A Riemannian Geometry: GR & Importance Summary

[binbagsss]

Hi

I've done a masters taught module in GR and from what I've learned these are two of some of the most important significance of needing a Riemannian Geometry:

1) If we consider the Lagrangian of a freely-falling particle given by $L= \int ds \sqrt{g_{uv}\dot{dx^u}\dot{dx^v}}$ and find the equations of motion, by the principle of least action this is the shortest path and so must be the definition of a geodesic.

The alternate way to define a geodesic is that the tangent vector of is parallel transported along itself :

$V^u \nabla_u V^a =0$

Then via the fundamental theorem of Riemannian geometry,( given a manifold equipped with a non-degenerate, symmetric, differentiable metric there exists a unique torsion-free connection such that $\nabla_a g_bc =0$), we can show that these two definitions of a geodesic are important

2) Due to the fundamental theorem of Riemannian geometry, equipped with a metric on the space-time, we can express important objects such as the Christoffel symbol and Riemman tensor in terms of the metric, and so the metric effectively encodes all the information about the space-time

Are there other important roles played by Riemannian geometry?

I find the first one pretty interesting- is it Palatini formalism that looks at when the geometry is non-Riemmanian and so the geodesics would not be the same?

Thanks in advance.

 

[romsofia]

Look into Vierbeins and Lorentz connection, also minimal coupling.

I'd recommend this book for questions like this: https://www.amazon.com/dp/8847026903/?tag=pfamazon01-20

 

[binbagsss]

So I have read that in Palatini formalism, where the metric and connection are treated independently, if one is to assume any torsion-free connection then metric compatibility comes out by varying the action, is torsion free connection or a non-symmetric connection most widely explored in Palatini formalism?

Also, metric compatibility in GR means we can add the cosmological constant term whilst conserving the energy-momentum tensor still. In a palatini formalism where one does not assume torsion-free, are there any discussions on this or the fact the geodesics defined above do not agree, and a physical interpretation, that anyone could link me to? thanks

 

[romsofia]

Yup, I read a paper on this a few months ago, here it is: https://arxiv.org/abs/1606.08756v5

I can recommend some other papers on the topic, but I think the book I mentioned above does go into this discussion as well in the appendix.To add into your question, I always see torsion free being explored because you can add torsion by just using a Dirac field as your source (which should have torsion). Also, if you want torsion and non-symmetry, you would be getting into a third theory which is Weyl's gravity.