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The following wikipedia articles are incompatible :

According to both articles, the equation describing the path of a photon in a Schwarzschild metric is :

$$\frac{d\varphi}{dr} = \frac{1}{r^2\sqrt{\frac{1}{b^2}-\left(1-\frac{r_s}{r}\right)\frac{1}{r^2}}}$$

with :

  • $\varphi$ the deviation angle
  • $r$ the distance between the photon and the mass
  • $r_s$ the Schwarzschild radius

and $b$ which is the distance of closest approach in the first article and the impact parameter in the second one.

So I have two questions :

  1. Which one is correct? (What is $b$?)

  2. If $b$ is the impact parameter, what is the formula of the distance of closest approach $r_0$ as a function of $b$ (or inversely, the impact parameter $b$ as a function of $r_0$)?

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  • $\begingroup$ I have seen $b$ defined as $L/E$ where $L$ is the angular momentum and $E$ is the energy. e.g. p.188 in Introduction to Black Hole Physics by Frolov. How does this make sense for a photon? Why does the impact parameter, essentially a function of the trajectory, depend on the angular momentum or the energy when the equations of motion don't depend on these things? $\endgroup$ Commented Nov 12, 2017 at 9:49

2 Answers 2

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The relationship between $b$ and $r_0$ for the Schwarzschild metric is:

$$ b = \frac{r_0}{\sqrt{1 - \tfrac{r_s}{r0}}} $$

where $r_s$ is the radius of the event horizon. See this paper for the gory details.

The equations given in the two articles are derived in the weak field limit i.e. $r \gg r_s$ so $b \approx r_0$ anyway and it doesn't make any real difference which you use. If I were writing the article I would describe $b$ as the impact parameter and not the distance of closest approach, but I don't feel strongly enough about it to want to edit the offending Wikipedia article.

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    $\begingroup$ $b$ is certainly the impact parameter. As you mention, in the relevant limit $b\approx r_0$ and hence I would say it's probably OK for the wikipedia article that's wrong to state that "the length-scale b can be interpreted as the distance of closest approach", which is already worded somewhat carefully. $\endgroup$
    – Danu
    Commented May 2, 2014 at 7:37
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The quantity $b$ in these equations is the impact parameter and is defined as $L/E$ (in units where $c=1$). It is called this because $L/E$ is the perpendicular distance between the trajectory of a photon at $r \gg r_s$ and a radial line through the origin, where the photon has linear momentum $E$ and angular momentum $bE$.

To see how this relates to the clostest approach it is better to write out the equation for the coordinate speed of light as $$\frac{dr}{dt} = \pm \frac{b}{r_s} \left(1- \frac{r_s}{r}\right)\left[ \left(\frac{r_s}{b}\right)^2 - \left(1 - \frac{r_s}{r}\right)\frac{r_s^2}{r^2}\, \right]^{1/2}\ . $$

The "turning points" are when the right hand side equals zero. One of these is when $r=r_s$, because light cannot cross he event horizon in Schwarzschild coordinates. Equating the contents of the square bracket to zero then gives a cubic equation in $r$ that determine the location of any other possible turning points $$ r_{0}^3 - b^2 r_{0} + b^2 r_s = 0\ .$$

If you are interested in light approaching from $r \gg r_s$ with $dr/dt<0$ then it is the largest of the three possible roots, with $r_0 \geq 3r_s/2$, that is of interest (the middle one corresponds to light travelling outwards from $r < 3r_s/2$, initially with $dr/dt>0$, and then falling back; the smallest root is unphysical). Real roots with $r_0 > r_s$ only exist for $b \geq 3\sqrt{3}r_s/2$. Smaller $b$ values mean that $dr/dt$ is always negative until $r \rightarrow r_s$.

The cubic equation above can also be rearranged to give $b$ as a function of $r_{0}$: $$ b = \pm \frac{r_{0}}{\sqrt{1 - r_s/r_{0}}}\ .$$ The plus and minus signs here can be interpreted as clockwise or anticlockwise orbits.

When $r_{0} \gg r_s$ then you can see that $b \simeq \pm r_{0}$ and it would be ok to use $b$ and $r_0$ interchangeably in these circumstances (e.g. the bending of light from stars around the limb of the Sun, where $R_\odot \gg r_s$).

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