For instance; life manages to develop on a planet orbiting a black hole. What are some methods that native life could use for perceiving the environment around them?
(e.g. echolocation, sonar, gamma-ray detection, etc.)
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Black holes are pretty strange environments. There's quite a lot going on around them, but it depends on exactly what kind of black hole there is and what's happening around it.
So, black holes exist in many different kind of environments. You can have single, stable black holes that function pretty much as dark suns, with a planet orbiting quite far away and not really noticing or caring that it is a black hole. This then pretty much boils down to many of the same themes of life on a super-dense planet.
However, there are lots of environments where it gets interesting and different. For example, black holes in areas where there are large amounts of matter falling into the black hole create accretion disks. These emit in X-rays and maybe gamma rays.
This creates some interesting effects in terms of sensory perception, because X-rays and gamma rays are quite penetrating. However, they're also likely to ionize any potential atmosphere. In Greg Egan's Incandescence, which is about small asteroid orbiting an accretion disk, the world is mainly vacuum and sight seems to be by X-rays, which quite happily penetrate through the rock several kilometers, which felt right to me.
Most (but not all) accretion scenarios involve a companion star that's emitting EM, but this isn't necessary (e.g. nebulas).
Stellar locality also matters. Black holes are more common near the galactic centre and there's also the super-massive black holes there. The neighborhood around there is teeming with stars and encounters are common. Any life will have to have evolved a way of dealing with a Sun/planet dropping into its black hole on quite frequent timescales (by evolutionary standards) (also explored in Incandescence). Towards the galactic edge, things are more boring, stellar encounters rare and, again, back to boring dark star routine.
Closer to black holes, gravity gets weird. It's difficult to talk about this, because I don't think a planet can form close enough to a black hole to experience any of this (unless it's been captured in its past). But, near black holes, gravity gradients become pronounced and you can feel differences between one side of a planet and another quite easily. Again, Incandescence goes into this. Indeed, the orbital mechanics gets so fast and messy that a planet would be going around so fast that the difference in centrifugal forces between the centre and outer sides would be noticeable and would probably rip the planet apart long before anything interesting happened. That's why Incandescence deals with asteroids.
Going back to locality, if two (or more?) black holes are together (e.g. binary), then you get gravitational wave emission. This does some interesting stuff as well, because in theory, something could pick those up. In practice, I find that doubtful, as GW detection is really hard and, compared to GWs, neutrinos are positively detectable (and neutrinos are the hardest particle bar the Higgs we've discovered).
However, GWs near black holes might be felt. The ripples in space time can cause gravity currents that you would feel. However, these would probably prevent planetary formation and, well, GW emission would cause the black holes to inspiral towards each other. I can't remember the timescale for that, but I think it's short by evolutionary standards.
Incandescence does talk about General Relativistic effects being easy to detect, however. In Incandescence, because of the proximity to a huge gravitational object, GR stuff occurs regularly and the characters can discover GR simply by running classroom experiments. This is true, and since it's all observable, they never bother with Newtonian approximations. This could be interesting for life, since all Earth life has been wired to deal with Newtonian physics because it makes sense for us.
Accretion environments might also have issues with strong magnetic fields and electrostatics, which can provide a lot of fun for creating strange and interesting senses. It's known that accretion disks in neutron stars, for example, have strong magnetic fields (and neutron stars that are pulsars certainly do, though I don't think black holes do). There's a field called MagentoHydroDynamics that tries to describe how fluids behave in strong magnetic fields, which most accretion disks follow. That could provide some fun for life, though I suspect any planet that survives near a black hole in an accretion disk would filter a lot of that out.
Finally, there might also be lots of strange particles being emitted and breaking down, which might have some sensory perception possibilities. We get bombed by lots of Mesons caused from interactions in our upper atmosphere from seriously high energy radiation. X-rays are still too low in energy to do most of the good stuff, but go higher and you can have some fun with short-lived particles like Pi mesons and muons as well as neutrinos.
The two methods you mentioned in your question are both very viable methods of "seeing" , but it completely depends on your universe, and I would definitely avoid vanilla methods like echolocation and go with something more creative, for example in your planet orbiting a black hole there could be some sort of way within the race who live there to detect black body radiation , like would be given off from a black hole, and the distortion that the environment would cause to that blackbody radiation. Perhaps the community would be able to help you more if you have specific details of the scenario?
This question has been marked as a duplicate to a question about seeing in a super dense atmosphere but I believe this to be a far more generic question, asking about alternative methods of seeing which could be applied in multiple scenarios , not a particular one.
A black hole, hmm...
To start off, I'll define what I imagine these creature's habitat to be like. Life needs energy to survive, and most life on earth gets that energy from the sun as light, but with a black hole you aren't going to get anything that convenient. (Though I suspect you'll run into all sorts of difficult-to-survive radiation... but that's not quite what the question is about.) So, what kind of life supporting habitat could we hope to find on a land without light?
Let's try in the ocean. On Earth, in the dark depths of the ocean, there are hydrothermal vents called black or white 'smokers' that range from 60 to 464 °C. (The water is kept from instantly boiling by the water pressure). According to NASA, such vents also likely exist on Jupiter's moon Europa, and Saturn's moon Enceladus. On Earth, these habitats are richly populated by Chemosynthetic bacteria and archaea, giant tube worms, clams, limpets and shrimp, so we know they can support life. You could hope to find a similar setup on a planet orbiting a black hole, or a large moon of such a planet, even if the ocean's surface was frozen over. (Ice on the planet's surface may even help to block some harmful radiation?)
Personally, I favor the idea of having the creature use both electroreception and biosonar so that it can both identify the specific type of organisms it encounters from a distance, and avoid running into rocks. Happy designing!