In our two previous articles, we have deduced the rather complicated expression of the Riemann curvature tensor, a glorious mixture of derivatives and products of connection coefficients, with **256 (=4^4) components** in four-dimensional spacetime.

But we have also demonstrated in our article Local Flatness or Local Inertial Frames and SpaceTime curvature that any arbitrary coordinate system could nullify all but **20 second derivatives of a given metric** in a curved spacetime. Our aim in this article is to demonstrate that the Riemann tensor has only 20 independant components and that these component are precisely a combination of these second not null derivatives.

The methodology to adopt there is to study the **Riemann tensor symmetries** in a Local Inertial Frame (LIF) - where as we know all the Christoffel symbols are null - and to generalize these symmetries to any reference frame, as by definition a tensor equation valid in a given referential will hold true in any other referential frame.

Using the definition of the Riemann tensor as seen in the precedent articles:

and knowing that all the Christoffel symbols are null at the origin of Local Inertial Frame, this expression get simplified to:

By applying the contraction mechanism as exposed in Introduction to Tensors, we can rewrite the Riemann tensor with all indices lowered:

We remember from our article Christoffel symbols in terms of the metric tensor how to write the Christoffel symbol with respect to the metric derivatives:

So that we can write

By substituting indices μ and ν, we get the second term of the Riemann tensor expression:

By substracting the two expressions we see that the middle terms cancel, so we’re left with:

This could obviously be written as:

This section of the article is only available for our subscribers. Please **click here** to subscribe to a subscription plan to view this part of the article.