Einstein Relatively Easy

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                       A geodesic of spacetime is a curve that is straight and uniformly parametrized, as measured in each local Lorentz frame along its way.

If the geosidesic is timelike, then it is a possible wordline for a freely falling particle, and its uniformly ticking parameter λ (called affine parameter) is a multiple of the particule's proper time, λ = κτ + μ.

This definition of geodesic translates into the abstract and coordinate-free language: a geodesic is a curve P(λ) that parallel-transports its tangent vector u = dP/dλ along itself.

 

 

Now defining a coordinate system {xα(P)), along with basis vectors eα = ∂/ ∂xα, we can define the tangent vector u and its components

 

 

and thus finally  the component version of the abstract geodesic equation definition becomes

 

CQFD

This geodesic equation can be solved (in principle) when both initial data xα and dxα / dλ have been specified.

 

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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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