Physical Paradoxes
we do have FTL travel and communication, but other laws of physics are the same
They can't be the same, FTL breaks known physics. A lot.
The speed of light isn't really the speed of light, it's the speed of information exchange and the speed of causality.
FTL probably breaks causality, meaning you can time travel to the past. If so, the best defense against a black hole attack is to time travel back and prevent the attack.
Your FTL tech could be used to defeat a black hole, but we'd need to know more about how it works. FTL has some pretty funky consequences for black holes.
A black hole is a region of space where gravity is so strong not even light can escape. Imagine trying to swim against a river that's flowing faster than your top speed. But if you can exceed the speed of light you can escape a black hole.
The event horizon of a black hole is not just the point of no return, it defines an area of space which can never interact with our own. Inside the event horizon is, effectively, outside our universe. Once inside a black hole you are in your own universe.
Except with FTL you can come back. This means you can bring information out of a black hole. With FTL, we can interact with another universe.
We do have teleportation tech
Without putting some careful limits on it, teleportation means infinite energy. Teleport something high in the air, extract energy from it as it falls, repeat.
One way to avoid this is to make the energy cost of teleportation equal to or greater than its potential energy gain.
But let's forget about that for the rest of the answer.
Original Answer
You have bombs you can drop anywhere, there's no defense, and they're infinitely powerful. Not too different from reality. Just as in reality, you don't fight overwhelming military force with military force. You conduct asymmetric warfare.
- Hide. You can't hit what you can't see.
- Make the environmental cost too high (it eats planets).
- Make the collateral damage too high (did we mention it eats planets).
But you obviously want a direct counter-measure. In order to do that we need to make some things clear about black holes.
Black holes don't have any more gravity than a normal object.
When a star collapses into a black hole, it doesn't suddenly suck everything in. It has exactly the same gravity as before. You wouldn't notice... gravitationally speaking. If you're not killed in the nova (which blows off mass reducing the gravity of the black hole) you'd freeze to death because your local star just went out.
This is the key to how we're going to get rid of this thing. A black hole can still be treated as a point mass. It's still subject to the law of gravity.
As an alternative weapon you could make neutron stars; they would do the trick just as well. Or you could create a normal star; just as much gravity, and it will burn you to a crisp from millions of miles away! It could be a wonderful terraforming device, or a hideous weapon.
Black holes evaporate.
The smaller the faster. Unfortunately your one meter black hole will take too long, probably longer than the age of the universe.
Black holes radiate tremendous energy.
...when stuff falls into them. This is called the accretion disc, and it's actually rather hard to fall into a black hole once you're orbiting it. To "fall" into a black hole you need to expend thrust to brake.
As it consumes stuff in our very, very cold outer solar system, our black hole will start to radiate energy from its growing accretion disc. This is how we can detect it, and we want to detect it while it's far away and small.
Black holes have only three properties.
Mass, charge and momentum. When matter falls into a black hole it retains these properties. This is very important for your question, especially momentum.
The Earth can be torn apart by its tidal forces.
The Earth doesn't need to cross the event horizon to be destroyed, it can be ripped apart by tidal forces. The side of the Earth facing the black hole is $13000 \; \text{km}$ closer to the black hole than the far side, meaning the gravitational pull is stronger. As the Earth gets closer to the black hole, this difference gets so strong it rips the Earth apart.
The point where a body is ripped apart by tidal forces is called the Roche Limit, and we don't want the Earth getting any closer to your black hole than that.
Now back to our problem.
The attack occurs outside our planet, but near or inside our solar system, and start very small (say a $1 \; \text{m}$ diameter, so it's needs a few time before eating an entire solar system).
Will this thing eat the entire solar system? Its gravity is related to its mass. So...
How much mass does a one meter diameter black hole have?
When you say "$1$ meter in diameter" I presume you mean the diameter of its event horizon. We need its mass. $R$ is the event horizon radius. $G$ is the gravitational constant. $M$ is the mass of the black hole, $c$ is the speed of light.
$R = \frac{2GM}{c^{2}}$
Solving for $M$...
$M = \frac{c^{2}\cdot R}{2G}$
Plug in the numbers, $R$ is $0.5\;\text{m}$, $G$ is $6.674\cdot 10^{-11} \; \text{N}\cdot \frac{\text{m}^{2}}{\text{kg}^{2}}$, $c$ is $3 \cdot 10^{8} \; \frac{\text{m}}{\text{s}}$.
$M = \frac{\left(3 \cdot 10^{8}\;\frac{\text{m}}{\text{s}}\right)^{2} \cdot 0.5 \; \text{m}}{ 1.3348 \cdot 10^{-10} \; \text{N}\cdot \frac{\text{m}^{2}}{\text{kg}^{2}}} = \frac{4.5 \cdot 10^{16}\;\frac{\text{m}^{3}}{\text{s}^{2}}}{ 1.3348 \cdot 10^{-10} \; \frac{\text{m}^3}{\text{kg} \cdot \text{s}^{2}}} = 3.37 \cdot 10^{26} \; \text{kg}$
That's a little more than half the mass of Saturn. Saturn is a big gravitational player, but it hasn't hoovered up the entire solar system. This thing is going to have to get pretty close to Earth to mess with it, and it's not going to be gaining much mass on the way.
What is the Roche Limit?
As mentioned earlier, a black hole can rip the Earth apart with tidal forces if it gets too close. How close? We need to calculate the Roche Limit.
$d = r \cdot (2 \frac{M}{m})^{\frac{1}{3}}$
Where $d$ is the Roche limit, $r$ is the radius of the Earth ($\approx 6300 \; \text{km}$), $M$ is the mass of the black hole ($3.37 \cdot 10^{26} \; \text{kg}$) and $m$ is the mass of the Earth ($6 \cdot 10^{24} \; \text{kg}$). I got $30400 \; \text{km}$.
Depending on its trajectory and speed, it will probably pass harmlessly through our Solar System, maybe mess up the orbits of a few planets that we can correct later. But let's assumed it's aimed at Earth. What can we do?
I'm going to ignore infinite energy and FTL and teleportation and anti-gravity and non-existent anti-particles, because once you have all that you might as well say "magic". Don't need them anyway, we have a perfectly fine way of handling rogue stellar objects using nothing more than Newton's Law of Gravitation.
Use a Gravity Tractor!
No, it's not a tractor beam.
Put a spacecraft in orbit around the black hole and, without touching it, use the spacecraft's gravity and thrust to change the black hole's trajectory to miss the Earth (and presumably go somewhere safe, like a parking orbit around a brown dwarf). Since we have infinite energy, it doesn't really matter how fast or how massive the black hole is, we can keep building bigger tractors.
Hmm... this black hole has the mass of Saturn. The Earth is about 100 times less massive than that. It would be easier to move the Earth (and anything thing smaller than Saturn) out of the way. Just be sure to remember to put it back.
This may sound crazy, but it's the most sensible way we currently have to defend against asteroids.
Knock it away by shooting it.
Since black holes have mass and momentum, you can treat them like a big, sticky billiard ball and smack it with another big billiard ball.
Let's say this $1 \; \text{m}$ black hole is coming toward Earth at high velocity and we detected it around the orbit of Neptune, about $30$ AU away. We'd probably notice distortions in the orbits of outer solar system objects. This gives us plenty of time to react. Even light takes four hours to reach the Earth from out there.
Momentum is mass x velocity. We have infinite energy to play with, and we don't want to make the black hole too much bigger, so we're going to fire a slug at near the speed of light. For best results, our anti-black-hole cannons are positioned above and below the plane of the solar system. That way the slug impacts at a 90 degree angle and bounces the black hole above or below the orbital plane where there's less for it to mess with.
How far do we have to knock the black hole? Will just one degree do? Let's say our slug impacts at $20 \; \text{AU}$ and knocks the black hole out of its course by $1$ degree. By the time it reaches the Earth it will be $\tan\left(1 \; \deg\right) \cdot 20 \; \text{AU} = 0.35 \; \text{AU}$ outside the plane of the solar system, or $52500000 \; \text{km}$. Safely outside the Roche Limit by three orders of magnitude.
What about shooting it with anti-matter?
Another answer said to shoot it with anti-matter, which is a really good idea, which is a really bad idea because it will just make the black hole bigger!
But for funsies, what happened if they did annihilate? How big an energy release would it be?
We know the mass of the black hole, $3.37 \cdot 10^{26} \; \text{kg}$, and we're going to shoot another object made of the same amount of anti-matter at it. Plug that into the old e=mc^hammer formula and we get... $3.03 \cdot 10^{43} \; \text{J}$. How much energy is that?
...going off inside the solar system.
Hmm.
Let's try the gravity tractor.
