Einstein Relatively Easy

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 As all the information about the spacetime structure is being contained in the metric, it should be possible to express the Christoffel symbols in terms of this metric.

The following calculation is a little bit long and requires  special attention (although it is not particularly difficult).

So far, we have defined both the metric tensor and the Christoffel symbols as respectively:

Let's begin by rewriting our metric tensor in the slightly different form gαμ:

 Now, in this second step, we want to calculate the partial derivative of gαμ by xν:

Now let's try to rewrite the Christoffel symbol by multiplying each part of the equation by the partial derivative of ξσ relative to xβ:

 We can now rewrite the partial derivative of gαμ by xν as follows:

 or we recognize from our previous article Generalisation of the metric tensor that

If we now do the operation (3) + (4) - (5) we get:

Finally the last step consists in multiplying  both sides of the equations by the inverse metric gβα to isolate the Christoffel symbol

Usually, we adopt the following convention for writing partial derivatives:





"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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