I think this mechanism would be largely indistinguishable from classic hubble expansion, at least on local scales.
Let me explain. When we talk about cosmology in general and the (accelerated) expansion of the universe in particular, we zoom out to very large scales and consider a homogenous model of the universe (a so-called FLRW spacetime[0]). Here, homogeneity means that mass density, hubble rate and so on are the same across the universe and only depend on time.
We then deduce that at this scale and under these assumptions we need to incorporate an additional parameter Λ, called cosmological constant aka dark energy, into our field equations / cosmological solution in order to match observations.[1]
Homogeneity was a simplifying assumption, though! Our universe is clearly not homogeneous! Next to your head, there is air, and inside your head evidently not :), and so the mass density is clearly not constant across space!
The same thing holds for black holes: Outside a black hole there's vacuum, inside a black hole there is… Well, we don't really know but the mass from which it formed has gotta be somewhere, right? So the mass density in black hole spacetimes is presumably not constant, either.
Interestingly, we know that spacetime near other celestial bodies (galaxies, stars, planets, moons, …) can be approximately described by one of the black hole spacetimes, too. (This is because outer region of black hole spacetimes describes not just black holes but any spherically or axially symmetric static/stationary spacetime.) So it's turtl—… uhh outer black hole spacetimes all the way down!
Anyway, in those cases of stars/planets/moons we have some massive object in the center of the spacetime region and vacuum outside – once more, the mass density is clearly non-zero!
So how do we square this with the homogeneity assumption of our cosmolical model (ΛCDM)? We can't but that's perfectly fine from a logical point of view. The reason is that we cannot simply take all those local spacetimes (of all stars, planets and black holes) and simply add them up to obtain the spacetime of the entire universe, because the field equations are not linear. Adding up two solutions does in general not yield a third solution! Conversely, we cannot simply zoom in on the FLRW spacetime and then compare a local, perfectly homogeneous "snippet" of FLRW with a black hole spacetime – this comparison does not make sense a priori.
Unfortunately, the reality is we simply don't know how to zoom in / zoom out between different spacetimes at different scales. So while we have a cosmological constant at large scales (when assuming homogeneity), there is no guarantee such a constant makes sense at smaller scales, i.e. that there is expansion at smaller scales.
This is because there are two possible ways to interpret the cosmological constant: One way is to interpret it as a fixed parameter in the field equations (i.e. it influences every solution), another way is to interpret it as a term in the specific large-scale spacetime solution of our universe (i.e. ΛCDM) and shove it into that solution's energy-momentum tensor.
Now, I think most physicists adhere to the first interpretation and assume that the constant is the same across all scales and possibly even homogeneous. Thus it should impact all spacetime solutions in the same way and at all scales, including black hole solutions.
For this reason people have introduced modified black hole solutions that, like our cosmological model, take into account a positive cosmological constant (= "de Sitter"), for instance
https://en.wikipedia.org/wiki/De_Sitter%E2%80%93Schwarzschil...
As you suspected, in these solutions you generally have both effects that counteract each other: The gravitational pull of the black hole (leading to an event horizon) and the expansion of the universe ("anti-gravitational pull") due to the cosmological constant, leading to a cosmological horizon in the region far away from the black hole.
Nevertheless, whether you believe in/consider a cosmological constant at small scales is a bit up to you. Most people I know seem assume that space at the level of atoms, planets, solar systems or even galaxies is not expanding and only very far away from gravitational systems you'll end up with expansion. This is supported by the fact that the cosmological constant is so tiny and, thus, in the aforementioned De Sitter black hole solutions, the cosmological horizon is far, far away from the center – so far indeed (111 (M/Msolar)1/3 parsecs[2]) that we know this outside region of the spacetime can no longer be valid/applicable.
To see the latter, remember that black hole solutions assume a perfect vacuum away from the central body (black hole/star/planet/…) but in reality no massive body is alone in the universe. This means that, before you reach a distance of 111 (M/Msolar)1/3 parsecs from a given body, you'll long have encountered another couple massive bodies which will modify your spacetime and cause additional gravitational pull. (Once again, how exactly they modify spacetime we don't know (short of maybe some numerical approximations), since we can't easily superpose spacetimes!)
Long story short: Only in very few situations it's worth considering a cosmological constant at small scales. Its effects are easily cancelled out / hidden by local gravity.
[0]: https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtr...
[1]: https://en.wikipedia.org/wiki/Lambda-CDM_model
[2]: https://ui.adsabs.harvard.edu/abs/2020AAS...23537904F/abstra...
Not in the standard black hole model with a singularity at the center, no. In the standard black hole model, the collapsing matter that formed the hole hits the singularity and is destroyed. The interior of the hole is vacuum, just like the exterior.
In the alternative "black hole" model being used in the proposed hypothesis for dark energy, no singularity is ever formed; instead, the collapsing matter undergoes a kind of phase transition (probably induced by quantum field effects of some sort) that changes its equation of state from that of normal matter to that of dark energy. This stops the collapse and creates an object that, at least on time scales much shorter than the Hawking evaporation time scale (since these objects will eventually evaporate by emitting Hawking radiation), looks from the outside like a standard black hole, though it isn't. The dark energy inside such objects, it is proposed, could drive accelerated expansion of the universe.
If the interior of a black hole becomes a vaccume, what continues to assert the intense gravitational pull that the now destroyed matter once created?
[0]: https://news.ycombinator.com/item?id=34898647
[1]: https://en.wikipedia.org/wiki/Mass_in_general_relativity#ADM...
But what this really means I haven't figured out yet.
The other person[0] responded to this thread by saying something like the curvature of space-time is curved, and there is no gravity in GR. So OK, but what causes the curvature if not mass / matter..?
The spacetime geometry of the hole. The "pull" you describe is a property of the spacetime geometry. Gravity is not a force in GR, so the "pull" is not being caused by an interaction with matter. Objects moving solely under gravity simply move on geodesics of the spacetime geometry.
Replace my question with space-time curvature:
"If the interior of a black hole becomes a vacuum, what continues to assert the intense space-time curvature that the new destroyed matter once created?"
1. The simplest black hole solutions in General Relativity are eternal black holes in vacuum. Yes, you read that right: There is no mass and yet they live forever (and have lived forever). There is no way to say why that is other than: The field equations permit these solutions, so they are possible (at least in principle).
The thing is: Nothing in the field equations says there has got to be matter for your spacetime to have curvature. The spacetime just has to fulfill the equations. If you set the energy-momentum tensor in the equations to zero (i.e. there is no matter), you end up with the equation Ric = 0, where Ric is the Ricci tensor, and it turns out that this equation has non-trivial solutions. "Non-trivial" here means: Non-trivial curvature (i.e. not flat) and/or non-trivial topology (e.g. not infinite volume like 4D Minkowski space).
For instance, apart from the curved black hole solutions, you could also have non-trivial flat solutions with the topology of a 4D torus/donut, i.e. with a finite volume.
Why do these solutions exist? Well, because each of them fulfills Ric = 0. Put differently, your question basically amounts to asking "Why do spheres exist if there are planes?" (Both are the solution to the 2D equation k = 0, where k is 2-dimensional (sectional) curvature.)
-- Intermezzo --
Let me approach your question from a slightly different angle: You are understandably surprised that vacuum black hole (i.e. curved) solutions exist because you associate curvature with gravity and have learned that curvature comes from matter ("matter curves spacetime and curvature is gravity"). But this only have the story:
No one ever said that curvature has to come from matter. What's more, gravity is a very narrow, human-invented term that comes from pre-relativistic times when apples were falling from trees. See, the fundamental thing about General Relativity is not that matter curves spacetime or gravity is curvature but that
1) the universe is a 4-dimensional object ("Lorentzian manifold" in math speech) which fulfills the Einstein field equations and which we call spacetime, and that
2) in the absence of other forces (i.e. in free fall), objects follow the equivalent of straight lines ("geodesics") in that spacetime.
In this sense, even the flat vacuum (Minkowski space) has "gravity": It's just very boring gravity because the geodesics are actual (Euclidean) straight lines: An object will stand still in 3D space and only move in time.
In short: Gravity doesn't care about the spacetime being curved or non-curved. It's always there. It's just that in human, practical terms, we call one situation (falling down from a tree) "gravity" and another one (standing still in empty 3D space) "absence of gravity" but from the perspective of General Relativity there is no real difference: Both are spacetimes with geodesics.
So is it surprising that there are curved spacetimes without any matter in them? I don't think so.
-- Intermezzo end --
Back to our eternal black hole solutions in vacuum: As I mentioned, not only do they live forever but they also have lived forever, so there are not exactly great models for reality as we have yet to encounter a black hole that's ∞ years old.
This brings us to the second case:
2. In realistic models of black holes (i.e. black hole formation) we don't really know what happens to the matter once it passes the event horizon. General Relativity predicts the matter will hit the singularity in finite time ("finite proper time") but in reality we have no clue what happens there.[0] Maybe it vanishes, maybe it does another thing entirely (because of quantum-gravitational effects or who-knows-what).
So I'm not sure I would go as far as saying once a black hole has formed, the situation is similar to one of those vacuum black holes where there's no matter whatsoever. (So I'm not sure your question applies here.)
But even if the situation is similar: We have learned in part 1) that spacetime can be curved for no reason.
Besides, as I mentioned elsewhere[1], in certain situations we can associate with curvature a mass, even if there is no matter present (i.e. we're in vacuum). So I think this answers your other question:
> If the blackhole is then a vacuum, surely it has no mass
Yes, it does!
Anyway, this means another possible answer to your question, why vacuum black holes (in theory) exist, is: They themselves have mass (in the aforementioned generalized sense) and therefore sustain themselves and their curvature.
Hope that helps!
The object that collapsed to form the black hole.
> If the interior of a black hole becomes a vacuum, what continues to assert the intense space-time curvature that the new destroyed matter once created?
Nothing has to. The curvature maintains itself once it is formed by the collapsing matter. This is an example of an effect of the nonlinearity of the Einstein Field Equation.
Sure, but that's a model for which we don't have any observational evidence whatsoever since it's the black hole's interior. That's why I phrased it that way.
It doesn't matter, though, even if you say the matter inside the black hole no longer exists (and I'm very happy to entertain that thought): Any sensible definition of mass of the vacuum black hole (i.e. not necessarily the ADM mass since it's defined at infinity but e.g. some definition of quasi-local mass) will still give a non-zero contribution to the mass density.
While this is true, it's not a very strong statement. Physicists extend models all the time into domains where we can't directly test them. The singularity theorems of GR guarantee that the key features of that model, the singularity and the collapsing matter getting destroyed in it, must be true as long as the collapsing matter has the equation of state of ordinary matter. That's why the phase transition I mentioned, to an equation of state corresponding to dark energy, is necessary to avoid forming the singularity and leaving only vacuum in the interior--because the dark energy equation of state violates the energy conditions that are premises of the singularity theorems.
> Any sensible definition of mass of the vacuum black hole (i.e. not necessarily the ADM mass since it's defined at infinity but e.g. some definition of quasi-local mass) will still give a non-zero contribution to the mass density.
Sure, but this "mass" is not due to the continuing presence of matter somewhere inside the hole. It's a property of the spacetime geometry.
Sure but it's not every day that we extend models into a domain (near the singularity) where we know our model must eventually break down somehow. So I think you'll see why I'm a bit sceptical about your claim that
> the collapsing matter that formed the hole hits the singularity and is destroyed
I don't even know what "destroyed" is supposed to mean. Is the claim that the matter simply disappears from the spacetime, with all its conserved quantum numbers and other conserved quantities?
As for:
> The singularity theorems of GR guarantee that the key features of that model, the singularity and the collapsing matter getting destroyed in it, must be true as long as the collapsing matter has the equation of state of ordinary matter.
Do they? The Singularity Theorems assume the existence of a trapped surface. Last I checked[0] (I didn't check deeply, though) the mathematical results on when trapped surfaces arise are very few and far between. Is there a clear proof that in realistic models of black hole formation we must have a trapped surface eventually?
[0]: This was in 2020 when the Nobel committee dubiously claimed that Penrose's Singularity Theorems prove that black hole formation and singularities "are a robust prediction of the general theory of relativity".
Yes, but that doesn't change what the model says. It just affects how likely we think it is that the model is actually realized in our universe. I agree that it's quite likely that the standard black hole model I described isn't realized in our actual universe. But we can still use it if we don't have any better model to replace it with, since even if it breaks down near the singularity, that still leaves the whole rest of the model with plenty of usefulness.
What's of great interest about alternate models for collapsed objects that have dark energy inside, like the Bardeen "black hole", is that they do hold out the promise of being a better model to replace the standard black hole model, that doesn't have a singularity anywhere and so would not be expected to break down the way we think the standard black hole model breaks down near the singularity.
> Is the claim that the matter simply disappears from the spacetime, with all its conserved quantum numbers and other conserved quantities?
The matter disappears, but conserved quantities do not. They remain embedded in the spacetime geometry that is left behind.
> The Singularity Theorems assume the existence of a trapped surface.
Yes, but the alternate "black hole" models with dark energy inside, such as the Bardeen "black hole", also have trapped surfaces, so this doesn't help to distinguish the models.
The singularity theorems also assume energy conditions. Those are the conditions that the models with dark energy inside violate, and which allow those models to not have singularities even though they do have trapped surfaces.
> Is there a clear proof that in realistic models of black hole formation we must have a trapped surface eventually?
I don't know about "proof", but there are plenty of numerical simulations of realistic collapses of massive objects like stars that show trapped surfaces forming. So I would say it's a robust expectation of any such collapse process, even if we don't have an ironclad proof that it must occur in every single case.
> the Nobel committee dubiously claimed that Penrose's Singularity Theorems prove that black hole formation and singularities "are a robust prediction of the general theory of relativity".
That statement was justified. But saying that it's a robust prediction if particular conditions are satisfied is not the same as saying that all of those conditions must be satisfied in our actual universe.
But that's not what they said. They said (or implied) it's a robust prediction of GR for our actual universe.
> But we can still use it if we don't have any better model to replace it with, since even if it breaks down near the singularity, that still leaves the whole rest of the model with plenty of usefulness.
But I never questioned the usefulness of the whole rest of GR? GR is a beautiful and much more satisfying, consistent and mathematically rigorous theory than e.g. QFT, so only because we know its predictions might not hold near singularities I wouldn't dare throwing out the baby with the bathwater. I merely said we don't really know what's happening with the matter once it's inside the black hole / close to the singularity. And it looks like we agree here.
Who is "they"? The people who published the singularity theorems didn't say that. They only said the theorems are mathematically valid given the assumptions, and they proved that by proving the theorems. They didn't say the assumptions had to be satisfied in our actual universe. In fact, most physicists say the opposite: that the mathematical validity of the singularity theorems shows that at least one of the assumptions they are based on must be violated in our actual universe, since it would be physically unreasonable for there to be singularities in our actual universe.
> I merely said we don't really know what's happening with the matter once it's inside the black hole / close to the singularity.
In the sense that most physicists believe a singularity is physically unreasonable, yes, I agree. But in the absence of a better model, that doesn't help very much. The nice thing about the hypothesis under discussion here is that it holds out the prospect of a better model, if it can be made to work.
> [link to wikipedia's de Sitter-Schwarzschild page]
Let's call it Schwarzschild-de Sitter (SdS, for short, and for ease of literature-searching).
More generally there is the McVittie family of metrics of a massive object in a dynamical spacetime. SdS is the limiting case of McVittie where the spacetime is stationary and the central mass is compact, spherically symmetric, 0-angular-momentum, and uncharged. (One could alternatively say that McVittie is a generalized time-dependent SdS.)
> black hole solutions assume a perfect vacuum
Not quite, but this is mainly a specialist quibble, since the most widely known theoretical black holes are vacuum or electrovac spacetimes. See for example the Kerr-Vaidya black hole solutions, which have either a incoming radiation ("null dust") field or an outgoing one (or both) falling onto resp. shining out of ("roughly Hawking") a spinning black hole. There are many other nonvacuum solutions with a compact central mass (which can look more or less black-hole-like), both exact and non-exact.
The important feature of these is asymptotic behaviour, as you touch on in your second-last paragraph, since if the influence of the central mass fades with distance, and the sources are kept distant from each other, that lets us ignore (but see below) the difficulties in combining two or more exact solutions of the Einstein Field Equations into a new exact solution.
Linearized gravity is usually applicable and sufficient, and if not one can obtain corrections using post-Newtonian theory. We don't really need numrel unless mass-ratios are small and compactness is extreme. See the handy diagram at <https://en.wikipedia.org/wiki/Post-Newtonian_expansion#/medi...>.
See also Ellis 2010 (Chapter 2, section 3 on inexact solutions, notably his complaint at the bottom of p. 34 to the top of p. 35) <https://doi.org/10.1017/CBO9780511622724.002>, which is handily also at s c y h o b. Re his complaint see also Visser 2014 on horizons: <https://arxiv.org/abs/1407.7295>.
> > black hole solutions assume a perfect vacuum
> Not quite
You're right, I should have been more precise here. I was merely trying to say: The spacetimes in the vicinity of celestial bodies can be modeled sufficiently well by vacuum black hole solutions (obviously as long as we don't get too close and don't need to consider accretion disks, radiation and what not).
Where modelling an astrophysical body this way tends to fall down is that as far as we can tell the central mass evolves and is in general not uniform in the sense that it has long-lasting multipole moments (that would quickly bald away for a black hole), frustrating the hopes of matching the object's interior with the exterior solution. In comparison, outer space is practically always empty enough of stress-energy that the non-physicality of the vacuum or lambdavac region is rarely the issue (and can be dealt with perturbatively, for example).
The other major problem is that binary (and triple) systems are surprisingly commonplace, and the N-body problem will drive one towards approximations of GR for tractability. This is the essence of this thread's cautions on the difficulties in superposing solutions to the EFEs. Worse, the two problems above can feed into one another, like when binary stars raise long-lived bumps on each other.
This reminds me to re-read the invigorating <https://link.springer.com/article/10.1007/s00190-016-0927-4> which compares the gravitation of Earth and its messy multibody neighbourhood with several exact vacuum solutions and a couple of formalisms. The first author is the redoubtable Michael Soffel <https://www.iau.org/administration/membership/individual/733...> A PDF of the paper can also be found at s c y h o b. The kicker is the two paragraphs before section 3.
The example in the talk was that “spacetime expanding” and “matter shrinking” are mathematically equivalent, and the choice is merely a preference between interpretations.
Said differently, how the expansion of the universe looks like at the human scale? Most probably the quetsion does not make sense (?)
No. That's not how the spacetime geometry around black holes works.