How to know what maximal atlas to use for spacetime manifold?

[Shirish]

I'm studying Liang's book on differential geometry for general relativity. There's a para in it talking about a maximal atlas - "Later on, when we talk about a manifold, we always assume that the largest possible atlas has been chosen as the differentiable structure, so that we can perform any coordinate transformation." A couple of questions based on this:

In the context of GR, spacetime is a 4D differentiable manifold. Is there some process of explicitly specifying the maximal atlas that we use? i.e. is there some widely (or even universally) agreed upon maximal atlas/differentiable structure that we use for general relativity?

If not, and if there are multiple possible maximal atlases, how do we choose which one to use? Is it dependent on what phenomena we're capturing or what problem we're trying to solve?

 

[pasmith]

The differentiable structure is determined by the metric.

A maximal atlas is an equivalence class of atlases, two atlases being equivalent if every chart of one atlas is smoothly compatible with every chart of the other atlas. Given an atlas $\mathcal{A}$, the atlas $$\bigcup_{\mathcal{B} \in [\mathcal{A}]} \mathcal{B}$$ is the largest possible atlas equivalent to $\mathcal{A}$.

The author is saying that once the differentiable structure is fixed by the metric, one can use any chart that is smoothly compatible with that structure.

 

[Shirish]

The differentiable structure is determined by the metric.

A maximal atlas is an equivalence class of atlases, two atlases being equivalent if every chart of one atlas is smoothly compatible with every chart of the other atlas. Given an atlas $\mathcal{A}$, the atlas $$\bigcup_{\mathcal{B} \in [\mathcal{A}]} \mathcal{B}$$ is the largest possible atlas equivalent to $\mathcal{A}$.

The author is saying that once the differentiable structure is fixed by the metric, one can use any chart that is smoothly compatible with that structure.

Thank you! So then physics-wise, the problem we're trying to solve or the object we're trying to study determines what metric we'll use - so indirectly it will determine the differentiable structure?

I say the above since I've heard different kinds of metric being used to study different kinds of black holes. For example near an uncharged, non-rotating black hole, the Schwarzchild metric is used - this induces a certain differentiable structure. For an uncharged but rotating black hole, the Kerr metric is used and that'll induce a different structure.

But doesn't this make the differentiable structure imposed on a spacetime manifold ambiguous? Seems like it's of one type near one kind of black hole and of another type near other kinds of black holes. I mean, what kind of differentiable structure do we actually assign to spacetime as a global 4-D manifold?

 

[pasmith]

The metric is found by solving the Einstein equation given the stress-energy tensor. Near a massive object one can make the simplifying assumption that the effects from any other massive objects are negligible and can be ignored.

 

[PeterDonis]

I've heard different kinds of metric being used to study different kinds of black holes.

Yes, because different kinds of black holes have different spacetime geometries.

doesn't this make the differentiable structure imposed on a spacetime manifold ambiguous?

No, it means that different kinds of black holes (and more generally different kinds of objects) have different spacetime geometries. To figure out the spacetime geometry of a particular black hole (or object), you need to measure it. Measuring it tells you the specific metric for that specific object, which in turn tells you the specific differentiable structure for that specific geometry.

what kind of differentiable structure do we actually assign to spacetime

The correct kind for the particular geometry of spacetime. So if you want to know the differentiable structure of the particular spacetime that is our universe, you need to measure its geometry.

 

[Ibix]

I think the point is that a coordinate chart covers some part of a manifold, a coordinate atlas is one or more charts that together cover the entire manifold, and the maximal atlas is every possible chart covering any part of the manifold.

Carroll comments that the utility of this is that we don't want to think of "flat Euclidean space with Cartesian coordinates" as different from "flat Euclidean space with polar coordinates". A maximal atlas includes both coordinate systems with every choice of origin, orientation and scale, and every other possible scheme to label the points on a plane too. So we think of a manifold as coming with a list of every possible coordinate scheme (even if we mostly only use one or two of them).

 

[Shirish]

Yes, because different kinds of black holes have different spacetime geometries.No, it means that different kinds of black holes (and more generally different kinds of objects) have different spacetime geometries. To figure out the spacetime geometry of a particular black hole (or object), you need to measure it. Measuring it tells you the specific metric for that specific object, which in turn tells you the specific differentiable structure for that specific geometry.The correct kind for the particular geometry of spacetime. So if you want to know the differentiable structure of the particular spacetime that is our universe, you need to measure its geometry.

So then the wrong assumption on my part is that there should be a global differentiable structure for the entirety of the spacetime manifold? Based on what you and @pasmith said, here's what I'm thinking and you can correct me wherever I'm wrong:

Whole spacetime is modeled as a 4D differentiable manifold. A 4D differentiable manifold is supposed to have a smooth atlas covering it. Depending on the mass energy content in some part of that manifold, we choose to model that part of spacetime as a manifold with a differentiable structure given by the appropriate metric. It's incorrect to say, "Near this black hole, spacetime has so-and-so differentiable structure, so everywhere else also it should have the same differentiable structure!" - since in this case I'm mistaking a mathematical model of spacetime for its global physical characteristic.

 

[PeterDonis]

So then the wrong assumption on my part is that there should be a global differentiable structure for the entirety of the spacetime manifold?

I don't know how you got that from what I said.

My point is that when you say "the" spacetime manifold, what do you mean? There is no single "spacetime manifold". There are an infinite number of different possible spacetime manifolds that are solutions of the Einstein Field Equation. Each different manifold has a different metric and therefore a different differentiable structure. So you can't just say "the" metric and "the" differentiable structure of "the" spacetime manifold. You have to first pick which of the infinite number of different possible spacetime manifolds you want to talk about.

Whole spacetime is modeled as a 4D differentiable manifold.

Each different spacetime (different possible solution of the Einstein Field Equation) is a different 4D differentiable manifold.

A 4D differentiable manifold is supposed to have a smooth atlas covering it.

Yes, a different atlas for each different 4D differentiable manifold.

Depending on the mass energy content in some part of that manifold, we choose to model that part of spacetime as a manifold with a differentiable structure given by the appropriate metric.

Not just "some part"--the entire manifold. If different parts of the spacetime have different stress-energy content, that information will dictate the corresponding solution of the Einstein Field Equation.

It's incorrect to say, "Near this black hole, spacetime has so-and-so differentiable structure, so everywhere else also it should have the same differentiable structure!"

Differentiable structures are local, not global, just like the Einstein Field Equation. A spacetime manifold can have a different metric and hence a different differetiable structure at each point. Indeed, even things that are normally described as a single "solution" of the Einstein Field Equation, for example a Schwarzschild black hole with a particular mass $M$, does not have the same metric and the same differentiable structure at every point: the metric depends on the $r$ coordinate, so it is different at different values of $r$, and hence so is the differentiable structure. There is no global metric or differentiable structure.

 

[Shirish]

My point is that when you say "the" spacetime manifold, what do you mean? There is no single "spacetime manifold". There are an infinite number of different possible spacetime manifolds that are solutions of the Einstein Field Equation. Each different manifold has a different metric and therefore a different differentiable structure. So you can't just say "the" metric and "the" differentiable structure of "the" spacetime manifold. You have to first pick which of the infinite number of different possible spacetime manifolds you want to talk about.

This I didn't know, and this hasn't been evident to me so far from what I've read in the texts. So thanks for clarifying this.

Differentiable structures are local, not global, just like the Einstein Field Equation. A spacetime manifold can have a different metric and hence a different differetiable structure at each point. Indeed, even things that are normally described as a single "solution" of the Einstein Field Equation, for example a Schwarzschild black hole with a particular mass $M$, does not have the same metric and the same differentiable structure at every point: the metric depends on the $r$ coordinate, so it is different at different values of $r$, and hence so is the differentiable structure. There is no global metric or differentiable structure.

This is also useful and also wasn't evident to me. The book quote I gave in the OP seems to imply an equivalence between atlas and differentiable structure. I thought, an atlas is specified for the entire manifold and is hence a global property. And since it was seemingly equivalent to differentiable structure, I thought the latter was also a global property

 

[PeterDonis]

I thought, an atlas is specified for the entire manifold and is hence a global property.

An atlas is a set of coordinate charts. A coordinate chart covers an open neighborhood, which is not necessarily the entire manifold. Every point in the manifold must be covered by at least one chart in the atlas, but it is not at all necessary that every chart must cover every point.

since it was seemingly equivalent to differentiable structure

No, it isn't. Differentiable structure can be specified in a coordinate-free manner, in terms of the covariant derivative operator. The book you are using chooses to use coordinate charts, but that is a choice, not a necessity.

 

[vanhees71]

A differentiable manifold is always defined by charts and atlasses. This scheme however makes this definition "coordinate-free" since the choice of coordinates in arbitrary charts covering open neighborhoods of any point is irrelevant, because the compatibility condition for overlapping charts builds an equivalence relation, and the abstract manifold is defined modulo the corresonding equivalence classes.

It should however be clear that the central discovery by Einstein is that there is not a predefined manifold that describes "spacetime" as in Newtonian or special-relstivistic physics but that it depends on the energy-momentum-stress content (described by continuum-mechanics for the matter and classical electromagnetism for the electromagnetic field and their interaction), i.e., the energy-momentum-stress tensor of matter and radiation on the right-hand side of Einstein's field equation.

That makes it so difficult to find analytic exact solutions, i.e., you have to solve the non-linear equations of motion for the matter, radiation, and the space-time-pseudometric (usually called "metric" by physicists) simultaneously. This usually is only possible for particularly symmetric situtations like the Schwarzschild solution, describing a spherically symmetric (and thus automatically static) compact object (or a black hole with a singularity), the Friedmann-Lemaitre-Robertson-Walker metric, describing a homogeneous, isotropic ideal fluid filling the entire universe (the description of the universe "coarase-grained" over sufficiently large regions) etc.

 

[Ibix]

My point is that when you say "the" spacetime manifold, what do you mean? There is no single "spacetime manifold". There are an infinite number of different possible spacetime manifolds that are solutions of the Einstein Field Equation.

Just for clarity, @Shirish, there is exactly one spacetime manifold that correctly describes our universe (to the extent that GR is a correct theory, anyway). That might legitimately be called "the" spacetime manifold. We don't know it exactly because we don't know the details of every last piece of matter in the universe.

There are a lot of different manifolds that are solutions to Einstein's field equations. Many of these are interesting because they are mathematically tractable and are good approximations to some part of reality. For example, all the text book black hole solutions like Schwarzschild, Kerr, etc, describe eternal black holes in otherwise empty universes. Like the Coulomb field in electromagnetism you will never see a true example in reality, but they are good enough for working out many things.

So when we talk about "the spacetime manifold", if we are talking about black holes we probably mean the Schwarzschild or Kerr manifold; if we are talking cosmology we mean the FLRW manifold. All of these are only models, and all have different structures and different maximal atlases.

 

[Shirish]

Just for clarity, @Shirish, there is exactly one spacetime manifold that correctly describes our universe (to the extent that GR is a correct theory, anyway). That might legitimately be called "the" spacetime manifold. We don't know it exactly because we don't know the details of every last piece of matter in the universe.

There are a lot of different manifolds that are solutions to Einstein's field equations. Many of these are interesting because they are mathematically tractable and are good approximations to some part of reality. For example, all the text book black hole solutions like Schwarzschild, Kerr, etc, describe eternal black holes in otherwise empty universes. Like the Coulomb field in electromagnetism you will never see a true example in reality, but they are good enough for working out many things.

So when we talk about "the spacetime manifold", if we are talking about black holes we probably mean the Schwarzschild or Kerr manifold; if we are talking cosmology we mean the FLRW manifold. All of these are only models, and all have different structures and different maximal atlases.

Thank you! This is very informatively written in an easy to understand way.