Can electromagnetic field be another manifestation of the spacetime? I know this is kind of a philosophical question in the risk to be opinion-based, but I would like to know what the people that know a lot more than me, think about this.
I'm asking myself about this since a lot of time now. I think everything started noticing that the maximum velocity for the propagation of an "information" is the speed of light and astonished me the fact that a gravitational wave moves at $c$.
In fact eletromagnetic field is usually considered "inside" the spacetime, not part of it and it's simple to imagine a Universe with just gravity and mass; in this sense "spacetime comes first" and $c$ should be called the "gravitation velocity" or something like this.
Electromagnetic field has different properties anyway, first of all the fact that its "quantum" has no mass and no charge, while the "graviton" has mass and interacts with other gravitons.
At the same time it's strange that cannot exists an electric charge without mass, while the opposite is perfectly possible (relating to the previous "spacetime comes first"), so that, in a way, electric charge is constrained to follow the movement of the mass to which is associated and cannot detach from it. But trying to imagine it, it's very hard, it's like several dimensions are interacting one with the other, such as a zip unifies the two sides of a (multidimensional) jumper.
So I'm asking myself, is it possible that originally spacetime has something like $8$ or more dimensions, but "doesn't know where to put all of them" and for some unknown reasons, make some of them "collapsing" immersed into others; and in this case, why spacetime seems to be the master space, above the others?
Sorry if I was so qualitative, I don't really know how to express myself better. Anyway I don't want it to be only a philosophical question, but also a physical one, with a formalism I didn't put (because of my limits), but that is more than well accepted.
Thanks, Rob.
 A: This sentiment is what Theodor Kaluza must have had when he unified gravity and electromagnetism into a 5 dimensional spacetime metric, just about 5 years after Einstein published his work on General Relativity! The Wikipedia page for this theory is a good read: Kaluza-Klein theory
You seem to take for granted the fact that gravity is the curvature of spacetime. This is not necessarily the case! The mathematics and intuition describing gravity may be explicitly involving curvature and distances of spacetime, but this could be a mere coincidence. The mathematical description of a theory does not imply it's reality.
Take for example thermodynamics. We have a pretty good idea that the math we use to describe gases is not the true underlying description of what is going on. We use quantities like pressure, temperature, entropy, etc. to explain the macroscopic properties of gases, but we know that a more accurate description of gases is that there are many particles moving around. In this case, we opt for a macroscopic description because the underlying one is too unwieldy.
If you want to take a pragmatic viewpoint of gravity, the only thing we can definitively say about gravity is that whatever it is, at low energies it can be described by a massless spin 2 excitation, just like electromagnetism is described by a massless spin 1 excitation.
The point is, gravity may truly be the curvature of spacetime, or maybe the gravitational field is a field on flat space just like electromagnetism, and gravitational forces lend themselves nicely to a spacetime curvature interpretation. At this point there is really no way to tell.
A: Your question is too general, trying to describe a Theory Of Everything, TOE.
Theoretical research is  ongoning, and the holy grail is a TOE, except the proposals are still at the embryonic stage.
My money is on string theories, which can embed the standard model of physics and  has quantization of gravity, and they do work with more than the four dimensions of the spacetime we live in. One has to study the way the higher dimensions and the strings end up into the forces we know and study in the laboratory and observations, but it has to be done with strict mathematics, a lot of study and graduate courses. String theories are still at the level of research, to determine the one that fits the universe we live in.
