Two galaxies aligned in a way where their gravity acts as a compound lens

If the lens curved light back toward us, could we see earth several million years ago?

Technically? But the image would be very very very small, so we'd need a detector bigger than the solar system (guesstimate) to see it. That's to see it: I can't imagine what it would take to resolve the image. The tricks in this paper are a start.

To zoom into a reflection on a lens or a water droplet?

From "Hear the sounds of Earth's magnetic field from 41,000 years ago" (2024) https://news.ycombinator.com/item?id=42010159 :

> [ Redshift, Doppler effect, ]

> to recall Earth's magnetic field from 41,000 years ago with such a method would presumably require a reflection (41,000/2 = 20,500) light years away

To see Earth in a reflection, though

Age of the Earth: https://en.wikipedia.org/wiki/Age_of_Earth :

> 4.54 × 10^9 years ± 1%

"J1721+8842: The first Einstein zig-zag lens" (2024) https://arxiv.org/abs/2411.04177v1

What is the distance to the centroid of the (possibly vortical ?) lens effect from Earth in light years?

/? J1721+8842 distance from Earth in light years

- https://www.iflscience.com/first-known-double-gravitational-... :

> The first lens is relatively close to the source, with a distance estimated at 10.2 billion light-years. What happens is that the quasar’s light is magnified and multiplied by this massive galaxy. Two of the images are deflected in the opposite direction as they reach the second lens, another massive galaxy. The path of the light is a zig-zag between the quasar, the first lens, and then the second one, which is just 2.3 billion light-years away

So, given a simplistic model with no relative motion between earth and the presumed constant location lens:

  Earth formation: 4.54b years ago
  2.3b * 2 = 4.6b years ago 
  10.2b * 2 = 20.4b years ago

Does it matter that our models of the solar systems typically omit that the sun is traveling through the universe (with the planets swirling now coplanarly and trailing behind), and would the relative motion of a black hole at the edge of our solar system change the paths between here and a distant reflector over time?

"The helical model - our solar system is a vortex" https://youtube.com/watch?v=0jHsq36_NTU

@Dang is there a version of /best but for comments? The thought experiment in this comment broke my mind.

No, because the light requires twice the time to travel there then back. If Earth did not move relative to the lens, it would work. Sadly we move, a lot, so what was here 2x ago was something not-earth.

To see earth, the lensing would been to be focused on where Earth was 2x ago. Still possible in theory, and you might even argue just as likely as a fully reflecting curve. But you'd not call it "back towards us". It would need to be "curved to where earth was".

Seems like if you could retrodict the position of past lenses, and predict their effects, perhaps it would somehow be possible to send a spacecraft to a specific location in order to observe Earth's past.

The idea being that a spacecraft traveling at 99% of light speed can't ordinarily catch up with light reflected by Earth. But if the light curves, and the spacecraft can travel directly towards where the light will end up (spacecraft traveling "as the crow flies"), it might be possible to catch up.

Same way I might be able to catch up with Usain Bolt at a track event if he's forced to run on the track, and I'm allowed to run across the turf in the middle.

I believe this could only happen around a black hole. In that case, yes: light that we emitted umpty-million years ago could be shot back at us. The problem is that there would be no focussing. At best it would be like looking at an Earth 2 x umpty-million light years away. I'm guessing that it would actually be worse, with the black hole dispersing light.

(IANAastronomer, but I have opinions on any given topic...)

Would this be the case even if you were moving toward or along-side the 'reflector' (black hole/other body)? For the sake of discussion assume we are at or beyond the focal point.