Degrees of freedom in the curvature tensor

[Orbb]

The Einstein field equations (EFE) in 4 dimensions have 10 degrees of freedom; The Riemannian curvature tensor in 4 dimensions has 20. If I understood this correctly, one can split up the curvature tensor and describe the remaining degrees of freedom by its traceless part, which is called the Weyl tensor.

I wonder now if these remaining degrees of freedom are actually determined by the EFE, because the metric is uniquely determined, and the full curvature tensor is defined completely by derivatives of the metric. So my question is, what is the solution to this (apperent only, i guess) contradiction 10 vs. 20 degrees of freedom?

And in case some degrees of freedom of the curvature tensor do remain undetermined by the EFE, can they have an observable effect on physics?

Thanks for your answers.

 

[atyy]

http://math.ucr.edu/home/baez/gr/ricci.weyl.html

 

[Entropia]

The degrees of freedom in the curvature tensor can be a bit confusing, so let's break it down. The Riemannian curvature tensor in 4 dimensions has 20 independent components, as you mentioned. However, not all of these components are independent degrees of freedom. Some of them are related by symmetries and can be expressed in terms of others. This is where the 10 degrees of freedom in the EFE come in - they represent the truly independent and physically meaningful components of the curvature tensor.

To understand this better, let's look at the Weyl tensor. This is the traceless part of the curvature tensor, meaning it contains the components that are not related by symmetries. The remaining 10 components of the curvature tensor can be expressed in terms of the Weyl tensor and the Ricci tensor (which is related to the energy and matter distribution in spacetime). So while the Riemannian curvature tensor has 20 components, only 10 of them are truly independent and can be determined by the EFE.

Now, to address your question about the apparent contradiction of 10 vs. 20 degrees of freedom - it's important to note that the EFE are not the only equations that govern the behavior of spacetime. There are other laws and principles, such as the conservation of energy and momentum, that also play a role. So while the EFE determine the 10 independent degrees of freedom in the curvature tensor, there are other factors at play that can affect the overall behavior of spacetime.

As for your question about the observable effects of the remaining degrees of freedom in the curvature tensor - it's difficult to say for certain. Some theories, such as string theory, suggest that these extra degrees of freedom may have observable effects at very small scales. However, at our current level of understanding, it's difficult to make concrete predictions about their effects. Further research and experimentation may shed more light on this topic in the future.