative to the supermassive black hole) does the event horizon have the same symmetrries as that of the isolated Schwarzschild supermassive black hole? If not, how is the symmetry broken? What shape does the event horizon take in the spacetime defined by the colliding neutron star and supermassive black hole system?"

Does this give you a different idea of what I'm trying to ask than you had from my OP?

 

[PeterDonis]

How precise do I have to be?

Lack of precision isn't the reason your question wasn't well-formed. It wasn't well-formed because it makes invalid assumptions: that the event horizon has a "shape" and the an object falling into the black hole will "impact" the horizon. Neither of those assumptions are valid, and without them your question as you asked it in the OP makes no sense.

From your latest post, you might be asking a somewhat different question, which at least has a well-defined meaning. See below.

if we rotate the solution in any way using the angular coordinates of the Schwarzschild metric, the shape of the event horizon is unchanged

I personally wouldn't use the word "shape" here, but the rotational symmetry of the Schwarzschild solution is well-defined, yes. (Although it does not depend on what coordinates you choose.)

if we model a neutron star approaching along a radial direction(that is, it has no angular momentum relative to the supermassive black hole) does the event horizon have the same symmetrries as that of the isolated Schwarzschild supermassive black hole?

I would not use a supermassive black hole for this thought experiment, because neutron stars have a maximum mass (somewhere between about 1.5 and 3 solar masses) which is so much smaller than the mass of a supermassive black hole that the neutron star falling into the supermassive hole can be treated as a "test object" with negligible mass and therefore negligible effect on the spacetime geometry. So the possible effect you are asking about would not be there in this case.

However, if you ask about, say, a black hole of 5 solar masses with a neutron star of, say, 2 solar masses falling radially into it, then the mass of the neutron star is not negligible and the spacetime geometry would be affected. I don't know if this sort of thing has been simulated numerically (although I would imagine it has been--I just don't know of any specific references). Intuitively, I would expect that,as the neutron star fell in, there would, heuristically speaking, be a "bump" on the event horizon in the direction the neutron star is falling in from; then, as the star falls below the horizon, the overall area of the horizon would increase as the hole's mass increased, but there would also be gravitational waves radiated--heuristically this can be thought of as the bump being radiated away and the horizon returning to spherical symmetry with a larger area. Something like this is described in Kip Thorne's popular book Black Holes and Time Warps.

The above is still probably an approximation, because strictly speaking, exact spherical symmetry would require that nothing else is present anywhere in the universe except the black hole (and that the hole is perfectly non-rotating, with exactly zero angular momentum). If a neutron star is falling in, that is of course not the case (never mind the rest of the universe). And even after the star falls in, the gravitational waves radiated would not be spherically symmetric either.

 

[The Bill]

I personally wouldn't use the word "shape" here, but the rotational symmetry of the Schwarzschild solution is well-defined, yes. (Although it does not depend on what coordinates you choose.)

By"shape" I mean the set of coordiantes in 4 dimensional spacetime that make up the event horizon. Also, I know that general relativity is coordinate-independent, I was just using the angular coordinates of the Schwarzschild solution as reference points, so I didn't have to try and fail to formulate a long-winded description of what "rotating" a solution means without referring to a coordinate system.

I would not use a supermassive black hole for this thought experiment, because neutron stars have a maximum mass (somewhere between about 1.5 and 3 solar masses) which is so much smaller than the mass of a supermassive black hole that the neutron star falling into the supermassive hole can be treated as a "test object" with negligible mass and therefore negligible effect on the spacetime geometry. So the possible effect you are asking about would not be there in this case.

My question and the questions asked by the person who I started thinking about this because of is and are explicitly dependent on the relativistic equivalent of "tidal forces" being low. I want to know what happens to the spacetime near the event horizon locally around the neutron star. Sure, if we were looking at a scale of tens or hundreds of millions of kilometers encompassing the entire supermassive black hole, we won't see an effect on that scale from a 20km neutron star approaching from outside. What I'm wondering, though, is if we will see a meters-and-kilometers-scale effect on the spacetime near the neutron star over the course of an event that starts out with the neutron star approaching the event horizon of an 88+ million km diameter black hole. And, if we do, then what form does that effect take?

 

[PeterDonis]

if we will see a meters-and-kilometers-scale effect on the spacetime near the neutron star over the course of an event that starts out with the neutron star approaching the event horizon of an 88+ million km diameter black hole.

If you insist on the hole being supermassive, then the effect on the spacetime geometry of the black hole is negligible. An observer on the neutron star will not even be able to tell that he is falling into a black hole; spacetime around him will look just the same as it has always looked. An observer hovering just above the horizon of the hole and watching the neutron star fall past will not be able to observe any change in the hole's geometry.

 

[The Bill]

If you insist on the hole being supermassive, then the effect on the spacetime geometry of the black hole is negligible.

Okay, can you point me to a citation for this? I can't prove it myself, but I should be able to mostly follow along with a proof at the level of Wald or MTW. I've found treatments of test masses and small objects made of regular matter falling into supermassive black holes, but nothing for anything denser like dwarf or neutron stars.

 

[PeterDonis]

can you point me to a citation for this?

I've found treatments of test masses and small objects made of regular matter falling into supermassive black holes, but nothing for anything denser like dwarf or neutron stars.

To a supermassive black hole, a neutron star is a test object. The neutron star itself has a non-negligible effect on the spacetime geometry around it, and that effect is still there when it falls into the black hole--which is why, to an observer falling in with the neutron star, things look just the same as they would if the neutron star were out in empty space far away from all other gravitating bodies (which means that the neutron star's own gravity continues to work). But the neutron star's mass is much too small to have a non-negligible effect on the hole's spacetime geometry.

 

[The Bill]

I'm not saying I don't believe you. I'm not asking you to explain further. I'm asking for an academic source for that result which I can cite and that isn't this thread

 

[PeterDonis]

I'm asking for an academic source for that result which I can cite and that isn't this thread

You have already said you have seen treatments of test objects and small objects of regular matter falling into black holes. Just substitute a neutron star for those objects. You should find that it makes no difference to the analysis, since the density of the objects does not play any role; only their mass as compared to the mass of the hole does.

If you can find a reference that says the density of the object (as opposed to its mass) makes a difference, I'd be interested in seeing it, because no such claim is made that I'm aware of in any of the textbooks I've read (I have read both MTW and Wald). They all say only the mass of the object as compared to the mass of the hole matters.

 

[The Bill]

Okay, put it this way. Show me a source where the amount of distortion of the event horizon due to a second mass is derived, so that I can plug in the numbers and see precisely how minimal the effect is.

Do you claim that the distortion is less than a meter? Less than a centimeter? Less than a nanometer? Not even measurable with a Planck length ruler?

Unless you define the limits you put on "negligible," I can't put any weight on your assertions.

I'm not at the point where I can prove things like this myself, but I can follow along with derivations at Wald/MTW level if I take my time and reference other sources along the way. So, I'd appreciate some math to back up your words. Link a source, put up a derivation that others more knowledgeable than me can critique, I don't care.

Please don't just continue asserting claims without backing them up with sources or verifiable math.

I don't believe that you're wrong, but I have no basis for determining on what scale and to how many decimal places you may be correct. Without that, I can't honestly repeat what you've said with any confidence.

 

[PeterDonis]

Show me a source where the amount of distortion of the event horizon due to a second mass is derived

Do the sources you refer to, that model a small object made of regular matter falling into a black hole, do this? Or do they give an argument for why they don't have to?

You keep asking me for references, but you yourself talk about sources that you've seen; what are they?

 

[The Bill]

Do the sources you refer to, that model a small object made of regular matter falling into a black hole, do this? Or do they give an argument for why they don't have to?

You keep asking me for references, but you yourself talk about sources that you've seen; what are they?

Sean Carroll's Spacetime and Geometry 3.4 defines a test particle as "a body that does not itself influence the geometry through which it moves." Do you know of a neutron star that is low enough in mass that no trajectories are affected by it, nothing can orbit it, etc.?

My notion of "negligible" depends on something acting like a test particle on the scale I'm looking at. The volume of space I'm interested in is on the order of 100km across, probably less. Would you say that the geometry of space around a neutron star is negligible at 50km away? Would you like a neutron star 50km away from your summer home? According to you, it's negligible on the scales I'm talking about, so why not?

 

[PeterDonis]

Sean Carroll's Spacetime and Geometry 3.4 defines a test particle as "a body that does not itself influence the geometry through which it moves."

In the case of the neutron star falling into the supermassive black hole, "the geometry through which it moves" would be the hole's geometry.

However, I wasn't asking for a reference for a definition of "test particle". I was asking for a reference for this statement of yours:

I've found treatments of test masses and small objects made of regular matter falling into supermassive black holes

What treatments? And note that you didn't just say "test masses", you also said "small objects made of regular matter". What treatments of the latter case are you referring to?

 

[SlowThinker]

I'm wondering if there's a simple formula for gravitational time dilation when the test clock is affected by 2 masses at the same time? Perhaps
$$t_0=t_f\sqrt{1-\frac{r_{s1}}{r_1}-\frac{r_{s2}}{r_2}}$$
where $r_{sn}$ is the Schwarzschild radius of the masses and $r_n$ is the distance?

If there is such a formula, it should be easy to find the point where time stops for various configurations of the two masses and the test clock. Of course time stops at the event horizon.

 

[The Bill]

What treatments? And note that you didn't just say "test masses", you also said "small objects made of regular matter". What treatments of the latter case are you referring to?

I was referring to treatments of tidal forces and "spaghettification," for example, section 11.8 of Hobson, Efstathiou, and Lasenby. In these treatments, one only cares about the effect of the Schwarzschild geometry on the object falling in. I've seen simulations of neutron stars falling into a black hole which, while much more massive, is of similar radius to the neutron star. I've seen black hole-black hole merger simulations.

What I have not seen is a simulation or calculation about something massive enough to not be considered a test particle falling into a supermassive black hole with no angular momentum. Hence this thread.

 

[PeterDonis]

I've seen simulations of neutron stars falling into a black hole which, while much more massive, is of similar radius to the neutron star.

And what did those simulations show the effect of the neutron star falling in on the black hole's horizon to be? In particular, what was the effect as a function of the ratio of masses of the neutron star and black hole? (Or more precisely: was it only a function of the masses, or did it also involve the density of the neutron star falling in?)

 

[The Bill]

I'm not in this thread to have the Socratic method used at me. I'm trying to find references, not answer questions with this particular topic.

 

[PeterDonis]

I'm not in this thread to have the Socratic method used at me. I'm trying to find references

And what I'm trying to find out is whether you already have references that can answer your question if looked at properly. I don't have any specific references that discuss the precise case you describe. But I strongly suspect that, if you tell me what references you already have looked at that describe the most similar cases--a neutron star falling into a black hole that is not supermassive, and an ordinary object, not as dense as a neutron star, falling into a supermassive black hole--I will be able to look at those references and see whether they contain enough information to generalize what they say to the case of a neutron star falling into a black hole. But I can't do that if you won't tell me what references you've already looked at. I don't understand why that's so difficult.

If you object to being asked questions, bear in mind that the question are aimed at seeing whether you have already considered the references you have looked at from the above viewpoint: can what they tell you be extended to the case of a neutron star falling into a supermassive black hole? If you haven't already done that, then you might consider it. If you have already done that, you could just say so and explain why doing that wasn't helpful. Or you could just give the references, as I've asked. Or you could keep objecting, in which case this thread is going nowhere and we might as well just close it. Your call.

 

[alantheastronomer]

A neutron star is itself relativistic and has an appreciable, non-negligible, effect on the spacetime around it. As a neutron star gets near the event horizon, be it a stellar mass or supermassive black hole, the distortion of the spacetime by the neutron star will also distort the spacetime near the event horizon. Of course the impact of the effect will appear less extreme on the event horizon of the supermassive black hole, than on the stellar mass one.

 

[The Bill]

If I were to say that I'v claimed something in this thread, it is that I am ignorant. I do not claim to have any particular knowledge about what the answer to my question is. A claim of ignorance is hardly extraordinary, and I feel it needs no evidence, references or mathematics to back it up

PeterDonis has made the claim that "If you insist on the hole being supermassive, then the effect on the spacetime geometry of the black hole is negligible." But has essentially refused to offer any scholarly evidence to back up this claim. Yes, when looking at the principles, I have a feeling that PeterDonis' claim may be correct or close to correct, but I have no idea where to start in determining how correct it may be in hard numbers.

As far as scientific substance, goes, then, I take PeterDonis' claim to be (so far) an extraordinary claim with no evidence presented to back it up, and hence devoid of substance if I wanted to repeat it.

[snip]
But I can't do that if you won't tell me what references you've already looked at. I don't understand why that's so difficult.
[snip]
Or you could just give the references, as I've asked. Or you could keep objecting, in which case this thread is going nowhere and we might as well just close it. Your call.

I own copies of MTW's Gravitation, Carroll's Spacetime and Geometry, Wald's General Relativity, and Hobson, Efstathiou, and Lasenby's General Relativity: An Introduction for Physicists.

The statements I made above are based on reading some of these texts and many other references I encountered over the past few decades. I don't have chapter and verse on that, so you may consider me ignorant as to the chapter and verse anything I said above is based on or potentially mis-remembered from, unless I already gave a section number or page number here.

I am here to answer a casual question asked by an acquaintance, so I don't intend to do a literature search and spend potentially hours poring over texts and other references to find exactly where I am remembering individual things from.

I haven't studied or applied any of this seriously for over a decade, so you may consider me very rusty, or essentially ignorant except for a passing familiarity with the basic concepts.

can what they tell you be extended to the case of a neutron star falling into a supermassive black hole?

I would consider that a fresh result. If it isn't worked out explicity in a reference I'm looking at, I would consider it to need expert verification before I would repeat it as true to the acquaintance who inspired the creation of this thread.

 

[PeterDonis]

The statements I made above are based on reading some of these texts and many other references I encountered over the past few decades.

You mentioned seeing numerical simulations of ordinary massive objects (but not dense ones like neutron stars) falling into supermassive black holes. Which specific reference did you get that from?

when looking at the principles, I have a feeling that PeterDonis' claim may be correct or close to correct, but I have no idea where to start in determining how correct it may be in hard numbers.

The heuristic I am using is that the effect of the infalling object on the hole's spacetime geometry near the horizon (and specifically the shape of the horizon) is roughly the ratio of the two masses. So, for example, if an object of one solar mass falls into a supermassive hole of a billion solar masses, the effect would be roughly 1 part in a billion.

I would consider that a fresh result. If it isn't worked out explicity in a reference I'm looking at,

Do any of the references you have looked at give formulas (possibly approximate ones derived from numerical simulations) for the magnitude of the effect of the infalling object on the hole's spacetime geometry, and specifically the shape of the horizon? If so, what do those formulas depend on? Do they depend on just the mass of the infalling object, or do they also depend on its density? That would give an indication of how reliable the heuristic I described above is (since the above heuristic only depends on the mass of the infalling object, so an ordinary star would be no different from a neutron star falling in).

I have read MTW and Wald, and the heuristic I gave above that depends only on the mass of the infalling object is based on what I've read there. But neither of those references, as far as I remember, describes numerical simulations of objects falling into black holes. (Both were published before computers became fast and capable enough to make such simulations feasible in quantity.) I have not read your third reference, so unfortunately I can't comment on what is in it.

 

[SlowThinker]

Perhaps someone could comment on this formula that I guessed?

I'm wondering if there's a simple formula for gravitational time dilation when the test clock is affected by 2 masses at the same time? Perhaps
$$t_0=t_f\sqrt{1-\frac{r_{s1}}{r_1}-\frac{r_{s2}}{r_2}}$$
where $r_{sn}$ is the Schwarzschild radius of the masses and $r_n$ is the distance?

The reasoning is that the time dilation is related to gravitational potential, and potential should be additive, so it might look something like this.

In the case it's true, the distance where the 2 horizons touch is $r_{s1}+r_{s2}+2\sqrt{r_{s1} r_{s2}}$ so for a 1 billion vs 1 solar mass gives a bulge of tens of thousands of kilometers (against a billion kilometer radius).

 

[nikkkom]

Please forgive the awkward title. "Supermassive black hole" uses up a lot of the title character limit.

Has anyone made a simulation of what would happen if a neutron star impacted a simplified (Schwarzschild) supermassive black hole? I've seen simulations of a neutron star colliding with a stellar mass black hole, and of two stellar mass black holes colliding, but my Google searches haven't found me this yet.

Specifically, would the event horizon temporarily change shape near the impact?

I believe such simulations are the domain of "numerical relativity".

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

I can't find it now, but I remember reading some papers with two black holes just before their merger having funny "beaks" on their event horizons.

 

[PeterDonis]

The reasoning is that the time dilation is related to gravitational potential

The concept of "gravitational potential" is not well defined in a spacetime with two black holes in it, since such a spacetime is not stationary.

potential should be additive

Not in GR. GR is a nonlinear theory. Only in the weak field approximation and a stationary spacetime (so potential is well-defined, see above) can you treat potentials as additive.