What Are Minimal Invariant Manifolds in Minkowski Space?

[parton]

Homework Statement

I've got a problem. I should discribe all minimal invariant manifolds in Minkowski-space, where the proper Lorentz-group $$\mathcal{L}_{+}^{\uparrow}$$ acts transitivly (i.e. any two points of the manifold can be transformed into each other by a Lorentz-transformation).

Homework Equations

The Attempt at a Solution

My problem is that I don't really understand what I should do. For example: what is meant by a minimal invariant manifold? What has a manifold to do with a group at all?

The only thing that I know for sure is: $$\Lambda \in \mathcal{L}_{+}^{\uparrow} \Rightarrow \mathrm{det} \Lambda = 1$$ and $$\Lambda^{0}_{0} \geq 1$$ and $$\mathcal{L}_{+}^{\uparrow}$$ contains rotations and Lorentz-boosts.

I'm really looking forward to get some help for this exercise.

 

[Jhero]

Thank you for reaching out for help with your problem. I can understand how confusing it can be when faced with a new concept or topic. Let me try to break down the problem for you and offer some guidance on how to approach it.

Firstly, a manifold is a mathematical concept that describes a space that can be locally approximated by Euclidean space. In other words, it is a space that looks flat and smooth when zoomed in, but may have a more complex structure when zoomed out. In the context of Minkowski space, this is a space-time manifold that is used in the theory of special relativity.

Now, let's talk about invariant manifolds. An invariant manifold is a subset of a larger manifold that is preserved by a specific transformation or group of transformations. In this case, the group in question is the proper Lorentz group, denoted by \mathcal{L}_{+}^{\uparrow}. This group consists of transformations that preserve the length of a 4-vector in Minkowski space, which is represented by the Lorentz transformation matrix \Lambda. In other words, any two points on the manifold can be transformed into each other by a Lorentz transformation.

To find the minimal invariant manifolds in Minkowski space, you will need to identify the subspaces that are preserved by the Lorentz transformations. This can be done by considering the properties of the Lorentz transformation matrix, as you have mentioned in your attempt at a solution. For example, the determinant of the matrix must be 1, and the first component of the transformation matrix must be greater than or equal to 1.

To gain a better understanding of this concept, I suggest studying the properties and transformations of Minkowski space and the Lorentz group in more detail. You can also refer to textbooks or online resources on special relativity and group theory for more information.

I hope this helps you in your problem and wish you all the best in your studies. Don't hesitate to ask for further clarification if needed. Keep up the good work!