I just finished reading Black Holes by E. Taylor and J. Wheeler. Throughout the book they use the SC metric for the the metric near a massive object(for radial motion only:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] d \tau^2 = \left( 1 - \frac{ 2M }{r} \right) dt^2 - \frac{dr^2}{\left( 1 - \frac{ 2M }{r} \right)} [/tex]

Where dt is measured very far away from the massive body, r is measured as the circumference of a sphere whose center is in the middle and whose outer shell reaches the point divided by two times pi. M,t and r are mearured in meters.

Then, in order to calculate to path of a particle near a massive object in between two events they use the metric to find the path that give the maximal proper time (the Principal Of Maximal Aging).

Now my question is, to use POMA you need to have initial and final events and then calculate the path between them via the POMA. But I want to prove that the SC metric also predicts that an object at rest next to a massive object will start falling in. To do that shouldn't I start at t0 and r0 and then calculate which r1 will give the maximal proper time for a giver t1? Shouldn't this r1 be smaller than r0 - meaning that in order to maximize proper time the object should start falling towards the massive object? I tried to do this but I got that for any r1=/=r0 the proper time is smaller for a giver t1 then if r1=r0. Is that right? If so, how does the SC metric predict that an object that starts at rest next to a massive body will move towards it?

Thanks.

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# Swartzchild metric and free fall

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