Question

In general relativity we have embedding diagrams of different slices of spacetimes. These can be quite helpful to understand the geometry of a given pseudo-Riemannian manifold (especially when the geodesics are drawn in). This way the curvature can quite literally be seen / understood intuitively.

Is there a similar way to visualise the curvature in the principal U(1) fiber bundle (caused by a charged point particle fx.)? If there is not, why?

Answer 1

The curvature of spacetime can be visualized using embedding diagrams, but there is no equivalent visualization for the curvature in a principal U(1) fiber bundle caused by a charged point particle.

Step-by-step explanation:

In general relativity, we can use embedding diagrams to visualize the curvature of spacetime caused by the presence of mass. However, there is no direct equivalent way to visualize the curvature in a principal U(1) fiber bundle caused by a charged point particle. This is because the curvature in a fiber bundle is different from the curvature in spacetime, and it cannot be easily represented in a visual diagram.

In general relativity, the curvature of spacetime is represented by the bending of geodesics, which are the paths that particles follow in spacetime. These geodesics can be visualized in the embedding diagrams to understand the geometry of the spacetime manifold. On the other hand, in a principal U(1) fiber bundle, the curvature is related to the connection and the curvature tensor, which are mathematical concepts that cannot be easily represented visually.

So, while embedding diagrams are a helpful tool to visualize the curvature in general relativity, there is no equivalent visual representation for the curvature in a principal U(1) fiber bundle caused by a charged point particle.