Christoffel Symbol vs. Vector Potential

[quickAndLucky]

As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative.

The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain constant from one point to another.

Does this mean that the vector potential A is necessary to define what it means for a classical field to remain constant from one point to another? This doesn’t seem right because we still have A in Minkowski space expressions.

 

[PeterDonis]

As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative.

I assume you mean the vector potential in the "covariant derivative" in electromagnetism? I.e., we replace the operator $\partial_\mu$ with the operator $\partial_\mu - i e A_\mu$?

Does this mean that the vector potential A is necessary to define what it means for a classical field to remain constant from one point to another?

No, it just means that there is a mathematical analogy between EM and gravity. You can view the EM "covariant derivative" as defined above as giving the "rate of change of a vector", but it isn't a rate of change in ordinary space; it's a rate of change in an abstract space that includes an extra "dimension" for the EM potential.

This doesn’t seem right because we still have A in Minkowski space expressions.

That's because the space that is "curved" in the EM version of the "covariant derivative" is not ordinary space, it's the abstract space described above.

 

[quickAndLucky]

Can you think of the field as living in a product space of the Minkowski and abstract spaces? Do you know of any reference that explains general yang mills theories geometrically in this way?

 

[PeterDonis]

Can you think of the field as living in a product space of the Minkowski and abstract spaces?

I think that's the basic idea behind Kaluza-Klein theory, yes.

Do you know of any reference that explains general yang mills theories geometrically in this way?

String theory more or less takes this approach, or at least one version of it does. It views all of the Standard Model fields, not just the electromagnetic field, as being just geometry in a higher dimensional abstract space (whereas the extra abstract space for EM alone is just a simple circle). These higher-dimensional abstract spaces are called Calabi-Yau manifolds; googling on that might turn up some references, although they might not be easily comprehensible.

 

[pervect]

As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative.

The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain constant from one point to another.

Does this mean that the vector potential A is necessary to define what it means for a classical field to remain constant from one point to another? This doesn’t seem right because we still have A in Minkowski space expressions.

I believe the usual analogy is to consider the metric $g_{\mu\nu}$ as a potential, making Christoffel symbol something else.

Wiki (linked_here) confirms this recollection.

In general relativity, the metric tensor (or simply, the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation.

If we accept that the metric is similar to the potential, the Christoffel symbols would be somewhat similar to the gradient of a potential, making it similar to a force. Some components (for instance $\Gamma^x{}_{tt}$ in an orthonormal basis) are similar to forces, but other components don't have a ready interpretation as a force, so caution should be used.

Going on, the Riemann tensor would be somewhat similar to the gradient of a force, i.e. a tidal force, which is also a common analogy. I don't have a more detailed reference of the issue at this point, except for Wiki which I used as a sanity check on my fallible memory.

The usual E&M analogy as I recall it relates the EM 4-potential (which includes the scalar potential $\phi$ and the vector potential A) to the rank 2 metric tensor $g_{\mu\nu}$. Both $g_{\mu\nu}$ and the 4-potential satisfy the wave equation for instance ((IIRC)).

 

[robphy]

Possibly interesting reading (I haven't read these in any detail):

http://math.ucr.edu/home/baez/gauge/
http://wwwphy.princeton.edu/~kirkmcd/examples/EP/wu_ijmpa_21_3235_06.pdf "Evolution of the Concept of Vector Potential..." by Wu & Yang
http://www.math.ucla.edu/~vsv/papers/paper.pdf "Vector Bundles and Connections..."
http://www.scholarpedia.org/article...A:_Connection-dynamics:_A_historical_anecdote

 

[haushofer]

For the analogy you should take the spin connection instead of the christoffel symbol. See e.g.

https://www.physicsforums.com/insights/general-relativity-gauge-theory/