How Do Direct Collapse Black Holes Form in the Early Universe?

[Chronos]

This paper http://arxiv.org/abs/1504.00263, Assessing inflow rates in atomic cooling halos: implications for direct collapse black holes, discusses formation of direct collapse black holes in the early universe. Supermassive black holes are the only reasonable explanation for quasers in the high z universe. A long standing question has been how can black holes achieve such staggering masses in such a short time after the BB? It is believed some SMBH could have been seeded via direct collapse, but, it has also been thought the UV background would suppress their numbers well below those necessary to explain high z quasar populations. The authors appear to have found this is not necessarily the case.

 

[marcus]

This paper http://arxiv.org/abs/1504.00263, Assessing inflow rates in atomic cooling halos: implications for direct collapse black holes, discusses formation of direct collapse black holes in the early universe. Supermassive black holes are the only reasonable explanation for quasers in the high z universe. A long standing question has been how can black holes achieve such staggering masses in such a short time after the BB? It is believed some SMBH could have been seeded via direct collapse, but, it has also been thought the UV background would suppress their numbers well below those necessary to explain high z quasar populations. The authors appear to have found this is not necessarily the case.

Chronos thanks for the pointer. The senior author Marta Volonteri is something of an expert specialized in Black Holes (massive and up). Good publication and citation track record. She got her PhD in 2003 and has over a hundred papers. She has co-authored with Joe Silk and other reputable people. Both authors are at Sorbonne and Paris CNRS Astrophysics. The paper seems to be based on numerical simulations of collapse under various conditions relevant to high z (early times). The junior author, Latif, may have been instrumental in the number-crunching. These are just random superficial observations but may help me see the paper in context. I think it could be important by steering people to a better understanding of how we got so many high-z quasars.

 

[Chronos]

I was a little disappointed no one else found this interesting enough to comment on - tx marcus!

 

[stedwards]

[...] Supermassive black holes are the only reasonable explanation for quasers in the high z universe. [...]

As well as black holes, are not massive objects where the bulk of the matter exists in pre-collapse, in a relatively thin layer outside the Schwarzschild radius also candidates?

 

[PeterDonis]

massive objects where the bulk of the matter exists in pre-collapse, in a relatively thin layer outside the Schwarzschild radius

Such an object cannot be stable; it will quickly collapse into a black hole (more precisely, in the state you describe it must be in the process of doing so).

 

[stedwards]

Such an object cannot be stable; it will quickly collapse into a black hole (more precisely, in the state you describe it must be in the process of doing so).

Which? Do you have a reference for instability that leads from collapse to a black hole in some sort of quick manner?

 

[Chronos]

Huh? What thin layer outside the Schwarzschild radius [of what?] are you talking about?

 

[Garth]

Thank you Chronos, I missed your original OP somehow.

Now that we can make SMBHs in order to explain http://www.nature.com/nature/journal/v518/n7540/full/nature14241.html#close all we have to do is make them bright.

Will an accretion disc subsequently form or will it be drawn in with the direct collapse?

Garth

 

[PeterDonis]

Do you have a reference for instability that leads from collapse to a black hole in some sort of quick manner?

It is impossible to have a stable equilibrium for any object with radius less than 9/8 of the Schwarzschild radius. Einstein proved this as a theorem in the 1930's. So any "thin layer close to the Schwarzschild radius" can't be stable; it must be in the process of collapsing.

 

[jerromyjon]

I'm not very familiar with the physics of the early universe. I'm not even sure what variables are calculated, but I'm curious about the "isolation" of gravity in the early universe.

If matter (as we know it now) was created everywhere then wouldn't there be greater attraction "everywhere locally". As the sphere of influence enlarges with time the effect of surrounding mass would reduce the local attraction? Laymen's terms as you start to see more stars around you then they would start to "pull apart" your local spacetime. Or would symmetry just typically cancel it out?

 

[PeterDonis]

If matter (as we know it now) was created everywhere

Matter wasn't "created everywhere" in the sense of stress-energy, which is the source of gravity, suddenly appearing where there was none before. The SET is locally conserved: it can't be created or destroyed, it can only change form. At the end of inflation, the SET changed form from the "false vacuum" inflaton field to ordinary matter and energy; but the "amount of stress-energy" was the same before and after the conversion, so the source of gravity did not change.

As the sphere of influence enlarges with time the effect of surrounding mass would reduce the local attraction?

On average, the universe is homogeneous and isotropic, so the "effect of surrounding mass" on a given piece of matter is zero.

would symmetry just typically cancel it out?

Yes. See above.

 

[jerromyjon]

Well that all confirms what I thought, by the time I typed "cancel it out" I was tempted to delete the post.

One last "stupid" thought, but from what I think I know if the universe was too smooth and evenly distributed galaxies wouldn't have formed. Some unexplained "balance" allows galaxies to form and group, yet loose enough to keep SMBHs from swallowing everything as this thread proposes was an evident epoch in the first billion years? I bet this is a bit "out there" but could constructive and destructive interference have an impact?

 

[Garth]

One last "stupid" thought, but from what I think I know if the universe was too smooth and evenly distributed galaxies wouldn't have formed. Some unexplained "balance" allows galaxies to form and group, yet loose enough to keep SMBHs from swallowing everything as this thread proposes was an evident epoch in the first billion years? I bet this is a bit "out there" but could constructive and destructive interference have an impact?

Not a stupid thought at all, but an interesting and difficult question that cosmology tries to explain.

The "balance" is provided by Inflation in the standard cosmological \LambdaCDM model, which makes the universe smooth yet with sufficient anisotropies, as seen in the CMB, to create the large scale structure of the universe. It does, however, require sufficient Dark Matter to accelerate that process.

Inflation and Dark Matter have not been discovered in 'laboratory experiments - but the LHC is having a good go at detecting something beyond the present particle physics standard model, as that model was completed by the discovery of the Higgs Boson.

We wait and see!

Garth

 

[Grinkle]

This is not at all the point of the thread, but the sidebar on mass outside the SR reminded me that I don't understand how black holes of differing masses can be observed unless the mass we are observing is all still on the verge of crossing the SR, since nothing (I think) can cross the SR (or maybe I mean the event horizon) in a finite amount of time according to GR.

Discussions I have seen on the topic tend to fall into two camps -

Singularities do form in finite time, we don't have math to describe a singularity, that is why its called a singularity

vs

We are observing matter in the process of collapsing, the singularity itself will inevitably form, but not in finite time.

At least that is how I summarize what I have read.

If anyone can point me to a good discussion on the topic, I'd appreciate it. Its befuddled me for a long time.

 

[Chronos]

This might be helpful http://mathpages.com/rr/s7-02/7-02.htm

 

[PeterDonis]

I don't understand how black holes of differing masses can be observed unless the mass we are observing is all still on the verge of crossing the SR, since nothing (I think) can cross the SR (or maybe I mean the event horizon) in a finite amount of time according to GR.

No, that's not what GR says. The problem is that statement "a finite amount of time". "Time" without qualification is not an invariant; it depends on your choice of coordinates. In Schwarzschild coordinates, yes, nothing reaches the horizon in a finite amount of coordinate time. But there are other coordinate charts in which objects do reach the horizon in a finite amount of coordinate time (for example, Painleve, Eddington-Finkelstein, or Kruskal).

What GR actually says is that coordinates don't have physical meaning; the physics is contained in the invariants, the things that don't depend on your choice of coordinates. For example, we can compute the proper time for an object to free-fall to the horizon from some finite radius; this computation gives a finite answer. Proper time along a given worldline is an invariant, so the finite answer is telling us something with physical meaning: namely, that objects can fall to the horizon, and on through it to the interior of the black hole. Similar computations for an object like a star that undergoes gravitational collapse show that, to an observer riding along with the collapsing matter, a horizon forms in a finite proper time, and the matter continues on inward and reaches $r = 0$ in a slightly longer finite proper time, where it forms a singularity. Again, these computations are of invariants, so they have physical meaning: they tell us that collapsing matter can form a horizon.

As for how we can observe holes of differing mass, even after the collapsing matter falls through the horizon, the reason is that the "mass" we observe is really an "imprint" on spacetime that is left behind by the matter even after it collapses. The way we measure the mass of a black hole, or any astronomical object, is to put test bodies in orbit about the object and measure the orbital parameters. What we are actually doing when we do this is measuring the spacetime curvature due to the object. But the curvature due to the collapsed object is static; once it forms, as the object collapses, it stays the same; the object does not need to be there continuously to produce it. (This is ultimately because the Schwarzschild spacetime geometry is a vacuum solution, i.e., no matter needs to be present to sustain it.) So the mass of the object is still measurable the same way even after it has collapsed to a black hole.

This might be helpful http://mathpages.com/rr/s7-02/7-02.htm

I'm not sure I like this presentation of the issue; it says some things in a way that appears to me to invite misunderstanding. Also, at least one statement it makes is simply wrong: it says "the event horizon is part of future null infinity", which is not correct.

 

[Jimster41]

I love the simulation approach. "Operational Dynamic Modeling" Of a sort. Enzo looks pretty amazing.

So, some random fluctuation to start accretion is assumed already. Then for direct collapse you need the right balance of UV flux to warm the in falling mostly H1 gas, dissipate angular momentum, make it less likely to fragment, and keep it ionized (non-molecular) to inhibit fusion ignition? But the most important piece is mass accretion rate. How exotic are the rates they are estimating?

They mention sink particles? Which I was a little confused by. These are protostars? So in the direct collapse fusion is inhibited but only part of the way down, and still you have a protostar (a really massive one, or just normal?) but then is that still going to be isothermal collapse? Are they saying a fairly massive protostar is there as usual but the density is low because it's from a primordial low metal cloud so the outward energy pressure is still low, or the outward pressure is normal for a massive protostar, but just gets overwhelmed by high mass in-flow? Total cartoon, but I guess I have hard time picturing the gravitational process (especially low density) blowing by the outward pressure from the protostar fusion ignition. I thought stellar fusion pressure was pretty effective at overwhelming the puny tug of gravity.

 

[stedwards]

Huh? What thin layer outside the Schwarzschild radius [of what?] are you talking about?

I am simply asking for a formation calculation. A reference would be fine. In the case of spherical symmetry, how much cosmological time is required such that a mass m will find itself fully within a radius 2m?

 

[PeterDonis]

how much cosmological time is required such that a mass m will find itself fully within a radius 2m?

What is "cosmological time"? If you mean proper time for an observer falling in with the collapsing matter, then for a star with the size and mass of the Sun, it takes about an hour to collapse to r = 2m. This was first calculated by Oppenheimer and Snyder in their classic paper on gravitational collapse in 1939. Misner, Thorne, and Wheeler has a good discussion.

 

[Chronos]

As noted in the abstract, the authors simulated primordial halos on the order of 10^7 M⊙ and assumed an accretion rate greater than 0.1 M⊙/year. This leads to a formation rate consistent with high redshift quasar populations.

 

[Jimster41]

Another paper on the direct collapse process...

http://arxiv.org/PS_cache/arxiv/pdf/0709/0709.0545v1.pdf

I think I see how the idea of a quasi-star (with fusion too puny to resist gravity) fits, and how the process results in lots of fuel remaining near the object, which does kind of help imagine where the crazy jets come from. It is a really different quasi star-like evolution.

So the Halo mass, over time, is divided into a portion that disappears into the black hole area, and a portion that gets converted to pure energy in the polar jets? There is no production of heavier elements through fusion and more typical stellar death.

The proper time relation between an observer watching the whole process and a particle that falls in only to be ejected as pure energy in the jet, somewhat boggles the mind. Same relationship as for a black hole "pop" I guess?

As I read more about this and also SMBH at galactic cores - the interaction between those and host galaxy dynamics, it seems so tempting to wonder about the role gravitational wave interference mechanisms could play in altering the constraints involved, or otherwise seeding dynamics.

 

[stedwards]

What is "cosmological time"? If you mean proper time for an observer falling in with the collapsing matter, then for a star with the size and mass of the Sun, it takes about an hour to collapse to r = 2m. This was first calculated by Oppenheimer and Snyder in their classic paper on gravitational collapse in 1939. Misner, Thorne, and Wheeler has a good discussion.

I errored. I should have referred to the elapsed time as measured by an observer stationed at asymptotic infinity, which is certainly different than proper time for an infalling observer. After all this is approximately the time scale implied is discussions unless otherwise clarified. Do MTW consider this?

 

[mfb]

I errored. I should have referred to the elapsed time as measured by an observer stationed at asymptotic infinity

How do you define "happens at the same time" for the observer at infinity and the matter falling into the black hole? This is not obvious, and different choices lead to different answers => not a meaningful physical quantity.

 

[PeterDonis]

Do MTW consider this?

Certainly. They just realize, as mfb pointed out, that there is no unique way to define "at the same time" for the observer at infinity and the matter falling into the black hole.

 

[stedwards]

No... It is not at all certain. This is, after all, why I asked if someone could provide something more substantial. Consider that light can reflex back and forth between a stationary (constant r) observer and a closer, infalling observer an uncountably large number of times. The same state of affairs may occur for the outer particles of a collapsing mass.

 

[PeterDonis]

light can reflex back and forth between a stationary (constant r) observer and a closer, infalling observer an uncountably large number of times

No, it can't. Once the infalling observer crosses the horizon, light from it can no longer make it back to the stationary observer. So there can be only a finite number of round trips before the light is trapped below the horizon.

 

[Grinkle]

In Schwarzschild coordinates, yes, nothing reaches the horizon in a finite amount of coordinate time.

With respect to any black hole far away from us, this is the human / Earth clock, yes? We are constrained to observe only things in the universe that happen within this particular slice of spacetime. So we can never observe any singularity forming unless we are close enough to it that our co-ordinate system is dominated by the local gravity around the matter forming the singularity. This is what I can't get past.

What GR actually says is that coordinates don't have physical meaning; the physics is contained in the invariants, the things that don't depend on your choice of coordinates. For example, we can compute the proper time for an object to free-fall to the horizon from some finite radius; this computation gives a finite answer. Proper time along a given worldline is an invariant, so the finite answer is telling us something with physical meaning: namely, that objects can fall to the horizon, and on through it to the interior of the black hole. Similar computations for an object like a star that undergoes gravitational collapse show that, to an observer riding along with the collapsing matter, a horizon forms in a finite proper time, and the matter continues on inward and reaches $r = 0$ in a slightly longer finite proper time, where it forms a singularity. Again, these computations are of invariants, so they have physical meaning: they tell us that collapsing matter can form a horizon.

No intuitive problems with this, as far as I can understand what you are saying, it makes sense to me. Still, I am constrained to my own co-ordinate systems with respect observations I myself am able to make.

IAs for how we can observe holes of differing mass, even after the collapsing matter falls through the horizon, the reason is that the "mass" we observe is really an "imprint" on spacetime that is left behind by the matter even after it collapses. The way we measure the mass of a black hole, or any astronomical object, is to put test bodies in orbit about the object and measure the orbital parameters. What we are actually doing when we do this is measuring the spacetime curvature due to the object. But the curvature due to the collapsed object is static; once it forms, as the object collapses, it stays the same; the object does not need to be there continuously to produce it. (This is ultimately because the Schwarzschild spacetime geometry is a vacuum solution, i.e., no matter needs to be present to sustain it.) So the mass of the object is still measurable the same way even after it has collapsed to a black hole.

And here is the crux of my befuddlement. How would our observations be any different if what we are seeing is gravity from matter that is in the process of collapsing into a singularity, but has not yet collapsed? And how can I be able to observe anything other than that unless I am on something other than a set of Schwarzschild co-ords? Am I, and I just don't realize it?

If the sun were to suddenly start collapsing, its gravity wouldn't change as long as its center of mass wrt to rest of the solar system didn't move, I think that is right? We'd still orbit it just the same. If our only data were the trajectory of our orbit around the sun, and we knew nothing about the radiation the sun emits or anything else about it, how could we tell if it were a neutron star or a normal star or a singularity with an EH or something else entirely?

 

[PeterDonis]

With respect to any black hole far away from us, this is the human / Earth clock, yes?

Coordinates aren't just a matter of a clock along a single worldline. Coordinates require a simultaneity convention: that is, a convention for which events spatially separated from us here on Earth (or whatever observer is at the spatial origin of the coordinates) happen "at the same time". But this is just a convention: there is no unique definition of simultaneity. Schwarzschild coordinates involve a particular choice of simultaneity which prevents them from covering events at or inside the horizon. But it is perfectly possible to pick other coordinates which match our human/earth clocks for events on Earth, but which also cover events at or inside the horizon of a black hole. Painleve coordinates are an example.

We are constrained to observe only things in the universe that happen within this particular slice of spacetime.

Yes, but that has nothing to do with coordinates; it has to do with what the possible paths of light rays are, which is a property of the spacetime geometry, regardless of what coordinates you choose.

we can never observe any singularity forming unless we are close enough to it that our co-ordinate system is dominated by the local gravity around the matter forming the singularity.

No; as above, what you can or can't observe has nothing to do with what coordinates you choose. Coordinates are a choice; they're not something that automatically comes with being in a certain region of spacetime.

I am constrained to my own co-ordinate systems with respect observations I myself am able to make.

No, you're not; see above. You're only constrained by the geometry of spacetime.

If the sun were to suddenly start collapsing, its gravity wouldn't change as long as its center of mass wrt to rest of the solar system didn't move, I think that is right? We'd still orbit it just the same.

Yes.

If our only data were the trajectory of our orbit around the sun, and we knew nothing about the radiation the sun emits or anything else about it, how could we tell if it were a neutron star or a normal star or a singularity with an EH or something else entirely?

We couldn't. But of course in any real situation we would have other data.

 

[Jimster41]

How do you define "happens at the same time" for the observer at infinity and the matter falling into the black hole? This is not obvious, and different choices lead to different answers => not a meaningful physical quantity.

Is it at all meaningful to define a third coordinate frame from which to view the relationship of these two proper times? I thought it was and was trying to picture a plot of that... Now I realize you could define a n infinite number of different coordinate frames in which to view the relationship. But then I can't figure out if there would be meaningful, "invariant" information across them...or whether the results of all of those would therefore be meaningless.

No wait, that's what a ST diagram does right?

 

[mfb]

At least in theory, it is possible to measure higher moments of the gravitational field of the sun. They are too small to have a notable effect on the planets, but that is just an experimental limit. Neutron stars and black holes would have different moments.

 

[PeterDonis]

Neutron stars and black holes would have different moments.

In the real world, yes, because in the real world none of these objects would be perfectly symmetrical (spherically symmetric for non-rotating or axially symmetric for rotating objects). But in the idealized case of perfect symmetry, there would be no higher moments; the only observables from a distance would be the conserved quantities, mass and angular momentum. In the absence of radiation being emitted (which is also an idealization, true), these would not change.

 

[Grinkle]

I think you are telling me this, Peter, but I'm not certain, so checking -

Is it correct then that GR does not preclude me from seeing the disk of an event horizon, and seeing matter cross an event horizon and then see that the disk of the event horizon is larger, all with Hubble 3.0 or Hubble 4.0 or whatever sensitive equipment I might need and however much luck I might need to be looking in the right spot into the just-right perfectly positioned field of stellar objects from right here on earth?

 

[PeterDonis]

Is it correct then that GR does not preclude me from seeing the disk of an event horizon

By "seeing the disk" I assume you mean seeing an area of sky that is black, with no light coming from it, correct? Yes, you will be able to see that with a sufficiently powerful telescope. You won't see the horizon itself, but you'll be able to tell that there isn't any light coming from a particular area. However, the area won't be precisely what you would calculate from the hole's mass; it will be somewhat larger. See below.

and seeing matter cross an event horizon and then see that the disk of the event horizon is larger

This is more problematic, because any light that passes close to the hole gets time delayed (this is called the Shapiro time delay); the closer it comes to the hole's horizon as it passes, the more it gets delayed. So the area of sky that appears to have no light coming from it, from very far away, will be somewhat larger than the area you would calculate from the hole's mass, because the dark region isn't just due to light being blocked by the hole altogether; at its edges it's also due to light being so time delayed from passing the hole that it simply hasn't gotten to you yet. And if the hole's horizon grows due to matter falling in, that just means more light now gets time delayed by enough that it won't have reached you yet. So it may take a very long time after you see the matter itself disappear, before you notice any difference in the size of the dark area due to the hole.

 

[Grinkle]

Thanks, Peter - you've given me a lot to delve into.

 

[Jimster41]

To analyze a single proper time relationship between a particle observing from the outside, and a particle falling in you would look at some Lorentz transformation results for the proper time observable at some coordinate frame "point". The coordinate frame chosen could be any coordinate frame. Is that correct?

Is there a way to analyze the evolution of some set of steps of proper time relationship? I was thinking that would be a curve in the "connection" space of the covariant derivative. Is that correct?

Lord, is ther much that is harder to understand than Relativity...

 

[Tanelorn]

In the early universe there would have been more hydrogen gas and at a much higher densities, so the first generation of stars would have formed very quickly and be much more massive.

When gas collapses would it not have to create a star before it can create a black hole? Or can these first stars continue to accumulate mass so quickly that some can quickly collapse into a black hole soon after their birth?

This page says that even a body of ordinary water can have a Schwarzschild Radius and thus be a black hole:
http://www.essayweb.net/astronomy/calculations/mathematica/schwarzschild_radius/blackhole.html

Could enough hydrogen gas accumulate in one place to produce a black hole without first forming a star?

 

[PeterDonis]

When gas collapses would it not have to create a star before it can create a black hole? Or can these first stars continue to accumulate mass so quickly that some can quickly collapse into a black hole soon after their birth?

I think this is an open question; the second case is certainly possible in principle, we just don't know to what extent it actually happened.

 

[PeterDonis]

To analyze a single proper time relationship between a particle observing from the outside, and a particle falling in you would look at some Lorentz transformation results for the proper time observable at some coordinate frame "point".

No, you wouldn't. Lorentz transformations are not valid in curved spacetime, except locally. And coordinate transformations in general do not have the direct physical interpretation that Lorentz transformations between inertial frames do in SR.

Is there a way to analyze the evolution of some set of steps of proper time relationship?

No. In a curved spacetime there is no such thing as "proper time relationship" in any invariant sense between spatially separated observers. It is purely a matter of convention.

 

[Jimster41]

No, you wouldn't. Lorentz transformations are not valid in curved spacetime, except locally. And coordinate transformations in general do not have the direct physical interpretation that Lorentz transformations between inertial frames do in SR.

I must somehow have misunderstood posts 28 and 35 on the page of the analyzing the twins using GR thread. I was imagining that you could define two locally flat spacetime regions on an otherwise curved spacetime, connect their inertial coordinate frames using the "connection" coefficient?, or a series of them, and this is essentially the same as defining a tensor field that connects them

https://www.physicsforums.com/threa...ity-to-analyze-the-twin-paradox.806102/page-2

@stevendaryl said this in that thread, and I admit latched onto it, as a helpful chain of relation between SR and GT... but I must have misunderstood.

"It's true that people lump considerations of curvature and nontrivial topologies to GR. Dealing with curved spacetime requires a lot of mathematical machinery that flat spacetime does not, but conceptually it doesn't seem that big a leap beyond SR. Conceptually, you break spacetime into little regions, and make sure that SR holds (approximately) in each region, and that solutions in neighboring regions are consistent in the overlap.

To me, the transition from SR to GR has a number of steps:

1. SR in Cartesian, inertial coordinates.
2. SR in curvilinear, noninertial coordinates.
3. SR in curved spacetime and nontrivial topologies.
4. The field equations relating curvature to the stress/energy tensor.

The transition from 1 to 2 is just mathematics, not physics, even though it's kind of difficult mathematics. But once you've got to step 2, you've already got most of the machinery needed to go on to step 3. Once you've allowed the components of the metric tensor to be nonconstant (which is what you need for curvilinear, noninertial coordinates), allowing spacetime to be curved is not a big leap. I think that what took Einstein so long in developing GR was the final step."

 

[PeterDonis]

I was imagining that you could define two locally flat spacetime regions on an otherwise curved spacetime, connect their inertial coordinate frames using the "connection" coefficient?, or a series of them, and this is essentially the same as defining a tensor field that connects them

If by "tensor field" you mean "Lorentz transformation" (they're not the same thing), then no, that's not what this does. The general term for what you are describing is "parallel transport"; given a vector at one point in spacetime, and a curve connecting that point to another point, you can use the connection to parallel transport the vector along the curve from one point to the other. You can then compare this vector to other vectors at the second point. However, all of this is independent of any coordinates, and it doesn't give you a Lorentz transformation from inertial coordinates in a small region centered on the first point to inertial coordinates in a small region centered on the second point.

Furthermore, the process of parallel transport in curved spacetime is path-dependent; the result you get depends on the curve along which you do the transport. So the correspondence induced between vectors at the two points is not even unique.

I admit latched onto it, as a helpful chain of relation between SR and GT... but I must have misunderstood.

It's a fair description of one chain linking SR to GR; but if you thought his step 3, "SR in curved spacetime", implied that you could somehow define Lorentz transformations in curved spacetime beyond a single local inertial frame, then yes, you misunderstood.

 

[Jimster41]

If by "tensor field" you mean "Lorentz transformation" (they're not the same thing), then no, that's not what this does. The general term for what you are describing is "parallel transport"; given a vector at one point in spacetime, and a curve connecting that point to another point, you can use the connection to parallel transport the vector along the curve from one point to the other. You can then compare this vector to other vectors at the second point. However, all of this is independent of any coordinates, and it doesn't give you a Lorentz transformation from inertial coordinates in a small region centered on the first point to inertial coordinates in a small region centered on the second point.

Furthermore, the process of parallel transport in curved spacetime is path-dependent; the result you get depends on the curve along which you do the transport. So the correspondence induced between vectors at the two points is not even unique.
It's a fair description of one chain linking SR to GR; but if you thought his step 3, "SR in curved spacetime", implied that you could somehow define Lorentz transformations in curved spacetime beyond a single local inertial frame, then yes, you misunderstood.

Thanks again Peter. It helps to be corrected in the moment.

 

[James Alton]

If by "tensor field" you mean "Lorentz transformation" (they're not the same thing), then no, that's not what this does. The general term for what you are describing is "parallel transport"; given a vector at one point in spacetime, and a curve connecting that point to another point, you can use the connection to parallel transport the vector along the curve from one point to the other. You can then compare this vector to other vectors at the second point. However, all of this is independent of any coordinates, and it doesn't give you a Lorentz transformation from inertial coordinates in a small region centered on the first point to inertial coordinates in a small region centered on the second point.

Furthermore, the process of parallel transport in curved spacetime is path-dependent; the result you get depends on the curve along which you do the transport. So the correspondence induced between vectors at the two points is not even unique.
It's a fair description of one chain linking SR to GR; but if you thought his step 3, "SR in curved spacetime", implied that you could somehow define Lorentz transformations in curved spacetime beyond a single local inertial frame, then yes, you misunderstood.

Given that time from our perspective effectively stops at the event horizon of a black hole, has any matter ever been consumed by a black hole? This stoppage of time also raises another question for me. When measuring the gravitational effects of a black hole, can we measure the portion caused by the mass contained within the EH or only what is outside the EH? Since gravity apparently travels at the speed of light, it would appear that the passage of time would be required for it to propagate through space, would it not?? A fascinating subject! Thanks for any insight. James

 

[PeterDonis]

Given that time from our perspective effectively stops at the event horizon of a black hole, has any matter ever been consumed by a black hole?

Yes. "Time from our perspective" is not a good standard of time to use when you are trying to figure out what happens to something falling into a black hole. There have been a number of recent threads on this in the relativity forum.

When measuring the gravitational effects of a black hole, can we measure the portion caused by the mass contained within the EH or only what is outside the EH?

A black hole is a vacuum; to the extent its mass is "located" anywhere, it's at the singularity at $r = 0$.

Since gravity apparently travels at the speed of light, it would appear that the passage of time would be required for it to propagate through space, would it not??

A black hole's gravitational field is static; nothing has to propagate. Or, to put it another way, the gravitational field of the hole is an "imprint" left on spacetime by the object that originally collapsed to form the hole; this "imprint" is static and does not change with time. To the extent that the gravity you feel as coming from the hole is "propagated" from anywhere, it's propagated from the past, from the object that originally collapsed to form the hole.

 

[James Alton]

Yes. "Time from our perspective" is not a good standard of time to use when you are trying to figure out what happens to something falling into a black hole. There have been a number of recent threads on this in the relativity forum.

Peter, Thanks for this, I will read through the recent threads you mentioned. A black hole is a vacuum; to the extent its mass is "located" anywhere, it's at the singularity at $r = 0$.

Understood, thanks. My question referred to the probable physical structure of a black hole.
A black hole's gravitational field is static; nothing has to propagate. Or, to put it another way, the gravitational field of the hole is an "imprint" left on spacetime by the object that originally collapsed to form the hole; this "imprint" is static and does not change with time. To the extent that the gravity you feel as coming from the hole is "propagated" from anywhere, it's propagated from the past, from the object that originally collapsed to form the hole.

Yes, I like the frozen star analogy, it makes sense. What troubles me however is that while the gravitational/ electromagnetic effects could be static (imprint) what is happening when the black hole is moving through space? Take the case of a binary star system where one of the stars becomes a black hole. Does the core of the imploded star continue to orbit it's companion as before? Would it's effects in that case not be dynamic on other objects affected by it's fields? The concept of mass in motion through space that is essentially frozen in time makes my head hurt a little. (grin) Wouldn't any movement through space of mass inside of a black hole violate c? Thanks for the insight. James

 

[PeterDonis]

What troubles me however is that while the gravitational/ electromagnetic effects could be static (imprint) what is happening when the black hole is moving through space?

Motion is relative; you can always view this using coordinates in which the hole is at rest, and other objects are simply moving in the static field of the hole.

Take the case of a binary star system where one of the stars becomes a black hole. Does the core of the imploded star continue to orbit it's companion as before? Would it's effects in that case not be dynamic on other objects affected by it's fields?

The hole's mass is unchanged, so from far away it behaves like any other object with the same mass.

The concept of mass in motion through space that is essentially frozen in time

A black hole is not "frozen in time". Again, the "time" of a distant observer is not a good standard of time when dealing with black holes. Someone who falls into a black hole would see time in their vicinity flowing perfectly normally.

Wouldn't any movement through space of mass inside of a black hole violate c?

No. The condition "nothing can move faster than light" is a local condition; no object can move faster than a light ray that is spatially co-located with it. But in curved spacetime, such as a black hole, there is no way to compare velocities of spatially separated objects--more precisely, such comparisons, while they can be done (by just comparing rates of change of coordinates), have no physical meaning. So while an object that falls into the hole could be moving "faster than light" in a coordinate sense, compared to an object far outside the hole, that comparison has no physical meaning; locally, the object inside the hole is moving slower than light rays in its immediate vicinity, and that is the only comparison that has physical meaning.

 

[James Alton]

Motion is relative; you can always view this using coordinates in which the hole is at rest, and other objects are simply moving in the static field of the hole.
The hole's mass is unchanged, so from far away it behaves like any other object with the same mass.
A black hole is not "frozen in time". Again, the "time" of a distant observer is not a good standard of time when dealing with black holes. Someone who falls into a black hole would see time in their vicinity flowing perfectly normally.

Peter, Thank you for the amazing discussion!
If it turns out that the Universe has a finite lifespan and goes through some sort of a phase transition in the future or ends in some other fashion, what would then happen to our black holes and the observer falling into them?
No. The condition "nothing can move faster than light" is a local condition; no object can move faster than a light ray that is spatially co-located with it. But in curved spacetime, such as a black hole, there is no way to compare velocities of spatially separated objects--more precisely, such comparisons, while they can be done (by just comparing rates of change of coordinates), have no physical meaning. So while an object that falls into the hole could be moving "faster than light" in a coordinate sense, compared to an object far outside the hole, that comparison has no physical meaning; locally, the object inside the hole is moving slower than light rays in its immediate vicinity, and that is the only comparison that has physical meaning.

 

[stedwards]

What is "cosmological time"? If you mean proper time for an observer falling in with the collapsing matter, then for a star with the size and mass of the Sun, it takes about an hour to collapse to r = 2m. This was first calculated by Oppenheimer and Snyder in their classic paper on gravitational collapse in 1939. Misner, Thorne, and Wheeler has a good discussion.

Do you have an MTW section number?

 

[PeterDonis]

Do you have an MTW section number?

I don't have my copy handy to check, but IIRC it's in one of the later chapters where gravitational collapse is discussed.

 

[Drakkith]

I was a little disappointed no one else found this interesting enough to comment on - tx marcus!

Personally, I rarely comment on papers posted here in the Astronomy or Cosmology forums because half the time it's like reading braille mixed with some strange form of hieroglyphs and it's incredibly taxing to get through them.