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What's the equation for a radial null geodesic?

  • #1
If we use the Schwartzschild metric with Eddington-Finkelstein coordinates what is the equation for a radial ingoing null geodesic in our new coordinates? With the substitution $$T = t + 2m \ln(r-2m)$$ we find our new line segment of the schwartzschild metric on the form
$$\textrm{d}s^2= \left(1-\frac{2m}{r}\right)\textrm{d}T^2-\left(1+\frac{2m}{r} \right)\textrm{d}r^2-\frac{4m}{r}\textrm{d}T\textrm{d}r=0.$$
$$\textrm{d}\theta=\textrm{d}\phi=0$$
My biggest challenge is to understand the question. What counts as the equation for a radial in-going null geodesic. Is it the line element that's the equation or is it the geodesic equation?
 

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