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Stream: theory: mathematics

Topic: Math and Physics News

John Onstead (Aug 06 2025 at 21:35):

Hi! This is a general thread where we can discuss interesting news in math and mathematical physics. Feel free to share any interesting news you might have come across (does not have to directly relate to category theory). Every so often I'll post an interesting article along with (perhaps open ended) discussion questions laced in.

John Onstead (Aug 06 2025 at 21:35):

I'll start it off with this article from one of my favorite science news sites, Quanta magazine (I'll probably be citing them a lot on this thread). In this article they cover ways of doing general relativity without calculus in what they call "non-smooth surfaces"- spacetimes where there can be edges and discontinuities like singularities. The goal is to try and extend results from GR (such as singularity theorems) to the non-smooth setting in the case that non-smooth geometry better describes the universe than smooth geometry (the article argues this to be the case, but I personally don't see why, since GR has achieved great success in its current smooth form). Though it is a little confusing that there'd be this research program when the major consensus among physicists is that singularities are mere mathematical artifacts and do not exist in any physical sense.

John Onstead (Aug 06 2025 at 21:35):

From the mathematical perspective, the main object covered is "non-smooth spacetimes", which apparently are a generalization of smooth ones (so there's an embedding from the category of smooth Lorentzian manifolds into the category of these spaces). Usual differential geometry does not apply, but the article cites how some calculus-like results are still possible in these spaces. This makes me wonder how this effort might relate to synthetic differential geometry- might there be some variant of that which applies to this category, or maybe this will eventually inspire a new field of "synthetic non-smooth geometry" as a generalization of synthetic differential geometry.
One proposed definition for "non-smooth spacetime" is given here (on the official website, Quanta does not usually go into the mathematical details). These are called "Lorentzian pre-length spaces". It doesn't seem immediately clear what (if anything) these have to do with Minkowski space. Ideally, even a non-smooth spacetime should still, in some sense "locally look like" Minkowski space. So I'm not sure how that manifests here- something to think about!

John Baez (Aug 06 2025 at 22:06):

Thanks for initiating this!

One thing worth noting is that there's no physical reason to expect solutions of the PDE in physics to be 'smooth' in the strict sense of infinitely differentiable. For example, for the vacuum Maxwell equations, the finite-energy solutions are typically not smooth: instead, the components of the electric and magnetic fields only need to be in $L^2(\mathbb{R}^3)$ at any given time. People have analyzed the hell out of the physically appropriate function spaces for solutions of the PDE of physics, using Sobolev spaces and such. It's particularly easy for the vacuum Maxwell's equations and other linear PDE, but people (like me, in my youth) have done a lot of work on nonlinear wave equations using this technology. Einstein's equations of general relativity are particularly challenging: e.g. Christodoulou and Klainerman won great acclaim for a 500-page book proving the stability of Minkowsksi spacetime, which necessarily uses Sobolev spaces rather than smooth solutions.

I haven't read the Quanta article yet, but I'm guessing that things like "Lorentzian pre-length spaces" go beyond the usual story.

Though it is a little confusing that there'd be this research program when the major consensus among physicists is that singularities are mere mathematical artifacts and do not exist in any physical sense.

They may or may not be "mere mathematical artifacts". There is not a unique definition of "singularity" in general relativity: there are many kinds of singular behavior, and it's far from obvious which are physically reasonable and which are not. Mathematical physicists approach theories of physics by studying the equations of physics, seeing what they predict. The question of which solutions are'"mere artifacts" slowly becomes easier to answer as we gain more knowledge. So there's been a huge research program of studying singularities in general relativity, starting with the Nobel-winning work of Penrose and parallel work of Hawking, and continuing to this day.

For example, an interesting question is whether singularities in general relativity are always hidden behind an event horizon. Various conjectures along these lines have been disproved.

John Onstead (Aug 06 2025 at 23:29):

John Baez said:

Thanks for initiating this!

No problem!

John Baez said:

For example, for the vacuum Maxwell equations, the finite-energy solutions are typically not smooth: instead, the components of the electric and magnetic fields only need to be in $L^2(\mathbb{R}^3)$ at any given time.

That's quite interesting (yet unexpected). A while ago on here I tried to learn some basic bundle theory, but it was all in the context of smooth manifolds since those were the usual setting of differential geometry. This makes me wonder which category is the one where these Sobolev spaces become relevant- IE, where the spaces of sections of bundles form Sobolev spaces. That becomes tricky if the functions aren't even required to be continuous. Maybe you have to make use of local sections- for instance, the sign function is discontinuous over the reals but not over the reals minus the point at zero.

John Onstead (Aug 06 2025 at 23:50):

John Baez said:

There is not a unique definition of "singularity" in general relativity: there are many kinds of singular behavior, and it's far from obvious which are physically reasonable and which are not.

From what I've learned, there's two kinds of singularity: coordinate and curvature singularities. A coordinate singularity is certainly a mathematical artifact even from within GR, since it isn't diffeomorphism invariant: you can change your coordinates to remove it (IE, the event horizon of a BH). A curvature singularity is the kind the article talks about, it's a diffeomorphism invariant point where GR predicts its own failure and "blows up". It's not technically an artifact in GR- it's a direct prediction of GR thanks to the singularity theorems. What I meant by "artifact" was that it seems (at least by my impression) to be a popular belief that the incorporation of quantum mechanics will resolve the singularity. For instance, in traditional QM, singularities are impossible because Heisenberg's uncertainty principle states it's impossible to infinitely localize something in space, hence it's not possible for there to be a point with infinite density and zero volume.

John Baez (Aug 07 2025 at 09:11):

John Onstead said:

John Baez said:

For example, for the vacuum Maxwell equations, the finite-energy solutions are typically not smooth: instead, the components of the electric and magnetic fields only need to be in $L^2(\mathbb{R}^3)$ at any given time.

That's quite interesting (yet unexpected). A while ago on here I tried to learn some basic bundle theory, but it was all in the context of smooth manifolds since those were the usual setting of differential geometry.

Differential geometers like to assume everything is infinitely differentiable since that makes the math easy. Mathematical physicists start with differential geometry but then go further, adding analysis. There are a couple of reasons:

1. There's no physical reason to think fields are infinitely differentiable; if we only assume some conserved quantities like energy have finite values, or expected values, then we can conclude only that fields like in various Sobolev spaces (or related spaces).
2. To prove that PDE have solutions, it's much easier to use Sobolev spaces, even if you ultimately are trying to prove they have infinitely differentiable solutions given infinitely differentiable initial data.

So, mathematical physicists, or even differential geometers who want to study solutions of PDE like the Yang-Mills equations or Einstein equations, can't work in the smooth context. The smooth context is just a first step before the real work starts.

John Baez (Aug 07 2025 at 09:24):

John Onstead said:

From what I've learned, there's two kinds of singularity: coordinate and curvature singularities.

There are dozens of kinds of singularities in general relativity. Coordinate singularities are merely the result of a bad choice of coordinates, so we avoid them by avoiding bad choices. But saying that the curvature "blows up" is very vague, and when we try to make it precise we discover there are many different concepts of curvature singularity.

Furthermore, there are ways for solutions of GR to "suddenly end" even without the curvature blowing up. In fact the famous Hawking and Penrose singularity theorems only show "null geodesic incompletness", not that curvature blows up. Can we find physically realistic hypotheses that together with Einstein's equations guarantee that the curvature need to blow up in some sense? This is an area of active research.

What I meant by "artifact" was that it seems (at least by my impression) to be a popular belief that the incorporation of quantum mechanics will resolve the singularity.

Yes, I understood you meant that. This is indeed a popular belief. However, physicists working on general relativity don't say "quantum mechanics will change what happens, so we won't study the classical Einstein's equations too hard". Instead they study the hell out of Einstein's equations to see what they can learn - and just because it's fun!

John Baez (Aug 07 2025 at 09:33):

By the way, if you check out Freed and Uhlenbeck's Instantons and Four-Manifolds you'll see that the classification of smooth 4-manifolds involves a lot of PDE and Sobolev spaces. Back then it was done using Yang-Mills theory; more recently the Seiberg-Witten equations have largely taken over.

(I mention this book not to be intimidating, but because it's rather pedagogical!)

David Michael Roberts (Aug 07 2025 at 11:06):

Even the study of the S-W equations uses a lot of Sobolev space theory. We once had a seminar on them, and I had to be baffled by things like a lecture on the/a Sobolev embedding theorem, when I presumed we were supposed to be doing geometry!

John Baez (Aug 07 2025 at 11:28):

Since I used to work on PDE, the Sobolev embedding theorem was my friend, back then. Seiberg-Witten is easier than Yang-Mills because it's 'less nonlinear', but even for linear elliptic PDE, Sobolev theory is how you prove the solutions are smooth.

John Onstead (Aug 07 2025 at 17:54):

John Baez said:

So, mathematical physicists, or even differential geometers who want to study solutions of PDE like the Yang-Mills equations or Einstein equations, can't work in the smooth context. The smooth context is just a first step before the real work starts.

That's quite eye-opening! I'll have to look more into Sobolev spaces then, they sound quite fascinating!

John Baez said:

But saying that the curvature "blows up" is very vague, and when we try to make it precise we discover there are many different concepts of curvature singularity.

Furthermore, there are ways for solutions of GR to "suddenly end" even without the curvature blowing up. In fact the famous Hawking and Penrose singularity theorems only show "null geodesic incompletness", not that curvature blows up

Ah, I wasn't aware. I haven't studied much GR, but from what I've heard it's quite complicated and at times unintuitive (though arguably it's probably more intuitive than QFT, but that's not saying much). It's on my list to learn at some point in the future though!

John Baez (Aug 08 2025 at 12:21):

There's something simple I forgot to say. If the only way for solutions of GR to "suddenly end" were for the curvature to blow up, this would be a strong hint that quantum gravity might save the day and eliminate this problem. After all, as curvature gets large we expect - by simple dimensional analysis - that quantum gravity effects should become large. But in fact, solutions can "suddenly end" without the curvature blowing up. This makes it more challenging to imagine how quantum gravity will save the day.

I was looking around for a good explanation of these issues, and I bumped into this quick but helpful passage:

In fact, there are different ways to define singularities. The most common ones include the curvature singularity and geodesic incompleteness. Curvature singularity usually means that the curvature invariants are divergent somewhere. Among all the invariants, seventeen curvature invariants, called Zakhary-Mcintosh (ZM) invariants, form a complete set. For spherically symmetric spacetimes, it has been shown that the Kretschmann scalar $K = R^{abcd}R_{abcd}$ and the Ricci square $R^{ab}R_{ab}$ are sufficient to determine whether all the seventeen ZM invariants are finite or infinite. Geodesic incompleteness refer to that the affine parameter along a timelike or null geodesic can not be extended to arbitrarily large values in either the future or past direction.

The two definitions are consistent in many cases, such as the central singularities in the Schwarzschild black hole and in the Kerr black hole. However, counterexamples have been found. For the well known C-metric, there exists a conic singularity while all polynomial curvature scalars are finite. On the other hand, [some] spacetimes with curvature singularities are found to be geodesically complete. This may seem incredible, since a metric usually cannot exist where a curvature singularity appears. In fact, a regular metric can generate a spacetime with curvature singularities and a series of solutions have been explicitly constructed. Unlike the usual curvature singularity where the metric cannot be defined, the metric is well defined where the curvature blows up. In such spacetimes, particles or light can reach or pass through the singularity.

John Onstead (Aug 11 2025 at 11:13):

John Baez said:

But in fact, solutions can "suddenly end" without the curvature blowing up. This makes it more challenging to imagine how quantum gravity will save the day.

John Baez said:

But saying that the curvature "blows up" is very vague, and when we try to make it precise we discover there are many different concepts of curvature singularity.

John Baez said:

For example, an interesting question is whether singularities in general relativity are always hidden behind an event horizon. Various conjectures along these lines have been disproved.

All these points have made me wonder about something. I could be completely off base with this so please let me know if I am... But it seems there's a gap between what we intuitively expect to be the case in GR and what can be the case- for instance, diverging notions of singularity (like the ones mentioned in the article above), existence of naked singularities, potentially any sort of scenario where CTCs might exist, and many similar occurrences. For brevity, I'll refer to these collectively as "pathological spacetimes" as an analogy to topology, where there's many "pathological spaces" that are perfectly self-consistent yet defy our personal intuitions. My question is then: why don't we just "restrict out" these spacetimes, in the same way we often restrict to less pathological topologies when doing topology?

John Onstead (Aug 11 2025 at 11:14):

To elaborate, it seems the reason these pathological spacetimes need to be taken seriously at all is because of an assumption I'll call $P \cong E$- that is, there is a bijection between the set of all physically possible spacetimes $S$ and the set of all possible spacetimes $E$ satisfying Einstein's field equations (EFE). But what if EFE were, in some sense, "too general", and $P$ is merely a subset of $E$, not bijective with it? Then perhaps $P$ only contains the non pathological spacetimes, which would then make the pathological ones a non-issue. After all, we've never experimentally/observationally observed a naked singularity or a condition where the different notions of singularity diverge. Thus, a theory (a modification to EFE) that predicts only the non-pathological spacetimes is just as consistent with the data as EFE, with the bonus that there's no "extra baggage" of having these pathological spacetimes.

John Onstead (Aug 11 2025 at 11:15):

As sort of an example of this, let's consider naked singularities. Physicists have spent much time trying to show these don't exist and have failed to do so consistently, as mentioned. But what if, instead, they were to take the option of simply declaring that naked singularities don't exist and going from there? Now, the work isn't to assume EFE is true and try to prove something that just isn't true within that framework, but instead to start with a physically reasonable assumption and derive the corresponding mathematical framework- in this case a modification to EFE that eliminates the possibility of spacetimes with naked singularities- and see what its consequences are and if this modification is reasonable to make. The only reason I currently can see this might fail is if it's impossible to avoid "throwing out the baby with the bathwater"- that is, the very act of trying to modify EFE to eliminate naked singularities also eliminates more "valid" solutions to EFE we might have already observed. My question is then ultimately: why don't (or can't) physicists do this- is it mathematically impossible for some reason, does it lead to a contradiction, does it lead to this "baby bathwater" conundrum, or is it just something they don't do for whatever reason?

John Baez (Aug 11 2025 at 11:36):

John Onstead said:

But it seems there's a gap between what we intuitively expect to be the case in GR and what can be the case- for instance, diverging notions of singularity (like the ones mentioned in the article above), existence of naked singularities, potentially any sort of scenario where CTCs might exist, and many similar occurrences. For brevity, I'll refer to these collectively as "pathological spacetimes" as an analogy to topology, where there's many "pathological spaces" that are perfectly self-consistent yet defy our personal intuitions. My question is then: why don't we just "restrict out" these spacetimes, in the same way we often restrict to less pathological topologies when doing topology?

Spacetimes with singularities are not pathological: in general relativity we intuitively expect them, essentially because as matter collapses its gravity keeps getting stronger making it collapse more.

We can formalize this: that's what the Penrose and Hawking singularity theorems are about, and that's why those guys are famous. I explained those theorems here:

• Struggles with the continuum - part 7: singularities in general relativity.

• Struggles with the continuum - part 8: cosmic censorship in general relativity.

Singularities are physically important, since our universe started with one, every star bigger than about 3 solar masses becomes one, the center of most galaxies contains one, and the future of our universe will be dominated by them.

(Please don't talk about how quantum gravity may change this story, because while that's true, we're talking about solutions of the equations of general relativity right now, and general relativity fits what we see well enough to be worth studying.)

Closed timelike curves are different: we often rule those out by focusing on 'globally hyperbolic' spacetimes:

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

For example, the Penrose and Hawking singularity theorems are about globally hyperbolic spacetimes.

John Baez (Aug 11 2025 at 11:48):

As sort of an example of this, let's consider naked singularities. Physicists have spent much time trying to show these don't exist and have failed to do so consistently, as mentioned. But what if, instead, they were to take the option of simply declaring that naked singularities don't exist and going from there?

If you have interesting theorems you can prove assuming your spacetime doesn't have a naked singularity, that's fine - nobody will complain. Of course you have to define a naked singularity, and there are various ways you might do that, but there are some well-known ways.

But if mathematical physicists had said "let's never think about naked singularities", then Preskill and Thorne probably wouldn't have found conditions under which spherically symmetric infalling matter creates a naked singularitiy, and Christodoulou probably wouldn't have proved that generically this does not occur for spherically symmetric solutions: i.e., the set of solutions without a naked singularity is open and dense.

So, we would be in the situation of closing our eyes so as to not see a scary thing, instead of investigating whether that scary thing is really likely to occur.

And I think we'd be poorer for that.

John Baez (Aug 11 2025 at 13:49):

By the way, @John Onstead, thanks for telling me about that Quanta article about general relativity on spacetimes that aren't manifolds. Digging into it, I was surprised to discover this work is related to entropy! The clearest explanation I've found so far is here:

• Andrea Mondino and Stefan Suhr, An optimal transport formulation of the Einstein equations of general relativity.

I still don't understand it, but it's intriguing. It provides a reformulation of Einstein's equations in terms of a kind of entropy function.

John Baez (Aug 11 2025 at 13:54):

I asked Ted Jacobson about this paper - he's the guy who discovered that Einstein's equations follow from the first law of thermodynamics, one of the most amazing true things I've ever seen - and he said he didn't see any clear connection to his own discovery. But I'm intrigued.

(I've seen a lot of amazing false things.)

John Onstead (Aug 11 2025 at 19:12):

John Baez said:

Spacetimes with singularities are not pathological: in general relativity we intuitively expect them, essentially because as matter collapses its gravity gets stronger making it collapse more in a feedback loop.

Ah, I think I should have been clearer. I didn't consider spacetimes with singularities in general to be pathological- only spacetimes with naked singularities, or singularities where the notions of singularity (infinite curvature, geodesic incompleteness, etc.) do not converge. I do think that singularities in general are important for exactly the reasons you describe!

John Onstead (Aug 11 2025 at 19:21):

John Baez said:

But if mathematical physicists had said "let's never think about naked singularities", then Preskill and Thorne probably wouldn't have found conditions under which spherically symmetric infalling matter creates a naked singularitiy, and Christodoulou probably wouldn't have proved that generically this does not occur for spherically symmetric solutions: i.e., the set of solutions without a naked singularity is open and dense.

So, we would be in the situation of closing our eyes so as to not see a scary thing, instead of investigating whether that scary thing is really likely to occur.

I see, that makes sense!

John Onstead (Aug 11 2025 at 19:28):

John Baez said:

I still don't understand it, but it's intriguing. It provides a reformulation of Einstein's equations in terms of a kind of entropy function.

John Baez said:

and he said he didn't see any clear connection to his own discovery. But I'm intrigued.

Maybe it can be related to Verlinde's theory of entropic gravity? Though we're not sure at the moment if that is a "true thing". Though there's also work done by a mathematical physicist named Gabriele Carcassi (if you haven't heard of him I recommend checking out his Youtube channel!) that suggest there's actually a generic relationship between entropy and geometry. That is, if something geometry-like occurs in mathematical physics, there's a good chance it can be reformulated into something involving an entropy-like quantity and vice versa. I believe it's one of his main research programs to make this statement more precise.

John Baez (Aug 11 2025 at 19:58):

Maybe it can be related to Verlinde's theory of entropic gravity?

As far as I can tell, Verlinde's work is much cruder than Jacobson's. For example Verlinde gets Newtonian gravity from some assumptions about entropy and gravity that aren't very clearly stated, while Jacobson gets Einstein's equation from a few clearly stated assumptions. But Verlinde's work got a lot more publicity.

Fernando Yamauti (Aug 12 2025 at 03:30):

Sorry to barge into this discussion with a naive question. Are those singularities like magnetic monopoles? In such a case, we are simply taking as probing spaces certain punctured spacetime patchs (usually globally hyperbolic), which results in non trivial holonomy. Is the case of singularities in general relativity similar to that or there are really authentic singular spaces in the game (like the ones we study in stratified geometry)?

John Baez (Aug 12 2025 at 07:33):

As discussed earlier in this thread, "singularities" in general relativity are quite diverse. Unlike magnetic monopoles, where we are solving Maxwell's equations on a pre-existing Lorentzian manifold $\mathbb{R}^4$ (spacetime) and considering solutions where the electromagnetic field is defined (and smooth) only on an open dense set, in general relativity the solution is traditionally none other than a choice of smooth Lorentzian manifold (with some auxiliary fields on it) obeying Einstein's equations. So it's not a stratified space, just a smooth manifold with a smooth Lorentzian metric on it. Instead, the "singularity" refers to the fact that this smooth Lorentzian manifold is smaller than we expect it to be, and there's no good way to extend it to a larger one obeying Einstein's equations.

Since "smaller than we expect it to be" and "no good way" can be made precise in various ways, there are various precise concepts of "singularity" in general relativity. As I explained earlier, two important kinds are curvature singularities and geodesic incompleteness:

John Baez said:

I was looking around for a good explanation of these issues, and I bumped into this quick but helpful passage:

In fact, there are different ways to define singularities. The most common ones include the curvature singularity and geodesic incompleteness. Curvature singularity usually means that the curvature invariants are divergent somewhere. Among all the invariants, seventeen curvature invariants, called Zakhary-Mcintosh (ZM) invariants, form a complete set. For spherically symmetric spacetimes, it has been shown that the Kretschmann scalar $K = R^{abcd}R_{abcd}$ and the Ricci square $R^{ab}R_{ab}$ are sufficient to determine whether all the seventeen ZM invariants are finite or infinite. Geodesic incompleteness refer to that the affine parameter along a timelike or null geodesic can not be extended to arbitrarily large values in either the future or past direction.

Geodesic incompleteness is the more important of the two, in that we know more theorems about it.

However, geodesic incompleteness is easily obtained by taking an open subset of a perfectly nice Lorentzian manifold like $\mathbb{R}^4$ with its Minkowski metric. This incompleteness can be easily "cured". So for a better concept of singularity one needs to supplement geodesic incompleteness by a condition saying that a Lorentzian manifold can't be extended to a larger one in a good way.

Fernando Yamauti (Aug 12 2025 at 10:49):

John Baez said:

As discussed earlier in this thread, "singularities" in general relativity are quite diverse. Unlike magnetic monopoles, where we are solving Maxwell's equations on a pre-existing Lorentzian manifold $\mathbb{R}^4$ (spacetime) and considering solutions where the electromagnetic field is defined (and smooth) only on an open dense set, in general relativity the solution is traditionally none other than a choice of smooth Lorentzian manifold (with some auxiliary fields on it) obeying Einstein's equations. So it's not a stratified space, just a smooth manifold with a smooth Lorentzian metric on it.

I thought a solution was a choice of a pseudo-metric on an already pre-existing manifold. If so, as I understand the lack of solution on certain points of such manifold could be seen in the same way as a monopole, no? If I recall correctly one can either consider a monopole either as a puncturing in a contractible space or by putting a delta function (as a field or observable? I don't remember...) and keeping the entire space.

John Baez (Aug 12 2025 at 11:39):

Fernando Yamauti said:

John Baez said:

**As discussed earlier in this thread, "singularities" in general relativity are quite diverse. Unlike magnetic monopoles, where we are solving Maxwell's equations on a pre-existing Lorentzian manifold $\mathbb{R}^4$ (spacetime) and considering solutions where the electromagnetic field is defined (and smooth) only on an open dense set, in general relativity the solution is traditionally none other than a choice of smooth Lorentzian manifold (with some auxiliary fields on it) obeying Einstein's equations. So it's not a stratified space, just a smooth manifold with a smooth Lorentzian metric on it.

I thought a solution was a choice of a pseudo-metric on an already pre-existing manifold.

I don't think that's a good way to think about it: one of the big lessons of general relativity is that the spacetime manifold is 'contingent', depending on events. But it's a subtle issue. What do you mean by 'pre-existing'?

A solution of vacuum general relativity is a pair consisting of a 4-manifold $M$ and a Lorentzian metric $g$ on $M$ obeying the vacuum Einstein equations. What does it mean to say $M$ is 'pre-existing'?

Here's how physicists think about it:

Suppose we start with some initial data on a connected 3-manifold $S$ ('space') and start solving Einstein's equations. The solution $g$ may wind up existing on some connected 4-manifold $M$ containing $S$, and not be smoothly extendible to any larger connected 4-manifold. $M$ can depend on the initial data.

But there are additional subtleties because general relativity is diffeomorphism-invariant. Given $M$ above, any manifold diffeomorphic to $M$ has the same properties.

This leads to interesting phenomena.

John Baez (Aug 12 2025 at 11:45):

If I recall correctly one can either consider a monopole either as a puncturing in a contractible space or by putting a delta function (as a field or observable? I don't remember...) and keeping the entire space.

Suppose you have a magnetic monopole sitting at rest at the origin of 3-dimensional space. What do we do with this? We solve Maxwell's equations on

$M = \mathbb{R}^4 - \{t \in \mathbb{R}, x = y = z = 0\}$,

$M$ is Minkowski spacetime with a line removed. That line is where the monopole is sitting. We solve Maxwell's equations in such a way that for any time $t$, the integral of the normal component of the magnetic field over any 3-sphere in space containing the origin $x = y = z = 0$ is the magnetic charge of our monopole. But the magnetic field is undefined at $x = y = z = 0$, and our solution doesn't even extend to a distributional solution on all of $\mathbb{R}^4$. The singularity is too bad.

John Baez (Aug 12 2025 at 11:54):

By the way, if we replaced the word "magnetic" with "electric" in my last comment, everything would still be true, and it would be more relevant to real-world physics because electrically charged particles exist while magnetically charged ones have never been seen.

Fernando Yamauti (Aug 12 2025 at 12:07):

John Baez said:

What do you mean by 'pre-existing'?

I was thinking something along the lines of this nlab page. So, perhaps, something a sheaf associating for each manifold the field of configurations given by some stack of pseudo-metrics. Though I don't know how the variation in time of field configurations is supposed to be given here.

I'm more used to the setting of AQFT, but there the (co)sheaf is from patches of a fixed spacetime, so I don't really know how people formulate gravity in such setting.

Fernando Yamauti (Aug 12 2025 at 12:24):

John Baez said:

But the magnetic field is undefined at $x = y = z = 0$, and our solution doesn't even extend to a distributional solution on all of $\mathbb{R}^4$. The singularity is too bad.

Really!? No distributional solution? I recall seeing a paper several years ago where I think it was claimed magnetic monopoles could be given on a contractible space by adding discontinuous potentials, but I can't remember now the exact claim.

David Michael Roberts (Aug 12 2025 at 13:48):

My limited understanding is that de Rham introduced currents (https://ncatlab.org/nlab/show/current+%28distribution+theory%29) as a way to talk about the source term J in Maxwell's equations where this could be something akin to a delta function (you can measure the current by integrating around it, but it has no actual values where the physical current is). He mentions this in the papers from the mid/late 1930s, where the theory was still rather crude, and only after Schwartz introduced distributions (also so-named because of electromagnetism! But this time distributions of charges) did de Rham recognise how to write his currents in terms of the dual of a space of compactly-supported smooth test forms.

David Michael Roberts (Aug 12 2025 at 13:51):

Since Maxwell's equations link the physical current to the (derivatives of the) E and B fields, these derivatives are themselves (de Rham) currents. I haven't thought about how E and B themselves may fail to "work" even as currents, at the singular support of J.

John Baez (Aug 12 2025 at 14:01):

Fernando Yamauti said:

John Baez said:

But the magnetic field is undefined at $x = y = z = 0$, and our solution doesn't even extend to a distributional solution on all of $\mathbb{R}^4$. The singularity is too bad.

Really!? No distributional solution?

Maybe I was wrong. I was thinking about how the function $1/x^2$ on $\mathbb{R} - \{0\}$ does not define a distribution on $\mathbb{R}$, since there's no way to make sense of

$\displaystyle{ \int_{-\infty}^{+\infty} f(x) \frac{1}{x^2} \, d x }$

for compactly supported test functions $f : \mathbb{R} \to \mathbb{R}$ that don't vanish at the origin. The $1/x^2$ singularity is just too nasty.

(But now I'm having doubts even about this. I suppose you could try to 'subtract off infinity' in some clever way, which amounts to a form of renormalization. Are you going to push me to figure this out? I used to know this stuff.)

It's worth comparing $1/x$, which is much better. We can define

$\displaystyle \int_{-\infty}^{+\infty} f(x) \frac{1}{x} \, d x$

to equal

$\displaystyle \lim_{\epsilon \downarrow 0} \int_{|x| \ge \epsilon} f(x) \frac{1}{x} \, d x$

which is called the principal value. Roughly speaking, here the divergence we get from integrating over all $x > 0$ cancels the divergence from integrating over all $x < 0$.

Anyway, now I believe the $\vec{B}$ field created by a monopole isn't as bad. If our monopole has charge 1, it's

$\displaystyle{ \vec{B}(\vec{x}) = \frac{\vec{x}}{4 \pi |\vec{x}|^3} }$

and while this blows up like $1/|\vec{x}|^2$, I guess that's not as bad in 3d space.

Another way to think about this is: we're trying to find a distributional solution of

$\vec{\nabla} \cdot \vec{B} = \delta$

I wrote down a solution above; the question is just whether that formula defines a distribution (or really a vector-valued distribution) despite the blowup at the origin.

John Baez (Aug 13 2025 at 10:13):

Okay, I see how to prove that

$\displaystyle{ \vec{B}(\vec{x}) = \frac{\vec{x}}{|\vec{x}|^3} }$

is a distributional vector field: i.e. if you take its dot product with any compactly supported smooth vector field, the integral of the resulting function is finite.

The basic idea is that while it blows up like like $1/r^2$, it has norm $1/r^2$ only on a sphere of area $\sim r^2$.

Fernando Yamauti (Aug 13 2025 at 20:35):

John Baez said:

Okay, I see how to prove that

$\displaystyle{ \vec{B}(\vec{x}) = \frac{\vec{x}}{|\vec{x}|^3} }$

is a distributional vector field: i.e. if you take its dot product with any compactly supported smooth vector field, the integral of the resulting function is finite.

The basic idea is that while it blows up like like $1/r^2$, it has norm $1/r^2$ only on a sphere of area $\sim r^2$.

I see. So the point is that your integral is on a 3d space and that sphere has measure 0.

Still, back to the original question. Is that nlab definition starting from a fixed manifold and, then, considering the Einstein-Hilbert functional on a fixed space of pseudo-metrics (over a fixed manifold) not the usual approach?

If we fix a 3d space $S$, as long as we assume that everything is globally hyperbolic, there would be no harm in considering the spacetime to be, as a manifold, $S \times \mathbf{R}$, right?

John Baez (Aug 13 2025 at 22:03):

Fernando Yamauti said:

John Baez said:

Okay, I see how to prove that

$\displaystyle{ \vec{B}(\vec{x}) = \frac{\vec{x}}{|\vec{x}|^3} }$

is a distributional vector field: i.e. if you take its dot product with any compactly supported smooth vector field, the integral of the resulting function is finite.

The basic idea is that while it blows up like like $1/r^2$, it has norm $1/r^2$ only on a sphere of area $\sim r^2$.

I see. So the point is that your integral is on a 3d space and that sphere has measure 0.

No, I was just stating the key fact needed to prove the necessary finiteness! If $\vec{v}(\vec{x})$ is a compactly supported smooth function, then

$|\vec{v}(\vec{x})| \le C$

for some $C$, and $\vec{v}$ is supported in some ball of radius $R$, so

$\left| \int_{\mathbb{R}^3} \vec{v}(\vec{x}) \cdot \vec{B}(\vec{x}) d^3 x \right|$

$\le C \int_{\mathbb{R}^3} |\vec{B}(\vec{x})| d^3 x$

$\displaystyle{ \le C \int_{\mathbb{R}^3} \left| \frac{\vec{x}}{|\vec{x}|^3} \right| d^3 x }$

so switching to spherical coordinates

$\displaystyle{ \left| \int_{\mathbb{R}^3} \vec{v}(\vec{x}) \cdot \vec{B}(\vec{x}) d^3 x \right| }$

$\displaystyle{ \le 4 \pi C \int_0^R \frac{1}{r^2} r^2 d r }$

$\le 4 \pi C R$

where I used that fact I mentioned in the second to last step.

John Baez (Aug 13 2025 at 22:20):

. Is that nLab definition starting from a fixed manifold and, then, considering the Einstein-Hilbert functional on a fixed space of pseudo-metrics (over a fixed manifold) not the usual approach?

That approach is mainly good for deriving Einstein's equation starting from an action, and proving some facts about solutions of Einstein's equation.

But suppose you want to predict what will happen if you know what's happening now. Then you need to solve Einstein's equation. You need to start with initial data, solve Einstein's equation, and find the maximal manifold (or maximal globally hyperbolic manifold) on which the solution exists.

John Baez (Aug 13 2025 at 22:26):

If we fix a 3d space $S$, as long as we assume that everything is globally hyperbolic, there would be no harm in considering the spacetime to be, as a manifold, $S \times \mathbb{R}$, right?

The manifold will be diffeomorphic to $S \times \mathbb{R}$ if it's globally hyperbolic. But as you solve Einstein's equation, you may have to cleverly 'slow down the rate of time' to keep from hitting a singularity, if you want to get a diffeomorphism to $S \times \mathbb{R}$.

For example, if you send two gravitational waves at each other, they may form a black hole, with a singularity. The curvature blows up as you approach the singularity. Someone falling into the black hole will 'hit the singularity' in finite proper time. But if you cleverly choose your coordinates, you can identify your manifold with $\mathbb{R}^3$.

This is the sort of thing I was alluding to earlier:

Suppose we start with some initial data on a connected 3-manifold $S$ ('space') and start solving Einstein's equations. The solution $g$ may wind up existing on some connected 4-manifold $M$ containing $S$, and not be smoothly extendible to any larger connected 4-manifold. $M$ can depend on the initial data.

But there are additional subtleties because general relativity is diffeomorphism-invariant. Given $M$ above, any manifold diffeomorphic to $M$ has the same properties.

This leads to interesting phenomena.

Fernando Yamauti (Aug 14 2025 at 21:02):

John Baez said:

The manifold will be diffeomorphic to $S \times \mathbb{R}$ if it's globally hyperbolic. But as you solve Einstein's equation, you may have to cleverly 'slow down the rate of time' to keep from hitting a singularity, if you want to get a diffeomorphism to $S \times \mathbb{R}$.

So it's enough to consider the on-shell space of field configurations to be the solutions of the action functional on the space of pseudo-metrics in $S \times \mathbb{R}$ quotiented by its diffeomorphisms (as a stack). Is that the conclusion? If so, that’s also what the nlab is pointing to.

Also, back to what I briefly wondered here before, how one does usually restate such field theory of gravity functorially? The usual approachs (for instance, AQFT or FQFT à la Atiyah-Segal) all have as domain either a cat containing as objects spacetimes with an already specified pseudo-metric (say, globally hyperbolic manifolds) or an euclidean variant (to be interpreted as a Wick rotated version).

John Baez (Aug 14 2025 at 21:46):

Fernando Yamauti said:

John Baez said:

The manifold will be diffeomorphic to $S \times \mathbb{R}$ if it's globally hyperbolic. But as you solve Einstein's equation, you may have to cleverly 'slow down the rate of time' to keep from hitting a singularity, if you want to get a diffeomorphism to $S \times \mathbb{R}$.

So it's enough to consider the on-shell space of field configurations to be the solutions of the action functional on the space of pseudo-metrics in $S \times \mathbb{R}$ quotiented by its diffeomorphisms (as a stack). Is that the conclusion?

That's correct if you're doing Lagrangian field theory.

In my initial remark, the one that led to our conversation here, I was taking more of an astrophysicist's perspective. If you're trying to see what happens when two neutron stars collide, you don't extremize an action functional over all Lorentzian metrics on a fixed manifold. You solve an initial value problem, where (very roughly) you're given the metric on space and its time derivative at $t = 0$, and you're trying to solve for the metric on spacetime at future times. Then, depending on how you do things, the manifold on which the solution is defined typically depends on the initial data.

John Baez (Aug 14 2025 at 21:47):

Also, back to what I briefly wondered here before, how one does usually restate such field theory of gravity functorially?

I don't know if people have studied this. They should, of course.

John Baez (Aug 21 2025 at 20:50):

I hope some astrophysics is allowed in this "math and physics news" thread:

Astronomers have found a truly huge black hole! It’s in the massive galaxy in the center here, called the Cosmic Horseshoe. The blue ring is light from a galaxy behind the Cosmic Horseshoe, severely bent by gravity.

This black hole is 36 billion times the mass of the Sun. It’s not just ‘supermassive’: any black hole over 10 billion times the Sun’s mass is considered ‘ultramassive’. Not many have been found.

To me the coolest part is that the Cosmic Horseshoe has swallowed all the other galaxies in its group: it’s part of something called a ‘fossil group’.

Our galaxy, the Milky Way, is part of a group too: the Local Group. Many smaller galaxies have already fallen into ours. Eventually the Milky Way and Andromeda may collide and form a single bigger galaxy. This will be an ‘elliptical galaxy’—too disorganized to have spiral arms. So it’s not surprising that if you wait long enough, galaxy groups form a single big elliptical galaxy which eventually eats the rest.

Some ancient galaxy groups have already done this, and they’re called 'fossil groups’. The Cosmic Horseshoe is the biggest galaxy in one particular fossil group: it’s 100 times heavier than the Milky Way. It’s surrounded by a halo of very hot gas, 10 million Kelvin, emitting lots of X rays. But most of its mass can’t be explained by stars, gas and dust, so we say 90% is dark matter. All this is completely typical of a fossil group, except the Cosmic Horseshoe is bigger than average.

Fossil groups show us what the future will be like. Big galaxies will eat the rest, and big black holes at the center of these galaxies will eventually eat most of the matter. Dark matter—whatever that is—takes longer to fall in. But there’s plenty of time.

Here’s the new paper about this ultramassive black hole. Luckily, it’s free to read:

• Carlos R. Melo-Carneiro, Thomas E. Collett, Lindsay J. Oldham, Wolfgang Enzi, Cristina Furlanetto, Ana L. Chies-Santos and Tian Li, Unveiling a 36 billion solar mass black hole at the centre of the Cosmic Horseshoe gravitational lens, Monthly Notices of the Royal Astronomical Society 541 (2025), 2853-–2871.

John Onstead (Aug 21 2025 at 21:23):

That's quite cool! It seems the black hole is so massive because maybe when the galaxies were merging, their supermassive black holes also ended up merging. I know there's some sort of "final parsec problem" but it's instances like these where it's clear supermassive black holes do end up merging with each other, somehow!

I also like the perspective that this might be somewhat like the fate of our own local group when the Milky Way and Andromeda form the Milkomeda galaxy.

John Baez (Aug 22 2025 at 06:57):

I'll have to read about the "final parsec problem". It makes sense: black holes aren't like vacuum cleaners that suck up everything; two black holes a light year apart will orbit each other and spiral down very slowly due to gravitational radiation unless there's enough matter around to radiate away energy.

Ruby Khondaker (she/her) (Aug 22 2025 at 13:42):

Yeah I'm not sure whether we've yet figured out how black holes this big form this quickly - I remember something from my astrophysics classes about this issue generally for large structures in the universe forming waaaaaay too rapidly...

John Baez (Aug 22 2025 at 13:59):

Of course people need dark matter to help explain the early formation of galaxies...