Einstein Relatively Easy

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To make the meaning of the equations of covariant differentiation seen in last article Introduction to Covariant Differentiation more explicit, we will consider the covariant derivative of vector V with respect  to θ in cylindrical coordinates (so x1=r, x2=θ, and x3=z).

Setting β=2 in the following equation, since we are interested in the covariant derivative with respect to θ:

we get

We know the values of the first two Christoffel symbol as we have already calcuted them in the previous article Christoffel symbol exercise: calculation in polar coordinates part I

so that we already  know that

We know also that since

 

all the symbols from the following form vanish

thus we end up with this equality

 

which says that a change in the r-component of vector V caused by a change in θ is caused both by a change in Vr with respect to θ and by a change in the basis vectors which causes a portion of V that was originally in the θ-direction to now point in the r-direction.

Likewise, for the change in Vθ as the value of θ is changed, we have

which says that a change in the θ-component of vector V caused by a change in θ is caused both by a change in Vθ with respect to θ and by a change in the basis vectors which causes a portion of V that was originally in the r-direction to now point in the  θ-direction.

 Thus finally the covariant derivative of vector V with respect to θ in cylindrical coordinates is

 

 

 

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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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