Live CEOing Ep 723: Language Design Review of General Relativity Functionality & More

Okay, hello everyone. We are here to do a kind of science meets technology discussion about gravitation, general relativity, Wolfram model hypergraphs and related topics. And we're going to talk about some of Jonathan's work on developing function repository entries and so on for doing general relativity calculations, basically on the basis of hypergraphy writing and so on. And I guess if we're lucky, we will manage to segue into some discussion about actual practical astrophysical experimental implications of our models. Okay, so Jonathan, if I let's, let's sort of aim for. I want to simulate the collision of two black holes. Where do I start?

Okay, so first thing to know is the first class object in this whole framework is metric tensor, right? So pretty much every other object that you and every other function that you would call in the GR framework has at least one metric tensor object attached to it somehow. So metric tensor, if you pull up the details, you'll see that metric tensor has a bunch of inbuilt metrics already available. And you know, we have plans to expand that library and it'll be, hopefully it'll become like a sort of axiomatic theory type function where.

Okay, so hold on, where is this now?

So if you look at the details section of the, of the documentation, then somewhere there. Okay, yeah, so that's a table of all the, all the inbuilt metric tensors that we currently support.

Okay, so let's take, you know, a curve with a mass.

Right.

So how do I then get.

So if you just call metric tensor. I mean, if you just call metric tensor and then it will, it will do the. It will do the general case of cur.

Okay, Look at this new tech.

Metric tensor.

Oops, very bad. Well, it's going to fail there. Okay, what is this?

So that's now a symbolic metric tensor object. So you can see by default it's. It's in covariant form. So both of its indices are lowered. And so you can look at its matrix representation. If you just take that thing and you call and you request property matrix representation.

There's no autocomplete for that.

That can be.

What the heck happened here with this metric tensor turning blue?

That. I don't know that. I don't think that should happen.

Well, it's happening because you've named a symbol and it's not in the system.

Context there, but except it shouldn't be a. That, that should be a string name.

Because this whole object should be a resource function. Is that correct?

Well, so the summary box is not like the summary box should not have metric symbolic metric tensors. It does not have symbolic metric tensors as its first argument. It has string metric tensors its first argument.

Okay, I don't understand that. Jose, do you happen to know how that works on Nick?

No, I don't know.

Okay, Can you make a note of that? So we follow up on that.

The reason for that is so if you have a function repository entry that returns just like a summary box with the symbolic name in it, then it renders really ugly because it will have some random context.

Yeah, sure.

So to get around that the, the WFR team said that what, you know the thing, the correct thing to do is just have this is have the string name in the summary box rather than.

Well okay, but so somehow the summary box is breaking itself open. That's a separate issue. Ok, so we've got a cur metric and so now what can I do with this?

Okay, so you, if you look at the full properties of metric tensor you can see the things you can do to it directly. But like so you know, mostly this is just different ways of visualizing and computing simple thing. You know, you can, you can look at this, you can look at the space time line element or you could, you know, you can get, you can extract out pure functions that allow you to.

What is the space time?

I think it's just called line element. I mean because in principle this doesn't have to be a space time. This could just be an arbitrary metric. Right?

Why does it say DS squared equals as opposed to just this being the line element?

I mean it could just, it could have the DS squared be implicit if you prefer.

Well, I just think then it's an equation here. This is not a line element, this is an equation.

I know, but you know, I don't know. Usually if you see someone say this is the line element for Kerr in a paper or something like that's what they mean.

Maybe it should say line element definition or something as opposed to just line element which I really think should just be the actual thing.

What do you think.

Joseph?

Sorry, sorry. Yes, I agree with Stephen. It would be more computable to have something that is. Is one of the sides of this equation.

Okay, okay, fair enough. I mean for what it's worth, the computable form of the line element would be length pure function.

I see.

The line element is supposed to be a kind of symbolic thing.

Where does that. Sorry, you have.

So yeah, length pure. It's the, it's the last but three or whatever of the of the properties. So now if you take that and you apply it to a four vector, it will give you the length of that four vector in cur.

Euclidean four vector. I mean a Minkowski four vector. What kind of a.

Well, yeah, A full vector with minus plus plus plus signature. There you go.

What's theta?

Sorry?

What's theta?

Oh, theta is the angular coordinate, so.

Oh, I see, I see, I see. Because this is. It varies with the position. It's not. It's exactly. Yeah, it varies with the radius and the position. Yeah, but it has only a cylindrical. It has cylindrical symmetry of some kind. So it doesn't have a phi.

Precisely. Precisely. And similarly, if you did angle pure function, that would give you a two argument pure function that would allow you to compute the angle between 24 vectors.

Okay, so this is the length of a vector at the tangent space at that point.

Yes.

Right. Okay, so one issue here. I mean, so, so this. The question of how to make a lightweight repository is an interesting question because that's what we really need for this. Although what about these parameterized ones? How do they work in repository? Do we have that issue with axiomatic theory?

Well, we do because we allow for operator renaming.

Oh, that's right. So it then becomes an association.

Yeah, I mean, in the original design that was why we decided against having it. I think the original plan was to have axiomatic theory be an entity store. And the reason we decided against that was because we couldn't find a nice way of doing the operator overloading.

This feels similar to coordinate charted data. And we have the same issue there with parameters.

Well, I mean, it's possible that this should actually be promoted one day to an actual system function of some kind. It's kind of weird here. I mean, these are some very standardized set of coordinate systems, aren't they? Like there's some number of curvilinear coordinate systems and we have all of them. Statement.

Yes.

So yes, these are the orthona as opposed to. Know what that means? Why does that. How do I get the. The analog of the metric? Oh, I can actually just ask for the metric. Okay, so in that case the metric is a function.

Yes, if you provide the third argument with coordinates list xyz.

Okay, Let's see if I say. Okay, but toroidal coordinates, for example, I have to give a parameter here. Aubrey, please report. This is 133 and it's pinking. It's this suggestions bar.

Got it captured. I'll report it.

What the heck? What's the R doing here? I think toroidal. Hey, what's this?

I don't know, to be honest. I can follow up on that.

What's going on here? Okay, well, any case. So this is a model. What is the relationship between coordinate charts? I mean, in. In general relativity, there are often multiple charts characterizing a given metric. Is that not true?

Yes. And so, okay. And this is. This is. Okay, this is a genuine design question. So right now, my. So, okay, to make this a little bit concrete, right, if you say metric tensor, Schwarzschild or something, by default, it will give you the Schwarzschild metric in Schwarzschild coordinates.

Okay?

But you can also say something like metric tensor, isotropic Schwarzschild, and it will give you the Schwarzschild metric and isotropic coordinates. Or you can say, you know, what else do we.

Eddington Finkelstein coordinates.

Yeah, yeah, exactly. So you could say Eddington Finkelstein if you wanted to, and it will give.

You where and how. So, okay, so this is. I can ask for Eddington Finkelstein, which is not just a metric, but it's also a coordinate system for the metric. Is that correct?

Right. And so by default, that's. That gives you. That gives you the Schwarzschild metric in Eddington Finkelstein coordinates. So it's. Okay. On the one hand, there's something a little bit uncomfortable about treating these matrix representation. There's something a little bit uncomfortable about treating these things as separate metric tensors when they are like, as you say, different coordinate charts for the same geometry. My reason for having them be separate metric or be treated as separate metric tensors is because they are, in some sense different manifolds. Right? There are bits of the. If you were to plot the isotropic Schwarzschild metric and you were to plot the Angels and Finkelstein version of the Schwarzschild metric, and you were to plot the Schwarzschild coordinates version of the Schwarzschild metric. They don't look the same, you know, on the interior. They have different geometries. And so my feeling was it makes. Even though they are just. Just different coordinate charts, it makes sense to treat them as different metrics.

But wait a minute. Physically, if you could probe. If you could penetrate the event horizon, would they be, in fact, different?

Well, I don't think that's a really meaningful question. Right, because the Schwarzschild coordinates only apply to an observer who is infinitely far away from the black hole, from the mass distribution.

So in other words, what you're saying is it's kind of like an interpolating function where in the region where the thing is supposed to be valid, they agree, and then it goes bonkers elsewhere.

Right. I mean, like an extreme version of that is Kruskal. You know, we support the Kruskal Sekires metric, right? Which is, you know, the Kruskal extension of the Schwarzschild metric. And so again, that's a very evocative example of where they are both defined. It agrees with the traditional Schwarzschild metric and Schwarzschild coordinates. But there are many pits where Schwarzschild is not defined, but where it is defined in Kruskal.

Zacharias, what's this plus minus one doing?

So that's because you can have both ingoing and outgoing Eddington Finkelstein coordinates. So if you say ingoing Eddington Finkelstein, that will change to just be a plus. If you change it to outgoing Eddington Finkelstein, it will change it to just be a minus.

Okay.

We have the same thing with things like ingoing and outgoing Ghostrand panlevier coordinates and so on.

Showing me what I should be doing for this toroidal thing. Oh, wow. So what does the Euclidean mean there? And how does that relate to what we're doing? What is the role of Euclidean there?

So.

We have metrics that could live on the sphere or different dimensions, etc. So this is a case in which there are other possibilities. I have to check with it.

I.

And it seems that we need a full specification in this case.

But this notion of Euclidean here, in a sense for all of these, we could have said Minkowski in addition to the. I'm. I'm confused. So coordinate chart data as compared to Eddington Finkelstein here, that's the same type of thing. Toroidal vs. Eddington Finkelstein is.

So, so confusing.

This is a labeling. Eddington Finkelstein is a labeling of the Schwarzschild manifold, so to speak.

There are two different things here. In the coordinate chart data case, we support coordinates for the sphere and for the flat space, the Euclidean space. In any dimension in relativity, the coordinates and the space are. So the metric and the space are mixed. And so it's a bit less clean. In the general relativity case, I think we have to say sphere.

It's autocompleting to spherical. So I don't know why I see it autocompleted a spherical, but maybe it doesn't support.

So not all coordinate systems are supported for the sphere, of course, but.

Right. A toroidal is pretty weird on the sphere. There's some weird homotopy thingy thingy going on there. Right. So again, I could try polar, for example, here. I'm confused.

Okay, so for example, we have stereographic for the sphere as well. Sphere change to a sphere, even though.

The autocompleter says spherical, which we need to look at. Yeah. Okay. All right. So say again what you were saying about the fact that in, I mean, because one could expect somewhere here, and Jonathan's probably going to explain, you know, that there could be a base here. A four dimensional Minkowski space is basically the thing.

But the problem is. Okay, so, so, and I guess when you say the basis and four dimensional Minkowski space, I guess you mean we're just treating this as a, you know, as a minus plus plus plus manifold.

Right. So a luxury that we do not have in the hypergraph case or really.

In the relativity case, if we're being honest. Right. Like so if you take one of these metrics, you can say, for instance, you know, Lorentzian conditions, and it will tell you what conditions on the coordinates are needed in order to guarantee that that happens. But like, you know, if you pass the, if you pass r equals 2m in the, you know, in the Schwarzschild geometry, then it's no longer Lorentzian.

What does that mean?

Okay, so that means that there's no coordinate, there are no, you know, ranges of coordinates in which that is truly Lorentzian.

How can that be? I don't understand that.

You might. Okay, hang on. We can maybe look at that in a sec. But okay, look at the Schwarzschild case, that those are the coordinate conditions you'd need.

Those first two conditions are the same.

Yes. It hasn't reduced those out. You can say reduce on that.

But okay, and how could this not be true?

Well, I mean, in principle that's greater than zero.

Is there a theta equals zero case?

Yes. And the metric does become singular there.

Huh. Okay, so.

So coming back for a moment to the structure of coordinate chart data. So that's trying to separate the geometry from the coordinates in which that geometry is being expressed. So in relativity we could.

Why didn't this work?

It probably would have done if you didn't have complete. So all the coordinate symbols are maximally formalized. If you change the second argument of Schwarzschild and just make it be. And have that argument be a list, say TRA1, A2 or something.

Why should it care that they're formalized here? I mean, I don't understand why it makes any difference? It's just. They're just symbols. Why does reduce care if you try.

It with non formalized symbols? I'm just. I don't know, I'm. That's my.

What do you think that's the condition? Well, how many variables are there here?

Four. Oh, and you might need to change. Oh, sorry. Try that. Okay. You may also need to say in Schwarzschild, change. Change that to a list of Schwarzschild comma, something for the mass. There we go.

That's just bizarre. What the heck? Why does it care? I mean, does it think it needs to go inside? Those are formal symbols, right? So if I look at the input.

Form, they might actually. I think they're stringed. They're stringified formal symbols.

Ya Kady Yakurda, I had my reasons. Why the heck do they have to be stringified? Well, that's going to break a lot of things. So that's highly undesirable that they're stringified. So it'd be nice to not have.

They're only stringified in the way that they get presented. They're not stringified in the, like in the, in the intestacies of the function. All the computations are done on non stringified versions.

Okay, but fine, but when they come out like this, we got to be able to evaluate them, right?

I mean, I can have them return a non stringified form. It's. Yeah.

Seems highly desirable. Okay. In any case. All right, so where do we go from? Okay, so these metrics are named metrics. And just so I'm trying to understand, you know, in the modern world of machine learning for everything and LLMs and all that kind of thing, you know, I can make a reasonable picture of a, of a rabbit in 3D by just using some text prompt. Is it conceivable that I could create a general relativity metric from a text prompt prompt?

An analytical one?

No. No, probably not. I mean, just like I can say, you know, if I say three black holes in gravitationally bound, this is, this is going to be unbelievably scary and ridiculous. But. But.

I doubt it. I mean, I don't know enough about LLMs, but.

No, I mean this is somewhat ridiculous, but.

It'S cool.

It's cool and amusing. But my point is what people are going to want is an easy way to create initial conditions for things.

Yes, yes, that much I agree with. And so I mean, if we want to talk about initial conditions, we probably need to talk about ADM decomposition, which is the function that really does that.

Okay, all right, but this is generating analytical condition, analytical metric tensors and things.

Exactly. So, so and if you, so if you wanted to do so, an initial condition would be a metric tensor object, but it would be a three dimensional rather than a four dimensional.

You know, it would have one dimensional for time. Okay, but so, so just so you know, let's say I'm trying to invent a great new solution to the Einstein equations.

Right.

Actually, for each one of these, presumably there's an energy momentum tensor that goes with it.

Right? Okay. So if you take one of these metrics that you've just, I mean you happen to have used only vacuum metrics, right? So that's pretty easy. We can just take one of those metric sensors. You can say, you can call resource function solve vacuum Einstein equations on any of these.

On the metric tensor.

On the metric tensor. And then you probably want to say, oh, okay, you probably want to say comma zero to say that you want zero cosmological constant. Otherwise it will assume non zero cosmological constant and then it won't work. Okay, so it's telling you that that is an exact solution to the field equations.

How do I tell that from this?

It says exact solution. True.

I don't understand this output. I mean normally I would expect. I'm a bit confused here. So this is saying this thing. So for example, let's say I put in the.

So it's arguable that this thing should not be called solve, it should be called reduce or something. But I. So what it's doing is you give it a metric tensor object, it will say it will determine a. Does this just directly solve the Einstein, you know, the Einstein equations? Exactly. And if not, are there conditions we could use that we could impose that would mean that they do that? It does. Okay. Yeah. So in that case, if you, if you get rid of the zero. There we go. Okay. So it's, it's a solution, but it assumes four field equations. So if you, if you extract, if you, if you extract out field equations from that.

Can you for example, compute the Einstein tensor?

Yes. Yeah. So if you call resource function Einstein tensor on a metric tensor, then you can compute. Yeah.

Because then using the Einstein equations, you could say this is equal module of constant to t minu.

Yeah, right.

Exactly. Yeah. Which is what solve Einstein equations does in generality. So yeah, if you take, so the Einstein tensor, so you can call that and say reduced matrix representation or something that will show you the matrix representation of that tensor with all sort of Known algebraic equivalences imposed reduced matrix.

What representation?

Reduced matrix representation. Just like. So you can call matrix representation or you can call reduced matrix representation, depending on whether you want it to be auto simplified or not. So that's the Einstein tensor for the Godel metric. And then you can set that equal to something. Right. So then you can say stress energy tensor.

Oh, okay.

So that should return a zero matrix. Schwarzdel has a W. Yeah, so it does.

Lovely. Okay, so what about something with. So this has a big cosmological term play. What about.

Well actually. So the girdle metric is really a. It's a. It's a dust metric, naturally. So if you say resource function stress energy tensor. Okay, so then, so we have again, we have a bunch of inbuilt. We have a small library of inbuilt stress. Okay, that's not going to work. You need to give it. So first of all, the first argument, you need to give it an actual stress energy distribution. So that could be, I mean for the girdle case, you'd probably want it to be dust or something.

This is too complicated. I think I get the point, but let's not get, get pulled into this hole here. I mean, I think, I think I'm understanding. I mean what we're looking at roughly is we've got, you know, a bunch of name metrics. But now if we ask the question of all. I mean. Oh, and you were now talking about the ADM decomposition. But as far as the. For all time metrics, which is what these are, aren't they?

Yes.

So just a sort of a learning moment. What for all time metrics. I mean, aren't there like, I don't know, Bianchi type something or other universes and things that represent sort of with given symmetry for all time metrics, or am I not right about that?

I think so. I mean, if what you're talking about is you can certainly parameterize classes of metrics. I mean, the Bianchi universes are one such parameterization.

Right. Is there a place where somebody has collected endless sort of exact solutions?

I know there are books that have tried to collate these things, but I don't know if Jose has.

Yes, those books are the main references. Some people have tried to collect them computationally, but there's not an accessible database. What you can go.

Well, you mean they've tried to collect them but haven't succeeded, or they've tried to collect them and they keep them to themselves? Or what's the.

They are adapted to some. I mean, because There are multiple packages for tensor computations. They are collections adapted to one of these packages.

So like for exact other, is there a collection?

We don't have an exact. We have, I mean, we have notebooks with examples, with collections of examples, but we don't have a systematic collection of metrics. It has been proposed, but nobody did it so far.

Well, is there, is there something that we could get, you know, is there, is this a project for some, for somebody to go and try and collect and then how many are there? Are there like 500? Are there 200? How many are there? Roughly? There are monkey of the stress energy tensors. Go ahead.

They can be very strange and sometimes they are classified by algebraic properties. So it's not that they correspond always to physical scenarios.

Okay, but so for example, I mean like the failure of a solution to the helium atom in analytical form. I suspect for more than one black hole, you're out of luck. For anything analytical that a true statement.

Certainly no analytical solution is known.

Yeah.

And it's widely believed that they don't exist. Well, I don't think anyone, I don't.

Think anyone approved the, the 21st century class of special functions. Yes.

Right.

Undecidability, it's either going to hit the wall of undecidability or there's going to be a 22nd century set of, you know, of special functions.

But so certainly in ADM decomposition we have, you can produce a symbolic metric sensor that corresponds to a black hole collision, to a binary black hole collision. But it involves things that you need to determine essentially by numerical differentiation, by doing numerical integration.

All right, so if I go here to ADM decomposition, I'm going to start seeing. Yeah.

So if you go in here again, so we have a subset of the, a subset of the things in metric tensor as sort of initial condition libraries. So if you say ADM decomposition, Schwarzschild or something, you don't get the same Schwarzschild metric that you would have got from metric sensor. You get a sort of more underdetermined version of it where you only know the Schwarzschild geometry on the initial space. Like hypersurface.

Yes.

So if you now say adm, if you take that thing and you say like spacetime metric tensorflow. As opposed to spatial metric sensor, it will give you the symbolic metric sensor object, which you can then say matrix representation of. So it's pretty, it's pretty easy if you put that in matrix form, by the way, just for visual purposes. But yeah, so you can see if you look at the three by three submatrix, apart from the first and second, you know, first row, first column. That three by three submatrix is just that. That's the geometry of the initial Schwarzschild hypersurface. That's just a spherically symmetric three geometry.

And these are the things you have to solve for.

Exactly. So those are then the gauge variables, alpha, beta 1, beta 2, beta 3. Those are your lapse and shift.

I see. But so in actuality, given that initial data that you've got there, there is a solution that is just the four full dimensional. Full four dimensional Schwarzschild solution.

Yes. And so although this function isn't in the function repository yet, it is in the GitHub repository. And so maybe when we switch over later I can show. So we have functions for doing, for solving the ADM equations in generality. And so there's a, there's a fairly easy way that you can, you can sort of, you know, starting from just this initial hypersurface version, you can reconstruct the analytical form of the Schwarzschild metric from that just by figuring out what, what, you know, what constraint, what, what conditions need to hold in order for the constraint equations to be satisfied and so on.

And that's the only solution? Yeah, the, the, the four dimensional Schwarzschild is the only solution, given that initial data.

Modulo some notion of equivalence. So you still remember, you still have a lot of coordinate freedom, you still have a lot of gauge freedom. There are many ways you can slice up the Schwarz geometry. And so it won't always lead you to exactly the same form of the metric tensors you would get in the analytical version.

Right. So you're saying there are different foliations corresponding to different, which will give you different analytical forms for the four dimensional metric tensor. Because essentially the time coordinate is, you know, the relationship at the time coordinate. Well, I'm not quite sure, given that you define the space like hypersurface, the initial space like hypersurface. Yeah. Subsequent space like hypersurfaces could be tipped, progressively tipped relative to that one.

Right.

Effectively defining different notions of simultaneity.

So the alpha function there is telling you what's the timelike distance between the neighboring space like Hypersurfaces and the Beta 1, Beta 2, Beta 3 is telling you effectively how do I relabel the spatial coordinates between the different hypersurfaces?

Okay.

And then if you look at, if you go back to the EDM decomposition documentation in the full list of properties, you'll see we have support for A bunch of inbuilt gauge conditions which effectively correspond to different foliation choices. So, okay, so those are the initial data, but in here. So, yeah, things like, so maximal slicing condition, harmonic slicing condition one plus log slicing condition, et cetera. Those are all different conditions you can put on the lapse function that tell you how you foliate the space time. And then things like normal coordinate conditions or harmonic coordinate conditions, those are conditions on the shift vector that tell you how you relabel the coordinate, the spatial coordinates.

Okay, okay. Okay, great. All right, let's. Let's dive below space, so to speak, and understand how we can. The things you've done for building these out of hypergraphs and so on.

Right. Okay, so for that you'll want the function discrete hypersurface decomposition. And again, as with any of these, there are more in the pipeline that will be in WFR soon. Discrete hypersurface decomposition. Oh, sorry.

I'm hoping that the search system will.

We only have one function with hypersurface in its description.

Wow, cool. Okay.

So what this thing does is you give it a symbolic metric tensor and it will slightly. It will do a spacetime foliation of it and it will produce a sequence of hypergraphs that correspond to different slices in that foliage.

Where those hypergraphs are not necessarily related by hypergraph, rewrite it.

So in the case where you are solving for the thing numerically, they will be related. So essentially what we're doing is.

The.

Key insight is that adaptive. Generalized adaptive mesh refinement is really just hypergraphy writing with a very weird event selection function. Your event selection function is saying progressively refine or course on the hypergraph at these points. So based on some criterion that has to do with curvature tensors and whatever. And so internally or formally, you can think of that as being a hypographic writing system just with a slightly odd event selection function.

Well, let me understand that for a second. So you've got an initial, essentially triangulation of space.

Yeah, it's not. It's not a triangle. It's less topologically structured than a triangulation.

But can we see what the initial one would look like here?

Yeah, so, okay, so if you take. All right, so that particular ADM metric, you know that's underdetermined, Right. That has alphas, betas and things. Yeah, if you take Schwarzschild.

Okay. With those symbolic parameters.

Well, those are the coordinates. There should be no symbolic parameters if we want a numerical thing.

Okay, fine. Then what does this mean?

So that's saying. Okay, that's telling you which part of the geometry you want to decompose. So it's saying between time between 0 and 1 radial coordinate between 0 and 4 and the theta coordinate between minus PI and PI. Then we're saying we want 200 vertices and with an initial sort of refinement scale of 1.5.

Okay, so how do I get the. Just the initial refinement, not the T equals zero refinement.

So I believe. Hang on, let me just quickly double check the right way to do this. Sorry. I believe you can just put in a time value into the property. But I just want to double check that. One sec.

Okay, but basically what this is. Hopefully we'll get some visualization in a moment.

Okay, so. All right. So, okay, if you were to just. If you were to take that and just say, what would you want? You want polar graph evolution colored?

Holograph evolution colored?

Yes. So polar graph colored will show you the final hypersurface only Polygraph evolution will show you all the hypersurfaces, including both the initial and final forms. Right now there isn't an option, a straightforward option to get just the initial one, but maybe I should add that. You can't use this.

Let's just get fewer of them and then we can get to. Then we'll pull out the initial one.

Right? There we go.

Let's get fewer of them and just get the initial one.

So if you give it a. In that property, if you give it a list, rather than saying polygraph evolution colored, you say the list. Polygraph evolution colored comma, however many steps you want to go. If you say zero, you would just get the initial. I think.

Okay, fine. List or just the zero there.

No, list. List and then zero.

Why is this taking so long? I mean, it's. Isn't it doing 200 steps? I don't need it to do 200 steps.

No, it's not. No, no. That's doing 200 vertices. Oh, okay. I'm not sure why that happened, but There we go.

Okay. 200 versus what's the 1.5?

That's the, that's the initial refinement level. So if you increase. So you would get a denser, you know, a more densely connected hypergraph for higher values of that.

Higher values.

For higher values.

So what is the actual computation going on here? Is the dimensional linear algebra. What is it? Is it going to be faster for everyone? I'm assuming the big GPU and so on?

It's a fair. Yeah, it's, it's largely linear algebra, multi linear Algebra, I guess.

All right, so this is. I mean, at some level I should be able to reconstruct, I mean, the traditional picture of the. Of the Schwarzschild solution with, you know, in a manifold, so to speak, this should limit to the manifold somehow.

Right. And if you say coordinated polar graph colors or something, then, then you'll get, you'll get that. Yeah, you'll get to see that with the vertex coordinates.

And the vertex coordinates here. This is projecting into. I mean, it's a 3D object.

Yeah. So it gets projected onto a plane. Effectively, the by default, whatever the third coordinate is, is, is going to be supp. So that is effectively the embedding diagram for the. That's a, you know, a discretization of the embedding diagram for the Schwarzschild metric.

Okay, so what's happening? Okay, there's nothing in the inside.

Yeah, so. So it gets truncated at a point, at a certain, you know, beyond a certain point. So if you were to, you know, if you were to extend this indefinitely, that interior bit would go on forever because that's where the coordinate singularity is.

Okay, but let me understand this. I mean, why in the far field? Let's imagine that instead of a Schwarzschild thing here, I was just doing flat space for my metric.

Right.

What will be the name for the flat space? So if I had a flat space metric, I just want to see what this hypergraph decomposer produces.

You'd want Minkowski.

Okay, so, okay, so here, instead of this, I want to use Minkowski.

Yeah, you may get some. I'm actually curious. You may get some weird division by zero error. I don't. I don't really know.

So I just say Minkowski here.

Yes.

And then what for these parameters?

I mean, you probably want XYZ or something.

TX parameters.

No, no, no, four. I'm just saying. I'm just saying that, you know, the RA1, A2 doesn't make sense. T should be the first.

Okay. Okay, so now I want to do the same thing here. I don't know whether I want that.

A.

What do I want to do with the TRX? I want.

Yeah, so T01 is fine. You probably want X between, I don't know, minus 2 and 2 or something. Y between minus 2 and 2. And then you don't want polar graph. You want Cartesian graph to indicate that you're using a Cartesian coordinate system.

That's a big graph. Go down.

Yeah, that you may want. Oh, you may know. You may want to reduce the, the 2.5. Make that one or something. Well, I don't know. That may be the problem.

Okay, so why are there two holes here?

I suspect that's a discretization artifact. If you run it again, those should not be there. I mean, there may be holes, but there'll be different holes.

Okay, okay, so then what I want to understand is this thing here. If I look at flat space. Okay, fine. So it's some weird thing there.

Yeah, because remember that the discretization process is, is essentially, is fundamentally random, right? We're doing a poisson. It's a Poisson point process.

So we're making. Okay, fine. Okay, so here, if I look at the judicial balls around these points, I would confirm that this is indeed flat Minkowski space.

Indeed. Yeah.

But I mean, this is pretty messy. Can I, Can I rerender this as to say. How would I rerender this? Can I just.

I mean, do you, do you want it to be coordinatized? It doesn't have to be coordinatized.

Right, well, let me for now make it coordinatized.

Okay?

Let's make it not tubular. And let's just. Let's just see what's going on. Let's make graph edge style arrow like gray or something. Do. Exactly. Oops. Why didn't that work?

It's a list.

Sorry. Okay. It ignored my edge style. I'm trying to understand, trying to see what this looks like because I'm trying to see this looks much more densely connected. Okay, let's not. Let's.

But it's only densely connected because you made it that way, so to speak.

Right?

If you lower the one, it will be less so.

Okay, so this purports to be flat space, even though this rendering doesn't make it look terribly flat, right? Well, it's flat in two dimensions, but I don't know whether that's the right thing, is it should be flat.

These, these are meant to be two dimensional projections of the hypersurfaces.

Okay, but, but I mean, this is kind of the null experiment. This is flat space time, right? Supposedly. And so if I were to evolve flat space time, it should not change.

Right?

But yet if I evolve the hypergraph corresponding to something which is non flat, I expect evolution of some kind.

Right? I mean, you'll still get quote unquote evolution in the flat space case, right? It's just that, you know, it'll be changed the flatness ergotic enough that.

Yeah, okay, but so in the case of the black hole solution, the Single Schwarzschild solution. What I expect again in the evolution is that we'll preserve that coarse grain Schwarzschild solution. Is that true?

That's true, yeah.

Does it work?

Yeah, I mean you see that above, right? Like the graphs, they flap around a bit, but the coarse properties are. I think you did a brief evolution. Yeah, there.

Yeah, there. Okay, but so what's a non stationary solution? What's a non stationary initial condition? Two black holes.

Two black holes, yeah, we can do. But that's complicated. FLRW with a dynamical scale factor would be a simple example of a non stationary thing. Although that's a bit boring because it's sort of conformally flat. Like it's not very exciting to look at.

What does it mean in these pictures that it's conformally flat?

I mean it will just look like flat space, but growing.

I see. Basically. Okay. All right, so. And what you found is that for a quite a wide class of hypergraph rewritings, you get things that's coarse grained correctly.

As long as you impose the right event selection function.

Right.

And that's, that's really where, I mean, in a sense it's, you know, it's a, that's, that's where we cheat, so to speak.

And the event selection function is defining foliations or what?

Yeah, exactly. And so, and so we just, we choose. There's enough freedom in how we select the events that we can just say, well, coarsen and refine the hypergraph. You know, apply, in other words, apply the hypergraphy writing rules in accordance with, you know, where the height regions of high or low curvature are.

But is there not something, wait a minute, the sort of naturally occurring event selection function, if we were to make essentially a token event graph out of this. Yeah. Would involve more events in places where there are in a sense a higher density of vertices, wouldn't it?

Yes. And so in the continuum limit that would correspond to normal coordinate conditions and geodesic slicing. And the problem with that is that that will stretch in any space time where there's a singularity or multiple. Like with Schwarzschild, it will mean that your foliation, the slices of your foliation will get hugely stretched around those singularities. And so it's actually, it's not a particularly good gauge choice.

So. And the reason for that is it's like the fact that the black hole never actually forms to the outside observer or something, that time is infinitely stretched.

Yeah. And, and very concretely it means that the, that the vast majority of your rewrites. In fact, in the limit, 100% of your rewrites are going to be concentrated around the event horizon.

Walk me through that. So that, that's because. What? That's the intuition behind that, the rewrites.

So because that's just where you have the highest density of hypergraph edges. Right.

No, I realize that, but I mean in terms of understanding time dilation. In terms of rewrites. Yeah, right. So what's actually happening? So the.

So that's, that's corresponding to the statement that an observer would. So the. The number of rewrites you would need to cross the event horizon would be infinite. In other words, you know, sort of.

And the interpretation of that is that to an outside observer, it takes an infinite time for something to cross the event horizon.

Yeah, precisely.

But why isn't there a co moving? Why is what matters. I see. It's because you're pinning somehow to asymptotic infinity. You're making some. Well, explain this. Is that because, I mean.

No. So if you did this with. In a different coordinate, if you didn't do it in Schrad coordinates, then that wouldn't be. Then, you know, then you're seeing the hypergraph evolution as observed by, you know, an infalling observer in Eddington, Finkelstein coordinates or something. Then it's a different. Then it's different. Then nothing special happens at the event horizon. You don't see a sudden spike in hypergraphy rights there.

I see. So the passage of more time correspond. The perceived passage of more time is the doing of more hypergraphy rights.

Right, right.

How should we understand the fact that the number of rewrites. There is an intrinsic number of rewrites that happen, but to different observers, they can bunch up the foliations or spread them out in such a way that they change the number of rewrites that happen per moment of foliation time. Is that a fair way to say it?

Yeah, I think so. I mean. So the important point is that with any of these space times, you know, the causal graph extends indefinitely. And so you can slice it apart in ways that take infinite time to reach certain parts, or infinite numbers of slices to reach certain parts, or you can slice it apart in ways that take a finite amount of time to reach certain parts, but in so doing, you're producing essentially infinite causal edge distances between those slices.

Can we generate the causal graph for this evolution?

Yes, but unfortunately that functionality is not yet in the function of the. That's in the pipeline.

But. So that causal Graph. What you're saying is these different coordinate systems sample the same causal graph. That is the same causal graph in the end, but they're sampling it differently.

Right? Exactly, exactly. And so the sort of the default event selection function, which is, as I say, the GD6 slicing normal coordinate conditions case for admission, is essentially the one where, if you like, we have, you know, all the slices are just distance one away from each other in some sense.

Right? Yeah.

And so inevitably that's not going to reach the, you know, the parts of the causal graph that are infinitely far away. It's not going to reach that infinite. It's not going to reach those in finite time.

But the parts of the causal graph, I mean, to a photon, it's just tooling along in some sense the causal graph with no time passing. So a photon, if I'm not mistaken. So a photon can successfully reach these distant parts of the causal graph.

Yeah, as can a massive object. Right. But the point is that the coordinate system, and therefore, you know, the gauge choice, and therefore the foliation observed by those things is not going to be the default distance 1. Foliation.

Yeah, I guess it's. But so to an observer. Let me think about this for a second. So different relativistic observers. I mean, I'm kind of wondering about mini observers that are based on actually doing rewriting for the observer. Does that make any sense? So in other words, rather than from the outside saying, here's how I slice the causal graph. The observer is seeing these slices. Yeah. Asking the question, could there actually be a little critter sort of being updated inside the. Where you identify a piece of causal graph as this is the observer.

Right. No, I mean, I think we should absolutely do that because it's useful for physics project purposes and other things. The reason that's not supported yet, at least from my perspective, is that it's. So there's a pretty well defined way of specifying continuum gauge choices and then discretizing them. We don't yet really have a way, a systematic way of specifying discrete gauge choices without just kind of straight up giving the answer chains in the causal graph.

I know this has been the long running, three plus year story of kind of how do you label, you know, its depth first, its breadth first? What are the reasonable choices of orderings?

Right.

Nick, I think if, if you, if Nick is still with us here, I think you've thought about this a bit.

Nick is ostensibly here.

Okay. And I think this is an old chestnut problem of what are the observer Reasonable choices of foliations.

Right. I mean, one question is how much inspiration can we take from the ADM case? In other words, how general are those coordinate systems or are those gauge choices? Like, does it make sense to use harmonic slices?

No, that's very technical ADM stuff. Let me look at that list. Let's see, we had. Okay, this is hypersurface. Okay. But by the way, when you're showing these, are these generated from hypergraph updates or are these generated from just taking slices of the, of the Einstein equation solutions?

The, the first is generated from taking slight. From taking an initial slice. The latter are generated by hypergraph updates.

Okay, and do we know which hypergraph updating scheme is the best at leading to a successful coarse grained?

You know, because as I say, the, the, the, the scheme is determined entirely by the curvature tensors. Right. So essentially all we're doing is we're saying we've got some very simple set of rewrites that we can apply that coarsen and refine the hypergraph. We compute the curvature at every point at every vertex. And if, you know, if the curvature is higher than some value, we apply a refinement there. If it's lower than some value, we apply a coarsening there.

Okay.

So because the purpose of this was essentially to do practical graduation. We do. It's a complete kind of backwards hack, if you see what I mean.

Right. So you compute curvature, but do you compute curvature intrinsically using hypergraph methods?

So if there's no analytical form of the curvature tensor. Yes.

And do you get reasonable results in that case? I mean, like if you, if you compare it with the case where there is an analytical form.

Yes, yeah. And yeah, I mean, we already showed numerically that, you know, the output of functions like Wolfram Ricci curvature tensor, which is sort of the analog of Ricci tensor for the discrete case that you get convergence to the right answer. And at some point I would quite like to write a kind of. I probably should do this with a student or something. Write a systematic exploration of the sort of convergence and stability properties of these discrete estimators.

Right, but this is convergence. I mean, the question of how well they converge is going to depend on the measure in the space of hypergraphs, because there will be hypergraphs on which they'll probably be crummy convergence. The question is whether the naturally generated hypergraphs generated by hypergraph rewriting have. Am I not right? I mean, you can probably make a pathological. It will be interesting to see. Is there a pathological hypergraph on which in fact this really good summer school project, are there pathological hypergraphs on which there is ultimately convergence? But it's really slow and silly.

Yeah, I agree. It's kind of, it's testing the limits of how stable GR is in discrete space time systems.

Right. Which is similar to looking at the continuum limit of fluid dynamics or something like that, and saying that there exist configurations of molecules such that the total density is correct and so on, but yet they're not with reasonable coarse graining procedures. They look wrong.

Right. But in some sense. Okay, so one way of interpreting the GR derivation, if you want to think, if you don't want to sort of assume weak ergodicity of hypergraphy writing, which of course we have to assume, another way to interpret that theorem is to say. Well, what it's actually saying is if you consider only causal invariant rewrites that preserve dimension, you know, it's preserved continuum dimension, then the class of rewrites that satisfy the Einstein equations is entropically dominant in that space, if that makes sense.

Sure, Right.

But of course, the pathological example, you know, just because it's entropically dominant doesn't mean that it's measure, you know, it's measure one.

Right, okay, but there are multiple things flapping around here. I mean, one thing is the, the existence of a continuum limit, AKA the analog of the second order of thermodynamics for, for space time. That's one thing. I mean, basically the statement that everything's going to be sort of reasonable is like a statement. The second order of thermodynamics for the structure of space time, I believe.

Sure.

And an interesting question is if we want to see through that, to actually see the discrete structure of spacetime, because in a sense, what to the numerical scheme is a pathology to the Brownian motion of space time? Searcher is a success.

Absolutely.

So question would be, where are the most likely deviations? We have talked about this a bit before. Where are the most likely deviations from the continuum limit, AKA second order of thermodynamics, AKA randomness, ergodicity, whatever, all of these, all of these similar concepts, where is there, where is there the most pathology, so to speak?

So my view is, I don't know that it's the most, but I think one place that's quite fruitful to look is cases where there's topology changes in the space like hypersurfaces. And a very basic example of that is in the collapse of a rotating fluid distribution or something. So you've got some spinning Fluid that's described by the Weyl Lewis Papapetry metric, it then collapses, it then undergoes gravitational collapse and it limits to extreme occur. But Wall, Lewis Papapetro and extreme will occur have different spatial topologies. And so there's some, you know, in the continuum case there's some quasi stationary transition between them. And so because they have different topologies and their vial tensors have different forms, there's a sort of. There's a relaxation time that's inherent there where you know, and that's where you see emission of gravitational waves.

Let me understand the different topologies. What are those different topologies?

So for, I mean a basic one is that with the, with extremal Kerr you have these two horizons, right? You have the interior and exterior horizons. With Varuis Papapetru you only have. There's only one sort of defined surface.

But now maybe Jose wants to comment on this. I mean, I think we. Jonathan, you and I talked about this a couple of months ago as well. To what extent is this just like a bifurcation phenomenon where there was or catastrophe theory like phenomenon of one of these kind of things where there was an initial sort of singularity and then it's split in two. Is that. Jose, do you know about this, this, this case?

Not much, to be honest.

No. Okay, so Jonathan, I think we talked about it before and we had some decent physical interpretation of what's going on at this place where there is this splitting into the two, you know, the inner and outer event horizon.

Right, right.

What was the physical interpretation that we had of that? I've now forgotten.

I can't remember either. I think you were the one who came up with it.

Okay, well that sounds like my kind of thing, but I unfortunately don't remember.

But anyways, my argument is only that what we see in the initial numerical simulations of that is that the. Okay, so the frequency for things like the frequency of the gravitational radiation that gets emitted in that case depends on the discretization scale of the space time. And so it's possible that we might be able to see some, you know, there might be some observable discrepancy between the continuum and discrete cases.

Wait, wait, what. What determines the frequency? I don't understand that statement. I mean, so you're saying this gravitational radiation emitted at the transition point between the 1 and 2.

During the relaxation phase. Right. Okay, so anytime you transition between space times with different topologies and crucially with different forms of the Weyl tensor. So that includes say spinning gravitational collapse, but it also includes like you know, if you have two black holes and they collide, right. You've gone from having two punctures to having one puncture, and the Weyl tensors are different. So in that case, because you're transitioning between two different forms of the Weyl tensor, there's a sort of finite relaxation time. And you know, in the black hole collision case, that's the ring down phase.

But hold on, with respect to the Weyl tensor and so on, is this not. There's some integral over of the surface integral of the Weyl tensor that. I mean, is this one of these counting things, essentially a topological counting thing like, I don't know, churn classes or any of these other things where you're evaluating some integral over some region and depending on how many holes there are, you get a different number?

Pretty much, yeah. Actually, the framework supports that. If you take some metric tensor object and you compute its Riemann tensor, which you can do with the function repository, then there's functionality for computing, not just you can compute its chernpontriagand scalar, you can compute its Euler scalar, and so on. And if you integrate those things, then you get things like the second Churn number, you get the instant on number, you get the Euler characteristic.

Okay, but let me understand this. So you're saying you've put. The Weyl tensor, you compute the Weyl tensor. And how should I understand this? What's a simple analog of this? I mean, if I'm, if I'm going from simple homotopy type thing where I have something which, you know, like the kind of the pair of pants type situation where you have a cylinder and it breaks into two cylinders.

Right.

What's an example of a thing like what you're describing where you compute a local, an integral of a local quantity, and then you see that somehow change as a result of a change of the overall topology?

Well, if you take that exact example, if you take the cobordism, the pants cobortism example, and you were to compute the Euler characteristic of those of that overall space, which you get by taking the Euler scalar and you just integrate it over the boundary, then yeah, then you'd see a sudden transition from that being whatever two to one or something, or you know, there's some discrete transition that you would see.

Okay. And the claim is that the gravitational waves are associated with some, you know, in gravitational waves, what you're seeing is vial tensor stuff. Is that correct?

Yes. And so for something like a black hole collision, I mean Typically for an inspiral at least, there are two different. There are two different sources of gravitational radiation, right? There's the, there's the gravitational radiation from the inspiral itself. It's produced by the quadrupole moment of the, the black holes. But you also get a second source of gravitational radiation that is from the merging, and that's the ring down. And that's effectively as say, the sort of the relaxation of the vial tensor.

Okay, so, so what are you saying here? So you're saying that if.

So that that relaxation, like the, the gravitational radiation that gets emitted during that relaxation seems to depend on the. I mean, if you. So like the value that you picked here, like, you know, whether it was 2.5 or 1.5, you know, the, essentially the, the underlying discretization scale of the space time, that seems to determine some shift in the frequency that seems to induce a shift in the frequency of the emitted gravitational radiation.

Let's just try to understand that. So, and, and what is the relationship if, if I look at the discretization scale and I look at the frequencies, is it that there is. I mean, what you would naively expect is that there's a cutoff in the frequencies that is, you know, that's on the order of the discretization scale, which is absurdly small. And therefore.

Yeah, so there is a one over the discretization scale, you know, to the power three or something cut off. But you also get a, what you seem to get is some kind of one over scale squared shift in the frequency.

Shift.

I mean, like rescaling, you see a pitch shift effectively.

Okay, why is that?

I mean, I'm not sure that I have a good physical explanation.

Let's figure it out. I mean, so what's happening, I mean, in, in this co boardism thinking thing where you go from the two legs to the one trunk, so to speak, in the merger, You're saying the gravitational waves, you're saying that that very merging produces gravitational waves because the far field file tensor has to change as a result of that merge. Is that correct?

Right, exactly.

And I mean, it's like the classic reason for radiation, which is you've got a near field thing like, like the classic reason that a, you know, an accelerated charge radiates is because you've got this kind of hedgehog of field lines for the charge sitting in one place. And there's a different hedgehog of, I shouldn't call it hedgehog. I mean the, the sort of outward radial lines from another place. The radial lines have Shifted. But at infinity, you can see that there's a difference between the radial lines from position one and position two. And that's something where you have to somehow transition from being the radial lines at position one to radial lines position two. And that means you have to have a wave go out. Right. To achieve that.

Right.

And I think what you're saying is the same thing is somehow true here, that the far field Weyl tensor is different for the single black hole as compared to the two black holes. And somehow to represent that transition, there has to be a gravitational wave that goes out. Okay, and now you're claiming that the frequency of that gravitational wave, so in the case of the charges, just, you go from here to there. It's just a single sort of update. There's no wave, there's no frequency.

Right.

In the case of ring down, what makes there be something periodic as opposed to just seeing a burst of gravitational radiation?

So my, my intuition has always been that it's, it's because it's a quadrupole moment.

Right.

It's because it's kind of wobbling in two different axes.

But that's not for the ring down phase. That's for the general inspiring phase.

But even during ring down, you see, I mean, I don't know. Jose probably, probably has a better intuition for this than I.

Is Jose with us?

Yes.

Do you. Can you save us from this male storm of gravitational radiation?

It's okay, so to summarize, I think this is a bit like a bell. We have a black hole that has been formed. It's not in its final state and it has to get rid of excess of various things that. In order to relax to the final care black hole, say. And these excess things will be emitted in the form of gravitational radiation. And it's a bit like a belt. Depending on the black hole that we have formed, these things will be emitted with different frequencies. It has its own, I mean, depending on the mass of the black hole, we will have various frequencies.

Right, but. So Jonathan's making the claim that there's a pitch shift associated with discretization.

I can believe that. I mean, if we have a different structure of space time at very small scale, that will affect probably the high.

Frequencies, part of, of the absurdly high frequencies. But look, in the bell analogy, what presumably affects the frequency of the ring down of the bell is the Young's modulus and so on of the bell, that is how deformable the bell is. So the question is, what's the Young's modulus? Of space time.

Well, sometimes the black hole is compared to some elastic surface. And so we have this surface that is trying to get rid of.

Like a bell.

Exactly. If it's not exactly axisymmetric or not exactly spherical for spatial, it would try to get rid of those bumps.

Right? Exactly. But that means that there's a question. What is the Young's modulus of space.

Time or I guess more broadly, what is the equation of state for spacetime?

Well, this will be related to the surface gravity of the black hole. So in the end we only have one parameter.

Right.

Or two. It's the final parameters of the black hole that is going to.

Well, actually, but the critical point is that in our model we have another parameter which is the, the elementary length. See, so any dimensional analysis that you do is going to be blown up by the fact that there's a new, new quantity involved.

It's important to mention, by the way, actually, at least in the preliminary numerical simulations I've done, you see two effects. Yeah. One is you see the frequency shift at the high frequency, you know, the high energy end. You also see that the ringdown phase occurs earlier than it sort of should, so to speak, if you assume the continuum approximation, simply because the higher the discretization scale, the sooner the transition happens between either the inspiral phase and the ring down phase, or the collapse phase and the ring down phase.

Well, basically the point that you're making there is to know that you have a single black hole. If there is a course of discretization, the black hole becomes a single one more quickly than if there's a finite discretization. Right. You can't distinguish two black holes if they're smaller than, you know, if they're distinguished by something smaller than the discretization scale.

Right, right. And so you see, there are two, if you, if you do the simulations, there are two discrepancies that you. Two isolatable discrepancies that you see between the discrete based amphibious and the continuous case. One is just.

Yeah, sorry, did you have a plot of the. I mean, you had some kind of plot of this?

I did, but, but it's very old now and I, you know, I want to do a follow up that looks specifically at that problem.

Do we need to blast it with lots of computer time?

I mean, that would always help. I, I mean, I, look, I ran it on a. I have access to university supercomputer classes and things so that, you know, I can, I can do that. But if we, if we can do better, somehow Then, Right.

Well, I mean, okay, but yeah, we're running out of time here. But, but, but the basic point is I think I understand how these simulations work. And the key thing and our best hope seems to be, you know, some sign of what's going on based on gravitational radiation at infinity. That seems to be about the only, I think before we were talking about gaps between optical phenomena and gravitational wave phenomena. I'm not sure.

Right. I mean, so one possibility, yeah, it remains a speculative possibility, is that, you know, we have some idea of how photon propagation works in a discrete space time. We have some idea of how gravitational waves work, but neither is complete yet. But it's at least possible that in these kind of multi messenger astrophysics scenarios where you have people looking for both say neutrino signatures and gravitational wave signatures or electromagnetic signatures and gravitational wave signatures, that they'll be. That there could be observable discrepancies in the arrival times of those different, you know, mess.

Well, let's understand why. I mean, the point is a photon. What's the difference between. Well, I mean, one would normally think all the action is happening at some kind of event horizon or near some kind of event horizon. Both, whatever, you know, optical phenomenon there is of matter being accelerated to some ridiculous rate and radiating and gravitational waves being emitted. It's all, you know, in the end they're all, they appear to be being emitted from the event horizon.

Right.

Right. But I mean, one huge difference in our models is that we do have an internal model for the black hole. We don't just say, oh, you know, it's an event horizon and we can't say what's happening inside. We only have an exterior solution. We have an interior. We have a full. Am I correct in saying that, I mean that we, we in a sense have a full solution for the complete structure of the space time even inside the black hole?

Yeah. If you take a sort of, I mean again to stress, these are sort of very synthetic hypergraphy writing systems that have been engineered to mirror certain spacetime geometries. If you take a sort of homegrown hypergraphy writing system though that contains a Schwarzenegger black hole, then yes, that will, you know, or that contains a black hole, then that will be the case. There won't be some, you know, there won't be some arbitrary cutoff point where you can't talk about it anymore. Right.

So one of the questions is, is our kind of event horizon porous in a way that one in continuum space time would not be? And if that's the case, is it possible to effectively get gravitational radiation if not photons? You know, I could imagine a situation where you can't fit a photon through, but you can fit something that essentially forms itself into gravitational radiation. So what's an example? To take an extreme example, you've got a harbor with a bunch of water and you've got water waves and they're flapping around on the, on the surface of the water and you've got a boat. And you could easily imagine a situation in which you can't fit the boat through some hole in the harbor wall, but you perfectly well can get these. I mean, this is a bad example because we really need something where the, where the intrinsic structure of the thing is defined by the water as opposed to.

But I get your point. I mean like. Or maybe one way to say it would be that sort of gravitons in inverted commas in this model by, you know, by which I just mean the carriers of gravitational information are just causal edges slash events. Right. There is no. Whereas photons, probably whatever they are, big fluffy things, likely have some internal structure. And so unlike most quantum gravity models, there probably is some large size discrepancy between the photon and the graviton.

Right. Or minimal graviton is.

Right.

I mean, what aggregates to something that is a full sort of particle, like propagating graviton. Yeah. I mean, Right. There's a core to the photon that defines its quantum numbers and so on. That requires a bunch of causal edges. Whereas gravitational effects happen even down at the level of a single quantity. Causal edge.

Right. And so therefore there are presumably obstructions that will apply to photons but won't apply to gravitons.

Right. In particular the shredding of space time. There can be a situation in which the, you know, gravitational effects can propagate through, but photons cannot. And it kind of reminds me of these cyclic universe things where, which have that feature, don't they, that they, the gravitational waves can get through the, the, the crunch and other things cannot.

I think so, yeah. Certainly in like, I don't know. I've heard Penrose claim that in CCC you can, you know, you can see gravitational wave signatures from prior eons. Somehow I don't.

Well, it's probably the same phenomenon as we're talking about here, except that because any. Well, all right, this is exciting. We need, I mean, we got further to go here. We're going to find, you know, it's super embarrassing as we've, as you know, and Brownian motion was discovered 100 years before people understood it was due to the presence of molecules. And I think we've discussed previously that it's quite possible that there's an astrophysical phenomenon sitting out there that's been around for a really long time that reveals the discreteness of space time and we just don't know it yet. And obviously the best candidate for that will be dark matter. And perhaps we should discuss a different time, the use of these simulations to ask questions. I mean, what would it take in this kind of simulation strategy? You know, we're dealing with the extreme case of black holes. What would it take to get not such an extreme case, but rather just a standard, you know, big black hole at the center of a galaxy rotation curve type thing.

Well, I mean for something like, I mean you'd probably still want to use Schwarzschild or something like, you know, like Schwarzschild is a good metric even for non black hole cases. Right. Just for massive objects.

I mean the rotation curve of a galaxy, we can naively we could just look at orbits in the Schwarzschild solution.

Right. And, and my, I, I don't, I don't know a huge amount about galactic astronomy. My understanding is that's a pretty good approximation. Like the, the, you know, the, the galactic cores are some often slowly rotating, sufficiently slowly rotating that Schwarzschild is a good approximation.

Oh yeah, I'm sure. But I mean there's also a bunch of. Look, there's only 10 to the 5 solar masses right at the center of our galaxy. And, but there's on the order of 10 to the 11 solar masses in the whole galaxy. Galaxy is not all ended up in the garbage can yet.

But yes, anyway, I mean, and there are again this functionality for doing. I mean it's actually quite nice. There's functions like discrete hypersurface geodesic which is again in the GitHub repository, but not yet in WFR, which essentially reduce solving the geodesic equation to solving for know graph G86 in the, in the hypergraphs. And so you see we can easily.

That's nice. So we can get the orbits.

Yeah, exactly.

But, but I mean the phenomenon that we're looking for is that there is larger effective gravity associated with the, with the external orbits than you would expect from the matter.

Right.

And the question that we could ask then we don't really have a great definition of matter. The only definition of matter we have is solve the Einstein equations and see what's there.

Right.

Well, although we do well we do have a definition of matter which has to do with. See, I know it's not clear because I mean, dark matter could be as effective in, you know, if we look at the flux of causal edges that could perfectly well include dark matter.

Yeah, I mean, my basic position about this is that, you know, okay, as in, in the true spirit of the equivalence principle, you know, the distinction between what is matter and what is space is not a particular, is, you know, is not defined a priori. Right. That, that's something which we have the, you know, we have the, the freedom to choose. And so, you know, we, whatever the dark matter story is in the Wolfram model case, my guess is that it's just something about, you know, there's some feature of space that we have, you know, we have chosen to interpret as a matter distribution, but it's really, you know, it's ultimately just some feature of space.

Right, yes. Or alternatively put, there's a zero in some kind of normalization renormalization story. There's a definition of what space is that is slightly off.

Right. But yeah, in a sense, the fact that there's no a priori way of saying this bundle of hyper edges is matter, this bundle of hyper edges is background space is precisely the statement that you can switch any positive term in the Einstein tensor to a negative term in the stress energy tensor and vice versa.

And also our expectation that the vast majority of causal edges are associated with knitting together the structure of space, not with providing the things that we identify as matter.

Right, right. Yeah. And that makes your questions about equations of state of space, time and so on, I think, quite well defined, even if you can't. Even if we don't know how to answer those yet.

Right. All right. Time to.

On a very mundane level though, I mean, one of the things I wanted to get out of this is just a sense of, is the interface to all this functionality with setting up symbolic metric tensors and symbolic tensors and, and plugging these into discrete hypersurface decomposition. Does that interface sort of make sense?

More or less. But you know, I have to say I feel like this whole thing is very much at the research phase and the uses are going to be leading edge research kinds of things rather than sort of, sort of industrial production type stuff. And so I think that the fact that you might have to read the documentation to figure out what's going on is probably less important.

Fair enough. I mean, my hope would be that at least the symbolic side, like the, at least the non numerical sides, might be useful. You Know, just as we've seen with the quantum framework being used in teaching and stuff, that it might potentially be useful.

No, I agree. I mean, this clarifies a lot of things about how things work in the. I think this needs another iteration for that. But I think that, that I mean, these things always. I mean, even my own little understanding of ADM and things like this is greatly helped by sort of concretizing these things in terms of functions and so on. And I think other peoples will be too. I mean, being able to see that, being able to see what it means. I mean, asking questions like what does energy mean in the, you know, in general relativity, we could probably compute that from all these things explicitly and see that, oh, there's a unknown 0 and etc. Etc. Etc. Which seems useful. I have got to go actually to another live stream, but this was a fine discussion and to be continued. And let's go find Brownian motion in space time. But I think we've still got a few more iterations to go. I think we should, we should talk about this thing about GD6 orbits defined by GD6 and how that relates to what is merely knitting together the structure of space versus. I mean, really what we want is something where, where there's some kind of overrun from knitting together the structure of space. It's, it's something kind of. Not like gravitational magnetism. It's some. You know what I bet it is? I bet there's something like viscosity in spacetime just like you, you can get. Or some effect that goes beyond. In other words, what we're, you know, the question is, is the identifiable mass in the galaxy all leads to gravity.

I mean, the, the presence of viscosity is. Should be fairly easy to test. Right. Because all we have to do is check whether the, you know, the continuum equations that describe the propagation of gravitational radiation, whether they are purely hyperbolic, as they are in the continuum case, in the, you know, in the normal continuum case, or whether they have a parabolic damping.

Yeah. Okay, so it probably. There isn't ordinary viscosity because that would actually damp out gravitational waves, right?

Yeah, you'd see gravitational wave dispersion, which I don't think is observed.

No, it is not. I mean, so far as. Well, at least the, the. Actually you just see damping. It depends on whether it's both viscosity, shear viscosity, what the kinds of. And gravitational waves anyway, are differently classified because they're neither long. You know, they have quadrupole deformations. Etc. Etc. Etc. All right, I better go to my other thing, but let's continue this, hopefully sooner rather than later because this was fun, exciting, and I think we're getting somewhere.

Sounds good. Thanks.

All right, thanks to folks on the live stream. Thank you, Jonathan, Jose, et cetera. See you all soon. Bye for now.