Gravitational binding energy and the TOV limit

[itssilva]

Disclaimer: to avoid giving the impression of speculative nature, I state the purpose of this thread is only to conflate known theory with my own understanding in a specific point and clarify where the disagreement lies; that is all.

TOV limit: since early research in black hole (BH) formation, it has been known that degeneracy pressure alone isn't enough to hold off the collapse of a sufficiently large neutron star. However, looking up referenced work (cf. the Appendix), I see that the initial assumption was that the degenerate fermion gas was free; now, while I understand modern(er) approaches include couplings with other interactions as well into the stellar equation of state (EOS) and get the same result, I cannot see that this effectively includes the gravitational binding contribution as well.

First, let me explain what I mean by "binding" here: consider a "jellium" or, simpler yet, a hydrogen atom under an external EM field: its electron couples with the the external source, naturally, but also with that of the field that binds it to the nucleus. Now let's try transferring this analogy to the case of the neutron star: in my understanding, I believe that the exterior field of the former corresponds more closely to that of the exterior Schwarzschild potential for the latter, while there seems to be no obvious analogue for the binding field responsible for the cohesion of the star - because, if the neutron gas was otherwise free of all other influences, it is not obvious (to me) that it should clamp itself into a ball-like object instead of merely diffusing away, as soon as its "nuclear fuel" is spent!

So I hope that I may make my point clear now. My question is: while undoubtedly the resulting interior field we get as a solution of the Einstein equations is correctly taken as stemming from a collapsing gravitational influence, whether account has ever been made of the minimally coupled GR field in the EOS as well, and it's seen that collapse occurs anyway. Why I'm asking this: because it could be a "loophole" to the established consensus (?) that big enough neutron stars invariantly collapse into BHs - however, the absence of mention of anything to this effect (which I believe would be all over the place) in the literature as well as popular accounts leads me to believe that there might be theoretical arguments that dismiss this possibility right outta the bat, but I ignore them entirely; so, either something in my intuition above is wrong, or there simply is no research done on the matter; which would it be?

 

[PeterDonis]

I cannot see that this effectively includes the gravitational binding contribution as well.

The "gravitational binding" contribution is in the spacetime metric.

the initial assumption was that the degenerate fermion gas was free

"Free" just means "doesn't interact with itself", i.e., the individual fermions don't interact with each other (apart from the obvious Pauli exclusion principle, if you want to call that an "interaction"). It does not mean there is no gravity. The whole calculation is being done in a curved spacetime metric, which includes all of the effects of gravity, including the part that prevents the neutron gas from flying off to infinity.

 

[PeterDonis]

modern(er) approaches include couplings with other interactions as well into the stellar equation of state (EOS)

That means they are including interactions between the fermions (other than the Pauli exclusion principle), and modifying the stress-energy tensor appropriately. Again, it does not mean there is no gravity; the calculation is still being done in a curved spacetime metric (which is related to the stress-energy tensor through the Einstein Field Equation--or, in this particular case, through the TOV equation, which is derived from the EFE), which includes all of the effects of gravity.

 

[itssilva]

The "gravitational binding" contribution is in the spacetime metric.

I'm not saying that it isn't - I'm asking whether it's not valid to ask whether the usual contribution of the gravitational potential (a.k.a. the metric) is incomplete, based on circumstantial evidence (cf. infra)

"Free" just means "doesn't interact with itself", i.e., the individual fermions don't interact with each other (apart from the obvious Pauli exclusion principle, if you want to call that an "interaction"). It does not mean there is no gravity.

In my understanding, "free" means "uncoupled from", and this applies to all physical interactions, gravity being no exception. If we had a truly "free" gas, it would not couple to the gravitational field in any manner, so that we would have a Minkowski potential as the solution, which, according to GR, is not "curved" (i.e., it's "flat"). It may appear I'm contradicting myself, since the final solution is clearly not the "flat" one, but it can be interpretated in terms of a partial specification of the total field: for instance, I can have two pieces of metal which interact electromagnetically with each other in any way (they're "external influences" to each other); however, this does not justify me "turning off" the EM field that binds the individual electrons and nuclei of each piece (or, in the case of "jellium", to make no account of electronic repulsion), because then I cannot make good predictions of all properties of any given metal always. I can stick two like-magnetic poles together if I push the magnets with powerful enough lasers; etc. So this is my point: if what makes work so that the neutrons stay bounded is the gravity due to the neutrons alone, it should appear as coupling to their pressure and density - just like electron-nucleus attraction and electronic repulsion alter the EOS of a real metal from that of a free gas of electrons and of "jellium", respec. Why this is not addressed is obscure, Einstein's GR does not difer from the rest of physics in this respect.

including the part that prevents the neutron gas from flying off to infinity.

Could you be more specific? In the O-V paper, they do say that they assumed as boundary condition for the surface of the star that the metric be that of Schwarzschild - and this is fine, really - but this alone does not explain why the whole thing should stay put - just like saying I compressed a gas inside an open container isn't enough to assure the gas will stay there. Sure, the explanation I think you're invoking is that what keeps the star together is its own collective weight - but again, this in itself does not exclude the possibility that the model neglects a contribution that could better describe realistic collapse, perhaps even preventing it (the electrons in metals do not just sit upon the nuclei; other things keep them at bay).

That means they are including interactions between the fermions (other than the Pauli exclusion principle), and modifying the stress-energy tensor appropriately. [...] which includes all of the effects of gravity.

That means they are including the couplings from the strong and electroweak interactions, but unless you can explicitily show a $g_{\mu \nu}$ or a $R_a{}^b{}_{cd}$ in some modern work's EOS, I don't think we're in a position to claim we've modified the stress tensor to account for everything. I can be rather pedantic about this and merely point out that Chandra's mass-shell relation (Eqs. 5 in the Appendix) is not written in covariant form as $g_{\mu \nu}p^\mu p^\nu = m^2$ - and I would point to other developments as quoted in MTW's Gravitation, ch. 22, and Landau, Lifshitz - Statistical Physics, Part 1, sec. 27 as well (though I can't comment on how fully their treatments deal with the gravitational minimal coupling; it seems more like a frame thing to me, which is what I think happens in the O-V paper as well...)

[Risking off-topicness, optional: nonwithstanding the above, you mentioned the exclusion principle: as a chemist, I think it's correct to describe some of its effects as if it were a "force", since e.g. in quantum chemistry exchange effects can be very relevant (not to mention the degeneracy pressure here), but there is more than merely that; so, are we saying that this fundamental principle is violated for the single case of neutron star collapse, or there are possible workarounds?]

 

[PeterDonis]

In my understanding, "free" means "uncoupled from", and this applies to all physical interactions, gravity being no exception.

Then your understanding of "free", at least as it is being used in the paper you reference, is incorrect. Models done in a curved spacetime include the effects of gravity by construction; you don't have to add any extra "interaction" for them, and since they are part of the spacetime geometry, they won't show up in any of the places you would expect non-gravitational interactions to show up.

unless you can explicitily show a $g_{\mu \nu}$ or a $R_a{}^b{}_{cd}$ in some modern work's EOS, I don't think we're in a position to claim we've modified the stress tensor to account for everything.

The metric $g_{\mu \nu}$ will always be there; you can't write down anything in curved spacetime without using it. If you don't see it in a particular EOS, it's because the EOS is being written in a local inertial frame centered on some particular event of interest, in which case the metric coefficients are all $\pm 1$ and drop out of the analysis. But that doesn't mean they aren't there; it just means coordinates have been chosen to make their effect as simple as possible. But as soon as you go beyond looking at just one particular event, or try to write things in coordinates that work globally instead of just in one local inertial frame, you will see the effects of the metric.

For example, in the TOV equation as presented on the Wikipedia page...

https://en.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_equation

... the factor of

$$
\left( 1 - \frac{2 G M}{c^2 r} \right)
$$

is from the metric; it would not be there in flat spacetime.

A coupling to the Riemann tensor $R_a{}^b{}_{cd}$ would be an explicit coupling to curvature, which is a particular kind of "interaction with gravity" and would not be expected to be present in most cases; certainly it isn't in standard analyses of neutron stars and gravitational collapse.

 

[PeterDonis]

[Risking off-topicness, optional: nonwithstanding the above, you mentioned the exclusion principle: as a chemist, I think it's correct to describe some of its effects as if it were a "force", since e.g. in quantum chemistry exchange effects can be very relevant (not to mention the degeneracy pressure here), but there is more than merely that; so, are we saying that this fundamental principle is violated for the single case of neutron star collapse, or there are possible workarounds?]

Please start a separate thread in the Quantum Physics forum if you want to discuss this.

 

[sweet springs]

For example, in the TOV equation as presented on the Wikipedia page...

https://en.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_equation

There total mass chapter shows delta M. More materials are contained inside the "radius" r sphere than non GR Euclid space case. Surplus matter energy is canceled by minus gravitational binding energy.

 

[itssilva]

Then your understanding of "free", at least as it is being used in the paper you reference, is incorrect. Models done in a curved spacetime include the effects of gravity by construction; you don't have to add any extra "interaction" for them, and since they are part of the spacetime geometry, they won't show up in any of the places you would expect non-gravitational interactions to show up.

My understanding of "free" is irrespective of the specific assumptions of the O-V paper - I'm basing myself on the general scheme of gauge theories as applied to physics. Gravity, like other gauge theories, is an expression of a symmetry of our equations: a "field strength", or "curvature", manifests itself dynamically due to a source term in the Lagrangian/Hamiltonian (say) - and frequently, this source term can be seen to arise explicitily from the minimal coupling that underlies said symmetry (e.g., like the current term in QED). So gravity cannot escape minimal coupling, although it is arguable that it does not manifest itself as clearly as the other interactions of the SM (that's why I'm sticking to metals and the EM field, which is also a curvature - of a U(1) p-bundle). And yes, I agree that, while the EM field of the "driven hydrogen" I mentioned is really just one field, it is merely convenient to split this holistic field into two portions - which does not justify us throwing away the other part. All that said, it appears to me you are not addressing the real issue here; to say that "curved spacetime include the effects of gravity by construction" is not wrong, but this doesn't fix "the spacetime geometry", since GR by construction fixed its own connection to be the Levi-Civita one - "geometry" here, as I expounded, really refers to "symmetry", which implies "coupling" - and the coupling scheme is the physical origin of any GR solution. (https://en.wikipedia.org/wiki/Gauge_theory)

If you don't see it [metric] in a particular EOS, it's because the EOS is being written in a local inertial frame centered on some particular event of interest, in which case the metric coefficients are all $\pm 1$ and drop out of the analysis. But that doesn't mean they aren't there; it just means coordinates have been chosen to make their effect as simple as possible. But as soon as you go beyond looking at just one particular event, or try to write things in coordinates that work globally instead of just in one local inertial frame, you will see the effects of the metric.

That's the most relevant piece you wrote so far. It is true - and is precisely my contention: see, things look "flat" from a local vantage point - however, we're not describing the effects on/by the metric for just one neighborhood, we're doing it for a big-ass (or big-mule) star (that is like, what, orders of km across?), and we must make sure everything is accounted for in the final solution: if this were a metal (again), we should take account of the extension of the Coulomb field, even if it's just a screened, effective or closest-neighbours approximation - but again, we're dealing with tremendous gravitational fields here, and so it is not obvious we can "screen off" fields so that locally everywhere inside the star the potential looks "flat". Coupling, always and again: for a single point in a local 'hood, let's say I describe gravity via a equation $$ \frac {d^2x^\lambda} {ds^2} + \Gamma^\lambda{}_{\mu \nu}\frac {dx^\mu} {ds}\frac {dx^\nu} {ds} $$ (where I don't even know what the $\Gamma$ are; if I'm using GR, they happen to be Christoffels). Now, in a regime of extreme gravity (like inside a neutron star), without solving the EFE, I can still guess that those $\Gamma$ things will have appreciable contribution to the whole problem - and indeed, it is something of this form I believe the correction I mentioned could be in the thermodynamics, one way or the other (if you write the equivalent expression for $T_{\mu \nu}$, I don't see why this is wrong, formalism-wise; it doesn't "double count" contributions or anything, it's a different problem altogether).

For example, in the TOV equation as presented on the Wikipedia page...

https://en.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_equation

... the factor of

$$
\left( 1 - \frac{2 G M}{c^2 r} \right)
$$

is from the metric; it would not be there in flat spacetime.

Yes, OK, that's what I stated in the previous post - still, the TOV authors were able to obtain this equation because they considered a problem with lots of symmetries, plus nice features such as the Schwarzschild boundary condition and the possibility of writing the EOS like $F(\rho,P)= 0$ without any dependence on the metric. These are features of their model, but by no means are proof all solutions of the EFE should assume any of those things.

A coupling to the Riemann tensor $R_a{}^b{}_{cd}$ would be an explicit coupling to curvature, which is a particular kind of "interaction with gravity" and would not be expected to be present in most cases; certainly it isn't in standard analyses of neutron stars and gravitational collapse.

This reiterates what I've been highlighting here: an "interaction with gravity" is already warranted in one way or the other, be it via an explicit appearance of $R_a{}^b{}_{cd}$ or other means. Since you claim such scenarios aren't expected in standard analyses, I assume that either 1) this has been explicitly tested or proved, and thus can be consulted in the literature, 2) it stems from the theoretical POV you expounded in these posts, or 3) it is simply not known because it's never been investigated. As it is, I'm OK with either 1) or 3), but 2) does not seem strong enough to dismiss the possibility (and thus risk a black swan situation here).

Please start a separate thread in the Quantum Physics forum if you want to discuss this

Uh, not really - you brought the exclusion principle up, so I thought you wanted to talk about it - hence "optional".

 

[PeterDonis]

I'm asking whether it's not valid to ask whether the usual contribution of the gravitational potential (a.k.a. the metric) is incomplete, based on circumstantial evidence

I don't see any suggestion of this in the references you provided. Is this just your personal idea? Or has this criticism been suggested in other papers? If so, can you reference them?

Gravity, like other gauge theories, is an expression of a symmetry of our equations

In the case of gravity, the gauge symmetry is diffeomorphism invariance; i.e., the laws are invariant under arbitrary continuous transformations of the coordinates.

gravity cannot escape minimal coupling

Of course not. GR is a metric theory of gravity with minimal coupling.

things look "flat" from a local vantage point - however, we're not describing the effects on/by the metric for just one neighborhood, we're doing it for a big-ass (or big-mule) star (that is like, what, orders of km across?), and we must make sure everything is accounted for in the final solution

That's what the global spacetime geometry does: it shows you how the individual locally flat viewpoints centered on particular events "fit together" globally. In the presence of gravity, the global spacetime geometry is curved; heuristically, the curvature is that associated with the connection between local inertial frames centered on different events. And the global curved spacetime geometry, as I've already said, accounts for all of the effects of gravity, including the effects that keep the neutron Fermi gas from flying off into space.

it is not obvious we can "screen off" fields so that locally everywhere inside the star the potential looks "flat".

You can certainly do it in any theory of gravity that obeys the equivalence principle, which GR certainly does. So your statement amounts to claiming that GR is incorrect. Do you have any evidence to support that claim?

In any case, if you aren't willing to accept GR as your theory of gravity, this thread will be closed as we have no other classical theory of gravity to use.

I don't even know what the $\Gamma$ are; if I'm using GR, they happen to be Christoffels

If you're not using GR, this discussion is off topic for this forum and the thread will be closed. See above.

Also, I'm surprised to even see this statement from you; what theory of gravity do you think the TOV paper, and all of the other papers that have studied neutron stars, were using?

the TOV authors were able to obtain this equation because they considered a problem with lots of symmetries, plus nice features such as the Schwarzschild boundary condition and the possibility of writing the EOS like $F(\rho,P)= 0$ without any dependence on the metric. These are features of their model, but by no means are proof all solutions of the EFE should assume any of those things.

Then you need to go find an analysis that doesn't assume them, and see what it looks like.

 

[PeterDonis]

Since you claim such scenarios aren't expected in standard analyses, I assume that either 1) this has been explicitly tested or proved, and thus can be consulted in the literature, 2) it stems from the theoretical POV you expounded in these posts, or 3) it is simply not known because it's never been investigated.

If you haven't looked at the classic reference by Shapiro & Teukolsky, I would strongly suggest doing so. If anything has been done along these lines, their book would probably talk about it (unless it was done recently).

Shapiro & Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects

 

[itssilva]

Disclaim: TL;DR

I don't see any suggestion of this in the references you provided. Is this just your personal idea? Or has this criticism been suggested in other papers? If so, can you reference them?

No, it isn't my personal idea - it's my question, the reason why I posted this. You do not find suggestions of these possibilities on any of the papers I ref'd - and apparently on any other - simply because none of which I know of seems to have talked about those things. Criticism it isn't, except of domain of validity: if it isn't clear up to now, the whole evening later, I am not putting into question the validity of all work that's based off the TOV approach, but rather whether it (or, more stringently, GR itself) exhausts all the possible scenarios for exploration of stellar collapse via the EFE - because, if it isn't rigorously proven by one way or the other, the (widespread, by laypeople and physicists alike) claim that "large neutron stars infalibly collapse into BHs" is not completely established, so it can't be marketed as such - I believe that's one of the tenets of this Forum, scientific rigour, isn't that so? That's what I'm trying to do with my vocabulary.

Gravity, like other gauge theories, is an expression of a symmetry of our equations
In the case of gravity, the gauge symmetry is diffeomorphism invariance; i.e., the laws are invariant under arbitrary continuous transformations of the coordinates.

OK, I know that's what Feynman says in his book on gravitation, along with many others, but if not wrong this statement is at best ambiguous: a "diffeomorphism" is nothing but a "bidifferential, bicontinuous map" (in informal terms), and "invariance under arbitrary coordinate transformations" is pretty much already guaranteed for smooth manifolds, by their definition; for definiteness, I'll say (and you may contest this claim in your own terms) gravity is the gauge theory associated to the frame bundle over a Lorentzian manifold; if you want to talk of its physics, cf. infra.

That's what the global spacetime geometry does: it shows you how the individual locally flat viewpoints centered on particular events "fit together" globally. In the presence of gravity, the global spacetime geometry is curved; heuristically, the curvature is that associated with the connection between local inertial frames centered on different events.

Again, you're not wrong; but then

And the global curved spacetime geometry, as I've already said, accounts for all of the effects of gravity, including the effects that keep the neutron Fermi gas from flying off into space.

And, as I've already said, this statement is problem-dependent: while it is true that if you set up the EFE as TOV did they will go and collapse, the EFE alone can't tell you what boundary conditions, stress tensor, etc., are physically relevant for that case; this is additional, having nothing to do with the EP holding or not.

it is not obvious we can "screen off" fields so that locally everywhere inside the star the potential looks "flat".
You can certainly do it in any theory of gravity that obeys the equivalence principle [EP], which GR certainly does.

That's really overstating the EP, which is inbuilt in GR due to its gauge nature; it seems to me you invoked a strawman to infer I'm attacking GR itself, while what I did with the quoted sentence was to point to the nature of the source term $T_{\mu \nu}$, which I argue is not fixed by GR in any way but by additional assumptions of physical nature regarding specific problems. The analogy with condensed matter systems should be perfectly transparent by the structure of the sentence.

So your statement amounts to claiming that GR is incorrect. Do you have any evidence to support that claim?

In any case, if you aren't willing to accept GR as your theory of gravity, this thread will be closed as we have no other classical theory of gravity to use.

This development was abrupt and did not follow from the discussion at hand; I'd like to politely ask you to refrain yourself from putting words such as "not willing to accept GR as your theory of gravity" in my mouth, specially if I didn't use them. As for "no other classical theory of gravity", other theories with their own advantages and disadvantages are the subject of peer-reviewed scholarly study, yet I'm not won't to delve into such topics in a subforum called "Special and General Relativity"; in other words, whenever I stop dealing with GR, I'll give a heads-up.

Also, I'm surprised to even see this statement from you; what theory of gravity do you think the TOV paper, and all of the other papers that have studied neutron stars, were using?

Also a rushed conclusion; by "I don't even know what the $\Gamma s$ are", I simply wanted to emphasize that the conclusion that they are to be taken by Christoffels is by no means essential, according to the differential geometry of the problem alone (cf. the rest of the SM, for which it is the connection itself the dynamized object); what GR does as a physical theory is to fix the connection (the "embodiment" of the EP) to be that of Levi-Civita, which exists and is unique (and also have other nice properties) - and in this case, $\Gamma s$ = Christoffels (see also Ohanian, Ruffini - Gravitation and Spacetime, for related discussions of the content of GR; ch. 7, if not mistaken). We are allowed to state this fact as it is, and use GR as it is, experimentally robust and all, but I am not making the presumptuous assumption that it must be this way, since that's not on-topic but would also be also disingenuous of me, without a proof in hand. So I still stick to GR, without loss of generality, insofar as it holds on its own.

Then you need to go find an analysis that doesn't assume them, and see what it looks like.

I was hoping someone here could point me to one, in case it's physically possible - and if not, to work that shows why it is not, be it a textbook or research paper. Another possibility is nothing; I keep repeating this, because I don't think it's a sin to ask whether this or that angle was covered yet - but it is to say that it has without appropriate backing. As for your book, I'll check it to see if it covers this particular angle, or whether it still leaves it an open question; thanks for the tip. I'll just throw books on differential geometry and related subjects I touched upon here for you to peruse at your leisure (so that, if you have any problems with anything they may have said, you can complain to their authors instead): Schutz - Geometrical Methods of Mathematical Physics; Bleecker - Gauge Theory and Variational Principles; Nakahara - Geometry, Topology and Physics; Kobayashi, Nomizu - Foundations of Differential Geometry, Vol. 1; Choquet-Bruhat, DeWitt-Morette, Dillard-Bleick - Analysis, manifolds and physics; Frankel - The Geometry of Physics - An Introduction; Thirring - A Course in Mathematical Physics, Vol. 2; and I believe there's some in Wald - General Relativity on fiber bundles, too). Finally, sorry for the overall negative tone - nothin' personal, etc.

 

[PeterDonis]

I am not putting into question the validity of all work that's based off the TOV approach, but rather whether it (or, more stringently, GR itself) exhausts all the possible scenarios for exploration of stellar collapse via the EFE

With regard to the standard TOV approach, I would agree that it might not exhaust all possibilities; in particular, it does appear to not consider any possible couplings to curvature. I don't think "coupling to the metric" in itself is not considered in the TOV approach, because I don't think there is a useful invariant notion of "coupled to the metric" vs. "not coupled to the metric"; for example, even if an equation of state appears to not depend on the metric when written in a local inertial frame, in general it will when written in global coordinates. But there certainly is a useful invariant notion of "coupled to curvature" vs. "not coupled to curvature". I'm not familiar enough with the literature to know if anyone has considered such a coupling, or if so, whether it makes a difference in the ultimate conclusions.

As far as "GR itself", as I've said, if you're not accepting GR as your theory of gravity, I have no idea what theory you would use. So I can't see any useful way to pursue a model that doesn't use "GR itself". Nothing in GR rules out couplings to curvature.

if it isn't rigorously proven by one way or the other, the (widespread, by laypeople and physicists alike) claim that "large neutron stars infalibly collapse into BHs" is not completely established

I think the claim that in the absence of couplings to curvature, large neutron stars infallibly collapse into BHs is completely established; we don' t know the exact maximum mass limit for neutron stars under that assumption, but we know there is one and we have narrowed down its range pretty well (IIRC the latest limits are somewhere between about 1.5 and 2.5 solar masses). So the question is whether couplings to curvature could make a difference. As I said above, I don't know if that possibility has been studied.

gravity is the gauge theory associated to the frame bundle over a Lorentzian manifold

This looks fine to me as a less ambiguous restatement of what I was saying.

while it is true that if you set up the EFE as TOV did they will go and collapse, the EFE alone can't tell you what boundary conditions, stress tensor, etc., are physically relevant for that case

Of course not. You don't get that stuff from the EFE; you get it from knowledge about what kind of matter you are dealing with and what its properties are. Nobody expects the EFE to tell them a priori what the stress-energy tensor is.

I'd like to politely ask you to refrain yourself from putting words such as "not willing to accept GR as your theory of gravity" in my mouth, specially if I didn't use them.

When you say things like "if I'm using GR", or "whether GR itself exhausts all possible scenarios", you should expect to be interpreted as at least raising the possibility of not accepting GR as your theory of gravity. And I'm just trying to make clear that that's not a possibility that's within scope for this forum.

As for "no other classical theory of gravity", other theories with their own advantages and disadvantages are the subject of peer-reviewed scholarly study

And all of them are ruled out by experiment except with particular values of their parameters that make them indistinguishable from GR. Since we are discussing whether real neutron stars, not imaginary or theoretical ones, must collapse to real black holes, theories of gravity that are different from GR but ruled out by experiment are irrelevant.

I simply wanted to emphasize that the conclusion that they are to be taken by Christoffels is by no means essential, according to the differential geometry of the problem alone (cf. the rest of the SM, for which it is the connection itself the dynamized object); what GR does as a physical theory is to fix the connection (the "embodiment" of the EP) to be that of Levi-Civita

See my remarks above about GR. How do you expect me to interpret this, if not as raising the possibility of not using GR?

We are allowed to state this fact as it is, and use GR as it is, experimentally robust and all, but I am not making the presumptuous assumption that it must be this way, since that's not on-topic but would also be also disingenuous of me, without a proof in hand.

Again, how do you expect me to interpret this, except as raising the possibility that GR might not be correct within the domain under discussion? (Which is not the same as saying that GR might not be the correct ultimate theory of gravity, covering all possible domains.)

 

[Dale]

When you say things like "if I'm using GR", or "whether GR itself exhausts all possible scenarios", you should expect to be interpreted as at least raising the possibility of not accepting GR as your theory of gravity. And I'm just trying to make clear that that's not a possibility that's within scope for this forum.

@itssilva this is also how I read your comments. If this is not your intention then you should rephrase because what you are saying is not communicating what you intend to multiple recipients. If it is your intention to raise the possibility of not accepting GR as the theory of gravity then we will need to close the thread.

 

[itssilva]

I don't think there is a useful invariant notion of "coupled to the metric" vs. "not coupled to the metric"; for example, even if an equation of state appears to not depend on the metric when written in a local inertial frame, in general it will when written in global coordinates. But there certainly is a useful invariant notion of "coupled to curvature" vs. "not coupled to curvature"

The invariance in the problem is enforced by the connection, so we don't need worry about it; as for whether matter couples to the metric (i.e., gravitational potential) or to the curvature (i.e., gravitational field), that can be indicated by other classical models, like - again! - the Maxwell theory; for instance, as we separate the full problem into a "bound" part and a "free" part.

I'm not familiar enough with the literature to know if anyone has considered such a coupling, or if so, whether it makes a difference in the ultimate conclusions

Then that makes two of us; my intuition as to how it could reverse the collapse is this simplified consideration: if the coupling to a strong field makes the degeneracy pressure increase monotonically (at least as the star approaches the Schwarzschild radius), it may be enough to hold the weight of the star and yield a stable object. Of course, the final equilibrium solution need be computed from the EFE to confirm this back-of-the-envelope logic, but in lieu of experimental results (as I assume) I'd be interested in comments on this rationale, at your own discretion - based, of course, on established physical/mathematical aspects of GR; those may rule out this paragraph as nonsense and I may not even realize it.

Nothing in GR rules out couplings to curvature.

So we seem to agree here; yet, in a previous post you noticed (as I understood) that this is not the case for standard analyses of this kind of GR problems. Did you have an underlying reason to rule them out, or it was just lack of studies on the subject? That wasn't too clear to me...

And all of them are ruled out by experiment except with particular values of their parameters that make them indistinguishable from GR. Since we are discussing whether real neutron stars, not imaginary or theoretical ones, must collapse to real black holes, theories of gravity that are different from GR but ruled out by experiment are irrelevant.

Ergo, advantages, disadvantages; this is of course another topic altogether, but if GR and some other theory are indistinguishable for a certain range of phenomena and I always keep to that range of application, it is immaterial whether I work under GR or the alternative - just like I don't throw Newtonian gravity in the garbage because GR exists. That much is obvious; only if by your comment you meant "no GR alternative can explain neutron stars as fully as GR", then it might have been relevant here; however, I'm keeping to GR, while making large use of simpler analogies to dissect its contents - and so far I've not detected you having a problem with any specific one I gave, so I guess this is OK?

Again, how do you expect me to interpret this, except as raising the possibility that GR might not be correct within the domain under discussion? (Which is not the same as saying that GR might not be the correct ultimate theory of gravity, covering all possible domains.)

as well other related remarks. I acknowledge it is a fair criticism, so I'll try to address it succintly: if I found the equation above in a loose piece of paper, nothing would warrant me to say "oh, this is the geodesic equation, from GR"; rather, it may be simply the expression you get when you describe a point particle moving along a Minkowskian geodesic according to arbitrary coordinates, if the $\Gamma s$ be defined appropriately; so I might be just using an accelerated point particle as the "alternative to GR" you seem to believe I am pushing for here, and this merely as another simple analogy (if you have an object that is the locus of several points, think what the equations look for each point rather than the global thing, before you graduate to that). This is also supposed to illustrate that the "curving" of spacetime is not an exclusivity of GR: however you wish to express your theory, the degree of "curving" depends on what those $\Gamma$ things are, not on the fact that "curving" is a thing. So, as I explained, if we talk in GR terms, those things have a definite structure to them - but this structure by no means relieves us from determining what they are via other considerations that the EFE cannot possibly fix. This is a long banter, but I hope it clarifies the usage here.

 

[PAllen]

I haven’t read this whole thread, but one discussion is the possibility of a law coupling directly to curvature. One formal embodiment of the POE is the principle of minimal coupling, which I believe rules this out. So any such theory would be outside GR as normally understood.

 

[PeterDonis]

if the coupling to a strong field makes the degeneracy pressure increase monotonically (at least as the star approaches the Schwarzschild radius), it may be enough to hold the weight of the star and yield a stable object

No, this won't work, because pressure gravitates. That's a key factor in the TOV limit: as pressure increases to hold the star up, attractive gravity also increases.

You might not be realizing how broad the TOV limit actually is. Here's how the limit arises, in a nutshell:

(1) As you add more mass to an object made of fermions, degeneracy pressure increases.

(2) As degeneracy pressure increases, the fermions become relativistic.

(3) As the fermions become relativistic, the adiabatic index decreases.

(4) As the adiabatic index decreases, the ratio of, heuristically, "pressure being able to hold the star up" to "pressure creating more attractive gravity" decreases.

(5) At some point, a limit is reached where the added attractive gravity of more pressure overbalances the outward support of more pressure, and the object collapses.

Furthermore, there is a theorem called Buchdahl's Theorem, which says that no static object of any kind can have a radius less than 9/8 of the Schwarzschild radius for its mass, regardless of its pressure. So even if we find some kind of stuff other than a neutron Fermi gas that resists compression better, so it takes longer for #5 above to occur, it's still going to have to occur at a radius greater than 9/8 of the Schwarzschild radius. There's no way to stave it off forever.

 

[itssilva]

@PeterDonis That's a nice overview, there, of the scope of TOV - but still, I would point to the use of the term "relativistic": it expressely refers to the free special-relativistic gas, in spite of a previous post in which you argue that this is justified due to the local Minkowskian character of GR, and to which I argued against in response. I will try to make this point clearer, like so: consider a (special-)relativistic gas of free protons: its pressure and density are formally just the same as that of the neutron gas we're talking about - and so can be said of its EOS. Now, let's introduce the (classical) Coulombic interactions between protons: certainly, this introduction will alter the form of the EOS (though I don't know what it would look like, it may likely be found in the condensed matter literature), and definitely alter other thermodynamic properties of this gas, as well. Now consider yet another gas model, in which the only modifications we make are proton $\rightarrow$ neutron and Coulombian potential $\rightarrow$ Newtonian potential; we've basically changed nothing (with due care for sign changes). However, consider that for some reason or the other (like a collapse pressure), the neutrons get really close together, so that the Newtonian potential gets "Schwarzschildian", and starts to markedly deviate from classical gravity - making the EOS deviate even further from the Minkowskian EOS. Each step of this development refines the description of the degenerate matter without invoking gauge invariance, and in a way that, I hope, shows clearly how different their predictions become. To better capture this picture, one may try solving the Maxwell problem of a charged ball of metal using different approximations for the metal (free-gas, jellium, full Coulomb) and see how pressure, density, heat capacity, etc. will be different; you don't need to trouble yourself with the gauge invariance imposed by the EM field to do this.

Another problem is that I believe the discussion only applies to materials that obey polytrope theory (https://en.wikipedia.org/wiki/Polytrope) - and, by the arguments in the previous paragraph, it does not seem likely it may apply to this "Schwarzschildian" matter I spoke of - unless one can prove, using very general assumptions, that polytrope theory is the only game in town for the star problem. Speaking of which,

Furthermore, there is a theorem called Buchdahl's Theorem, which says that no static object of any kind can have a radius less than 9/8 of the Schwarzschild radius for its mass, regardless of its pressure. So even if we find some kind of stuff other than a neutron Fermi gas that resists compression better, so it takes longer for #5 above to occur, it's still going to have to occur at a radius greater than 9/8 of the Schwarzschild radius. There's no way to stave it off forever.

I believe the Theorem in question is from this work? As I skimmed it through, I found two quotes that may be of note:

[There are obtained] general results for arbitrary static fluid spheres in the form of inequalities, the distributions being subject only to some limitations of a general kind.

What would the set of assumptions be? Are they restricted to

the possibility is taken into account of an a priori restriction upon the pressure to density ratio, such as that which arises from the postulate that the trace of the energy-momentum tensor be non-negative.

? So the only thing that keeps the validity of this result in check is an energy condition? (I have more to say on ECs, that being the case).

Also, mention is made (sec. 7) of a solution "singularity-free" in the origin; I presume this refers to the "ordinary" kind of singularity that "blows up", rather than that from geodesic incompleteness, is that the case?

 

[PeterDonis]

I would point to the use of the term "relativistic": it expressely refers to the free special-relativistic gas, in spite of a previous post in which you argue that this is justified due to the local Minkowskian character of GR, and to which I argued against in response

Your arguments were basically that GR might not be valid. That's not discussible here for reasons which have already been explained.

Your other argument was that there might be a coupling to curvature. @PAllen has given a good response to that.

Now consider yet another gas model, in which the only modifications we make are proton $\rightarrow$ neutron and Coulombian potential $\rightarrow Newtonian potential$; we've basically changed nothing

Yes, we have; gravity between neutrons is enormously weaker than the Coulomb interaction between protons.

Also, neither of these models are like the models that are driving the TOV limit. The TOV limit is not dependent on the Coulomb potential; even if we were talking about a Fermi gas of protons instead of neutrons, the Coulomb repulsion between the protons would only be a small correction at neutron star densities. The primary effect is the Pauli exclusion principle.

consider that for some reason or the other (like a collapse pressure), the neutrons get really close together, so that the Newtonian potential gets "Schwarzschildian", and starts to markedly deviate from classical gravity

You obviously have not bothered to run any numbers. I strongly suggest that you do so before making any suggestion, even a hand-waving one, that this is in any way relevant at densities anywhere near neutron star densities (or, for that matter, any density much less than the Planck density).

I believe the discussion only applies to materials that obey polytrope theory

No, it doesn't. The TOV limit does not assume that the star is a polytrope. Shapiro & Teukolsky discuss this.

What would the set of assumptions be?

Basically that the spacetime is stationary (there is a timelike Killing vector field) and spherically symmetric.

So the only thing that keeps the validity of this result in check is an energy condition?

Not to my knowledge. I think the quote you give just means the theorem covers this possibility (i.e., of an energy condition being assumed), not that it is the only possibility the theorem covers.

mention is made (sec. 7) of a "singularity-free" solution in the origin; I presume this means the "ordinary" kind of singularity that "blows up", rather than that from geodesic incompleteness, is that the case?

I think it means the latter (i.e., that there is no singularity of the kind that occurs at $r = 0$ in a Schwarzschild black hole). But I am not certain.

However, even if Buchdahl's original paper had some restrictions, the theorem is known to be valid in a wider range of cases than the ones he treated. See, for example, this paper:

https://arxiv.org/pdf/gr-qc/0605097.pdf

 

[itssilva]

Your arguments were basically that GR might not be valid. That's not discussible here for reasons which have already been explained.

That's not what I said at all, and your apparent unwillingness to accept that I am discussing here under the assumption GR is valid is starting to make me suspect you're trying to discredit my viewpoint as naïveté or something. OK, to make sure we're talking about the same devil, let's define GR: [ ] := [the frame bundle that I mentioned previously, plus the connection fixed to be Levi-Civita's, plus the metric tensor dynamically given by a set of PDEs called the EFE]. For all intents and purposes, that's what I've considered as "GR" here. Now, let's define an object { } := {[ ], (applications of [ ]), (reformulations of [ ]), (alternatives to [ ]), etc.}; this I call "The Theory of Gravitation". So, when you state things like that, please make sure to consider thoroughly whether you had in mind [ ], { } or something else; I, for myself, have been trying to point to the extent of (applications of [ ]), using analogies as crutches.

Your other argument was that there might be a coupling to curvature. @PAllen has given a good response to that.

The response was

[Of] the possibility of a law coupling directly to curvature. One formal embodiment of the POE is the principle of minimal coupling, which I believe rules this out. So any such theory would be outside GR as normally understood.

This is not terribly reassuring, and it's a point we've been latching onto for a while - not to mention the aforementioned business with [ ]/{ }; informally, I believe you fear that the problem as I've posed leads to a kind of unphysical "double counting" of the minimal coupling; again, I'll use my Maxwell crutch ("free" vs. "bound", etc.) to support my position that this is not the case, and in no part during the making of this movie a [ ] was harmed - plus the literature I mentioned already, naturally.

Yes, we have; gravity between neutrons is enormously weaker than the Coulomb interaction between protons.

The point was in comparing the form of the relevant formulae, not the strength of the interactions/couplings - but even this formal comparison is not relevant for the final conclusion (cf. infra).

Also, neither of these models are like the models that are driving the TOV limit. The TOV limit is not dependent on the Coulomb potential; even if we were talking about a Fermi gas of protons instead of neutrons, the Coulomb repulsion between the protons would only be a small correction at neutron star densities. The primary effect is the Pauli exclusion principle.

True - but more specifically, the Pauli principle mandates filling of energy shells up to the Fermi surface, whose detailed structure is dependent on the Coulomb potential. Now suppose we introduce, in place of the Coulomb well, some arbitrary interaction; then it is not obvious that it won't deform this surface in a dramatic way, making it "blow up" - unless you know of other developments that prohibit so (you mentioned something in that way, but cf. infra).

You obviously have not bothered to run any numbers. I strongly suggest that you do so before making any suggestion, even a hand-waving one, that this is in any way relevant at densities anywhere near neutron star densities

If hand-waving makes you happy, I'll try to make the most hand-waving calculation I possibly can: say I have a neutron star of Solar mass $M_{\odot}$ bounded by its Schwarzschild radius $R_*$, and neutrons with mass $m_n$ and radius $r_n$. If I assume all mass and volume of the star comes from adding up those of the neutrons, I get it contains $N = \frac {M_\odot} {m_n} \approx 10^{57}$ neutrons, each with $r_n = \frac {R_*} {N^{1/3}}$, which is comparable to the radius of a free neutron (give or take an order of magnitude), but falls stupidly off its Schwarzschild radius $R_n = \frac {R_*} {N}$ - meaning relativistic effects negligible. So there you have it: a lame model yielding a lame outcome. Yet the lame model cuts short on many details, such as the ridiculous pressure felt by the neutrons as they're crushed into oblivion - not to mention that even a Newtonian treatment of neutron-neutron coupling is still technically "Schwarzschildian", if not dramatically so.

No, it doesn't. The TOV limit does not assume that the star is a polytrope. Shapiro & Teukolsky discuss this.

Basically that the spacetime is stationary (there is a timelike Killing vector field) and spherically symmetric.

I'll have to sit with your book for a while - however, it seems to me that you may be overselling those results; for instance, while in the original 1939 work it's true that the authors started from the most general static line element exhibiting spherical symmetry, but later on they used the non-dependance of the EOS on $g$ (or $R$) to integrate their eq. 5; each assumption - even innocuous-looking ones, such as that of a perfect fluid - should be carefully accounted for, since the case considered is unusual.

I think it means the latter (i.e., that there is no singularity of the kind that occurs at $r = 0$ in a Schwarzschild black hole). But I am not certain.

Wait a sec, your terminology is confusing the heck outta me; I hold that the Schwarzschild hole contains a "blow-up" singularity at the origin, since the non-vanishing components of the gravitational field go as $\frac {M} {r^3}$ (MTW, Gravitation, eq. 31.4b), but therein there's also one of the Penrose type; so which one do you mean? Wasn't the point of the Buchdahl result to show the Penrose one is unavoidable if the static star is massive enough? The way you stated it it only says it prohibits stable radii within a certain range - and that under suitable conditions (to be found in the paper).

However, even if Buchdahl's original paper had some restrictions, the theorem is known to be valid in a wider range of cases than the ones he treated.

Even so, I don't see that Andréasson's extensions rule out the specific manner of coupling we've been discussing as a form to prevent the collapse (although I'll give it it's much more congenial in the placement of things it assumed). In any case, Buchdahl or Andréasson, one has to be careful to highlight the limits of such "generalist theorems"; I understand the latter claims that

in the present paper no field equations for the matter are used

, but that doesn't mean other conditions, implied or not, will automatically be met in the scenario proposed - specially those that are justified by numerical experiments.

 

[PeterDonis]

let's define GR: [ ] := [the frame bundle that I mentioned previously, plus the connection fixed to be Levi-Civita's, plus the metric tensor dynamically given by a set of PDEs called the EFE]. For all intents and purposes, that's what I've considered as "GR" here.

Ok so far.

Now, let's define an object { } := {[ ], (applications of [ ]), (reformulations of [ ]), (alternatives to [ ]), etc.}; this I call "The Theory of Gravitation".

And you keep on bringing it up, even though I've repeatedly asked you to limit discussion to GR. Why do you keep bringing it up if you don't want to talk about it?

The point was in comparing the form of the relevant formulae, not the strength of the interactions/couplings

But the strength of the couplings is crucial if you're trying to figure out whether any of this is relevant to the actual case under consideration: whether neutron stars must collapse to black holes. Bringing up something that is 38 orders of magnitude or so too small to affect that in any way is irrelevant, regardless of how similar the form of the equations looks to you.

Now suppose we introduce, in place of the Coulomb well, some arbitrary interaction

Irrelevant if the interaction is 38 orders of magnitude or so too weak to be relevant in the case under consideration.

I'll try to make the most hand-waving calculation I possibly can

Which just underscores how irrelevant the gravity of individual neutrons is in the case under consideration. And if even a hand-waving calculation tells you this, there's no point in looking further.

in the original 1939 work

Which was roughly 4 decades before Shapiro & Teukolsky was published. Physicists learned an awful lot about the subject in those four decades.

I hold that the Schwarzschild hole contains a "blow-up" singularity at the origin

Yes; more precisely, the maximally extended Schwarzschild spacetime is geodesically incomplete at $r = 0$, since curvature invariants increase without bound as that limit is approached.

therein there's also one of the Penrose type;

That's the same one.

Wasn't the point of the Buchdahl result to show the Penrose one is unavoidable if the static star is massive enough?

No. Buchdahl's Theorem says nothing about what happens to an object that is more compact than the limit given in the theorem, except that it can't be static. It can be collapsing, or it can be exploding, but it can't be static.

In the collapsing case, one would expect that the object would meet the Penrose condition that predicts that a singularity will form at a finite time in the future (trace of extrinsic curvature everywhere negative) fairly soon after becoming more compact than the Buchdahl limit; but Buchdahl's Theorem itself does not go into that.