# I Curved Spacetime for E&M

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1. Nov 26, 2017

### dreens

So General Relativity explains the force of gravity as mass/energy induced curvature of spacetime. This correctly predicts gravitational time distortion, nonlinear geodesics and gravitational lensing, the anomolous precession of planetary orbits, the schwarzchild metric, and so on.

Could the other forces be thought of in a similar way? For example the electromagnetic force. What if it acted by curving some effective spacetime for charges objects? I’m not talking about E&M on a curved gravitational background, but E&M itself as a kind of curvature somehow.

Is this already a thing? Is there some obvious reason that this couldn’t ever work? Any recommended reading material?

Thanks!
Dave

2. Nov 27, 2017

### PeroK

After he published the theory of General Relativity, Einstein worked on a UFT (Unified Field Theory) that tried to unify gravity, EM and the other fundamental forces. You might start here:

https://en.wikipedia.org/wiki/Unified_field_theory

3. Nov 27, 2017

### Delta²

The various fields (EM field, gravitational, strong and weak nuclear field, fermionic fields) cannot be unified in the sense that they all are different aspects of the same underlying field.

They can be unified only in the way our brain processes them via mathematics and physics, that is for all fields an action functional can be defined (which is different for each and every field) and the equations of motion (the equation that describe the dynamics of the field, which are also different for every field) of the field can be derived by considering stationary points of the action functional.

But even so, gravity possesses a special place among fields, according to general relativity and relativistic quantum physics, gravity is the only field that changes the curvature of spacetime.

4. Nov 27, 2017

### Demystifier

5. Nov 27, 2017

### George Jones

Staff Emeritus
Yes, the other the field strength interactions are curvatures of "internal spaces", not curvature of spacetime. But these ideas are not Beyond the Standard Model, as they part of the standard model of the physics of elementary particles.

At what level? Mathenatical? Non-mathematical?

A nice introduction to these ideas for the example of electromagnetism is the elementary but still very technical book "Electricity and Magnetism for Mathematicians: A Guided Path from Maxwell's Equations to Yang-Mills" by Thomas A. Garrity, which is intended for undergrad math students,

https://www.amazon.com/Electricity-...107078202/ref=mt_hardcover?_encoding=UTF8&me=

Use Look Inside to see the table of contents. The explanation starts with Chapter 16, which begins with "The goal for the rest of this book is to understand the idea that Force = Curvature"

6. Nov 27, 2017

### Urs Schreiber

As Mr. Demystifier says, what you are after is called Kaluza-Klein theory, one of the tantalizing subjects of 20th century physics.

The Kaluza-Klein mechanism, named after Theodor Kaluza and Oskar Klein, is the observation that pure gravity on a product spacetime $X \times F$ with fixed metric $g_F$ on $F$ looks on $X$, as an effective field theory, like gravity coupled to Yang-Mills theory – Einstein-Yang-Mills theory – for gauge group $G$ the Lie group of isometries of $(F,g_F)$. In particular for $F = S^1$ the circle, it yields electromagnetism coupled to gravity (and a dilaton) – Einstein-Maxwell theory.

Since in general relativity also the size and shape of the fiber $F$ is dynamical, generically effective field theories arising from KK-compactification contain spurious fields parameterizing the geometry of $F$. In the simplest case this is just the dilaton, encoding the total volume of $F$, more generally these fields are often called the moduli fields. Since these moduli fields are not observed in experiment, naive KK-models are generically phenomenologically unviable. However, in variants of gravity such as higher dimensional supergravity there are possibilities for the moduli to obtain masses and hence for the KK-models to become viable after all. This is the problem of moduli stabilization.

For more see also the PF Insights article Spectral Standard Model and String Compactifications.

7. Nov 29, 2017

### dreens

Wow everyone, thanks for the awesome responses.

I'm going to go grab this one from the library right now. It sounds perfect since I have a math theory degree with coursework in algebraic topology and have missed doing math lately. (I'm a physics PHD now, but not theoretical, I like building things too much).

You know, this is actually ringing a bell from high school- I'm pretty sure someone told me or I read in a book about E&M arising from a single additional looped dimension. At the time I didn't rightly know what E&M was or any abstract algebra, so its nice to re-encounter this with a more solid backing.

Thanks again,
Dave

8. Nov 29, 2017

### haushofer

Zee's Einstein Gravity in a nutshell is also a nice reference.

I believe Pauli was working on obtaining Yang-Mills theory from KK-compactification when he attended Yang's talk at Princeton. That's how he knew the difficulties concerning the mass of the gauge fields.

9. Jan 17, 2018

### dreens

Okay so I finished Garrity’s E&M for mathematicians. Very fun! I now know how to think about and use wedge products and forms, integrate them on n-d manifolds, define connections for vector bundles over a manifold, etc.

Or at least I’ve been thoroughly reminded, since I did sleep through a class on manifold analysis that I supposedly got an A on once, and I did Carroll’s GR grad class once and presumably knew how to parallel transport along a geodesic back then too.

Regarding understanding E&M as curvature however I’m not quite there. The book ended with some hasty ideas. One was this idea that was also mentioned above- that for an appropriately defined curvature encoded in the form of a connection specifying how tangent bundles evolve across a manifold, E&M results. Specifically, one uses the potential four vector to define the connection for a tangent bundle of vectors in U(1) over Minkowski space.

I get all the definitions but I’m missing some things about how to actually use this. If I wanted to say compute the path of a charged particle between two points... I could do this classically via the Lagrangian formulation using the electromagnetic two-form (or tensor, whatever. Sorry math purists!). But now thanks to my new connection picture I can do what exactly. Use the lagrangian picture to get the trajectory but with only the kinetic term in the Lagrangian because it’s all in the curvature now? What if I wanted to use the Lagrangian to get the fields themselves. I guess that can’t be done in this picture because I need the fields in advance to define my connection?

Also what is the space over Minkowski. Is this the Quantum mechanical phase of some test particle? Or does the wave function itself get defined over Minkowski as a map from it into the tensor product space defined by attaching a phase to each point. If so why couldn’t I attach an amplitude to each point as well.

Feel free to answer only one of my many questions or just suggest your favorite source on the subject for further investigation. Urs Schrieber i think I’m not quite ready to understand the nice web source you linked above but maybe getting close?

10. Jan 19, 2018

### strangerep

The trouble with all these so-called unified field theories is that there's exactly zero experimental evidence for any of them. KK theory and Einstein-Mayer rely on extra dimensions (unobserved). The nonsymmetric UFT that Einstein was working on before his death also ended up in a fruitless desert.

A common feature of these attempts at geometrizing E&M is that they are unable to give a useful explanation of how electric charge is somehow embedded as a property in (4D) spacetime. They tend to focus on the EM field, since that's an interesting mathematical problem of a gauge field. But the electron spinor field is charged, and interacts with the EM field. A deeper insight into that interaction always seems to be absent, and merely assumed by writing down a Lagrangian. And let's not forget that EM is nontrivially intertwined with the weak force, so any attempted unification that ignores the weak force is probably wrong.

Last edited: Jan 19, 2018
11. Jan 20, 2018

### Urs Schreiber

For what it's worth, in KK-reduction of 11d supergravity (here), the charged fermions in 4d arise from the odd-graded part of the super-geometry in11d.

12. Jan 20, 2018

### strangerep

I'm not sure I'd call that a "deeper insight", since 7 unphysical dimensions have been assumed, and the fermionic properties have been put in my hand via a superalgebra, (rather than being derived by a spin-statistics theorem in ordinary 4D QFT).