Probably there is no explanation. When I speak of a group of transformations that preserve the volume S2xT2 , I mean that the group of linear transformations that preserve the area of the torus is isomorphic to SO(1,1), and the group of transformations that preserve the area of the sphere is isomorphic to SO(3) and therefore we have the group SO(1,3).

I'm not sure how you get SO(1,3) out of those. There seems to be an implication in what you are saying that the direct product SO(1,1) x SO(3) is isomorphic to SO(1,3). This can't be the case. SO(1,3) has 6 generators. SO(1,1) has 2 generators and SO(3) has 3, giving 5 generators for the direct product and its minimal embedding would be in the group SO(1,4).

Tom Lawrence, Firstly, the group SO(1,1) is one-parameter, that is, it has one generator, and secondly, my message does not mention the direct product of the groups SO(1,1) x SO(3) . As an illustrative example, we can refer to a group that preserves the torus area S1xS1, since it is easy to notice that SO(1,1) not equal SO(1)xSO(1) , where SO(1) is a group that preserves the lengths of these circles.

On the first point, apologies - I was thinking "one generator" but somehow I put 2. This means that the direct product has 4 generators, so still can't be isomorphic to SO(1,3).

This brings me to your second point. You've put " the group of linear transformations that preserve the area of the torus is isomorphic to SO(1,1), and the group of transformations that preserve the area of the sphere is isomorphic to SO(3) and therefore we have the group SO(1,3)". What do you mean by "therefore we have the group SO(1,3)"? If you're not combining SO(1,1) and SO(3) in a direct product, how are you combining them to get SO(1,3)?

Tom Lawrence, If we take an arbitrary pseudo-Euclidean plane in the Minkowski space, then by mapping the isotropic lines on the circles we can easily map it to a torus, but since the space-like unit vector of the pseudo-Euclidean plane of the Minkowski space lies (more accurately draws or runs) on the sphere (if hyperbolic rotation is excluded), the Minkowski space itself is mapped onto the product of the torus and the sphere. Thus, the Lorentz group arises due to the transformational properties of the Minkowski space, and therefore it is not necessary to combine the subgroups into a direct product.

Ah I think I start to understand what you are talking about. Let me see if I've got this correct. The isometry group on a (flat) Minkowski space contains the Lorentz group - that is, vector lengths are invariant under this group. You are then constructing a map from, for example, the (x,t)-plane in this space to a torus and a map from its complementary subspace, the (y,z)-plane, to a sphere. Is that right? And then you are saying that the isometry groups for those spaces - the groups which preserve the lengths of vectors tangent to those spaces - are SO(1,1) and SO(3). And you're asking how the action of SO(1,3) induced on S2xT2 by the maps relates to the SO(1,1) and SO(3) transformations?

Something like this, but I'm not talking about local symmetry groups (nothing to do with tangent vectors), we are talking about global symmetry groups, for example, a group that preserves the area of a sphere, this is a group of transformations that preserve its radius.

What is the action of these groups? That is, what are they acting on, and how do you represent this action mathematically? And following this, how do you define area-preserving?

We have already talked about the action of the group preserving the area of ​​the sphere — these are the movements of Euclidean space, since the movements preserving the radius of the sphere coincide with the movements preserving the Euclidean length of the vector. As for the group preserving the area of ​​the torus, these are motions of the pseudo-Euclidean plane, since the motions preserving the product of the lengths of the defining circles coincides with the hyperbolic rotations (due to the fact that (x-t)(x+t)=x2-t2). Now about a group that preserves the volume of a product of a sphere by a torus. We presume that it is generated by successive movements that preserve the areas of the sphere and torus separately. Then, using the product of the Euclidean rotation and the hyperbolic rotation, we obtain the Lorentz group.

Thanks. It sounds like you're much more familiar with the geometry and transformations of T2 than I am. It's not something I really have much knowledge about at all. So here's the best response I can give you.

To start with, note that the action of SO(3) on S2 can be defined without reference to a three-dimensional space: it need not have any "interior" for us to understand its geometry. However, embedding it in R3 makes the analysis much clearer. The action of any one-parameter subgroup of SO(3) on any chosen point on S2 traces out a great circle. The action of the group as a whole on any chosen point traces out the entire surface. (That is, every point on the surface is contained in a single orbit under this action.)

As I've suggested above, what you are asking is essentially what is the induced action of the Lorentz group on S2xT2 under a map from Minkowski spacetime to this Cartesian product manifold. The mathematical machinery you need for this is that of embeddings.

Part of the map you are considering is a map from an R2 subspace of the Minkowski spacetime (e.g. the (y,t) plane) to S2. Now bear in mind that if you are putting coordinates on S2 you are constructing a map from S2 to R2. There are an infinite number of coordinate systems, so there are an infinite number of such maps. What you are seeking to do is the reverse - essentially constructing the inverse of one of these maps.

My notes here should help you a lot:

https://www.researchgate.net/project/Non-linearly-realised-O3-symmetries

These notes demonstrate three different maps between S2 and R2: spherical polar coordinates, projection onto the equatorial plane, and "standard coordinates" (those based on the parameters of the SO(3)/SO(2) coset space). They will give you a sense of how to use embeddings to identify the induced action of a group under maps between different manifolds.

You'd then need to do something similar for the map to the torus.

This would allow you to identify the resulting SO(3) transformations on S2 and the SO(1,1) transformations on T2, i.e. transformations in the group SO(3) x SO(1,1). This is actually a direct product group, due to the independence of S2 and T2. It would give you a map from elements of SO(1,3) to elements of SO(3) x SO(1,1), that is, a homomorphism between the groups. (It can't be an isomorphism for the reasons set out in the earlier posts above.)

I think that actually what we're talking about here does relate to the action on tangent vectors. I don't know about the torus, but SO(3) is definitely the isometry group for the two-sphere. It is area-preserving precisely because it preserves the inner product of two vectors (and therefore a parallelogram formed from them).

Also, I think it's worth pointing out that the way you have distinguished between local and global symmetries above does not align with the usual meaning of these two terms. A global symmetry is one in which the same element of the symmetry group is applied to every point of the carrier space. A local symmetry, on the other hand, is one in which different points in the space may be acted on by different elements of the group. So, for example, if you imagine a cloud of dust particles, under a global SO(3) rotation, they all rotate through the same angle about the same axis. Under a local SO(3) rotation, they would in general be rotated through different angles about different axes. A global symmetry can still be applied to vectors. For example, if you take a map of the wind across a country, the wind velocity at a particular location is a vector at that point. A global rotation could be applied to that map - it could, for example, be a rotation of 90 degrees clockwise. Then every arrow pointing north would end up pointing east, every arrow pointing east would end up pointing south, and so forth. Under a local rotation, the velocity vector at one town would be rotated differently to that at another town.

I hope that helps.

Best wishes

Tom

Tom Lawrence, probably, in terms of tangent vector fields, this mathematical construction can also be implemented. See, for example, the section " Lie algebras of rotating tori " of the Deleted research item The research item mentioned here has been deleted

Thanks - I'll have a look at that. Sorry I never got back to you. A few days after you posted that last message, I attended my first physics conference in about 20 years: http://193.40.2.20/~telegrav2020/index.html . This made me realise that there was a lot of appetite for the research I had done on symmetries in teleparallelism. Consequently, I have been tied up over recent weeks with several revisions to Tangent Space Symmetries, as a result of conversations at the conference, during it and afterwards.

I am now starting to work on a second preprint. I had a Zoom conversation with another researcher on this site a few days ago about the material this will cover. I put some notes together for it which will form the basis of a preprint. They mention the group SO(1,1)xSO(2) as a maximal pseudo-orthogonal subgroup of SO(1,3). I'm about to update them and will share them with you, along with a link to another paper you might find interesting.

Tom Lawrence, I must admit that the head post is erroneous, since in fact the Minkowski space should be wrapped around the product of a three-dimensional sphere and a circle.

It's worth being aware of the following. Sn is diffeomorphic to SO(n+1)/SO(n); the SO(n) is linearly realised on Sn but the remaining symmetries (those associated with the coset space parameters of SO(n+1)/SO(n) ) are non-linearly realised. Furthermore, when SO(n+1) is gauged and compactification takes place on Sn, the non-linearly realised gauge symmetries drop out - the gauge fields have zero field strength and can therefore be eliminated by a gauge transformation. This was shown by Volkov et al: D.V. Volkov, Dmitri P. Sorokin, V.I. TkachTheor.Math.Phys.56(1984)746–751,Teor.Mat.Fiz.56(1983) 171–179.

In my version of spont comp, using a vector field to break the symmetry, a similar thing happens. With an S2 internal space, it has an SO(3) isometry, but the phi-coordinates and phi-components drop out of the solution, leaving just an SO(2) symmetry.

So while your S2 space has an SO(3) isometry, it's possible that only the SO(2) symmetry is physically relevant. The product of this SO(2) and the SO(1,1) may be found within the SO(1,3) as a maximal pseudo-orthogonal subgroup.

I understand that you do not want to abandon physical models based on the pseudo-orthogonal subgroup of the Lorentz group. Of course, this is your right, but nevertheless, I note that I prefer the physical model, which is based on the dynamics of vector fields in an 8-dimensional space with a neutral metric, obtained as the Finsler product of a doublet of Minkowski spaces with an inverse metric. The gauge symmetries of this model are hidden in the dual Minkowski space (4-momentum space), which is wrapped around S1×S3 . The breaking of gauge symmetries in this model is associated with closed curves that lie in two Minkowski spaces at once, and the length of a closed curve in the Minkowski space-time corresponds to the mass of an elementary particle.