# Could electromagnetic charge curve something like spacetime in analogy with general relativity? [duplicate]

Newtonian gravity and electrostatics have the same form; this analogy is extended when we look at full dynamic electromagnetism, and correspondingly "gravitomagnetism". We are quite capable of observing higher-order general relativistic effects, because mass (and energy) attracts more of itself and we get very large quantities of it. It would be difficult to isolate a net charge on planetary scales, though -- the whole Sun has a charge of 77 Coulombs. How plausible is it to consider that classical electromagnetism is just the low-charge limit of a larger theory, that curves spacetime (or something like that -- in some way that only affects charge) in a more complicated, dynamical way?

I'm aware that, as electromagnetic waves move through space, they would carry energy themselves, bending spacetime somewhat. I lack the GR background to understand how that behaves, though. My first thought is that - if a single photon moved through space, its energy is so small as to only bend spacetime a very small amount, with radii of curvature much less than its own wavelength. As the energy of the photon goes up, and the radius goes down, so does the wavelength. So to see nonlinear behavior would require enormous numbers of coherent photons.

## marked as duplicate by ACuriousMind♦, Qmechanic♦Jul 3 '15 at 7:18

• – ACuriousMind Jul 2 '15 at 21:56
• Thank you for the references, I didn't know what to search. This is largely a duplicate, but one thing I wasn't able to get from the links was part of my question: what kinds of physical predictions do these broader theories make, and how testable/tested are they? – Alex Meiburg Jul 2 '15 at 22:08
• The pure geometric formulation as a $\mathrm{U}(1)$-gauge theory isn't broader, it's an equivalent way to do electromagnetism. – ACuriousMind Jul 2 '15 at 22:12
• Alex: for an analogy, imagine you're standing on a headland overlooking a flat calm sea near an estuary. You see a single wave, and notice its path curves a little because of the salinity gradient. The radius of curvature of this "geodesic" is about 100km. Now look at the surface of the see where the wave is. It's curved. And the radius of curvature is about 10m. Then see what Percy Hammond says here: "We conclude that the field describes the curvature that characterizes the electromagnetic interaction". – John Duffield Jul 2 '15 at 22:36

Yes and no. Charges and currents curve the $U(1)$ gauge connection. We experience this curvature every day so we even have a special name for it: an electromagnetic field. Just like spacetime curvature is called gravity. However, the choice of the word 'curvature' is somewhat unintuitive here due to the fact that it is not our spacetime that gets curved.

You can think of the gauge connection as of a way to do parallel transport. You can transport only some special structures, however. Those are elements (vectors) of representations of the $U(1)$ group. In General Relativity, the affine connection lets you do parallel transport of tangent-space vectors. In this sense GR has a beautiful geometrical interpretation.

• The geometric interpretation depends on the equivalence of inertial and gravitational mass, which we don't have for electromagnetic charges, doesn't it? To me that's closer to a no, even though the gauge connection is a beautiful thing in itself. – CuriousOne Jul 2 '15 at 23:00
• The principle of equivalence has to do with the geometrical interpretation of GR, but little to do with curvature. So I don't think it's either 'yes' or 'no'. Just what I described in the answer above. – Solenodon Paradoxus Jul 3 '15 at 1:54
• The U(1) gauge connection is an abstract thing, Hindsight. See Einstein talking about field theory here: "It can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds". What state are we talking about? IMHO the answer is curved, and curved spacetime is inhomogeneous space. – John Duffield Jul 3 '15 at 12:42
• See this and this. When an ocean waves moves through the sea, the sea waves. When a seismic wave moves through the ground, the ground waves. When an electromagnetic waves moves through space, space waves. I kopw of no waves where something doesn't wave. So I'm with Maxwell: light consists of transverse undulations. – John Duffield Jul 3 '15 at 12:53
• @JohnDuffield electrodynamics does not curve space. Gravity does. Are you implying the opposite? – Solenodon Paradoxus Jul 3 '15 at 23:16