# Can an image formed by gravitational lensing be corrected for the inevitable aberration?

Carl Zeiss would not be impressed with your average gravitational lens. Compared with familar optical lenses that are generally used to form sharp undistorted images, gravitational lenses make quite a mess of the background object. A compact mass makes for a nice circularly symmetric lens, but the deflections are proportional to $$1/r$$ rather than $$r$$ like a conventional lens. (here $$r$$ would be the distance the light ray passes from the center of the lens).
To what extent can images like Einstein's ring or Einstein's cross be corrected by optics and/or electronics and/or image-processing? Presumably, if one can correctly recognize which parts of the image correspond to each other, one can combine the appropriate energies (number of photons). But one can do much better in signal-to-noise if the actual wavefront distortions (delays) can be exactly compensated. For a distant galaxy are these differential delays (e.g. the four images in Einstein's cross) measured in nanoseconds or in years? Nowadays we can measure time with extreme accuracy and can demodulate coherently at optical frequencies. So this type of wavefront correction does not seem entirely out of the question.

• I am not posting this as an answer because I am not sure of its correctness. But I believe that the time delay will strongly depend on the type of lens. For a galaxy, I would not be surprised if we're talking delays on the order of thousands of years or more, if the object being imaged is not exactly on the reverse side of the galaxy, by simple (a)symmetry arguments, plus the scales of galaxies. Commented Jan 3, 2021 at 4:50
• (Or in SI units, on the order of $10^{12}\ \mathrm{s}$, most likely.) Commented Jan 3, 2021 at 4:53
• I see the Einstein Cross subtends 1.6 arcseconds or 8 microradians. The foreground galaxy is 400 million light-years away. So the difference in delays will be of the order of 400-400cos(8 microradians) = 400,(1/2).(8.10^-6)^2 = 1.28e-8 years = 0.4 seconds or 120,000 km. Light has a wavelength a bit less than 1 micron, so this means we have to do the wavefront correction with an accuracy better than 1 part in 10^15 - which sounds barely possible. Commented Jan 3, 2021 at 6:52
• Um, you made a lil mistake there. That's 1.28e-8 MEGAyears (Myr), because you left off the "million" on 400. Or 1.28e-2 years. 6 orders of magnitude, 120 Tm, and you need 1 part in $10^{21}$. So it's absurdly small, and that's the answer to the question. Getting phase when $10^{20}$ish cycles of the wave have separated things is mighty hard even for astronomy equipment. Commented Jan 3, 2021 at 7:02
• That said, I do wonder about other methods. Since we're talking a galaxy, there is no reason that observation must be carried out only in visible light. We can use other wavelengths - perhaps much longer ones, like radio waves, and we could still have near-perfect theoretical resolution. All you need to do is go from micrometer waves to meter waves, and now you've compensated away that $10^6$ factor again. And maybe you can use the information from that, figuring out the "texture" of the gravitational lens, to then remap an image taken in other wavelengths, i.e. use measurements Commented Jan 3, 2021 at 7:20