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Do black holes really exist? If yes, why we have not found them moving around neighboring galaxies?
Answer 1:

They do! Or at least, we have very, very good evidence that they do.

Part of the problem with detecting black holes is that they, well, black: they emit no light, so a black hole by itself is impossible to see (unless you get close enough to it to detect the very strong gravity from it). However, if any ordinary matter gets close enough to a black hole, it will spiral into it and heat up due to friction with itself. Because of the black hole's intense gravity, the matter can hear up a lot, to the point of emitting very strong jets of radiation. These objects are called quasars, and we have observed them at the centers of many galaxies. Although the black hole causing them is impossible to see directly, the evidence very strongly suggests that they are caused by black holes.

In addition, we've also observed some of the stars near the very center of our galaxy, and discovered that they seem to be orbiting a very, very massive invisible object - to the best of our knowledge, this invisible object is a black hole. In fact, evidence seems to suggest that every single galaxy has a supermassive black hole at its center.

Finally, there is hope that eventually we'll be able to detect black holes directly in the coming years through detection of their gravitational waves. It turns out that according to Einstein's theory of gravity, accelerating massive objects should emit gravitational waves - these are basically ripples in spacetime. There are currently experiments running (the biggest one being LIGO - the Laser Interferometer Gravitational-wave Observatory) which are designed to detect these gravitational waves when they pass through the Earth. One of the possible sources for these gravitational waves is the collision of two orbiting black holes, and LIGO scientists hope to be able to detect these waves in the near future.

I hope that clears it up a bit!

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