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.

 

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"Five or six weeks elapsed between the conception of the idea for the special theory of relativity and the completion of the relevant publication" Einstein to Carl Seeling on March 11, 1952

"Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."
David Hilbert

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