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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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1 Answers1

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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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