Is EM theory in curved spacetime the same as unification ?

[pellman]

Is EM theory in curved spacetime the same as "unification"?

I am wanting to learn about classical EM theory in curved spacetime (just curious) and I found this old thread containing some references https://www.physicsforums.com/showthread.php?t=3950

The historical topic of unification of gravitation with electromagnetism in the Einstein direction takes one down a trail that hasn't survived the explosion of theory for fundamental physics since 1960. But for a while, it was almost Einstein and his associates alone who credibly pushed the idea of unification with gravitation. That was because of Einstein's enormous reputation

Is simply talking about the EM field in curved spacetime the same thing as unified theory? Isn't it valid to ask what the Maxwell equations look like in curved spacetime?

Anyway, I'd be interested if further recommended on this topic as well.

 

[JesseM]

Is simply talking about the EM field in curved spacetime the same thing as unified theory? Isn't it valid to ask what the Maxwell equations look like in curved spacetime?

I think that when dealing with electromagnetism in GR (whether classical or quantum field theory in curved spacetime), the usual approach is to ignore the contribution of the field itself to the curvature of spacetime, which might be a reasonable approximation in the case of low energies but wouldn't be a true unified theory. On the other hand there is also the Kaluza-Klein model which tries to explain electromagnetism in a way analogous to the explanation of gravity in GR, by saying it's really curvature in a spacetime with an additional compact (rolled-up) spatial dimension...my understanding is that this approach has somehow been incorporated into string theory.

 

[bcrowell]

I found the following online:

http://en.wikipedia.org/wiki/Classical_unified_field_theories
Hubert F. M. Goenner, "On the History of Unified Field Theories," http://www.livingreviews.org/lrr-2004-2
Eddington, "Space, time and gravitation: an outline of the general relativity theory," http://books.google.com/books?id=uU1WAAAAMAAJ&pg=PA167#v=onepage&q=&f=false

I thought the chapter from the Eddington book (which was written for a general audience) was very readable.

I think that when dealing with electromagnetism in GR (whether classical or quantum field theory in curved spacetime), the usual approach is to ignore the contribution of the field itself to the curvature of spacetime, which might be a reasonable approximation in the case of low energies but wouldn't be a true unified theory.

I could be wrong, but I don't think you're right about this. I think electrovac solutions do include the curvature produced by the mass-energy of the EM field. See, e.g., http://en.wikipedia.org/wiki/Electrovacuum_solution

As the WP article makes clear, there have been many decades worth of attempts in the direction of classical unified theories, and many different approaches have been tried. Therefore it's probably not possible to make categorical statements about all of them. But in the case of Weyl's theory, described in the Eddington book, the flavor of what they're trying to accomplish is not that they want to be able to create a self-consistent theory that includes both gravity and electromagnetism; that was already accomplished as soon as GR was published. They wanted to explain things like why the classical electron radius has the value it does, and why spacetime is four-dimensional. They also wanted to make E&M fully geometrical, rather than throwing it into the source terms of the Einstein field equations.

According to Goenner, there were hopes of making a theory in which electrons and protons (the only fundamental particles known at the time) would pop out of the theory naturally, rather than having to be put in by hand.

 

[bcrowell]

If anyone wants to read Eddington's description of the Weyl theory and discuss it, I'd be interested in talking about it. One thing that seems odd to me about it is the following. He posits that parallel transport around a closed path can change a vector's length. This seems to me to be different in an important way from the usual GR idea that transport around a closed path can change a vector's direction. Let's call the length effect L and the direction effect D. D works the same for all vectors, regardless of what type of vector it is. But it seems to me that the same can't be true for L. L involves changing a scalar. Suppose every scalar were to change by the same amount on transport around a given loop -- regardless of the type of scalar. This rapidly leads to a mathematical contradiction, since for any scalar x, 1/x is also a scalar, but you can't have x and 1/x both scale the same way. As far as I can tell, L can therefore apply only to norms of spacetime displacement vectors, i.e., to meter-sticks and clocks. To me, this seems ugly and contrary to the purely geometrical spirit of parallel transport in normal GR.

 

[JesseM]

I could be wrong, but I don't think you're right about this. I think electrovac solutions do include the curvature produced by the mass-energy of the EM field. See, e.g., http://en.wikipedia.org/wiki/Electrovacuum_solution

OK, it looks like I was wrong about classical electromagnetism, but I'm pretty sure I've read that quantum field theories on curved spacetime (including quantum electrodynamics) aren't able to take into account the fact that you should actually be summing over different spacetime curvatures when summing over Feynman diagrams with virtual particles of different energies...

 

[jfy4]

You can look these up on Wikipedia, and maybe this isn't what you are looking for, but here are Maxwell's equations in curved space-time.

$$\frac{\partial F_{\rho\sigma}}{\partial x_{\tau}}+\frac{\partial F_{\sigma\tau}}{\partial x_{\rho}}+\frac{\partial F_{\tau\rho}}{\partial x_{\sigma}}=0$$

and

$$\frac{\partial}{\partial x_{\nu}}\left(\sqrt{-g}g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}\right)=\mathcal{J}^{\mu}$$

these are two of my favorites of all time!

 

[bcrowell]

OK, it looks like I was wrong about classical electromagnetism, but I'm pretty sure I've read that quantum field theories on curved spacetime (including quantum electrodynamics) aren't able to take into account the fact that you should actually be summing over different spacetime curvatures when summing over Feynman diagrams with virtual particles of different energies...

I think quantum field theories would be a whole different ballgame. The OP was asking about classical unified theories.

 

[Mentz114]

If anyone wants to read Eddington's description of the Weyl theory and discuss it, I'd be interested in talking about it. One thing that seems odd to me about it is the following. He posits that parallel transport around a closed path can change a vector's length. This seems to me to be different in an important way from the usual GR idea that transport around a closed path can change a vector's direction. Let's call the length effect L and the direction effect D. D works the same for all vectors, regardless of what type of vector it is. But it seems to me that the same can't be true for L. L involves changing a scalar. Suppose every scalar were to change by the same amount on transport around a given loop -- regardless of the type of scalar. This rapidly leads to a mathematical contradiction, since for any scalar x, 1/x is also a scalar, but you can't have x and 1/x both scale the same way. As far as I can tell, L can therefore apply only to norms of spacetime displacement vectors, i.e., to meter-sticks and clocks. To me, this seems ugly and contrary to the purely geometrical spirit of parallel transport in normal GR.

Yes, the length of a vector changing on transport around a loop is hard to imagine.
Where's the best place to read about this theory ?

There are at least two ways that do work, curvature and torsion, in GR and translational gauge gravity ( aka Teleparallel). In both of these theories the deficit is a wedge product i.e. a commutator product. There are probably mathematical reasons why a scalar deficit can't work as a repository of gravity.

 

[bcrowell]

Yes, the length of a vector changing on transport around a loop is hard to imagine.
Where's the best place to read about this theory ?

I liked Eddington's description -- see #3. The book is public domain and available online at that link.

There are probably mathematical reasons why a scalar deficit can't work as a repository of gravity.

In the Weyl theory it's used as a geometric description of the electromagnetic field, not gravity.

 

[Mentz114]

In the Weyl theory it's used as a geometric description of the electromagnetic field, not gravity.

OK. But a scalar won't work surely because we need a vector potential. I'll check it out, thanks.

 

[pellman]

You can look these up on Wikipedia, and maybe this isn't what you are looking for, but here are Maxwell's equations in curved space-time.

$$\frac{\partial F_{\rho\sigma}}{\partial x_{\tau}}+\frac{\partial F_{\sigma\tau}}{\partial x_{\rho}}+\frac{\partial F_{\tau\rho}}{\partial x_{\sigma}}=0$$

and

$$\frac{\partial}{\partial x_{\nu}}\left(\sqrt{-g}g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}\right)=\mathcal{J}^{\mu}$$

these are two of my favorites of all time!

The trouble I have at first is the ambiguity in what is the EM tensor: is it $$F_{\alpha\beta}$$ or $$F^{\alpha\beta}$$ ? One of these is the "bare" EM tensor and the other is the EM tensor mixed up with the metric tensor, right? In SR the distinction doesn't matter because they only differ by a sign. But in GR the metric depends on x.

Now I know that someone is going to say that which one we choose is arbitrary. But that's not so obvious to me. One of them may more obviously coincide with what we think of as the EM field. I'm just beginning to look at this so none of it is obvious to me.

 

[bcrowell]

OK. But a scalar won't work surely because we need a vector potential. I'll check it out, thanks.

The change in the scalar is determined by a four-vector, which is interpreted as the vector potential.

 

[bcrowell]

Now I know that someone is going to say that which one we choose is arbitrary. But that's not so obvious to me. One of them may more obviously coincide with what we think of as the EM field. I'm just beginning to look at this so none of it is obvious to me.

You can always transform into a local Minkowski frame, so I don't think that's a big issue.

 

[pellman]

Thanks for the replies. Very helpful.