In Wheeler and Taylor's 'Exploring Black Holes', on pages 3-12 and 3-13, the bookkeeper measure of radial velocity (i.e. radial velocity as measured from infinity) is derived. Basically the equation for 'Energy in Schwarzschild geometry' is established-(adsbygoogle = window.adsbygoogle || []).push({});

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

The book states-

'From the energy equation and the Schwarzschild metric, we can find an expression for [itex]dr/dt[/itex], the rate of the change of the r-coordinate with far-away time t for a stone starting from rest at a very great distance. To obtain this derivative, square terms on either side of the right-hand equality, multiply through by [itex]d\tau^2[/itex], and equate it to the Schwarzschild metric equation for [itex]d\tau^2[/itex] in the case of radial fall [itex](d\phi=0)[/itex]:

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

Divide through by [itex]dt^2[/itex], solve for [itex]dr/dt[/itex], and take the square root to obtain

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

I'd appreciate if someone could show the process of derivation between the second and third equation.

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# Deriving radial velocity as observed from infinity

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