Einstein Relatively Easy

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In this article, our goal is to show that the geodesics for a two-dimensional Euclidean space are straight lines.

In Cartesian coordinates and in two dimensional space, as there is no z coordinates, the Euclidean line element there becomes:

dl2 = dx2 + dy2

Therefore, the corresponding metric is - we are using Latin indices as we are not working in spacetime:

We also know from our previous article Geodesic equation and Christoffel symbols that the geodesic equation can be written as

But the Proper time is clearly not a convenient parameter in the case of the propagation of photons (the proper time is not defined for massless particles)

We should better use a so called affine parameter λ, as per below:

In order to calculate the eight Christoffel symbols (2*2*2 in 2D space), we need to use the equation given in Christoffel symbols in terms of the metric tensor

 But as the values of the metric are constant (equal to 0 or 1 as pointed out above), the partial derivatives gij,k = 0 for all values of i, j and k. Therefore Γjki = 0 for all values i, j, k and the geodesic equation simply becomes :

The function xi = aλ + b where a and b are constants is obviously a solution to this equation as when derived twice it gives 0.

As we are using Cartesian coordinates where xi equals x and y, the above equation becomes then

 Solving for λ gives:

which is the equation of a straight line with gradient c/a and constant (ad-bc)/a.




"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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