Struct parallels between curvature behavior/abstract geometric cycles?

[Thor Jackson]

I’ve been studying the way curvature evolves in general relativity, especially in scenarios involving increasing density, trapped‑surface formation, and singularity development. While reading, I noticed that the sequence(1) intensification of sources → (2) curvature growth → (3) collapse → (4) singularity → (5) possible continuation seems to appear repeatedly in GR discussions.

My question is not about proposing a new model, but about understanding whether this pattern has an established name or interpretation in the GR literature.

Specifically:

• Is there a standard geometric or structural framework in GR that describes the progression from curvature amplification to trapped surfaces and singularity formation as a unified cycle?
• Do GR textbooks or papers treat these stages as a single structural sequence, or are they usually discussed separately (e.g., collapse theory, singularity theorems, cosmic censorship, etc.)?
• Are there known mathematical formalisms that treat “post‑singularity continuation” (bounces, extensions, white‑hole regions) as part of a general geometric pattern, even if only in speculative or classical contexts?

I’m not proposing any new physics — I’m trying to understand whether GR already has a recognized structural interpretation of this sequence, or whether it’s just a convenient way to think about the standard results.

Any references or clarifications would be appreciated.

 

[PeterDonis]

While reading, I noticed that the sequence(1) intensification of sources → (2) curvature growth → (3) collapse → (4) singularity → (5) possible continuation seems to appear repeatedly in GR discussions.

Please give specific references. It's not clear from your description what actual physics this "sequence" is referring to.

 

[Thor Jackson]

Thanks — yes, I can give specific references. The “sequence” I’m referring to isn’t a new model; it’s just the standard progression described in GR when discussing gravitational collapse and singularity formation. The steps I listed correspond to well‑known results:

1. Increasing density → curvature growth This is discussed in any treatment of relativistic collapse. For example:

• Misner, Thorne & Wheeler, Gravitation, Ch. 32–33 (curvature growth in collapsing matter)
• Wald, General Relativity, Sec. 9.2 (curvature invariants increasing under collapse)

2. Formation of trapped surfaces This is the classical criterion for collapse:

• Penrose (1965), Phys. Rev. Lett. 14:57 — trapped surfaces as the onset of irreversible collapse
• Wald, Sec. 12.2

3. Singularity theorems The transition from trapped surfaces to geodesic incompleteness:

• Hawking & Ellis, The Large Scale Structure of Spacetime, Ch. 8
• Wald, Ch. 9

4. Extensions / post‑singularity regions Not physical predictions, but mathematically standard:

• Kruskal extension of Schwarzschild (Wald, Sec. 6.2)
• Maximal analytic extensions in MTW, Ch. 31

So my question is simply whether GR literature treats these as a single structural progression, or whether they’re normally discussed as separate results (collapse → trapped surfaces → singularity → possible extensions). I’m not proposing a new theory — just trying to understand how unified this sequence is considered in standard GR treatments.

 

[PeterDonis]

yes, I can give specific references.

Thanks, this is very helpful. Some comments below.

Increasing density → curvature growth This is discussed in any treatment of relativistic collapse. For example:

Your bolded text here appears to imply a causal relationship, but there isn't one. Both increasing density and curvature growth are effects of the collapse, and the reason the matter is collapsing has to be put into the model at the start--for example, because it's a star in which fusion reactions can no longer provide enough kinetic pressure to support it against its own gravity.

Formation of trapped surfaces This is the classical criterion for collapse

I don't understand what you mean by "the classical criterion for collapse". A collapsing object doesn't necessarily form a trapped surface. For example, an ordinary star can end up collapsing to a white dwarf or a neutron star, neither of which have trapped surfaces.

If you mean "the classical criterion for irreversible collapse", see below.

Penrose (1965), Phys. Rev. Lett. 14:57 — trapped surfaces as the onset of irreversible collapse

If we assume that the matter satisfies appropriate energy conditions, yes, once a trapped surface is formed, the collapse is irreversible and will end in a singularity, at least according to classical GR, because of the singularity theorems. However, it is also true that most physicists believe that classical GR breaks down (its predictions cease to be accurate) before the point of formation of a singularity is reached in the model, and that some other model, such as a quantum gravity model (which we don't currently have), will be needed to make correct predictions of what happens in this regime.

Singularity theorems The transition from trapped surfaces to geodesic incompleteness

There is no such "transition", if by that you mean an actual, physical change. The singularity theorems do not say that, once a trapped surface is formed during a collapse process, geodesics become incomplete. They say that, given certain assumptions, any spacetime that contains a trapped surface also contains incomplete geodesics. The spacetime is one geometric object: it doesn't "transition". It already contains the entire history of the physical system being modeled.

Extensions / post‑singularity regions Not physical predictions, but mathematically standard:

I don't understand what you mean by "post-singularity regions". The Kruskal extension of the Schwarzschild geometry, which is the one discussed in both of your references, does not extend anything "post-singularity".

my question is simply whether GR literature treats these as a single structural progression, or whether they’re normally discussed as separate results (collapse → trapped surfaces → singularity → possible extensions).

I don't understand what you mean by "a single structural progression". Perhaps the above comments might help to clarify some things.

 

[Thor Jackson]

Thanks — this is helpful clarification. Let me respond point‑by‑point using standard GR language.

1. On “increasing density → curvature growth”​

You’re right that I shouldn’t imply a direct causal arrow. A better way to phrase it is:

• In many collapse models, increasing density and increasing curvature are correlated consequences of the same underlying dynamical process.

For example, in Oppenheimer–Snyder collapse, both the matter density and curvature scalars (like RμνρσRμνρσ) increase as the collapse proceeds, but neither “causes” the other. They’re both outcomes of the evolving geometry.

So I appreciate the correction — I’m not trying to imply a new causal mechanism.

2. On “classical criterion for collapse”​

Yes, I should have been more precise. What I meant was:

• A trapped surface is the classical criterion for irreversible collapse leading to a singularity, assuming the usual energy conditions.

I agree completely that not all collapse leads to trapped surfaces — white dwarfs and neutron stars are good counterexamples. I’m specifically referring to the regime where collapse cannot halt.

3. On Penrose (1965) and irreversibility​

Yes — exactly. Under the standard energy conditions, once a trapped surface forms, the singularity theorems guarantee geodesic incompleteness.

And I also agree with your final point: classical GR is expected to break down before the singularity itself. I’m not making any claims about what happens “at” the singularity — only about the classical sequence up to the point where GR ceases to be reliable.

1. On “transition from trapped surfaces to geodesic incompleteness”​

Yes, agreed. “Transition” was not the best word. I’m not implying a physical change or temporal process. A better phrasing is:

• Given the assumptions of the singularity theorems, the presence of a trapped surface implies that the spacetime contains incomplete causal geodesics.

As you say, the theorems are global statements about the entire spacetime manifold, not dynamical transitions. I appreciate the correction.

2. On “post‑singularity regions”​

You’re right — I should clarify what I meant.

I’m not referring to physical evolution “after” a singularity. I’m referring to the fact that some exact GR solutions (like Schwarzschild) admit maximal analytic extensions that include regions not present in the original coordinate patch.

So instead of “post‑singularity,” a better term is:

• “additional regions appearing in maximal analytic extensions of certain solutions.”

I’m not claiming these are physically realized; only that they are part of the mathematical structure of the extended spacetime.

3. On “single structural progression”​

Let me try to phrase this more clearly.

I’m not suggesting that GR contains a literal sequence or physical chain of events beyond what the standard theorems say. What I’m asking is:

• In the GR literature, are the following elements typically treated as conceptually linked parts of a single framework for understanding gravitational collapse?
  
  • conditions for collapse
  • formation of trapped surfaces
  • singularity theorems
  • (mathematical) maximal extensions of solutions

Or are these usually treated as separate topics that are not grouped together under a single conceptual heading?

Restating my actual question​

Given all that, what I’m trying to understand is simply:

Do GR texts or papers treat the progression collapse → trapped surface → singularity theorems → (mathematical) extensionsas a single structural sequence, or are these usually treated as separate results?

I’m not proposing anything new — just trying to understand how unified this chain is considered in the standard literature.

 

[PeterDonis]

Let me respond point‑by‑point using standard GR language.

Your rephrasings all look fine to me.

What I’m asking is:

• In the GR literature, are the following elements typically treated as conceptually linked parts of a single framework for understanding gravitational collapse?
  • conditions for collapse
  • formation of trapped surfaces
  • singularity theorems
  • (mathematical) maximal extensions of solutions

Or are these usually treated as separate topics that are not grouped together under a single conceptual heading?

You referenced MTW and Wald, both of which are classic works that treat this subject (and you reference specific places where they do). How would you answer your own question for those two references?

(For the "singularity theorems" part, Wald is probably a better reference to use as a test, since those theorems were still pretty new when MTW was published--Wald was more than a decade later.)

 

[Thor Jackson]

Thanks, that’s a fair question.

For Wald and MTW, I would say the following.

Wald​

In Wald, the pieces I listed are clearly logically connected, but they’re not packaged under a single explicit heading:

• Conditions for collapse / trapped surfaces / singularity theorems These show up together in the context of global structure and gravitational collapse—particularly in Ch. 9 (singularity theorems) and later when discussing black holes and collapse scenarios. Trapped surfaces are the bridge between “local collapse conditions” and the global conclusions of the singularity theorems.
• Maximal extensions These are treated earlier (e.g., Schwarzschild/Kruskal in Ch. 6) as part of the general study of spacetime structure, not specifically as “post‑collapse” evolution, but they obviously interact conceptually with collapse models and horizons.

So my tentative answer for Wald would be:they are conceptually linked as part of the general framework of global spacetime structure and gravitational collapse, but they are presented in separate chapters/contexts rather than as a single named framework.

MTW​

In MTW, the pattern feels similar:

• Collapse, trapped surfaces, and singularity theorems are all part of the broader story of gravitational collapse and the fate of massive bodies.
• Maximal analytic extensions (like Kruskal) are treated in the context of exact solutions and global structure, again not under a single unifying label with collapse.

So for MTW as well, I would say they are tightly related conceptually, but not grouped under a single explicit conceptual heading.

What I’m really trying to check​

Given that, I’m basically asking:

• Is there any standard terminology or recognized “umbrella framework” in the GR community that explicitly treats these ingredients—collapse conditions, trapped surfaces, singularity theorems, and maximal extensions—as one coherent structural package?

Or is it more accurate to say that, while they are obviously related, they are usually handled as distinct topics under the broader themes of global structure and gravitational collapse, without a single unifying label?

 

[Dale]

I noticed that the sequence(1) intensification of sources → (2) curvature growth → (3) collapse → (4) singularity → (5) possible continuation

I think this is all covered in the generic term “collapse”, with maybe step (5) excluded.

 

[PeterDonis]

Is there any standard terminology or recognized “umbrella framework” in the GR community that explicitly treats these ingredients—collapse conditions, trapped surfaces, singularity theorems, and maximal extensions—as one coherent structural package?

From what I know of the literature, the presentations in MTW and Wald are fairly representative. So I would say that your answers for those two references, which basically amount to what @Dale posted (the "collapse" topic covers your (1) through (4), but your (5), maximal analytic extensions, is usually considered a separate topic) are probably reasonable answers for the literature as a whole.

Note that the topic of maximal analytic extensions is by no means limited to spacetime geometries that fall under the general "collapse" topic, so there are good reasons to treat it as a separate topic.

 

[Thor Jackson]

I think this is all covered in the generic term “collapse”, with maybe step (5) excluded.

Thanks — that helps clarify how you’re using the term.

When you say that all of those steps fall under “collapse,” I want to make sure I’m understanding the scope correctly. In the GR literature, “collapse” often refers to the physical process leading to trapped surfaces and curvature growth, but the later stages — singularity formation and questions about possible extensions — are sometimes treated in different chapters or contexts.

So the reason I separated the steps was not to claim they’re independent, but to ask whether GR texts typically treat:

1. intensification of sources
2. trapped surfaces
3. curvature growth
4. singularity formation
5. breakdown of the original spacetime and possible continuation

as one conceptual arc, or whether they’re usually discussed as distinct topics that happen to be related.

If your view is that (1)–(4) are all naturally grouped under “collapse,” and (5) is the only one that sits outside that umbrella, then that already answers part of what I was trying to understand. It suggests that GR does have a kind of unified conceptual structure for the pre‑singularity part of the story, even if the post‑singularity continuation is handled separately.

 

[PeterDonis]

I’m referring to the fact that some exact GR solutions (like Schwarzschild) admit maximal analytic extensions that include regions not present in the original coordinate patch.

One note about this: your usage of the term "original coordinate patch" here assumes that it was a particular one that only covers a portion of the spacetime. But in every case I'm aware of where the maximal analytic extension is known, a single coordinate chart is known that covers all of it. (For the Schwarzschild spacetime geometry, that's the Kruskal chart.)

 

[Thor Jackson]

From what I know of the literature, the presentations in MTW and Wald are fairly representative. So I would say that your answers for those two references, which basically amount to what @Dale posted (the "collapse" topic covers your (1) through (4), but your (5), maximal analytic extensions, is usually considered a separate topic) are probably reasonable answers for the literature as a whole.

Note that the topic of maximal analytic extensions is by no means limited to spacetime geometries that fall under the general "collapse" topic, so there are good reasons to treat it as a separate topic.

Thanks — that helps clarify how the literature separates the topics.

One thing I’m still trying to understand is whether the separation between “collapse” and “maximal extensions” is purely organizational, or whether it reflects a deeper conceptual boundary. In other words, when we look at the collapse sequence:

• physical conditions →
• trapped surfaces →
• curvature focusing →
• geodesic incompleteness

is that treated as a self‑contained structural arc because of the physics involved, or simply because that’s how the textbooks choose to present it?

And similarly, when maximal extensions are treated separately, is that because they are conceptually unrelated to collapse, or because they belong to a more general toolkit that happens to apply after collapse in some spacetimes?

I’m not trying to merge the topics — just trying to understand whether the boundary between them is conceptual or editorial.

 

[PeterDonis]

In the GR literature, “collapse” often refers to the physical process leading to trapped surfaces and curvature growth, but the later stages — singularity formation and questions about possible extensions — are sometimes treated in different chapters or contexts.

Can you give specific examples? Your own answers in post #7 say that, at least for singularity formation, this is not the case for the two references you've given so far.

Also, you're shifting the meaning of "questions about possible extensions" here from what you said in post #5. Maximal analytic extensions, as you described them there, are not extensions "past a singularity", and are not a "later stage" of a collapse process (in the sense that singularity formation is "later" than trapped surface formation). Now you appear to be thinking that they are.

 

[PeterDonis]

I’m still trying to understand is whether the separation between “collapse” and “maximal extensions” is purely organizational, or whether it reflects a deeper conceptual boundary.

I would say the latter. The last paragraph of my post #9 explains why.

 

[Thor Jackson]

One note about this: your usage of the term "original coordinate patch" here assumes that it was a particular one that only covers a portion of the spacetime. But in every case I'm aware of where the maximal analytic extension is known, a single coordinate chart is known that covers all of it. (For the Schwarzschild spacetime geometry, that's the Kruskal chart.)

Thanks — that’s a good clarification. I didn’t mean to imply that the “original coordinate patch” was somehow privileged; only that many presentations introduce the spacetime in a chart that doesn’t cover the full analytic extension. Your point about Kruskal (and similar global charts in other cases) is well taken.

What I’m trying to understand is whether the conceptual role of maximal analytic extensions is tied to the fact that certain coordinate presentations only reveal part of the spacetime, or whether extensions are viewed as a more general structural tool that GR uses independently of how the spacetime is first introduced.

In other words, is the need for an extension seen as:

• a coordinate‑presentation issue,
• a deeper feature of the spacetime’s global structure,
• or some combination of both?

I’m not assuming any particular answer — just trying to get a clearer sense of how GR practitioners think about the relationship between local presentations and the global analytic structure.

 

[PeterDonis]

when maximal extensions are treated separately, is that because they are conceptually unrelated to collapse

In the sense of the last paragraph of my post #9, they are, yes.

they belong to a more general toolkit

Yes.

that happens to apply after collapse in some spacetimes?

What do you mean by "after collapse"? I explained in post #4 why maximal extensions are not "post-singularity", and you agreed with me in post #5. Now you seem to be shifting your ground.

 

[Thor Jackson]

Can you give specific examples? Your own answers in post #7 say that, at least for singularity formation, this is not the case for the two references you've given so far.

Also, you're shifting the meaning of "questions about possible extensions" here from what you said in post #5. Maximal analytic extensions, as you described them there, are not extensions "past a singularity", and are not a "later stage" of a collapse process (in the sense that singularity formation is "later" than trapped surface formation). Now you appear to be thinking that they are.

Thanks, that’s a fair point — I see that I’ve been a bit sloppy in how I grouped things, and that’s part of what I’m trying to get clearer about.

On the first part: you’re right that in both Wald and MTW, singularity formation (in the sense of geodesic incompleteness) is not treated as something separate from collapse, but as part of the same overall story once the relevant conditions (e.g., trapped surfaces, energy conditions, etc.) are in place. So I should not have implied that singularity formation is usually handled in a completely different “later” context; that was imprecise on my part.

What I was trying to get at is more about emphasis and organization than strict separation: in some presentations, the physical picture of collapse, trapped surfaces, and curvature growth is developed first, and the singularity theorems are then brought in as a more formal, global result. But I agree that, conceptually, they belong to the same arc in those treatments.

On the second part: you’re also right that I blurred two different things when I talked about “possible extensions” as a “later stage.” Maximal analytic extensions, as you said, are not extensions past a singularity, and they’re not a temporal continuation of a collapse process. They’re a global structural tool that can be applied in many contexts, including but not limited to spacetimes that arise from collapse.

So let me restate more carefully what I’m trying to understand: within the literature, is there a sense in which

• collapse conditions,
• trapped surfaces,
• curvature focusing,
• and singularity theorems

are treated as a coherent package in the collapse context, while maximal analytic extensions are treated as part of a more general discussion of global structure that is conceptually adjacent but not “later in time” along the same process?

If that’s still not a good way of carving it up, I’m happy to adjust — I’m mainly trying to get a clearer feel for how GR practitioners mentally organize these ingredients, not to impose a particular grouping.

 

[Thor Jackson]

Thanks — I see why that phrasing raised a red flag. I’m not trying to reintroduce the idea that maximal extensions are “post‑singularity” or part of a temporal sequence. I agree with what you said earlier: maximal analytic extensions are not continuations past a singularity, and they aren’t a “later stage” of collapse.

What I meant by “after collapse” was not “after in time,” but simply “conceptually downstream” in the sense that discussions of maximal extensions often appear in the literature after the material on collapse, trapped surfaces, and singularity theorems — not because they are temporally connected, but because they are part of the broader topic of global structure.

So let me restate more precisely:I’m trying to understand whether the reason maximal extensions are treated separately is that they belong to a different conceptual category — global analytic structure — whereas the collapse → trapped surfaces → curvature focusing → geodesic incompleteness chain belongs to the gravitational collapse category.

I’m not trying to merge them or imply a temporal progression. I’m just trying to understand how GR practitioners mentally organize these topics: whether the separation is conceptual (different kinds of questions) or simply organizational (different chapters).

In the sense of the last paragraph of my post #9, they are, yes.


Yes.


What do you mean by "after collapse"? I explained in post #4 why maximal extensions are not "post-singularity", and you agreed with me in post #5. Now you seem to be shifting your ground.

 

[PeterDonis]

let me restate more carefully what I’m trying to understand

As far as I can see, your question has already been answered in this thread--by you yourself. @Dale and I have simply concurred with your assessment of how MTW and Wald treat these issues, and I've said that, as far as I know, those treatments are fairly representative. I'm confused at this point about what more we can tell you.

I’m trying to understand whether the reason maximal extensions are treated separately is that they belong to a different conceptual category — global analytic structure — whereas the collapse → trapped surfaces → curvature focusing → geodesic incompleteness chain belongs to the gravitational collapse category.

Again, as far as I can see, this question has already been answered in the thread. I don't see the point of continuing to repeat the same things.

 

[Thor Jackson]

As far as I can see, your question has already been answered in this thread--by you yourself. @Dale and I have simply concurred with your assessment of how MTW and Wald treat these issues, and I've said that, as far as I know, those treatments are fairly representative. I'm confused at this point about what more we can tell you.


Again, as far as I can see, this question has already been answered in the thread. I don't see the point of continuing to repeat the same things.

Thanks — I think I finally see the distinction more clearly. Let me try to articulate it in a way that checks my understanding.

It seems like GR contains two very different kinds of structural questions:

(1) Evolution under intensifying constraints This is the collapse arc:energy conditions → trapped surfaces → focusing → geodesic incompleteness.These ingredients are linked because each one strengthens the next; they form a chain driven by the dynamics of the Einstein equations.

(2) Global analytic structure of the manifold Maximal analytic extensions belong here.They aren’t “later in time” relative to collapse — they’re a different kind of question entirely. They’re about how far a given metric can be analytically continued, regardless of how the spacetime was originally presented.

If that’s the right way to understand the conceptual boundary, then the separation in the literature makes sense: the first group is about how constraints accumulate along physical evolution, while the second is about the global geometry that exists independently of that evolution.

I think that resolves what I was trying to sort out.

 

[Thor Jackson]

As far as I can see, your question has already been answered in this thread--by you yourself. @Dale and I have simply concurred with your assessment of how MTW and Wald treat these issues, and I've said that, as far as I know, those treatments are fairly representative. I'm confused at this point about what more we can tell you.


Again, as far as I can see, this question has already been answered in the thread. I don't see the point of continuing to repeat the same things.

Thanks — I think I understand the taxonomy now.What I’m still curious about is the reason GR ends up with this particular division of concepts.

In collapse scenarios, the ingredients form a chain where each step intensifies the next.In global structure questions, the tools are independent of that chain.

So here’s the deeper thing I’m trying to understand:

Is this separation simply historical and pedagogical, or does it reflect something fundamental about how GR organizes information — that is, a distinction between “local evolution under accumulating constraints” and “global analytic structure of the manifold”?

I’m not assuming any particular answer.I’m trying to understand whether GR itself enforces this split, or whether it’s just how we’ve chosen to present the theory.

 

[Thor Jackson]

I think I finally understand the distinction. GR seems to separate two very different kinds of structural behavior:

(1) Local evolution under accumulating constraints This is the collapse arc: energy conditions → trapped surfaces → focusing → geodesic incompleteness.Each step intensifies the next.

(2) Global reorganization of the manifold This is where maximal analytic extensions live.They’re not “later in time” — they’re a different kind of structural question entirely.

What I’m trying to understand now is whether this split reflects something deeper about how GR organizes information: that is, a distinction between constraint‑driven evolution and global structural reconfiguration.

If that’s the case, then the way the literature separates these topics makes a lot more sense.