How Does Tetrad Formalism Relate to the Hilbert-Einstein Action?

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In summary, the conversation is about using the tetrad formalism to write the HIlbert-Einstein action and the difficulty in proving that it is equivalent to the scalar curvature multiplied by the determinant of the metric. The person is seeking help and mentions a relevant paper by Rovelli and Vidotto. They also mention having a conversation with the authors about the solution being found in their book.
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Andre' Quanta
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I would like to write the HIlbert-Einstein action using tetrad formalism.
I have seen in a paper that the lagrangian is the contraction between a levi civita tensor, against two veilbeins and a two form curvature.
The problem is that i can' t prove that this is the same as the scalar curvature moltiplied by the determinant of the metric.
Could you help me? I am sure of the result, but i can' prove it
 
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See section 3.2: "Covariant Loop Quantum Gravity", by Rovelli and Vidotto, published by Cambridge University Press.
 
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WannabeNewton said:
See section 3.2: "Covariant Loop Quantum Gravity", by Rovelli and Vidotto, published by Cambridge University Press.
Thanks you really much :)
I had a launch with Carlo Rovelli and Francesca Vidotto, they were really nice: it s a surprise to see the solution in their book
 
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Related to How Does Tetrad Formalism Relate to the Hilbert-Einstein Action?

1. What is the Einstein action with tetrads?

The Einstein action with tetrads is a mathematical formulation of Einstein's theory of general relativity that uses tetrads, also known as vierbeins, instead of the usual metric tensor. Tetrads are four linearly independent vector fields that can be used to define a local coordinate system at each point in spacetime.

2. How is the Einstein action with tetrads different from the traditional formulation of general relativity?

The traditional formulation of general relativity uses the metric tensor to describe the curvature of spacetime. In contrast, the Einstein action with tetrads uses tetrads to define the geometry of spacetime. This allows for a more flexible and intuitive description of gravity, particularly in the presence of matter and energy.

3. What is the significance of using tetrads in the Einstein action?

Using tetrads in the Einstein action allows for a more direct connection between geometry and physical quantities. This leads to a more intuitive understanding of the effects of gravity on matter and energy. Additionally, the use of tetrads can simplify calculations and provide insight into the underlying structure of spacetime.

4. Can the Einstein action with tetrads be applied to other theories of gravity?

Yes, the Einstein action with tetrads can be extended to other theories of gravity, such as teleparallel gravity and f(R) gravity. In fact, the use of tetrads can provide a unifying framework for different theories of gravity, allowing for a deeper understanding of the connections between them.

5. What are some current research areas related to the Einstein action with tetrads?

Current research on the Einstein action with tetrads includes investigations into its applications in cosmology, black hole physics, and quantum gravity. There is also ongoing research on the development of numerical and computational methods for solving the equations of motion derived from the Einstein action with tetrads.

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