State of "matter" of a singularity

[Mitcher]

18 hours ago, joigus said:

I suppose what you mean is that a 2-sphere cannot be mapped with just one chart. I fail to see how this is relevant here.

 

You're still trapped in the mirage of singularities of the coordinate chart. By the same token, you'd probably think there's a problem at the north pole of a sphere... 

I rest my case.

Not at all, I was talking about the change of coordinates Kruskal did specifically to avoid the appearance of a singularity at 2m.

On 8/19/2022 at 11:37 PM, Markus Hanke said:

If the embedding diagram terminates in a throat, then that means the coordinate chart isn’t continuous at that point. This doesn’t imply anything about spacetime itself.

The geodesics don't have to end at the throat, which could be seen, with the appropriate metric, as a space bridge, linking two 4D spaces so the geodesics could be prolonged contiuously in the adjacent field. Time and mass would seen to invert there. As a bonus, the central singularity disappears and negative mass could explain negative energy, which we seem to be badly needing now in cosmology.

[joigus]

7 hours ago, Mitcher said:

Not at all, I was talking about the change of coordinates Kruskal did specifically to avoid the appearance of a singularity at 2m.

Polar coordinates on the plane: (r,theta)

Change of coordinates: 

x=r*cos(theta)

y=r*sin(theta)

Singular @ r=0, as J(x,y/r,theta)=0 there.

Is something fishy going on at r=0?

No! A plane is a plane is a plane.

(Sigh)

7 hours ago, Mitcher said:

and negative mass could explain negative energy, which we seem to be badly needing now in cosmology.

Why?

[Mitcher]

1 hour ago, joigus said:

Change of coordinates: 

x=r*cos(theta)

y=r*sin(theta)

Singular @ r=0, as J(x,y/r,theta)=0 there.

Is something fishy going on at r=0?

No! A plane is a plane is a plane.

Is this supposed to be Kruskal's new coordinates ? It looks like Weyl's ones to me, leading to dsigma^2 = hd(sigma)^2 + r^2 d(theta)^2 which is about the geometry described by the rotation of a parabola of equation z = (8a(r - 2a))^1/2 in euclidian space with ortho coordinates x, y, z. But as Weyl puts it there is "nothing" inside the sphere with radius 2a.

1 hour ago, joigus said:

Why?

Cosmic acceleration implies the action of a negative pressure and hence of negative energy (energy density by unit volume) so it would not be completely nonsensical  to include negative energy states in QFT.

[joigus]

11 hours ago, Mitcher said:

Is this supposed to be Kruskal's new coordinates ? It looks like Weyl's ones to me, leading to dsigma^2 = hd(sigma)^2 + r^2 d(theta)^2 which is about the geometry described by the rotation of a parabola of equation z = (8a(r - 2a))^1/2 in euclidian space with ortho coordinates x, y, z. But as Weyl puts it there is "nothing" inside the sphere with radius 2a.

No. It's our old friend the Cartesian plane \( \mathbb{R}^{2} \). Flat, trivial manifold, with Riemann tensor identically zero, and (0,2) signature. It's totally isomorphic to its tangent space. Yet there are singular charts, and singular changes of different charts. I argued, in a way that you either have been unable or unwilling to understand that those singularities say nothing about the plane. They are singularities in the parametrization. It's plain to see now that you don't understand what a differential manifold is. That's why you're making incorrect statements about Schwarzschild spacetime over and over.

And over.

And over.

And over...

11 hours ago, Mitcher said:

Cosmic acceleration implies the action of a negative pressure and hence of negative energy (energy density by unit volume) so it would not be completely nonsensical  to include negative energy states in QFT.

Yes, it would. 

 

Edited August 29, 2022 by joigus
minor correction

[joigus]

And by the way. It's an illuminating exercise to write down the equations of GR for a completely silly, trivial, flat spacetime, by calculating the Christoffel symbols, the geodesic equation, etc. in curvilinear coordinates. Of course, all the components of the Riemann tensor will be identically zero. But every definition and procedure for curved ST goes through.

Here's a suggestion: Try and use your imagination, and write down a set of curvilinear coordinates that cover most of flat space, you can fill this totally dumb, seamless, featureless spacetime with (non-existent) "singularities" that all disappear once you introduce the proper set of (singular) coordinate changes that remove all your (non-existent) "singularities".

Your toy model of field equations is the Laplace equation with the obvious boundary condition of all fields vanishing at spatial infinity.

The genius of Kruskal was to realise that's kinda similar to what's happening with the Schwarzschild space time during a time when many people working on GR were still just chasing shadows.

Stop chasing shadows, please.

Now I do rest my case.

---

One final caveat. If you insist on thinking of so-called vacuum energy in tems of energy content of some stuff, in the sense that it can be expressed as a density of "something", you're gonna run into problems. We call it "dark energy" for lack of a better word. But it's not really a local energy density. Dark energy does not dilute when you stretch your spacetime. This doesn't bode well with the proposal that it's due to a density of stuff filling in spacetime.

And QFT certainly needs not negative energies. Those "cores" of negative energy would be quantum-mechanically unstable, it would violate causality, etc. You need a Hamiltonian bounded below for good reasons.

[studiot]

1 hour ago, joigus said:

Dark energy does not dilute when you stretch your spacetime. This doesn't bode well with the proposal that it's due to a density of stuff filling in spacetime.

In fairness I don't think conventional mass density 'dilutes' with expansion either, but please correct me if I am wrong about that, I am not a cosmologist.

However +1 for the calculations first half of your post.

[joigus]

1 hour ago, studiot said:

In fairness I don't think conventional mass density 'dilutes' with expansion either, but please correct me if I am wrong about that, I am not a cosmologist.

However +1 for the calculations first half of your post.

Thank you, Studiot. Standard cosmology separates different terms for these local densities. I hesitate to use this word "local" on account of how much confusion it's created through the years. People distinguish several terms on the RHS of Einstein's eqs. within the FRWL model according to the different stuff that fills the universe, which leads to the standard "epochs" post-big-bang: radiation dominated, matter dominated, and vacuum-energy dominated (present cosmological epoch):

Radiation-dominated: energy density proportional to a^-1 

Matter-dominated: energy density proportional to a^-3

Vacuum-energy-dominated: energy density = constant (a-independent)

where a is the scale factor.

It's in this sense that I say the vacuum-energy density does not dilute when the universe expands. As everything else would dilute (a^-1,a^-3), on account of no new matter or radiation being formed, while the vacuum energy density is kept constant (which leads to the exponentially growing solution with time that characterises vacuum-driven universes, also known as De Sitter universes) that's why I say it seems difficult to conceive of a model made of stuff --never mind it coming from negative-energy interiors of BHs.

It doesn't seem plausible to me. It's the least I can say.

 

Edited August 29, 2022 by joigus
link added

[studiot]

3 minutes ago, joigus said:

Standard cosmology separates different terms for these local densities.

Thank you. I haven't kept up with cosmology, standard or otherwise, for many years, decades even, as theories seem to change almost as frequently as I change my socks.

 

I had understood that the original reason for (now discredited) ' the continuous creation of steady state theory' because the mass density did not appear to be going down, despite inflation.

[joigus]

43 minutes ago, studiot said:

Thank you. I haven't kept up with cosmology, standard or otherwise, for many years, decades even, as theories seem to change almost as frequently as I change my socks.

 

I had understood that the original reason for (now discredited) ' the continuous creation of steady state theory' because the mass density did not appear to be going down, despite inflation.

Yes. This is the Bondi steady-state model. I grew up with that theory being a serious contender. But that theory came long before inflation, and actually before Penzias and Wilson's discovery of the expanding universe. Hermann Bondi wanted to account for a stationary universe, which we don't believe in anymore. It purported that the density of matter in the universe is constant. It also implied that the universe is eternal.

It sounds similar, but it's quite different.

[studiot]

2 hours ago, joigus said:

Yes. This is the Bondi steady-state model. I grew up with that theory being a serious contender. But that theory came long before inflation, and actually before Penzias and Wilson's discovery of the expanding universe. Hermann Bondi wanted to account for a stationary universe, which we don't believe in anymore. It purported that the density of matter in the universe is constant. It also implied that the universe is eternal.

It sounds similar, but it's quite different.

So why did Bondi and Hoyle propose 'continuous creation', if not because the universe was known to be expanding ?

Surely that would lead to a steadily increasing mass density, which equally surely can't be called 'steady state' ?

[Ghideon]

3 hours ago, joigus said:

But that theory came long before inflation

 

48 minutes ago, studiot said:

So why did Bondi and Hoyle propose 'continuous creation', if not because the universe was known to be expanding ?

(Emphasis mine)

As far as I know, in cosmology Inflation is not the same as Expansion. So You are discussing two different aspects of cosmology, introduced at different times?

Expansion; Hubble 1930's
Steady State, Hermann Bondi, Thomas Gold, and Fred Hoyle publication: 1948
Inflation; Alan Guth in 1979

Edited August 29, 2022 by Ghideon
Emphasis & clarification

[joigus]

25 minutes ago, Ghideon said:

(Emphasis mine)

As far as I know, in cosmology Inflation is not the same as Expansion. So You are discussing two different aspects of cosmology, introduced at different times?

Sure. Very different things. Vacuum energy is very different even from the initial big bang theory before inflation was started by Alan Guth. So it was a contender of the big bang theory. The big-bang theory prevailed after Penzias and Wilson found CMB (1964), and inflation was introduced to solve some of its puzzles.

1 hour ago, studiot said:

So why did Bondi and Hoyle propose 'continuous creation', if not because the universe was known to be expanding ?

Back in the 40's the idea that the universe was eternal and essentially the same at all times in its history was still popular. That had been Einstein's motivation for introducing a cosmological constant in 1917. He wanted a static universe. He failed at this, as his cosmological solution would have been unstable, not static.

[studiot]

29 minutes ago, joigus said:

Sure. Very different things. Vacuum energy is very different even from the initial big bang theory before inflation was started by Alan Guth. So it was a contender of the big bang theory. The big-bang theory prevailed after Penzias and Wilson found CMB (1964), and inflation was introduced to solve some of its puzzles.

Back in the 40's the idea that the universe was eternal and essentially the same at all times in its history was still popular. That had been Einstein's motivation for introducing a cosmological constant in 1917. He wanted a static universe. He failed at this, as his cosmological solution would have been unstable, not static.

That would seem to be at variance with the history I grew up with.

 

Quote

https://www.britannica.com/science/steady-state-theory

steady-state theory, in cosmology, a view that the universe is always expanding but maintaining a constant average density, with matter being continuously created to form new stars and galaxies at the same rate that old ones become unobservable as a consequence of their increasing distance and velocity of recession. A steady-state universe has no beginning or end in time, and from any point within it the view on the grand scale—i.e., the average density and arrangement of galaxies—is the same. Galaxies of all possible ages are intermingled.

 

The theory was first put forward in 1948 by British scientists Sir Hermann Bondi, Thomas Gold, and Sir Fred Hoyle. It was further developed by Hoyle to deal with problems that had arisen in connection with the alternative big-bang hypothesis. Observations since the 1950s (most notably, those of the cosmic microwave background, which was predicted by the big-bang model) have produced much evidence contradictory to the steady-state picture and have led scientists to overwhelmingly support the big-bang model.

 

The experiments that led to the expanding universe predated the 'steady state theory'

So any proposal of steady state had to account for the lack of mass density change.

Quote

On March 15, 1929 – Hubble published his observation that the farthest galaxies are moving away faster than the closest ones. This is the insight that initially became known as Hubble's law.

 

[joigus]

39 minutes ago, studiot said:

That would seem to be at variance with the history I grew up with.

 

 

The experiments that led to the expanding universe predated the 'steady state theory'

So any proposal of steady state had to account for the lack of mass density change.

 

Actually, you're right. I stand corrected. Hubble --> Expansion; Penzias & Wilson --> CMB

Those are different things.

Confirmation of CMB is considered the last stepping stone in proving the standard cosmological model (previous to inflation), but the expansion was observed first.

Thank you.

Nevertheless, the Bondi-Gold-Hoyle model is to do with a steady universe. The idea was, if I remember correctly, that a tiny amount of matter is produced at a constant rate uniformly throughout the universe, compensating for the dilution of matter due to expansion. My details are hazy, as it's been such a long time since it was made irrelevant by experimental observation.

We're getting farther and farther away from the topic of singularities though. Maybe a split is in order?

[Markus Hanke]

On 8/28/2022 at 1:10 AM, Mitcher said:

If a mere change of variables allows for central or spherical singularities to disappear then what is real ?

But that’s the thing - you can remove the singularity at the horizon by a simple change in coordinates (while all curvature tensors are regular there), so it isn’t a physical singularity, merely a coordinate one. At the same time you can not remove the central singularity in the same way, because the region is geodesically incomplete (all curvature tensors diverge or become undefined there).

‘Real’ in this sense is that which does not depend on choice of coordinates, ie covariant quantities such as tensors, but not coordinate charts and coordinate singularities.

On 8/28/2022 at 9:35 PM, Mitcher said:

The geodesics don't have to end at the throat, which could be seen, with the appropriate metric, as a space bridge, linking two 4D spaces so the geodesics could be prolonged contiuously in the adjacent field.

While it may conceivably be possible to construct such a solution (eg as a special case of Ellis-Bronnikov spacetime), this would not be Schwarzschild spacetime, but a different type of geometry. Using the boundary conditions for the Schwarzschild solution, it can be shown that the central singularity is in fact inevitable.

On 8/28/2022 at 9:35 PM, Mitcher said:

Time and mass would seen to invert there.

Schwarzschild black hole, in its collapsed state, is a vacuum solution, it assumes an entirely empty spacetime.

On 8/28/2022 at 1:10 AM, Mitcher said:

ST would be flat very far away from the Sun if it was not for the planets.

Yes, that was my point - in the real world there are all kinds of distant sources, so asymptotic flatness cannot actually occur, meaning Schwarzschild ST is just an approximation that cannot be found in the real world. It’s still very useful though.

[Mitcher]

On 8/30/2022 at 6:23 AM, Markus Hanke said:

Yes, that was my point - in the real world there are all kinds of distant sources, so asymptotic flatness cannot actually occur, meaning Schwarzschild ST is just an approximation that cannot be found in the real world. It’s still very useful though.

Specially that BH, if they would exist, would be formed from rotating neutron stars and the rpm's would just instantly go over the hedge. Everything rotates in the Universe. They will say we have to take into account primordial BH's, but since nobody know how they could have formed...

[Markus Hanke]

16 minutes ago, Mitcher said:

Specially that BH, if they would exist, would be formed from rotating neutron stars

Yes.

I should clarify here though that my comments were specifically about the Schwarzschild solution - this wasn’t meant to imply that BHs don’t exist at all in the real world. It’s just that they would be of a different kind than Schwarzschild. At a minimum you have to consider angular momentum and the absence of asymptotic flatness - which leads to Kerr-Vaidya spacetime as a starting point. That’s a considerably more complex geometry than Schwarzschild.

[Mitcher]

On 8/29/2022 at 11:36 AM, joigus said:

If you insist on thinking of so-called vacuum energy in tems of energy content of some stuff, in the sense that it can be expressed as a density of "something", you're gonna run into problems.

However one of the main problem in Physics is the discrepancy of 120 order of magnitude between the vacuum energy density at the cosmological scale expressed by the cosmological constant and the quantum vacuum energy density, at the Planck scale. The first one is given in g/cm3 and comes from the content of the Universe. The fact that they are being compared is not insignificant, I'am not sure we can speak about a density of nothing, at least it's not easy to grab.

57 minutes ago, Markus Hanke said:

I should clarify here though that my comments were specifically about the Schwarzschild solution

Just curious, hve you read BOTH papers from Schwarschild, with the external but also the internal solution ?

[joigus]

16 minutes ago, Mitcher said:

However one of the main problem in Physics is the discrepancy of 120 order of magnitude between the vacuum energy density at the cosmological scale expressed by the cosmological constant and the quantum vacuum energy density, at the Planck scale. The first one is given in g/cm3 and comes from the content of the Universe. The fact that they are being compared is not insignificant, I'am not sure we can speak about a density of nothing, at least it's not easy to grab.

Agreed that the cosmological constant problem is probably the most significant problem faced by modern physics. I'm not so sure it's related with singularities. Singularities seem to have to do with very strong local fields. Vacuum energy seems to have to do with global properties of ST. In any case, I think it's a topic for discussion in a separate thread. Why don't you open such a thread and present your concerns there?

[Markus Hanke]

9 hours ago, Mitcher said:

Just curious, hve you read BOTH papers from Schwarschild, with the external but also the internal solution ?

I haven’t read any of Schwarzschild’s originals papers, since to me these are only of historical interest, and I’m not much into the history of science.

My main focus is on the contemporary foundations of GR - its underlying symmetries, why the field equations have the specific form they do, what kinds of solutions they admit and how to find and classify them, what kind of covariant objects can “live” on spacetime, possible candidate models for quantum gravity, generalisations of GR etc etc. Things of that nature. 

I acquired all my knowledge from a wide range of contemporary texts and sources on physics and maths, with Milner/Thorne/Wheeler being the oldest of them. Both the internal and external Schwarzschild solutions can be found in depth in nearly all these texts, presented in different ways, and maximally extended metrics that cover the entirety of this particular spacetime are also found in many of these texts. I find it unwise to rely too heavily on a single source or author for one’s understanding of GR, so I always make sure I consult multiple sources.

[Mitcher]

On 9/1/2022 at 9:56 AM, Markus Hanke said:

I haven’t read any of Schwarzschild’s originals papers, since to me these are only of historical interest, and I’m not much into the history of science.

My main focus is on the contemporary foundations of GR - its underlying symmetries, why the field equations have the specific form they do, what kinds of solutions they admit and how to find and classify them, what kind of covariant objects can “live” on spacetime, possible candidate models for quantum gravity, generalisations of GR etc etc. Things of that nature. 

I acquired all my knowledge from a wide range of contemporary texts and sources on physics and maths, with Milner/Thorne/Wheeler being the oldest of them. Both the internal and external Schwarzschild solutions can be found in depth in nearly all these texts, presented in different ways, and maximally extended metrics that cover the entirety of this particular spacetime are also found in many of these texts. I find it unwise to rely too heavily on a single source or author for one’s understanding of GR, so I always make sure I consult multiple sources.

I find it strange that you read multiple authors in the domain of BH but not the original, first one of them. It's not at all about historical interest, it is to make sure Schwarshid was not misunderstood. He was, and the recently translated papers can now easily be found.

On 9/1/2022 at 12:00 AM, joigus said:

Agreed that the cosmological constant problem is probably the most significant problem faced by modern physics. I'm not so sure it's related with singularities. Singularities seem to have to do with very strong local fields. Vacuum energy seems to have to do with global properties of ST. In any case, I think it's a topic for discussion in a separate thread. Why don't you open such a thread and present your concerns there?

My concern there was about previous comment concerning a "density of some stuff". Properties of ST are mainly properties of mass. Most simplest models of BH, with their "incompressible sphere of matter" or 'point-mass' do not take matter itself into account, only the effects it creates on SP and fields. BH is about matter and energy content: no matter, no BH, hence the density towards their center should probably have something to do with Planck's density, at least in theory since my point here is that matter-energy in its known states cannot realistically exist past the horizon. And central singularities even less as it can be cancelled according to some theories.. I read and assimilated past comments though.

[Markus Hanke]

2 hours ago, Mitcher said:

I find it strange that you read multiple authors in the domain of BH but not the original, first one of them.

And I find it a bit strange that you are so fixated on historical papers that are 100+ years old. Would you go and solely rely on Maxwell’s original publications when wanting to learn electromagnetism? Would you read Newton when wanting to learn calculus?

To be clear, there’s nothing wrong with doing this, but you shouldn’t just disregard all the progress that’s been made since. We now know a lot more about GR than Schwarzschild ever did.

But regardless, I’ve done something better than just read about it, whatever the source - I’ve done the derivation of the Schwarzschild metric myself, using pen and paper, and some little extra help from MAPLE with some of the more complicated differential equations. So I understand the boundary conditions, how these give rise to the spacetime, and what the features of this geometry are. I thus have no need to rely on anyone’s words with regards to this subject, since I’ve acquired the tools and knowledge to do the maths myself. It’s certainly quite tedious and takes time, but there’s no mystery left in it. It’s quite straightforward, really. 

So no, Schwarzschild hasn’t been “misunderstood” - there’s literally nothing there to misunderstand or interpret, it’s all just standard differential geometry. You are essentially just working out the solution to a system of differential equations, that’s all.

2 hours ago, Mitcher said:

He was

Well, we’ve been over this within the past three pages of discussion, so I think we may just have to disagree on this. There’s no grounds for any misunderstanding or ambiguity in the Schwarzschild solution, so far as I (and the physics community in general) am concerned. Saying the spacetime somehow terminates at the horizon (which seems to be what you’re imying? Please correct me if I’m wrong) is like covering Earth with a chart that terminates at the equator, and then claiming the Southern Hemisphere doesn’t exist.

2 hours ago, Mitcher said:

Properties of ST are mainly properties of mass.

The properties are of a geometric nature. Even within the energy-momentum tensor, ‘mass’ does not directly appear.

2 hours ago, Mitcher said:

since my point here is that matter-energy in its known states cannot realistically exist past the horizon

The Schwarzschild black hole spacetime (post-collapse) has T=0 and thus R=0 everywhere, so it’s a pure vacuum spacetime. There is no matter of any kind anywhere. The mass term M appearing in the metric is a global property of the entire spacetime, and is technically just a selection parameter for a 1-parameter family of metrics.

Nonetheless, matter and radiation can exist just fine while they are in the process of falling in - but you need a different spacetime geometry to model this accurately, such as Vaidya and its generalisations.

Edited September 4, 2022 by Markus Hanke

[Mitcher]

40 minutes ago, Markus Hanke said:

Saying the spacetime somehow terminates at the horizon (which seems to be what you’re imying?) is like covering Earth with a chart that terminates at the equator, and then claiming the Southern Hemisphere doesn’t exist.

On the contrary I keep saying that geodesics are not bounded at the EH. And the Earth sphere is an apt exemple since its Poles are not a central singularity either. What's the point of those sophisticated geometries since most seem to possess one inside, which it is completely unphysical. It cannot be real that billions of stars will keep falling endlessly at the speed of light into some sort of Gabriel's horn or something.

[MigL]

1 hour ago, Mitcher said:

On the contrary I keep saying that geodesics are not bounded at the EH. And the Earth sphere is an apt exemple since its Poles are not a central singularity either. What's the point of those sophisticated geometries since most seem to possess one inside, which it is completely unphysical. It cannot be real that billions of stars will keep falling endlessly at the speed of light into some sort of Gabriel's horn or something.

You don't seem to realize that you've just given an example of a co-ordinate singularity.

If you use latitude and longitude as your co-ordinate system, the North Pole and South Pole are singular. You cannot go any further North ( or South ) from those points, yet there is no 'edge', and nothing stops you from going further ( around the globe ).

That is the case with the Schwarzschild solution; a co-ordinate singularity ( at the horizon ), but nothing stops you from going further. It is not an 'edge' either.

[studiot]

8 hours ago, MigL said:

You don't seem to realize that you've just given an example of a co-ordinate singularity.

If you use latitude and longitude as your co-ordinate system, the North Pole and South Pole are singular. You cannot go any further North ( or South ) from those points, yet there is no 'edge', and nothing stops you from going further ( around the globe ).

That is the case with the Schwarzschild solution; a co-ordinate singularity ( at the horizon ), but nothing stops you from going further. It is not an 'edge' either.

Lovely and simple that even my dog can understand it. +1

But please note that whilst all lines of longitude are geodesics, the equator is the only line of latitude that is one.
So lat andm long constitute a mixed coordinate system, not a pure geodesic one.

@Mitcher

This reminds me of another mathematical oddity.

The Hairy Ball Theorem and the Mermin-Ho Theorem and their relation to singularities and Index Theory.

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

https://www.parabola.unsw.edu.au/files/articles/2000-2009/volume-43-2007/issue-2/vol43_no2_3.pdf

http://www.maths.adelaide.edu.au/hang.wang/resources/What-is-Index-theory-1.pdf

https://physics.stackexchange.com/questions/656025/how-does-the-mermin-ho-theorem-handle-singularities-arising-from-the-hairy-ball