# Geometric interpretation of Electromagnetism

For gravity, we have General Relativity, which is a geometric theory for gravitation.

Is there a similar analog for Electromagnetism?

• Comment to the question (v1): Do you mean something like (i) EM in curved spacetime, (ii) Kaluza-Klein theory, or perhaps (iii) something along the lines of this Phys.SE question? Sep 5 '13 at 17:52
• @Qmechanic: I think the OP means something like curving the $U(1)$ bundle. Sep 5 '13 at 17:54
• Not sure if this helps, but: "However, ordinary Riemannian geometry is unable to describe the properties of the electromagnetic field as a purely geometric phenomenon." (Wikipedia) Sep 5 '13 at 17:57
• @aufkag: Keyword here is bundles. Sep 5 '13 at 18:00
• @Qmechanic: Hm, in this case, I think the qed tag is quite relevant because of the $U(1)$ bundle. Sep 14 '13 at 6:20

2. Gauge theory. As mentioned by DImension10 Abhimanyu PS, electromagnetism can be described by a gauge theory whose gauge group is $U(1)$; the electromagnetic potential becomes a connection, and the electromagnetic field the curvature associated to the connection. It is in fact the symmetry group of the fourth dimension in Kaluza-Klein theory. For mathematicians, a gauge theory is described in terms of principal bundles, which, if the gauge group is $U(1)$, are in fact 4+1 dimensional spaces, satisfying symmetry conditions like in the Kaluza-Klein theory. So, mathematically, they are equivalent, although there are variations of the Kaluza-Klein theory which cannot be described by a standard gauge theory.
The geometric interpretation comes from the similarity of the covariant derivatives between general relativity and quantum field theory -- $\nabla_\mu -\partial_\mu$ in general relativity gives you the Christoffel symbol (although it contracts in an actual operation), which is the gravitational field strength. Similarly in QED, $\nabla_\mu -\partial_\mu = ig_sA_\mu$, which measures the electromagnetic field strength.
This gives rise to the geometric interpretation of electromagnetism in which it is the curvature on the $U(1)$ bundle (and in general for a gauge theory, the gauge force is a curvature on its corresponding gauge group's bundle).