Integrating Geodesic Equations: Kevin Brown

[exmarine]

Kevin Brown, in his excellent book "Reflections on Relativity" p. 409, "immediately" integrates 2 geodesic equations:

\frac{d^{2}t}{ds^{2}}=-\frac{2m}{r(r-2m)}\frac{dr}{ds}\frac{dt}{ds}

\frac{d^{2}\phi}{ds^{2}}=-\frac{2}{r}\frac{dr}{ds}\frac{d\phi}{ds}

to get:

\frac{dt}{ds}=\frac{kr}{(r-2m)}

\frac{d\phi}{ds}=\frac{h}{r^{2}}

Does anyone understand that? I certainly don't.

 

[Bill_K]

\frac{d^{2}\phi}{ds^{2}}=-\frac{2}{r}\frac{dr}{ds}\frac{d\phi}{ds}

\frac{d\phi}{ds}=\frac{h}{r^{2}}

They both go pretty much the same way. For the second one,

\begin{eqnarray*}\frac{\frac{d^2 \phi}{ds^2}}{\frac{d \phi}{ds}} &amp;=&amp; - \frac{2}{r}\frac{dr}{ds}\\<br /> \frac{d}{ds}(\ln(\frac{d \phi}{ds})) &amp;=&amp; -2 \frac{d}{ds} \ln(r)\\<br /> \ln(\frac{d \phi}{ds}) &amp;=&amp; -2 \ln(r) + const\\<br /> \frac{d \phi}{ds} &amp;=&amp; \frac{h}{r^2}\end{eqnarray*}