Our aim is to get more familiar with the Riemann curvature tensor and to calculate its components for a two-dimensional surface of a sphere of radius r.
First let's remark that for a two-dimensional space such as the surface of a sphere, the Riemann curvature tensor has only one not null independent component.
Actually as we know from our previous article The Riemann curvature tensor part III: Symmetries and independant components, the first pair and last pair of indices must both consist of different values in order for the component to be (possibly) non-zero. Therefore, in two dimensional space where each indice could only take the values 0 and 1, the only possibility for each pair is to contain these distinct indices 0 and 1, which represent the coordinates θ and φ in polar coordinates.
So, over 2^4 = 16 components of the Riemann tensor in two-dimensional space, only one component is independent and not null.
If we choose this component to be Rθφθφ for example, then we can easily deduce the three other not null components by using the Riemann tensor symmetries:
Therefore, we only have to calculate the first term Rθφθφ
Indeed, we recall from our article The Riemann curvature tensor for the surface of a sphere that the spacetime interval on the surface of a sphere of radius r in polar coordinates is:
ds2 = r2dθ2 + r2sin2θdΦ2
So that we get as the corresponding metric gij:
which means that gθφ=0 and that gθθ=r2
As the expression of the Riemann tensor as deduced in The Riemann curvature tensor part II: derivation from the geodesic deviation is given by
If we substitute the indices for Rθφθφ ,the above equation becomes
We now sum over dummy indice m to give
Now it is just a matter of calculating all these connection coefficients. We are lucky as we have already calculated them in our previous article Christoffel symbol exercise: calculation in polar coordinates part II; we have found that the eight Christoffel symbols at a given point on the surface of the sphere were:
We can therefore simplify our Riemann tensor expression to
So that finally
We can now find the Ricci tensor
So we get by summing over indices a and b
And finally the last two components of the Ricci tensor:
The last quantity to calculate is the Ricci scalar R = gabRab
We sum over the a and b indices to give
meaning the Ricci scalar decreases as the radius increase and tends to zero for large radii.