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Introduction:

Consider the EM field. There was a time when the field was defined in a similar manner to that of the gravitational field. This changed when the view on gravitation evolved to this new idea which suggested it was a consequence of space-time curvature.
Now consider the equivalence principle, it was due to this principle that the view on gravitation was changed from that of a classical field to one which suggested a more geometric nature. I cannot help but think that we should of owed the same respect to the EM field since it doesn't seem unreasonable to think that an some equivalence principle of some kind may be assigned EM field. If we may assign an equivalence principle to the EM field then perhaps we can define the electromagnetic field in some geometric way as well.
I understand and certainly agree that EM energy stored in a volume of a space will contribute to the mass-energy content in that volume and hence curvature but this is not what I am talking about, rather I am considering the possibility of a newer description of the EM field which may complement the geometric description of the gravitational field in a more explicit way.

Consider gravity before it was described by Einstein, I claim that one could derive an gravitational tensor in the same way as one can derive an electromagnetic tensor $F^{\mu\nu}$. I say this because gravity possesses a gravito-magnetic property. This isn't surprising to me since magnetism is nothing more than a relativistic effect of static fields. What I am trying to say is that before Einstein, one could in principle derive a gravitational tensor which was analogous to the electromagnetic tensor yet after getting to this point it would still remain a special-relativistic compatible classical description of a gravitational field (meaning it doesn't necessarily take gravitational time dilation into account); in this sense the electromagnetic tensor describes the EM field classically. Of course this is not important anymore since we DO have a more powerful set of machinery for describing the gravitational field. All this to me only suggests that the EM field could be better described at the macroscopic scales.

Conclusions:

1) The principle of equivalence made it possible to assign a geometric description to the gravitational interaction.

2) If gravitational fields may be defined in a geometric way then perhaps the EM field can be as well (if a principle of equivalence may be defined for the EM field).

Questions:

1) Is it impossible to assign an equivalence principle of some kind to the EM field? If not, why?

2) If it is possible to assign an equivalence principle to the EM field then what may be the first steps in constructing the EM field in a more geometric way?

3) Where is my logic / thought process flawed?

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You mean something like the Maxwell stress tensor and the EM stress-energy tensor? –  Kyle Kanos May 23 at 16:04
    
No I do not. I just made some edits to hopefully add more clarity to my question. –  SpacetimeEngineer May 23 at 16:18

3 Answers 3

up vote 6 down vote accepted

To give a short answer: There is a huge geometric framework behind electromagnetism. This framework is gauge theory!

The leading idea is that you have electromagnetism as the gauge theory of a $U(1)$ Lie group. To keep the theory invariant under local $U(1)$ transformations, you introduce a connection (the gauge field $A_\mu$ which is identical to the four-potential) and a gauge curvature.

This curvature is exactly the $F_{\mu\nu}$ in QED. This corresponds to the Riemann tensor.

Another way to look at it is to see gravity as the gauge theory of the Lorentz group, but that would lead off-topic here.

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The fact that gravitational field can be simulated/canceled by inertial forces relies upon the following elementary but fundamental fact.

The gravitational coupling constant of a given body, i.e. its gravitational mass,$M$, coincides with the other universal constant associated with that body, appearing in the general law of motion, i.e. the inertial mass $m$. So if a gravitational field $\vec{g}(t,x)$ is given, the equation of motion of a body with mass $m$, immersed in that field of acceleration is, $$m\frac{d^2\vec{x}}{dt^2} = M\vec{g}(t,\vec{x})$$ and, since $m=M$ $$\frac{d^2\vec{x}}{dt^2} = \vec{g}(t,\vec{x})\:.$$ The motion, therefore, depends only on initial position and velocity of the body but not on other properties. Exactly as geodesics do in a spacetime. So, a description in terms of geodesics in spacetime is allowed this way and the geometrization of gravitational theory enters physics.

Referring to Electromagnetic field, this story stops at the first step. Indeed the corresponding of gravitational mass is the electric charge $q$ and, evidently, $q \neq m$ and so, $$m\frac{d^2\vec{x}}{dt^2} = q\vec{E}(t,\vec{x})\:,$$ whose solution depends on the ratio $q/m$, not only on the initial position and velocity.

This is the reason why there is no equivalent of equivalence principle for electric forces and any attempt to geometrically describe electromagnetic interaction must be constructed following other approaches (gauge theories) without involving things like metrics and geodesics.

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I'm under the impression that geodesics depend only on the end positions, and not the velocity since this is just a rotation in space-time and won't affect a geodesic. –  Larry Harson May 30 at 21:51
    
I was referring to geodesics in spacetime, the stories of particles. To fix one of them, you have to give the initial event and the initial tangent vector. The latter, referring to the proper time as parameter describing the geodesic is the four velocity which, in turn, is equivalent to the standard velocity in a any fixed reference frame. The fact that you need these two types of initial conditions is mathematically evident form the fact that the differential equation of geodesics is of second order. –  Valter Moretti May 30 at 22:16

It is possible to treat electromagnetism in a directly geometric way. This theory is known as Kaluza-Klein theory and it works by applying Einstein field equations in a 5 dimensional spacetime, and then taking the 5th dimension to be 'small' in some sense.

A nice review can be found here:

Christopher F. Chyba: "Kaluza–Klein unified field theory and apparent four‐dimensional space‐time", Am. J. Phys. 53, 863 (1985)

Of course, as the paper states: "Whether the theory represents more than an elegant curiosity remains unclear...".

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