# Christoffel Symbol vs. Vector Potential

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• quickAndLucky
In summary, the Christoffel symbol in the expression of the Connection in general relativity is analogous to the vector potential A in the definition of the covariant derivative. However, the analogy is not perfect and caution should be used when interpreting the Christoffel symbol as a force. Additionally, the spin connection may be a better choice for the analogy. Some references for further reading on the topic include the works of Wu & Yang, the UCLA paper "Vector Bundles and Connections...", and the Scholarpedia article on Ashtekar variables.
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.

quickAndLucky said:
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##?

quickAndLucky said:
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.

quickAndLucky said:
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?

quickAndLucky said:
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.

quickAndLucky said:
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.

quickAndLucky
quickAndLucky said:
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.

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)).

## What is the difference between Christoffel Symbol and Vector Potential?

The Christoffel Symbol is a mathematical concept used in differential geometry to represent the curvature of a space. It is closely related to the metric tensor and plays a crucial role in Einstein's theory of general relativity. On the other hand, the Vector Potential is a mathematical quantity used in electromagnetism to represent the magnetic field in terms of the electric current flowing in a conductor. It is closely related to the magnetic vector potential and is used to calculate the magnetic field in a given space.

## How are Christoffel Symbols and Vector Potential calculated?

The Christoffel Symbol is calculated using the metric tensor, which describes the properties of the space being studied. It involves taking derivatives of the metric tensor and using them to construct a set of coefficients. On the other hand, the Vector Potential is calculated using the Biot-Savart law, which relates the magnetic field to the electric current. It involves integrating the current distribution over the entire space to determine the vector potential at a given point.

## What are the applications of Christoffel Symbols and Vector Potential?

Christoffel Symbols are primarily used in the field of differential geometry and general relativity. They are used to study the curvature of spacetime and its effects on the motion of objects. On the other hand, Vector Potential is used in the study of electromagnetism, particularly in calculating the magnetic field in a given space. It is also used in the design of electromagnetic devices and systems.

## Can Christoffel Symbols and Vector Potential be used interchangeably?

No, Christoffel Symbols and Vector Potential are two distinct mathematical concepts that cannot be used interchangeably. They have different applications and are calculated using different methods. However, they are both important in their respective fields and play significant roles in understanding the properties of space and electromagnetic fields.

## How do Christoffel Symbols and Vector Potential relate to each other?

Christoffel Symbols and Vector Potential are related through their use in the study of curved spaces and the effects of curvature on physical phenomena. In general relativity, the Christoffel Symbols are used to describe the curvature of spacetime, while in electromagnetism, the Vector Potential is used to describe the magnetic field caused by the curvature of spacetime. Both concepts are essential in understanding the fundamental principles of modern physics.

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