The singularities of gravitational collapse and cosmology

[Cerenkov]

This should be discussed in a separate thread. I can spin off that portion of your post to a new thread if you want.

Ummm... there seems to be no need, Peter. But thank you for offering. Your other response is where I'll reply, thank you.

Cerenkov.

 

[PeterDonis]

there seems to be no need, Peter

I'll go ahead and give a brief response on the white dwarf/neutron star question anyway. The short answer is, yes, if objects are found above the currently accepted limit then the limit needs to be re-assessed because at least one of its assumptions must be violated. But in the case of the Chandrasekhar limit for white dwarfs and the corresponding Tolman-Oppenheimer-Volkoff limit for neutron stars, there are two obvious assumptions that are idealizations and might need adjustment: perfect spherical symmetry (i.e., exactly zero rotation) and the equation of state (the relationship between density and pressure). A third factor might also be involved, magnetic fields, which were not included in the standard calculations of the limits. For white dwarfs, the equation of state is not really open to question, but the other factors could be different and could affect the limit; for neutron stars, all of the factors, including the equation of state, could be different (we don't have a good understanding of the equation of state of strongly interacting matter at neutron star densities and above). These issues are understood by astronomers and continue to be researched.

 

[Cerenkov]

They're the same thing; "conditions" as I was using the term is just another word for "assumptions" (still another word that is often used is "premises" of a theorem).

Ok, thanks for putting me straight there.

There's no need to go looking for other theorems whose assumptions might not be true in the real universe. We already know of possible cases where the assumptions of the Hawking-Penrose theorem would not be true in the real universe.

Here are two of them: dark energy (i.e,. a cosmological constant) and a scalar field (such as is used in the simplest models of inflation). Both of these violate the energy conditions, which are key assumptions of the Hawking-Penrose theorem. So it's perfectly possible to have, for example, inflation models of the early universe with no initial singularity (such as "eternal inflation" models). And quantum fields under certain conditions have an effective stress-energy tensor similar to dark energy, and there are proposals for gravitational collapse that invoke this property to avoid singularities inside black holes, by having some kind of quantum "bounce" happen during the collapse process, after a horizon is formed but before a singularity is formed.

Right. Two examples where the energy conditions are violated. Where the key assumptions of the H-P theorem would be violated. Thank you for that insight.

But what of the assumptions/conditions of the H-P theorem itself, Peter? If you look back to message #4, you corrected me about the 'strongest' gravitational fields GR had been tested under. (Strength being more properly called, spacetime curvature.) Binary pulsars and the LIGO detection of black hole mergers being the two cases you cited. So far, the H-P assumptions seems to be holding up in both?

But when it comes to the cosmological (initial) singularity in the H-P theorem, isn't the spacetime curvature assumed to become infinite? This state of affairs puzzles and troubles me in equal measure! Unlike binary pulsars and black hole mergers, which have been observed, the initial singularity (should such an entity exist) has not.

So how can the assumptions which lead to it being predicted by the theorem ever be tested? By inference, I presume. Just as we cannot see past the event horizons of black holes and have to infer what might be within them, so I imagine that we are forced to do the same with the initial singularity. To my knowledge it's cloaked by the CMBR and the only way we could infer anything about it might be with the help of primordial gravitational waves. It all seems very tenuous to me and I really can't see how the assumptions of the H-P (unlike those underpinning the Chandrasekhar limit) can ever be tested.

(Hmmm... sorry for this over-wordy message. Enough rambling!)

Thank you.

Cerenkov.

p.s.
Thanks also for the info on neutron stars and the Chandrasekhar limit. Very interesting! :)

 

[Cerenkov]

I'll go ahead and give a brief response on the white dwarf/neutron star question anyway. The short answer is, yes, if objects are found above the currently accepted limit then the limit needs to be re-assessed because at least one of its assumptions must be violated. But in the case of the Chandrasekhar limit for white dwarfs and the corresponding Tolman-Oppenheimer-Volkoff limit for neutron stars, there are two obvious assumptions that are idealizations and might need adjustment: perfect spherical symmetry (i.e., exactly zero rotation) and the equation of state (the relationship between density and pressure). A third factor might also be involved, magnetic fields, which were not included in the standard calculations of the limits. For white dwarfs, the equation of state is not really open to question, but the other factors could be different and could affect the limit; for neutron stars, all of the factors, including the equation of state, could be different (we don't have a good understanding of the equation of state of strongly interacting matter at neutron star densities and above). These issues are understood by astronomers and continue to be researched.

Aha! Thank you, Peter.

So would I be so very far off the mark if I were to suggest that, like the Tolman-Oppenheimer-Volkoff limit, the assumptions made by Hawking and Penrose carried with them a degree of idealization? Because in the 60's and 70's their singularity theorems were attempting to describe things that were beyond all possibility of testing?

Cerenkov.

 

[PeterDonis]

But what of the assumptions/conditions of the H-P theorem itself, Peter?

Um, the energy conditions are assumptions/conditions of the H-P theorem itself.

So far, the H-P assumptions seems to be holding up in both?

The binary pulsar case doesn't test any conditions similar to the H-P theorem or its assumptions.

Black hole mergers at least test physics close to an apparent horizon (trapped surface), which is one of the key conditions in the H-P theorem. But the LIGO black hole merger detections were of vacuum black hole mergers, not highly dense collapsing matter, so they don't test anything about the stress-energy tensor of highly dense collapsing matter, which is what we would need to probe to test the H-P theorem energy condition assumptions. A vacuum solution trivially satisfies the energy conditions because the stress-energy tensor is zero everywhere, but that's not a very helpful test.

when it comes to the cosmological (initial) singularity in the H-P theorem, isn't the spacetime curvature assumed to become infinite?

It's not "assumed", it's derived as a theorem. A singularity is spacetime curvature becoming infinite (at least, if we're willing to accept that level of sloppiness of language).

how can the assumptions which lead to it being predicted by the theorem ever be tested? By inference, I presume

That's one way, yes. Another way is to look for cases of highly dense collapsing matter, such as supernovas, and collect as much data about them as we can. Still another way would be to try to create such conditions in the laboratory, but it's going to take a lot longer to develop the capability to do that.

 

[PeterDonis]

So would I be so very far off the mark if I were to suggest that, like the Tolman-Oppenheimer-Volkoff limit, the assumptions made by Hawking and Penrose carried with them a degree of idealization?

Not really, no. Spherical symmetry is an idealization; we don't expect anything in the real universe to satisfy it exactly. The energy conditions and the trapped surface assumptions of the H-P theorem are not idealizations; we know that all the ordinary matter and radiation we observe satisfies the energy conditions, and we have strong evidence for the existence of trapped surfaces.

The main limitation of the H-P theorem is that it is not constructive; it predicts singularities under certain conditions, but it tells you nothing at all about the specific spacetime geometry around those singularities. So even if the H-P theorem is true of a system in our actual universe, we still have a lot of work to do to figure out what solution of the Einstein Field Equation actually describes that system.

 

[Cerenkov]

Um, the energy conditions are assumptions/conditions of the H-P theorem itself.

The binary pulsar case doesn't test any conditions similar to the H-P theorem or its assumptions.

Black hole mergers at least test physics close to an apparent horizon (trapped surface), which is one of the key conditions in the H-P theorem. But the LIGO black hole merger detections were of vacuum black hole mergers, not highly dense collapsing matter, so they don't test anything about the stress-energy tensor of highly dense collapsing matter, which is what we would need to probe to test the H-P theorem energy condition assumptions. A vacuum solution trivially satisfies the energy conditions because the stress-energy tensor is zero everywhere, but that's not a very helpful test.

This is a fascinating insight Peter.
It's too easy (at least for me) to forget that a black hole is not composed of highly dense collapsing matter.

It's not "assumed", it's derived as a theorem. A singularity is spacetime curvature becoming infinite (at least, if we're willing to accept that level of sloppiness of language).

Thank you.

That's one way, yes. Another way is to look for cases of highly dense collapsing matter, such as supernovas, and collect as much data about them as we can. Still another way would be to try to create such conditions in the laboratory, but it's going to take a lot longer to develop the capability to do that.

I see. Thank you, again.

 

[Cerenkov]

Not really, no. Spherical symmetry is an idealization; we don't expect anything in the real universe to satisfy it exactly. The energy conditions and the trapped surface assumptions of the H-P theorem are not idealizations; we know that all the ordinary matter and radiation we observe satisfies the energy conditions, and we have strong evidence for the existence of trapped surfaces.

The main limitation of the H-P theorem is that it is not constructive; it predicts singularities under certain conditions, but it tells you nothing at all about the specific spacetime geometry around those singularities. So even if the H-P theorem is true of a system in our actual universe, we still have a lot of work to do to figure out what solution of the Einstein Field Equation actually describes that system.

Thank you for these insights, Peter.
You've given me a great deal to consider. I knew something of the the successes of the H - P theorem, but next to nothing about it's problems. With that in mind, I'd like to draw this thread to a close by asking just one more thing of you.

Given my basic level of understanding, where may I find out more about two things?

* The stress-energy tensor
* How the H-P theorem falls short in describing the spacetime geometry around the singularities

My hope is that I'll arrive at a more balanced view of the H-P theorem.

With thanks in advance.

Cerenkov.

 

[PeterDonis]

The stress-energy tensor

Short of tackling one of the classic textbooks, I believe Sean Carroll's online lecture notes on GR give a decent treatment of this.

How the H-P theorem falls short in describing the spacetime geometry around the singularities

I don't know that there's much more to say about this other than the fact that the theorem says nothing whatever about the spacetime geometry.

 

[Cerenkov]

Short of tackling one of the classic textbooks, I believe Sean Carroll's online lecture notes on GR give a decent treatment of this.
I don't know that there's much more to say about this other than the fact that the theorem says nothing whatever about the spacetime geometry.

Thank you for this, Peter.
I'll check out what Carroll has to say.

Cheers,

Cerenkov.

 

[PeterDonis]

Thank you for this

You're welcome!