Einstein Relatively Easy

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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 = r22 + 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 

 

Ricci tensor

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:

 

 

Ricci scalar

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.

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"The essence of my theory is precisely that no independent properties are attributed to space on its own. It can be put jokingly this way. If I allow all things to vanish from the world, then following Newton, the Galilean inertial space remains; following my interpretation, however, nothing remains.."
Letter from A.Einstein to Karl Schwarzschild - Berlin, 9 January 1916

"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice."
Einstein to Max Born, letter 52, 4th december 1926

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