A A Question on the TOV Limit: Continuous Forces vs. a Discrete Trigger

[euquila]

TL;DR Summary

We explain neutron star collapse with continuous concepts: GR's curvature & QM's pressure. Yet the trigger is discrete: the single (N+1)th fermion. Is the language of "mass contribution" missing the point of this fundamental counting event?

Hello everyone,

I've been thinking about the standard physical picture of a neutron star reaching the TOV limit, and I've run into a conceptual question that I can't quite shake. I'd appreciate your perspective on it.

The textbook explanation is a beautiful balance of our two great theories. On one side, General Relativity describes gravity as an inward pull, a consequence of the smooth curvature of spacetime sourced by a continuous mass-density. On the other side, Quantum Mechanics provides the stabilizing outward force of degeneracy pressure, which arises from the Pauli exclusion principle applied to continuous fermion wavefunctions. The "collapse" occurs when the former overwhelms the latter.

But if we zoom in on the idealized, final moment, the trigger for this event is profoundly discrete. A star with N fermions is stable. The entire catastrophic process is initiated by the arrival of just one more particle, the (N+1)-th fermion.

This is the dissonance I'm grappling with. We explain the process using the language of continuous fields, pressures, and curvatures. But the deciding factor that settles the "battle of forces" isn't a gradual increase in pressure - it's a discrete, integer event.

Describing this pivotal moment in terms of the "mass contribution" of that single, final neutron feels like we're using the wrong language. It seems to miss the significance of the event being fundamentally about a number.

Does anyone else feel a tension between the continuous nature of our physical explanation and the discrete nature of the physical trigger? I'm finding it hard to reconcile the two descriptions in a satisfying way.

 

[Dale]

A star with N fermions is stable. The entire catastrophic process is initiated by the arrival of just one more particle, the (N+1)-th fermion.

I am pretty skeptical about this. Do you have a reliable source for this claim?

 

[euquila]

I am pretty skeptical about this. Do you have a reliable source for this claim?

You've raised a very fair point, and I should have been clearer about the context for my claim. I'm not describing a realistic astrophysical event, but rather a highly idealized thought experiment designed to probe the nature of the boundary itself.

I was envisioning a purely theoretical "clean" system: a cold, non-rotating, uncharged neutron star. In this setup, I imagined a process where we could add one neutron at a time and allow the entire star to settle into perfect equilibrium after each addition. In such a controlled sequence, I assumed that a maximum stable number of particles, N_max, must exist.

My question was aimed squarely at the physics of that very last step (the N_max + 1 transition) and whether our focus on that single particle's "mass" is the only to describe why such a discrete action triggers the entire event.

 

[Dale]

I assumed that a maximum stable number of particles, N_max, must exist.

I doubt that is the case. Do you have any reputable sources that support this assumption? Even for the ideal case?

describe why such a discrete action triggers the entire event

Before we worry about why such a discrete action triggers it, we should determine if such a discrete action triggers it.

First, I doubt that a neutron star near the TOV limit is an eigenstate of the fermion number, even in the ideal case. Second, I doubt that the likelihood of collapse is a unit step function of the fermion number. Without the first you cannot claim that it has N fermions, and without the second you cannot claim that it collapses at N+1 fermions.

I should have been clearer about the context for my claim.

I should have been clearer about my request. I am looking for a peer reviewed paper in a reputable scientific journal that supports this idea that there is a quantum mechanically sharp transition from stability to instability at the TOV limit.

 

[PeterDonis]

The textbook explanation

Do you know the textbook explanation? The classic textbook on this subject is Shapiro & Teukolsky. Can you give any references from it that support your claim?

 

[euquila]

Thank you for the helpful feedback. You asked for a source for my N+1 particle claim. I don't have one. But your comment made it clear why that premise is too simplistic. Perhaps there exists a function P(collapses) = f(N) that describes this reality (sigmoid?). The "when" of the collapse is uncertain, much like the timing of the atomic decay. In this picture, the fundamental "why" of the collapse is still tied to the countable nature of its constituent particles (the total neutrons brought in from infinitity), even though the timing is probabilistic.

I looked into Shapiro & Teukolsky and found Buchdahl's inequality 2M/R < 8/9. I find this inequality to represent the endpoint to the Newton-Einstein line of reasoning.

 

[PeterDonis]

You asked for a source for my N+1 particle claim. I don't have one.

Then you should not have made the claim in the first place.

the fundamental "why" of the collapse is still tied to the countable nature of its constituent particles

No, it 's not. I would ask you for a source for this claim, but I expect you don't have one. And again, that means you should not be making the claim in the first place.

I looked into Shapiro & Teukolsky and found Buchdahl's inequality

Which has nothing to do with the TOV limit (or the Chandrasekhar limit).

For a discussion of what actually drives these limits, along with references to the relevant sections of Shapiro & Teukolsky, see this Insights article:

https://www.physicsforums.com/insights/why-there-are-maximum-mass-limits-for-compact-objects/

 

[Dale]

the fundamental "why" of the collapse is still tied to the countable nature of its constituent particles

Do you have a reference for that claim?

It seems to me like a clearly incorrect claim since the star is not an eigenstate of the fermion number. You really need to stop asserting claims that are just personal speculation.

 

[Dale]

I'm finding it hard to reconcile the two descriptions in a satisfying way.

I think that is probably because your picture of quantum mechanics is mistaken. You seem to be envisioning quantum mechanics in a “little chunks of classical mechanics” mode. Like there are a definite number of particles that are definitely in the star at any defined time. Quantum mechanics just doesn’t work that way.

You shouldn’t try to reconcile the two descriptions. You should discard the semi-quantum description altogether.

 

[euquila]

I appreciate your comments and guidance and accept that the premise was off and not respresentative of the intricacies of quantum mechanics. I made some bad assumptions without realizing it.

Thank you for referencing that book and your article. I learned of many interesting things like Buchdahl.

 

[ersmith]

One more thing to note is that even if we disregard quantum mechanics and treat the neutron star as a collection of classical particles, whether or not adding another particle causes a collapse would depend on the energy of the incoming particle, not just on its rest mass. So even in this unrealistic scenario the situation is still continuous.