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I think what I'm looking for here is some sort of a bridge between the very material terms and mental images that I have access to and more of a pure math understanding. My deepest exposure to abstractions in math is constructing vectors from their properties in linear algebra--so I understand what it looks like to uncover the form of an object from the way it interacts with other elements of math. I don't have a connection between that way of thinking and physics, though; my deepest connection to physics essentially amounts to curve-fitting. So I suppose these kinds of questions are hard to ask in an interesting way, and if there's a mathematical foundation that would help me think productively about what it means or doesn't mean to describe physics--I'd appreciate hearing about it.

My understanding of GR is that it details how gravity can be modeled by describing spacetime with geometric equations and deriving motion from those:

$$R_{\mu \nu} - {1 \over 2}R \, g_{\mu \nu} + \Lambda g_{\mu \nu}= {8 \pi G \over c^4} T_{\mu \nu}$$

the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of space-time with the distribution of matter within it. (Wikipedia)

Not just matter, but the above equation appears to relate the energy of other forces, i.e. electromagnetism, as directly contributing to the curvature of space and therefore motion due to gravitation.

Meanwhile, nowhere in the equations for electricity and magnetism do I see any geometric relation to spacetime; they appear to be properties in space rather than properties of space.

If GR relates electromagnetic energy to spacetime curvature, why do we not have any geometric description of the electromagnetic forces like we have for gravity? Does this imply that

  • all of these forces are abstractly distinct objects, whose behavior can't be generalized together instead of separately; or
  • there may be a description for electromagnetism similar to GR, but we haven't found it?

I interpret the unification of physics as implying that there is an abstract object of which all of these are properties, and that those separate properties can be derived from a more general description of that unitary object's nature. If that's an accurate picture, do we have any reason to think that it's true? Or will finding such a unifying principle be the sole reason to suspect that it's possible to generalize across these forces?

About Relativistic Electromagnetism,

... the aspiration, reflected in references for this article, is for an analytic geometry of spacetime and charges providing a deductive route to forces and currents in practice. (Wikipedia)

If a geometric relation between charges and spacetime is an aspiration, does that imply that such a description is so distinct from GR that developing GR gives us no additions to theories of electromagnetism? Are there problems in electromagnetism that could be solved by such a theory, or does it appear that there is no missing description of those other forces?

I suppose my question boils down to: is gravity really fundamentally distinct, as we can tell at this point, from the other forces? Are the other forces then also really distinct from one another, in that the foundation of a good model for electromagnetism will be totally different from good models for the strong and weak forces? Why do we need differential geometry to think about gravity, but not the others? Does it appear that this distinction will be a persistent pattern in physics?

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Well, scientists and physicists alike once tried mapping electromagnetism with the curvature of space time in an extra spatial dimension actually. It sort of worked. However that theory was disbanded and it gave rise to quantum electrodynamics. For the question whether gravity is distinct: I should say it sort of is. It is actually one of the weakest forces of nature. There is also the possibility gravity might not be distinct, further research is still needed, but as far as scientists are concerned gravity might be distinct. This can be seen by how difficult it is to make quantum gravity and introduce the "graviton". As for the other forces, they are pretty similar. All the other forces require the force carrier(aka: the particles that mediate the other forces). However out of electromagnetism, strong, and weak force; electromagnetism is the most different out of them. Though not by a lot. Well we need differential geometry mainly because right now we are lacking a good theory of gravity, but until then we will most likely stick to differential geometry to explain gravity. As for the other forces, their force carriers have been discovered and explains almost everything. We could use geometry and explain the other forces with vectors but it is just easier to use particles. (Yes sometimes they are described with vectors, it is for quantum field theory.) Well gravity might always be different and this pattern might be shown, but again physicists are still working on quantum gravity. Until then we can not know for sure whether we are going to see gravity as a different force or whether it is like the other forces.

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