Can single black holes be observed by its lensing effect?

A comment on a question I asked left me confused. I asked about the possibility of observing a binary black hole by examining the lensing the binary produces. A binary has a different lensing effect as a single hole. My idea of lensing is turned upside down. I assumed it to be possible to see a hole by lensing. Which it is not, according to the comment. The hole is tiny indeed (only 3 kilometers in the case of a Sun mass hole) but does this mean the image of the stars behind doesn't change? Of course the hole must be moving wrt to the stars as seen from here. Is that not the case maybe?

• 3 km at a distance of 1000+ light-years is pretty small. ;) en.wikipedia.org/wiki/List_of_nearest_black_holes Aug 1 at 17:24
• @PM2Ring Yes, thats true...I guess there is not enough light entering from whatever direction on the hole to be observable here. But what about the claim that BHs are mirrors? Are only big ones involved? Aug 1 at 17:49
• Do the “ pictures” of the event horizon telescope count? Aug 1 at 22:42
• @mmeent I think the image you're referring to is of the shadow of the black hole ias.edu/press-releases/2020/eht-gr-test Aug 2 at 1:01
• @DaddyKropotkin The Event Horizon Telescope pictures absolutely show lensing by the BH; the arc at the "top" is light from the accretion disk behind the BH, bent as it passes near the BH. Aug 2 at 10:57

In principle, sure. The basic phenomenon of lensing occurs on scales of the "Einstein radius", which is a function of the mass of the lens (e.g., star or black hole) only, not its compactness. So-called "microlensing" occurs when the relative motion of a lensing mass (the "lens") and a more distant star (the "lensed" object) brings the star within the projected Einstein radius of the lens, which produces multiple images of the background star (or, if the lens and background star line up almost perfect, a ring of light)[1]. We can't resolve the multiple images, so what we see is the background star getting brighter (and then fainter as it passes out of the Einstein radius).

Microlensing of stars (e.g., stars in the Galactic Bulge) has been observed many times; people have even detected secondary amplifications due to a massive planet orbiting around the lensing star producing its own microlensing signature.

The only problem is that the microlensing signature due to a star being the lens is pretty much identical to the signature of a BH being the lens. The only practical way to tell what's going on would be to get a mass measurement of the lens. If it were, say, $$\sim 10$$ solar masses and there was no bright star corresponding to the lens (an actual star that massive would be very bright, and probably easier to detect than the background star!), then you could conclude the lens was a dark object with that mass, and thus almost certainly a BH (since $$10 M_{\odot}$$ is much too large for a white dwarf or neutron star).

This is discussion of a 2016 paper that tried a clever technique involving looking for shifts in the position of the background star as a way of getting a better measurement of the lensing object's mass, since anything other than perfect alignment would produce off-center lensed images). (They didn't really find anything, but maybe in the future...?)

[1] This has nothing to do with the "photon sphere" of a black hole; an "Einstein ring" is much, much larger and doesn't require a black hole as the lens.

What you are referring to here is called microlensing https://en.wikipedia.org/wiki/Gravitational_microlensing when a lens is a compact object (like a black hole).
There are some simple equations in the article so one can calculate the separation between the black holes in the binary BH system required in order to be observed. There are two manifestations of the microlensing: changing position of the lensed object and changing of its "brightness". To detect the former effect, I'd suggest that it is required to observe distant compact quasars to be lensed at radio wavelengths with VLBI technique https://en.wikipedia.org/wiki/Very-long-baseline_interferometry. This might give sufficient resolution to distinguish between a lonely BH lens and a binary BH lens. The effect was detected already, but afaik with no chance to resolve the possible binary nature of the lens. Check this out: https://www.aanda.org/articles/aa/pdf/2013/07/aa21484-13.pdf