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

[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.

 

[Garth]

Do you have an MTW section number?

MTW Box 24.1, C page 620 - Chapter 32, page 842 'Gravitational Collapse', Chapter 33, page 872 'Black Holes' - and then Chapter 34, page 916 'Global Techniques, Horizons and Singularity Theorems'.

The Oppenheimer and Snyder paper was ''On Continued Gravitational Contraction' Physical Review, Sept. 1939, Vol 56.

I hope this helps,
Garth

 

[stedwards]

MTW Box 24.1, C page 620 - Chapter 32, page 842 'Gravitational Collapse', Chapter 33, page 872 'Black Holes' - and then Chapter 34, page 916 'Global Techniques, Horizons and Singularity Theorems'.

The Oppenheimer and Snyder paper was ''On Continued Gravitational Contraction' Physical Review, Sept. 1939, Vol 56.

I hope this helps,
Garth

Thanks, Garth