# Physical interpretation of radial null geodesics in Schwarzschild geometry

(Note: $$c =1$$ throughout) The Schwarzschild metric is $$ds^2 = (1- \frac{2m}{r})dt^2 - \frac{1}{1-\frac{2m}{r}}dr^2 - r^2 d\Omega ^2,$$ with $$d\Omega^2$$ being the square of the solid angle element and $$m = GM$$, where $$M$$ is the mass of the object. Radial null geodesics in this geometry are given by $$t_\pm(r) = \pm(r+2m\log |r-2m|+C),$$ where $$r = \pm k\lambda$$, $$\lambda$$ an affine parameter, with the plus sign indicating that the particle is outgoing, and the minus sign indicating that it is ingoing. My question is: what does this physically represent? After thinking about it for some time, I have considered that it might be the time taken for a light particle to reach a distance $$r$$ from the singularity, but where is it falling from? If anyone could clarify my confusions, it would be greatly appreciated.

Edit: The photon would be falling from an initial position $$r_0$$, where $$\mp (r_0 + \log|r_0 -2m|) = C$$ as mentioned by Triatticus and myself.

• Wouldn't you determine where it is falling from by determining the constant C? Seems like the information on the starting point is there. Oct 30, 2022 at 1:13
• Yes I thought of that after posting the question. However, it seems that problems arise when trying to compute $t(C)$ (which should be $0$), so should the formula only apply for $r \not = C$? Oct 30, 2022 at 1:36
• I actually just found the answer to the question I posed in the last comment. The constant is not $r_0$, but rather $\mp (r_0 + \log|r_0 - 2m|)$. Oct 30, 2022 at 1:39
• Your metric has three problems: The second term on the right isn’t infinitesimal, and the second and third terms on the right are dimensionally incorrect. Oct 30, 2022 at 2:15
• The Schwarzschild metric in polar coordinates is: $${ds}^{2} = \left(1 - \frac{2m}{r} \right) \,dt^2 - \left(1-\frac{2m}{r}\right)^{-1} \,dr^2 - r^2 d\Omega^2$$ Where $d\Omega^2$ denotes the spherical metric induced by the Euclidean on a $2$-sphere: $$d\Omega^2 = d\theta^2 + \sin^2\theta \, d\varphi^2$$ Oct 30, 2022 at 6:56

The radial null geodesic $$t_\pm(r)$$ indeed represents the time taken for the light to reach a radial coordinate $$r$$ from the frame of reference of a distant observer (or an "observer at infinity"). The constant $$C$$ encodes the initial position and must be chosen in such a way that $$t_\pm(r_0) = 0$$. This implies that $$C = \mp(r_0 +\log|r_0 - 2m|)$$, in which case, all of the problems mentioned above (in both the question and the comments) are solved. For further reference see https://www.reed.edu/physics/courses/Physics411/html/411/page2/files/Lecture.31.pdf.