One of the central ideas of general relativity is that freely movin...

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One of the central ideas of general relativity is that freely moving objects travel along geodesics in spacetime, which are paths of minimum proper time. Suppose we have an object moving under the influence of a uniform gravity field g. We are given that the object travels from (x,y)=(0,0) to (x,y)=(d,0) along a path that moves in x at a constant rate, and follows some parabolic path in y. There are infinitely many different parabolic paths that pass through the end points: 1(x)=R(2-xd) where K is some constant. Using the proper time integral T = dt find the value of K that minimizes the proper time of a clock traveling along with the object. Discuss the significance of this path.

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