This came up in another thread. I thought I'd make a few notes about them.(adsbygoogle = window.adsbygoogle || []).push({});

It should be spelled Painleve, I can't edit the typo in the title :-(

Google finds http://www.physics.umd.edu/grt/taj/776b/hw1soln.pdf, which, being a homework solution, will probably disappear soon.

The line element is:

[tex]

-{d{{T}}}^{2}+ \left( d{{r}}+\sqrt {{\frac {2M}{r}}}d{{T}}

\right) ^{2}+{r}^{2} \left( {d{{\theta}}}^{2}+ \sin ^2

\theta d {{\phi}}}^{2} \right)

[/tex]

The following coordinate transformation will map PG coordinates into Schwarzschild coordinates:

[tex]

t = T -2\,\sqrt {2\,M\,r}+4\,M\,\mathrm{arctanh} \left(

\sqrt {{\frac {r}{2 \, M}}} \right)

[/tex]

[add]

arctanh(x) is defined only for x<1, while arctanh(x) = 1/2 (ln(1+x)-ln(1-x)), more work needs to be done to deal with the sign issues that arise when making x > 1.

The above

The metric is not a function of T, therfore [itex]u_0 = g_{0i} u^i =[/itex] = (-1+2M/r) [itex]dT/d\tau[/itex] + sqrt(2M/r) [itex]dr/d\tau[/itex] = constant, a conserved energy-like quantity of the orbit.

Here [itex]u^i[/itex] is the 4-velocity [itex](dT/d\tau, dr/d\tau, d\theta/d\tau, d\phi / d\tau)[/itex]

Similarly, since the metric is not a function of [itex]\phi[/itex]

[itex]u_3 = g_{3i} u^i[/itex] = r^2 sin^2 [itex]\theta[/itex] [itex]d\phi / d\tau[/itex]= constant

representing a conserved angular momentum-like quantity of the orbit.

Generally, the orbit will be taken to be in the equatorial plane, [itex]\theta = \pi/2[/itex] and the above two conserved quantites plus the metric equation will be sufficient to calculate orbital motion.

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# Painleve-Gullstrand coordinates

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