I am trying to formulate an integral representing Tau between two r-values for radial motion in the Schwarzschild solution.(adsbygoogle = window.adsbygoogle || []).push({});

There are a few possibilities:

1. Free fall from infinity with zero initial local velocity (v_{0}=0 and r_{0}-> infinity)

2. Free fall from infinity with a given local velocity (v_{0}=initial velocity and r_{0}-> infinity)

3. Free fall from a certain r-value with a given velocity (v_{0}=initial velocity (including 0) and r_{0}= r value of the initial velocity)

4. Free fall from a certain r-value with a given velocity that is negative (v_{0}=initial velocity (including 0) and r_{0}= r value of the initial velocity)

I am able to describe all but case 4 when the velocity is directed away from the center of gravity.

This is the integral I came up with:

[tex]\LARGE \int _{{\it ro}}^{{\it ri}}-{\frac {1}{\sqrt {{\frac {-rr_{{s}}+r{v_{{0}}}^{2}r_{{0}}+r_{{s}}r_{{0

}}}{r_{{0}}r}}}}} {dr}[/tex]

r_{s}= Schwarzschild radius

ro = Outer radius

ri = Inner radius

r_{0}= Start value of free fall

v_{0}= Start velocity of free fall

Now how do I include the case for a negative local velocity, because by using negative v_{0}I get complex times (by replacing v_{0}^{2}by v_{0}*|v_{0}|)

I suspect I need to split up the integral into two parts one for each direction and totaling the results.

Any help?

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# Towards a Generic Integral for Tau in Schwarzschild Radial Motion

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