# Unraveling the Mystery of Spin Connection in an Expanding Universe

• JorisL
In summary, the conversation discusses the use of spin connection and tetrad one-forms in calculating in a spatially flat, expanding universe. The speaker has trouble understanding some of the identities shown by Carroll and seeks clarification. Through using the antisymmetry of indices, they are able to understand the relations and continue their work.
JorisL
Hello,

I've worked through most of Carroll's appendix on the non-coordinate basis.
I see and agree how the spin connection and tetrad one-forms are useful while calculating.

However as an example he sets out to apply the formalism to a spatially flat, expanding universe.
$$ds^2 = -dt^2 +a^2(t)\delta_{ij}dx^idx^j = -e^0\otimes e^0 + \sum_i e^i\otimes e^i$$

The choice of the vielbein one-forms is clear in this case as is the calculation.

He states however that by using raising and lowering indices on the spin connection ##\omega_{ab}## we can show the following identities by using the antisymmetry. (I copied the expressions verbatim, i and j are different from 0)

$$\omega^0_{\,\, j} = \omega^j_{\,\, 0}$$

Here I wrote ##\omega^0_{\,\, j} = \eta^{0a}\omega_{a j} = - \eta^{0a}\omega_{j a} = \omega_{j}^{\, \, 0}##
This is similar but not the same.

$$\omega^i_{\,\, j} = -\omega^j_{\,\, i}$$
For the other relation I can do exactly the same and get a similar result.
With the same difference in index position.

So I don't get the same relation. Is he abusing notation here?
Or am I overlooking something? It bothers me without measure whenever I encounter such a problem.

Thanks,

Joris

bcrowell
JorisL said:
Here I wrote ##\omega^0_{\,\, j} = \eta^{0a}\omega_{a j} = - \eta^{0a}\omega_{j a} = \omega_{j}^{\, \, 0}##

##-\eta^{0a}\omega_{ja} = -\eta^{00}\omega_{j0} = \omega_{j0} = \omega^j{}{}_0##

Thanks, I don't know why I couldn't get it.
But now I can continue to do the mixmaster universe in the tetrad formalism.

## 1. What is the significance of spin connection in an expanding universe?

The spin connection is a fundamental concept in physics that describes the relationship between the geometry of space and the spin of particles. In an expanding universe, the spin connection plays a crucial role in understanding the evolution of matter and the formation of structures like galaxies and clusters of galaxies.

## 2. How does spin connection affect the expansion rate of the universe?

The spin connection is related to the curvature of space-time, which in turn affects the expansion rate of the universe. In fact, the spin connection is thought to be one of the factors that contribute to the acceleration of the expansion of the universe, along with dark energy and dark matter.

## 3. What are the current theories and models explaining the spin connection in an expanding universe?

There are various theories and models that attempt to explain the spin connection in an expanding universe. Some of the most prominent ones include the Standard Model of particle physics, General Relativity, and the Inflationary Model. However, there is still ongoing research and debate in the scientific community about the exact nature of the spin connection.

## 4. How do scientists study and measure the spin connection in an expanding universe?

Scientists use a variety of tools and techniques to study and measure the spin connection in an expanding universe. This includes observations from telescopes and other instruments, as well as mathematical and computational models. Additionally, experiments at particle accelerators can also provide valuable insights into the spin connection.

## 5. What are the potential implications of unraveling the mystery of spin connection in an expanding universe?

Understanding the spin connection in an expanding universe has significant implications for our understanding of the laws of physics and the origins and evolution of the universe. It could also potentially lead to new technologies and advancements in fields such as cosmology and particle physics.

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