What is the distance of closest approach in this Wikipedia article?
I can't seem to find its definition, and this other question doesn't have an answer I can understand.
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The Schwarzschild spacetime has Killing vector fields $\partial_t$ and $\partial_\phi$ that give conservation of specific energy and angular momentum, respectively: $$\begin{eqnarray*}e = \left(1-\frac{2M}{r}\right)\frac{\mathrm{d}t}{\mathrm{d}\lambda}\text{,} &\quad\quad& l= r^2\sin^2\theta\frac{\mathrm{d}\phi}{\mathrm{d}\lambda}\text{,}\end{eqnarray*}$$ although one can also obtain those constants of motion by integrating the components of geodesic equation directly. Here $\lambda$ is any affine parameter for the geodesic, which for timelike ones can be taken to be the proper time.
The parameter $b$ is simply the ratio $b = \left|l/e\right|$.
It is commonly called the impact parameter for the following reason. For $r\gg 2M$, spacetime is approximately Minkowski. Let the light ray be in the equatorial plane ($\theta = \pi/2$) and take Cartesian axes such that the light ray is parallel to one of them at infinity with $d$ the distance from the ray to that axis, $$b \approx \frac{r^2\mathrm{d}\phi/\mathrm{d}\lambda}{\mathrm{d}t/\mathrm{d}\lambda} = r^2\frac{\mathrm{d}\phi}{\mathrm{d}t} = r^2\frac{\mathrm{d}\phi}{dr}\frac{\mathrm{d}r}{\mathrm{d}t} \approx d\text{,}$$ where we approximated $\phi \approx d/r$ and $\mathrm{d}r/\mathrm{d}t \approx -1$ in the distant limit.
Stan Liou
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