Why can we explain gravity geometrically but not EM force?

[Ontophile]

Why is a magnetic field "a field-of-force in space" while a gravitiational field is "not a field in space, but a curvature of space itself"? Why can't we describe and explain EM repulsion and attraction the way we explain gravitational attraction? Why don't we say that the presence of a charge distorts the space around it the way we say that the presence of mass distorts the space around it? Why can't an Alcubierre drive work just as well (in theory) by using the EM force, as it would (in theory) but using gravity?

 

[Matterwave]

The EM force doesn't act on all particles like gravity does. Given a charge, there are particles which are attracted, repelled, or unaffected by it. If you describe it as space-time curvature, then you have to explain why different particles given the exact same initial conditions move differently. In other words, one given source would produce 3 different "curvatures", one that applied for a positive charge, one that applied for a negative charge, and one for a neutral particle.

Gravity acts on all particles in the same way: attractive. Therefore, you can describe it with curvature.

 

[haushofer]

Well, you could describe EM via Kaluza-Klein theory as geometry. The problem is that you have to introduce an extra dimension, which you have to compactify in a small circle. Because spacetime is dynamical, it's not guaranteed that this circle stays small. And in general, it turns out that it doesn't. This problem is now known as "moduli stabilization".

 

[Ben Niehoff]

As Haushofer mentions, the Kaluza-Klein mechanism gives a way to geometrize electromagnetism.

You mentioned that EM is special because it only acts on charged or current-carrying objects. The way the KK mechanism implements this is that the "charge" of an object turns out to be equal to its momentum around the compact circular direction (in appropriate units). This in turn provides a natural way to quantize charge; as know from quantum mechanics, momentum around a circle always gets quantized.

There are two main problems with the KK mechanism. One, as Haushofer mentions, is that the extra circular direction doesn't always remain "small". A second problem is that the KK mechanism actually gives us more than what we wanted: in addition to giving us Maxwell's equations, it also gives us a scalar field. In order to reproduce GR + EM, we have to artificially set this scalar field to zero.

The KK mechanism is actually quite simple to work out mathematically. Start with the metric ansatz

$$ds_5^2 = a^2 (d\chi + A)^2 + ds_4^2$$
where $ds_5^2$ is the 5d metric, $\chi$ is the coordinate around the extra circular dimension, $ds_4^2$ is the 4d metric, a is a constant and A is a 1-form having components only along the 4d directions (and whose components are functions only of the 4d coordinates).

You should find that the 5d vacuum Einstein's equations are just the 4d Einstein's equations coupled to an EM field having A as its gauge potential. Also, the 5d geodesic equation becomes the 4d geodesic equation coupled to the Lorentz force sourced by A.

 

[pervect]

Why is a magnetic field "a field-of-force in space" while a gravitiational field is "not a field in space, but a curvature of space itself"? Why can't we describe and explain EM repulsion and attraction the way we explain gravitational attraction? Why don't we say that the presence of a charge distorts the space around it the way we say that the presence of mass distorts the space around it? Why can't an Alcubierre drive work just as well (in theory) by using the EM force, as it would (in theory) but using gravity?

One of the first clues about the curvature of space and space-time (the two are related, but not identical, topics) is gravitational time dilation.

At the time it was proposed, it was hard to observe, but experiments soon confirmed it. Nowadays, it's something that's hard to avoid, and something that has to be routinely dealt with for accurate timekeeping.

The gist of the matter is, that if you have a clock at a lower gravitational potential, it ticks more slowly, as seen by static observers comparing the clocks via static paths with a constant propagation delay.

Furthermore, if you compare the effects of gravitational potential to EM potential by looking at a clock at a lower electromagnetic potential, the EM potential doesn't have any observable effect on how fast it ticks. This makes gravity somehow different than E&M. (Actually, one does expects the electromagnetic potential to have additional gravitational effects, but not only are the effects small, they are due to the tiny gravity caused by the EM field, rather than directly by the EM field itself).

The relationship of this to curvature isn't evident, until you start trying to draw parallelograms on a space-time diagram. Then you realize that somehow you've got a parallelogram where opposite sides of the parallelogram (the time sides, this is a space-time diagram) are parallel, but the lengths are different (as seen by the time dilation). This strongly suggest curvature, as Euclidean parallelograms have opposite sides of equal lengths.