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I watched a video on YouTube analyzing some anime fight scene where a guy used a black hole as a weapon or something, and I immediately thought, "how could someone actually use a black hole as a weapon?"
Would it be possible to create and "fire" a black hole from some super-satellite/space station and fire it a few miles above the Earth, above your target, causing massive, if not total destruction?

The reason black holes might be interesting weapons is that they work against all kinds of matter, even dark matter. The reason they might be useless is that black holes are small, too hot, and hard to make.

In principle one can fire a black hole by giving it an electric charge and surround it by electromagnetic fields. This leads to the first problem: an easily accelerated black hole will have a low mass.

The second problem is that the capture cross section, essentially how "broad" the black hole is, is about $(27pi) G^2M^2/c^4$. That is very small for a low mass black hole.

Sure, the hole will attract matter in its vicinity and accelerate it, but most matter just swing around the black hole without hitting it. This acceleration is actually what would do most damage by ripping apart nearby objects and irradiating the vicinity by x-rays from accretion.

If we consider a 100 ton black hole (that can be moved with a cannon-like device) it has a radius of $1.4852cdot 10^{-22}$ meter and a cross section of $4.6775cdot 10^{-43}$ square meter. That is about a trillion times smaller than a proton's cross section. It is very hard to absorb matter this way. It does produce a force of about 6 N at a distance of 1 mm, but that is likely too fine calibre to be useful.

What will do damage is Hawking radiation. The black hole will radiate $3.5609cdot 10^{22}$ Watt - about 1/10,000 of the total solar output. It will radiate away its entire mass in 0.0841 seconds. That is going to be an impressive weapon - but it will also damage the cannon, and even when thrown at lightspeed it has just range of 25,206 km. So the third problem is handling the Hawking radiation.

Throwing a big black hole that can actually rip apart macroscopic objects reduces the Hawking radiation somewhat. A billion ton black hole will exert 66 N at 1 m distance, and shine with 84 exawatt of power. But it is still not going to eat planets (since the radiation keeps matter away from it, and it is very rare for a particle in the plasma around it to hit the tiny hole). So this is still damaging to the cannon, doesn't make that big holes, and is now very hard to accelerate.

The fourth problem is of course how to make the black hole. Somehow enough mass-energy can be compressed into a tiny volume. But if you have that ability, why not just throw the energy at your target?

Gravity is an extremely weak force. You need lots of mass to do anything worthwhile - and that entirely eliminates any thought of using black hole guns as personal weapons.

There's another problem. Gravity affects everything - you can't "shield" against gravity. It's always attractive, so you can't even "negate" gravity. In the end, if the weapon had enough attractive force to do any serious damage to your target, it would also kill you, long before you could fire the weapon. The same thing that makes it interesting (affecting every kind of matter and even particles without mass) also makes it very impractical as a weapon.

The closest thing to a gravity weapon that would be practical at all would probably be some space-based "gravity tractor" - a relatively massive spaceship that would keep station "above" an asteroid to disrupt its trajectory to make it hit a planet. Would black holes be better? Not really. Black holes have the same gravity as a non-black hole object with the same mass, as long as you're outside of the radius of that object. And worse, they radiate lots of energy away as they "evaporate" - not only making small black holes extremely unstable, but also once again posing great danger to the operators of the weapon.