I Nash embedding theorem & curved spacetime

[BWV]

Curious, is there any useful reason to translate the 4d curved Lorentzian manifold in GR to, if i read this right, either a 46 or 230 dimensional flat Euclidian space, depending whether the manifold is compact or not? (although another source listed a 39 dimensional flat embedding).

( from https://en.wikipedia.org/wiki/Nash_embedding_theorem)
The technical statement appearing in Nash's original paper is as follows: if M is a given m-dimensional Riemannian manifold (analytic or of class Ck, 3 ≤ k ≤ ∞), then there exists a number n (with n ≤ m(3m+11)/2 if Mis a compact manifold, or n ≤ m(m+1)(3m+11)/2 if M is a non-compact manifold) and an injective map ƒ: M → Rn(also analytic or of class Ck) such that for every point p of M, the derivative dƒp is a linear map from the tangent space TpM to Rn which is compatible with the given inner product on TpM and the standard dot product of Rn in the following sense:

for all vectors u, v in TpM. This is an undetermined system of partial differential equations

 

[Orodruin]

A Lorentzian manifold is not Riemannian (it is pseudo-Riemannian).

 

[BWV]

Yes but most of the theorems that apply to Riemannian manifolds work on Lorentzian manifolds, including this one - the reference for the theorem was in an introductory book on GR

 

[Orodruin]

No it doesn’t. It is quite obvious that you cannot embed a Lorentzian manifold into Euclidean space as the induced metric would be positive definite and therefore Riemannian and not pseudo-Riemannian.

That being said, the corresponding thing for a pseudo-Riemannian manifold would be to embed it into a pseudo-Riemannian affine space. Regardless, this in itself has no bearing on the actual physics of GR.

the reference for the theorem was in an introductory book on GR

Please give actual references instead of vague descriptions of where you are getting information from.

 

[martinbn]

I think there are analogs of Nash theorem for isometric embedding of Lorentzian manifolds in Minkowski spacetime, but why would you think that it would be useful?

 

[BWV]

This is the reference

“According to this theorem(Nash) , 4-dimensional curved spacetime can be isometrically embedded in a flat spacetime of 39 dimensions or less1.”

— Essential Differential Geometry: The Langauge of General Relativity (Fiat Lux) by Naveen Balaji Umasankar, Sujan Kumar S

i have no idea whether it would be useful, which is why i asked the question in the OP

 

[martinbn]

This is the reference

“According to this theorem(Nash) , 4-dimensional curved spacetime can be isometrically embedded in a flat spacetime of 39 dimensions or less1.”

— Essential Differential Geometry: The Langauge of General Relativity (Fiat Lux) by Naveen Balaji Umasankar, Sujan Kumar S

They are being sloppy. Nash's theorem is specific to the Riemannian case. For any extension they need to give a different reference.

i have no idea whether it would be useful, which is why i asked the question in the OP

But why do you think that it might be useful? Otherwise why ask the question? For instance you are not asking the same question about Whitney's theorem. Is it useful to view manifolds as submanifolds in $\mathbb R^N$?

 

[BWV]

here is another source: https://arxiv.org/pdf/0812.4439.pdf

Abstract. In this article, the Lorentzian manifolds isometrically embeddable in LN (for some large N, in the spirit of Nash’s theorem) are characterized as a a subclass of the set of all stably causal spacetimes; concretely, those which admit a smooth time function τ with |∇τ| > 1. Then, we prove that any globally hyperbolic spacetime (M,g) admits such a function, and, even more, a global orthogonal decomposition M = R × S, g = −βdt2 + gt with bounded function β and Cauchy slices.
In particular, a proof of a result stated by C.J.S. Clarke is obtained: any globally hyperbolic
N
Keywords: causality theory, globally hyperbolic, isometric embedding, conformal embedding 2000 MSC: 53C50, 53C12, 83E15, 83C45.
spacetime can be isometrically embedded in Minkowski spacetime L . The role of the so- called “folk problems on smoothability” in Clarke’s approach is also discussed.

so given that its feasible, my OP was simply asking if there was any practical reason to do this, such as perhaps to make some computation easier

 

[Auto-Didact]

A consequence of the generalized Nash embedding theorem is that any 4D Lorentzian manifold can be represented as a subspace of a Minkowski space of 231 dimensions.

A reason for doing this would be to avoid explicitly dealing with curvature i.e. it treats the curved problem using simpler mathematics, but whether doing this actually makes the calculation easier is another question altogether.

 

[Orodruin]

whether doing this actually makes the calculation easier is another question altogether.

In my experience it just complicates things (even if you are just considering simpler things such as 4D de Sitter embedded in 5D Minkowski).

 

[Auto-Didact]

I think the utility depends on the specific application, i.e. usually GR or some difficult problem in some field of research, where the issue that they are dealing with can best be classified as belonging to applied mathematics.

If one is able to write a simple program, has a large amount of computing power, no experience in differential geometry and one is just simply trying to churn out answers numerically, then I can certainly see the advantage; in my experience, this profile applies to the large majority of data scientists and graduate students that pursue such applied problems.

 

[PAllen]

A consequence of the generalized Nash embedding theorem is that any 4D Lorentzian manifold can be represented as a subspace of a Minkowski space of 231 dimensions.

A reason for doing this would be to avoid explicitly dealing with curvature i.e. it treats the curved problem using simpler mathematics, but whether doing this actually makes the calculation easier is another question altogether.

Well, as stated, this is false, as noted in the referenced paper. For an arbitrary, 4-D pseudo-Riemannian manifold (with 1 minus sign, s=1 in the parlance of the paper) in the signature, you cannot embed in any higher dimensional Minkowski space (taking Minkowski space to have s=1, as done in the paper). The reason is simply CTC cannot be smoothly isometrically embedded in such a case. You have to allow the embedding target flat manifold to have s > 1, thus not be just a higher dimensional Minkowski space. The main thrust of the referenced paper is then to identify precisely the subset of 4-d pseudo-Riemannian manifolds that can be embedded in higher dimensional Minkowski space (i.e. keeping s=1).

 

[martinbn]

I think the utility depends on the specific application, i.e. usually GR or some difficult problem in some field of research, where the issue that they are dealing with can best be classified as belonging to applied mathematics.

If one is able to write a simple program, has a large amount of computing power, no experience in differential geometry and one is just simply trying to churn out answers numerically, then I can certainly see the advantage; in my experience, this profile applies to the large majority of data scientists and graduate students that pursue such applied problems.

The embbeding is not going to be given explicitly, so what calculations can be done just because an embbeding exists.

 

[Auto-Didact]

Well, as stated, this is false, as noted in the referenced paper. For an arbitrary, 4-D pseudo-Riemannian manifold (with 1 minus sign, s=1 in the parlance of the paper) in the signature, you cannot embed in any higher dimensional Minkowski space (taking Minkowski space to have s=1, as done in the paper). The reason is simply CTC cannot be smoothly isometrically embedded in such a case. You have to allow the embedding target flat manifold to have s > 1, thus not be just a higher dimensional Minkowski space. The main thrust of the referenced paper is then to identify precisely the subset of 4-d pseudo-Riemannian manifolds that can be embedded in higher dimensional Minkowski space (i.e. keeping s=1).

This is almost completely irrelevant; you are focussing too much on particulars (e.g. pseudo-Riemannian, having CTCs) and not on the general content of the theorem which goes far beyond this direct implementation with which you find issue; apart from CTC's being unphysical and therefore de facto uninteresting in the context of analysis for instability reasons, the particular technicality you have a problem with can trivially be transformed away for all practical purposes by applying a sequence of conventional applied mathematics techniques.

The core content of the embedding theorems is that the intrinsic view of manifolds from non-Euclidean geometry and the extrinsic view of these manifolds being embedded in a high dimensional Euclidean space are in fact equivalent; this is an extremely far-reaching unification in geometry and mathematics more generally for which Nash is not nearly celebrated enough. The unification occurs through the form of the embedding, namely PDEs, making this a unification of intrinsic and extrinsic geometry through objects belonging to the theory of analysis.

The proof depends on a particular mixture of classical pure mathematical concepts leading to a miraculous answer: the embedding - a geometric concept - is done using a large system of equations - essentially algebra - which have the certain property that the derivatives tend to vanish, based on the differentiability class of the equations - i.e. a topic belonging to analysis - while simultaneously using a rapid convergence technique - numerical analysis - to counteract the rate of vanishing of the derivatives.

In modern terminology this isn't a topic of pure mathematics anymore (NB: due to formalism having become the dominant school of thought among pure mathematicians) but instead the topic has gotten the reputation of being a topic belonging to applied mathematics (NB: because applied mathematicians now work on these problems), namely it is a geometric study of the analytic properties of a large system of nonlinear differential equations, i.e. doing geometry, by doing algebra, by doing analysis, both analytic and numerical at the same time. This field of research is called nonlinear dynamical systems theory and it unifies almost all areas of sophisticated applied mathematics in the most natural manner imaginable.

The embbeding is not going to be given explicitly, so what calculations can be done just because an embbeding exists.

The embedding theorem is an existence proof for certain nonlinear PDEs, which can actually be used to construct solutions to other nonlinear PDEs if used correctly in conjunction with a host of different techniques and methodologies - KAM theory, Morse theory, bifurcation theory, non-perturbative analysis, numerical analysis, the method of characteristics, integral transforms and so on; different combinations of these are used to construct an ansatz and go from there.

The applications of this to all the sciences - both natural and social - seems almost limitless both directly for mathematical modelling and indirectly for empirical analysis, especially given that topological data analysis is slowly but steadily becoming a more common tool in data science. Indeed, in the words of Nash (1958) himself: The open problems in the area of non-linear partial differential equations are very relevant to applied mathematics and science as a whole, perhaps more so than the open problems in any other area of mathematics, and this field seems poised for rapid development. It seems clear, however, that fresh methods must be employed.

 

[PAllen]

This is almost completely irrelevant; you are focussing too much on particulars (e.g. pseudo-Riemannian, having CTCs) and not on the general content of the theorem which goes far beyond this direct implementation with which you find issue; apart from CTC's being unphysical and therefore de facto uninteresting in the context of analysis for instability reasons, the particular technicality you have a problem with can trivially be transformed away for all practical purposes by applying a sequence of conventional applied mathematics techniques.

The core content of the embedding theorems is that the intrinsic view of manifolds from non-Euclidean geometry and the extrinsic view of these manifolds being embedded in a high dimensional Euclidean space are in fact equivalent; this is an extremely far-reaching unification in geometry and mathematics more generally for which Nash is not nearly celebrated enough. The unification occurs through the form of the embedding, namely PDEs, making this a unification of intrinsic and extrinsic geometry through objects belonging to the theory of analysis.

The proof depends on a particular mixture of classical pure mathematical concepts leading to a miraculous answer: the embedding - a geometric concept - is done using a large system of equations - essentially algebra - which have the certain property that the derivatives tend to vanish, based on the differentiability class of the equations - i.e. a topic belonging to analysis - while simultaneously using a rapid convergence technique - numerical analysis - to counteract the rate of vanishing of the derivatives.

In modern terminology this isn't a topic of pure mathematics anymore (NB: due to formalism having become the dominant school of thought among pure mathematicians) but instead the topic has gotten the reputation of being a topic belonging to applied mathematics (NB: because applied mathematicians now work on these problems), namely it is a geometric study of the analytic properties of a large system of nonlinear differential equations, i.e. doing geometry, by doing algebra, by doing analysis, both analytic and numerical at the same time. This field of research is called nonlinear dynamical systems theory and it unifies almost all areas of sophisticated applied mathematics in the most natural manner imaginable.

What you wrote is wholly irrelevant to what I wrote, and there is no plausible reasoning by which you could construe any disagreement (or agreement - thus irrelevant) between what I wrote and what you wrote.

I corrected a statement that, as given, was false. My reply itself included two ways it could be made correct: allowing embedding in flat manifolds more general than Minkowski, that is s > 1. Alternatively, restricting the 4-D pseudo-Riemannian manifolds to those that are globally hyperbolic (that's the gist of the paper linked a few posts before mine).

A non-disngenuous reply could have been simply: "Ok, but I consider only globally hyperbolic spacetimes to be physically reasonable". To which I mostly agree.

Also, I mostly agree with your irrelevant post as to its math/physics content, but strongly disagree with its schematic imputation of motives and opinions to large groups mathematicians.

 

[Auto-Didact]

What you wrote is wholly irrelevant to what I wrote, and there is no plausible reasoning by which you could construe any disagreement (or agreement - thus irrelevant) between what I wrote and what you wrote.

There is; I simply changed the context of the discussion on a metacommunicative level by going back to the OP's question, framing the fundamental relevance of the embedding theorems in spite of any contingent counterarguments and then giving an answer to the OP, namely of utility in general in terms of actual applied mathematics based on an historico-sociological descriptive analysis specifically described in such a fashion which has pedagogical value for those unfamiliar with the material.

I corrected a statement that, as given, was false. My reply itself included two ways it could be made correct: allowing embedding in flat manifolds more general than Minkowski, that is s > 1. Alternatively, restricting the 4-D pseudo-Riemannian manifolds to those that are globally hyperbolic (that's the gist of the paper linked a few posts before mine).

This counterargument is formally correct, but as I say contingent and therefore may actually be disregarded depending on what the goal of the discussion is: in the context of the OP's question this clearly seems to fall in the category "statistically significant, but not practically relevant"; for anyone enquiring into this topic to choose to stop here would be losing sight of the forest because of the trees.

Disregarding the utility of a technique or method based on such contingent formal counterarguments tends to be a highly premature decision, based purely on a cognitive bias, typically of wanting to conform to strict formal rigour, when such strictness may be unwarranted at this stage of inquiry; this is an instance of logical positivist thinking occurring in the practice of mathematics, and is a widely studied and documented phenomenon.

In the practice of almost any subject - including (applied) mathematics - taking the logical positivist attitude w.r.t. some topic is in the bigger picture almost always the incorrect decision to make - or even stronger an incorrect frame of mind to have - which only hampers finding solutions or actually understanding the topic more in depth, apart from having a mere surface level inconsistency-free procedural and declarative knowledge.

A non-disngenuous reply could have been simply: "Ok, but I consider only globally hyperbolic spacetimes to be physically reasonable". To which I mostly agree.

Also, I mostly agree with your irrelevant post as to its math/physics content, but strongly disagree with its schematic imputation of motives and opinions to large groups mathematicians.

The intention is not to be disingenuous; instead time is limited and therefore on the battlefield of online discussion, instead of addressing all points directly by addressing their content as if one were doing homework or writing a report - which would take a lot of time - instead properly sectioning off and framing targets, making directed calculated attacks and then getting back to the core of the discussion tends to be a better strategy, with the explanation only coming later if deemed necessary.

In any case, glad to hear we agree more than we disagree. Please excuse my paroxysmal ascerbic tone; it is not meant to be personal but instead meant to encourage or discourage others reading w.r.t. what is actually of priority in the context of this discussion, both w.r.t. OP's inquiry as well as in actual practice.

 

[PAllen]

...

The intention is not to be disingenuous; instead time is limited and therefore on the battlefield of online discussion, instead of addressing all points directly by addressing their content as if one were doing homework or writing a report - which would take a lot of time - instead properly sectioning off and framing targets, making directed calculated attacks and then getting back to the core of the discussion tends to be a better strategy, with the explanation only coming later if deemed necessary.

...

And this gets to the core of our different approach - you think in terms of battle, targets, attacks.

I prefer mutually respectful discussion.

[edit: I bolded something crucial: you don’t want to address the content of what someone says, by your own words. Instead, some overall battle is more important. I feel PF is all about discussing content, not agenda and purported motivations ]

 

[PeterDonis]

I simply changed the context of the discussion on a metacommunicative level...

Words, words, words...

The intention is not to be disingenuous

What you intended is not the point. You are hijacking someone else's thread with argumentative irrelevancies. You are therefore banned from further posting in this thread.