# Spin connection vs. Christoffel connection

• JosephButler
In summary, the spin connection and the Christoffel connection are related in that the Christoffel connection is the connection on an associated bundle induced by a (spin) connection on the principal bundle.
JosephButler
Hi everyone -- I have a question about the relation between the spin connection and the Christoffel connection.

The spin connection comes from the local (gauge) Lorentz symmetry of how we orient vielbeins at each point in space, it contains a manifold index and two tangent space indices. The Christoffel connection comes from the (also local/gauge) diffeomorphism invariance of general relativity, carrying three manifold indices.

My question is how are these two connections related, or are they both independent degrees of freedom?

My background is in particle physics rather than relativity, so I prefer to think about these connections as gauge fields. In this case, I'm confused about the degrees of freedom that people work with when they do perturbative quantum gravity. Why is it that people work with perturbations on the metric rather than perturbations of the Christoffel symbol, which seems to be the "actual" gauge field? (I'm told this has something to do with the Palatini formalism which connects the two?) Further, why don't people treat the spin connection as a physical gauge field?

Thanks!
Joe

My knowledge in this field is limited but there are all sorts of quantum gravity theories that choose different variables as "basic" quantities.

As for the relation between spin and Christoffel connections, you can think of the Christoffel connection as the connection on an associated bundle induced by a (spin) connection on the principal bundle.

JosephButler said:
Hi everyone -- I have a question about the relation between the spin connection and the Christoffel connection.

The spin connection comes from the local (gauge) Lorentz symmetry of how we orient vielbeins at each point in space, it contains a manifold index and two tangent space indices. The Christoffel connection comes from the (also local/gauge) diffeomorphism invariance of general relativity, carrying three manifold indices.

My question is how are these two connections related, or are they both independent degrees of freedom?

...
Thanks!
Joe

See for example Chamseddine hep-th/0511074 (2005) and the discussion in "Beyond the standard model" page 2, Martin Kober ...

I have made notes from various sources on this subject if you're interested
http://www.mathematics.thetangentbundle.net/wiki/Differential_geometry/spin_connection
http://www.physics.thetangentbundle.net/wiki/Gravitational_physics/fermions_in_curved_space
I had to study this stuff to work with superstrings and superspace where the spin connection is graded, but the same sort of ideas hold. Hope you find this useful.

Last edited by a moderator:

## 1. What is the difference between spin connection and Christoffel connection?

The spin connection and Christoffel connection are two different mathematical objects used in the study of spacetime geometry. The spin connection is a mathematical quantity that helps describe how spinors (quantities that have both magnitude and direction) behave in curved spacetime, while the Christoffel connection is used to describe the curvature of spacetime itself.

## 2. How are spin connection and Christoffel connection related?

The spin connection and Christoffel connection are related through the concept of parallel transport. Parallel transport is a mathematical operation that involves moving a quantity (such as a vector or spinor) along a curved path without changing its direction. Both the spin connection and Christoffel connection are used in the equations of parallel transport, but they serve different purposes and have different mathematical properties.

## 3. Can you explain the physical significance of spin connection and Christoffel connection?

The spin connection and Christoffel connection are both important in the study of general relativity, which is a theory that describes how gravity affects the curvature of spacetime. The spin connection is used to describe how spinors (such as particles with spin) interact with this curved spacetime, while the Christoffel connection helps us understand how the curvature of spacetime is related to the distribution of mass and energy in the universe.

## 4. How are spin connection and Christoffel connection used in the study of black holes?

The spin connection and Christoffel connection are both used in the study of black holes, which are mysterious objects in space that have such strong gravitational pull that even light cannot escape from them. These mathematical quantities help us understand how the extreme curvature of spacetime near a black hole affects the behavior of matter and energy, and how matter and energy can escape from or fall into a black hole.

## 5. Are there any real-world applications of spin connection and Christoffel connection?

While the spin connection and Christoffel connection may seem abstract and theoretical, they actually have many practical applications in physics and engineering. For example, they are used in the design of space probes and satellites, which need to take into account the effects of gravity and curved spacetime in order to navigate accurately. They are also used in the development of new technologies, such as quantum computing, which relies on the principles of general relativity and the behavior of spinors in curved spacetime.

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