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Gravitational lensing is everywhere! because it falls off so slowly with $r$:

$$\Delta \phi \approx \frac{4GM}{c^2r_0}.$$

That's the first order term. For a nice derivation see Viktor Toth's The bending of light by gravity "...square rooted expression in the denominator can be simplified to first order..."

Wikipedia's gravitational lensing breaks it up into three general categories

Questions:

  1. What's the largest angle that light has been "seen to bend" by gravity? (of one object by a separate object)
  2. Was it large enough that a higher order term in this expansion was at all significant?

I'm going to arbitrarily exclude seeing the accretion disk behind a black hole or other strong local phenomenon like that, so I've added "of one object by a separate object" to the title, otherwise the now famous Event Horizon Telescope image seems like a winner. (To my understanding the dimmer part across the top is light from the accretion disk behind the black hole "bent over the top and towards us" Is the angular size of the black hole in the movie "interstellar" completely overblown?) As an aside, certainly the first order equation above would no longer apply in this case.

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    $\begingroup$ This question makes little sense, citing the deflection angle of a point mass. Extended mass distributions have deflection laws that don't look like point masses anywhere where significant deflection occurs. An isothermal sphere, for example, has constant deflection angle. So "higher order terms", being a bad choice of name, are important for almost all gravitational lenses. Besides, it's not clear whether the first question asks about the largest physical deflection angle (i.e. kink in light ray), or the largest apparent displacement of an object seen (i.e. largest Einstein ring observed). $\endgroup$
    – ntessore
    Commented Jan 30, 2022 at 21:20
  • $\begingroup$ @ntessore If you don't like Viktor Toth's use of "first order" in the derivation I've linked to, you can take it up with them. Of course this is for a point object, I've simply used the equation to illustrate why "Gravitational lensing is everywhere! because it falls off so slowly with $r$" Now check the question again; this asks is about the largest deflection angle that has been observed rather than calculated. So the mathematical aspect will not be the focus of answers. $\endgroup$
    – uhoh
    Commented Jan 30, 2022 at 22:12
  • $\begingroup$ Fair enough, although the way the question is written, it sounds as if you are asking about strong lensing (exclude [...] strong local phenomenon like that). There you need large, extended masses to get large deflection angles, which don't deflect like point masses where they are deflecting significantly. So the premise of the question seems odd. $\endgroup$
    – ntessore
    Commented Jan 30, 2022 at 23:45
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    $\begingroup$ In comments: "It seems that observation of light from the accretion disk behind a black hole is a good candidate, why not take a moment and post something like that as an answer?" In the Q: "I'm going to arbitrarily exclude seeing the accretion disk behind a black hole or other strong local phenomenon like that". Confused. $\endgroup$
    – ProfRob
    Commented Feb 8 at 8:56
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    $\begingroup$ @uhoh Sorry, I can focus on the mathematical aspects, but I don't know much about potential record-breaking lensing observations. $\endgroup$
    – PM 2Ring
    Commented Feb 14 at 8:31

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