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In the book Exploring Black Holes in the second chapter they say over and over again that "The Principle of Maximal Aging tells us that a falling stone moves so that its summed wristwatch time is maximum across every pair of adjoining spacetime patches along its worldline." They imply that this is proven by the global spacetime metric presented a few pages earlier.

The global spacetime metric is nothing more than the Pythagorean theorem in multiple dimensions. I can't figure out how the statement is proven by the metric. Can someone explain it in plainer English for me.

foolishmuse
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  • Look at my answer to this question https://physics.stackexchange.com/q/609546/ – Valter Moretti Jul 18 '22 at 18:40
  • Global spacetime metric by itself does not prove anything, and the authors do not imply that it does. Metric is an object that is used in formalization of the principle of maximal aging, but the principle is not “proven” by such formalization. – A.V.S. Jul 18 '22 at 19:06
  • Related: https://physics.stackexchange.com/q/708626/2451 – Qmechanic Jul 18 '22 at 19:08
  • @ValterMoretti I think the answer is somewhere in there, but you missed the part of my question "in plainer English". I am not looking for a proof, but for an explanation. – foolishmuse Jul 18 '22 at 19:23
  • @Qmechanic I am not looking for a new proof of the concept. Rather I am looking for an explanation of how the global spacetime metric proves it in itself (if it does so). – foolishmuse Jul 18 '22 at 19:28
  • @A.V.S. perhaps you can explain how the metric is used in formalization of the principle. This might be what I am looking for. – foolishmuse Jul 18 '22 at 19:46
  • @foolishmuse Sorry I do not know that book, so I cannot help you. – Valter Moretti Jul 18 '22 at 20:30

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It's discussed in section 1.6 (The Twin "Paradox" and the Principle of Maximal Aging) and then generalized in section 2.4 (Motion of a Stone in Curved Spacetime). The metric isn't used in the formulation of the principle; rather, the metric is used in applications of the principle (in order to find the maximal time path, you need to use the metric).

The principle itself is basically the generalization of Newton's first law (a body in motion continues to move in a uniform straight line unless acted upon by a force). In curved spacetime the path to be followed is called a "geodesic", which is a curve with extremal length (maximum time, or minimum distance, depending on whether it's timelike or spacelike).

Eric Smith
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  • Be careful, spacelike geodesics do not minimise their length in a spacetime and timelike ones maximise only locally in general. – Valter Moretti Jul 18 '22 at 20:25
  • When I look at the Quintuplet Paradox in Figure 5 of chapter 1, I come to understand that quint 1 follows the shortest possible distance from A to B. The other quints follow longer paths, so in order to reach B at the same moment, they must travel faster. Therefore they would face kinetic time dilation and age slower than quint 1. This is the application of the metric that you are speaking of? Quint 1 follows the "natural" path which has the slowest time. – foolishmuse Jul 18 '22 at 20:58
  • I haven't looked at it for a while but that sounds roughly right -- although I'd say Quint 1 has the "longest" time rather than "slowest". The application of the metric is that you can integrate the length (determined by the metric) over the path to find the total length. I suspect Taylor and Wheeler have a simpler way of expressing this, but again, I haven't looked at it for a little while. – Eric Smith Jul 19 '22 at 00:01