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Passionflower
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I am trying to formulate an integral representing Tau between two r-values for radial motion in the Schwarzschild solution.
There are a few possibilities:
1. Free fall from infinity with zero initial local velocity (v0=0 and r0 -> infinity)
2. Free fall from infinity with a given local velocity (v0=initial velocity and r0 -> infinity)
3. Free fall from a certain r-value with a given velocity (v0=initial velocity (including 0) and r0 = r value of the initial velocity)
4. Free fall from a certain r-value with a given velocity that is negative (v0=initial velocity (including 0) and r0 = 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]
rs = Schwarzschild radius
ro = Outer radius
ri = Inner radius
r0 = Start value of free fall
v0 = Start velocity of free fall
Now how do I include the case for a negative local velocity, because by using negative v0 I get complex times (by replacing v02 by v0*|v0|)
I suspect I need to split up the integral into two parts one for each direction and totaling the results.
Any help?
There are a few possibilities:
1. Free fall from infinity with zero initial local velocity (v0=0 and r0 -> infinity)
2. Free fall from infinity with a given local velocity (v0=initial velocity and r0 -> infinity)
3. Free fall from a certain r-value with a given velocity (v0=initial velocity (including 0) and r0 = r value of the initial velocity)
4. Free fall from a certain r-value with a given velocity that is negative (v0=initial velocity (including 0) and r0 = 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]
rs = Schwarzschild radius
ro = Outer radius
ri = Inner radius
r0 = Start value of free fall
v0 = Start velocity of free fall
Now how do I include the case for a negative local velocity, because by using negative v0 I get complex times (by replacing v02 by v0*|v0|)
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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