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In Hartle's book 'Introduction to General Relativity', he says that straight-line paths between two time-like separated points are the longest paths. He uses this in the case of Twin paradox, where the moving observer does not have a straight path between the same points, hence its 'distance' or the proper time is shorter. This is understood clearly using the time-dilation formula, but how are straight-line in non-Euclidean space longer? Any geometrical or mathematical explanation would do.

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  • $\begingroup$ This is more a problem of Lorentzian vs Riemannian as opposed to Euclidean or not Euclidean but still Riemannian. $\endgroup$
    – AHusain
    Jan 7, 2017 at 5:13
  • $\begingroup$ Hi Maryam. There is some discussion of this in What is time dilation really? and What is the proper way to explain the twin paradox? though neither are a direct answer to your question. $\endgroup$ Jan 7, 2017 at 6:38
  • $\begingroup$ Quanta repeated the longest time bit, but without specifying timelike.. (!). They just write about two points in space-time, so naturally it made no sense to me. This question was the answer I needed :) $\endgroup$
    – tux3
    Nov 27, 2022 at 0:18

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The "distance" between two points(events) in space-time is given by,

$ds^2 = dt^2-d\bar{x}^2$

Where $d\bar{x}^2$ is the spatial part of the space-time. Now moving in a timelike straight line you can always choose a co-ordinate such that this spatial part is zero. And then in that co-ordinate $ds^2 = dt^2$. Now for any other path between this two events the extra term $d\bar{x}^2$ comes with a negative sign and therefor decreases the distance. Thus the longest distance will be straight line one.

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The "length" is the Lorentzian length: For a world line $\gamma$ with velocity $v^\mu$, we have $L(\gamma)=\int \sqrt{-g_{\mu\nu}v^\mu v^\nu}\,dt$. A geodesic is a critical point of the length functional. To see that a geodesic (straight line) is not a minimum, we need to show that there is a nearby curve with shorter length. This construction is standard: We can perturb $\gamma$ slightly to make it "almost spacelike", which decreases the total length since $g_{\mu\nu}v^\mu v^\nu$ becomes arbitrarily close to $0$. To do this, we can zig-zag a curve very close to $\gamma$ so that each "straight" part is very close to being null. This clearly fails in the Euclidean/Riemannian case. If there is no conjucate point along the geodesic, then it is in fact a maximum of length.

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