Part 4a: Speeding Up
Ok, so how do you get up to 0.9 c in the first place?
The common answer is "antimatter", but antimatter actually isn't that good for missions that are extremely relativistic. This is because of the Tsiolkovsky Rocket Equation, which applies anytime you're carrying your energy source and propellant onboard. Fortunately, it's very simple. . is your change in velocity. is your exhaust velocity. And is the mass of the rocket full of propellant, with being the mass of the rocket without propellant.
Eyeballing this, we see that the exhaust velocity gives you a decent approximation to how much you can change your velocity by, if you've got about 2 parts propellant to 1 part mass. Getting more velocity change requires an exponential rise in your mass ratio, and very rapidly gets to not be worth it, as pretty much no rocket has a mass ratio greater than about 20. Also, for stuff going really fast, the energy delivered is high, but the momentum isn't nearly as high, so high-specific-impulse rockets that whip their exhaust up to relativistic speeds emit an awful lot of energy, but have the sort of thrust typically associated with a fleet of asthmatic hummingbirds because they're very fuel-efficient and have a low mass-loss rate.
There are relativistic adjustments, of course, but the same basic behavior applies. Also antimatter annihilation has the problem of spending about 40% of its energy as gamma rays which just go in all directions and can't be used for thrust as a result, so you have to adjust the equation to account for that inefficiency.
So, even for a beam-core antimatter rocket, it's a bit more disappointing than you'd think. The classic example of this is the Frisbee Antimatter Starship, a hilariously ambitious starship design that is about 700 km long, has about 160,000 tons of antimatter aboard, blasts out 100 terawatts of power, and achieves a measly 0.25 c. There have been notable improvements in beam-core engine design since then, but it's still too much for too little speed. And eyeballing it, the dust shield looks pretty puny, because it's sized for erosion from relativistic protons and small dust grains, not the cruise-missile-level dust grains I'm worried about.
Certainly insufficient for an intergalactic mission at high-relativistic velocities.
Edit: Found another beam-core antimatter starship design that does a lot better, the Valkyrie starship. Apparently the titanic size is partly an artifact of trying to squeeze high thrust out of an antimatter beam core so it doesn't take millenia to get up to full speed, and the antimatter beam core innately has very little thrust, so you have to crank up the power to enormous extremes. It's also partly an artifact of having a gamma-ray shield that's a lot bigger than strictly necessary with a different configuration, which requires massive radiators, which means you have more dry mass to push, which means you have to make everything else bigger to compensate, which includes the rocket and the shield and you make the radiators bigger again... The Valkyrie claims the ability to get up to 0.92 c and back down with a 20:1 antimatter to ship ratio by mass, or 2,000 tons of antimatter, which is a bit much, especially because solid (anti)hydrogen isn't very dense, and if it's possible to go a given speed without getting destroyed by dust, the upcoming pair of approaches seems to be strictly better than any given antimatter rocket design because they completely dodge the rocket equation and the pesky ln term interfering with high-relativistic speeds, and also don't require enormous amounts of antimatter.
But if you think about it, why does the energy source for getting the ship to go fast have to be onboard?
A far superior solution is a lightsail, a gigantic and very thin and very reflective sheet that you can fire a laser at to get your payload up to speed. Now, I didn't really design this part to a high degree of detail, and I think there might be issues with having a sufficiently large sheet not crumple like tissue paper under the stress of its launch. A spot for future work on making a realistic design if someone wants to take it up. You also need a gigantic floating laser lens to shoot the thing up to a distance of about 40 lightyears.
However, assuming you've got a dyson swarm available, you have more than enough energy on tap to bring whatever you'd want up to high-relativistic speeds. I was getting numbers that were something like an exawatt per ship (and again, we'd need 30 of them). So you'd need astronomical levels of energy-harvesting and lasers, and especially heat radiators, and this wouldn't be an immediate pulse, but you'd be cranking out multiple exawatts for decades at a time. Fortunately, assuming the ability to devote enough resources towards firin an astronomically large lazor, this is peanuts compared to the energy that's available from a star, and it lets you skip the rocket equation completely! No reason to be powered by antimatter when you've got the fury of a star-powered laser at your back, launching you unto the cosmic void.
Of course, transhuman technology might find something better, or a great refinement on the basic idea, but I'm still pretty confident that they'll skip designs subject to the rocket equation, that is a pretty punishing aspect.
But how do you slow down? That will take just as much energy as speeding up....
Part 4b: Slowing Down
Once upon a time, Robert Bussard had an idea for a starship. Interstellar space isn't empty, it has a very thin misting of protons in it. If you could do proton-proton fusion, you could have a giant magnetic scoop that funneled the interstellar medium into the rocket, where it'd fuse it for energy, then shoot it out the back, so it'd be gathering its own propellant and energy source as it went, and could get up to very relativistic speeds.
It captured the popular imagination, and then more calculations were done. It turned out to be bad. Really bad. So hilariously bad that it managed to achieve the elusive feat stated in Reversed Stupidity is not Intelligence about how a broken car couldn't go 200 mph in reverse, even if it was really broken.
The magnetic field produced a lot of drag. A hell of a lot of drag. In fact, the basic insight of "set up a large magnetic field in the interstellar medium" is the currently known best way to come to an absolutely screeching halt from relativistic speeds and is plausibly going to be an indispensable part of any serious interstellar mission. It produces so much drag that it is used to drop my starship design from 90% of lightspeed to 2% of lightspeed in 1.5 lightyears, pulling 1.5 g's of deceleration at the peak. This is a lot.
It turns out the way to decelerate from relativistic speeds doesn't take a rocket, it just takes a big loop of superconducting coil towed behind you and which slows down by dumping kinetic energy into violently shoving interstellar hydrogen away.
Now, there's a caveat. My design actually doesn't shed most of the kinetic energy. The analogy is that if you've got a crashing plane and a passenger on it, you're much better off attaching the parachute to the passenger than the crashing plane. Yes, you're going very fast, but you're only going through a lightyear and a half, so much less dust shielding is needed because you're much less likely to get hit in that space interval, so you can just separate from most of the dust-shielding block, let it streak through the galaxy at 0.9 c, (and get spectacularly wrecked in a violent kaboom by a piece of gravel at some point), and keep dumping shielding-mass as you slow, which makes it even easier to slow you down, and it feeds on itself until most of the remaining mass is actually in the superconducting coil.
There's some further details, one is about how to slow down from 2% of light speed (magsails don't slow you much at nonrelativistic velocities, but this is still far beyond the capabilities of almost all rockets that aren't antimatter, but there's another way to cheat this without propellant), and the other is about how you probably can't hit a specific star from 200 million lightyears away so you'll need some extremely beefy engine to get about 0.1% of lightspeed of on approach so you can boost sideways to aim at a specific star that looks promising, but those are implementation details that I'll go over later.
Part 5c: Power Sources and the Proper Use of Antimatter
Wait, didn't that previous stuff about needing an energy source for the final deceleration and the magnetic parachute, imply the use of power?
To a first approximation, there's exactly three energy sources that are compact enough for space missions (that we know about given present technology). There's antimatter, which releases about 100% of itself as energy. There's fusion, which releases about 1% of itself as energy. And fission of radioactive elements, which releases about 0.1% of itself as energy.
Obviously you'd want to use antimatter, right?
Well, it depends on how much you're using. You see, antimatter annihilation has extremely penetrating decay products. There's a bunch of very high-energy gamma rays (low hundreds of megaelectronvolts, MeV). There's a bunch of charged pions with a similar energy range, which go about 60 m or 60 ft (I forgot) before decaying into muons and neutrinos. Both muons and charged pions are really penetrating. We regularly find notable levels of muon radiation from cosmic rays 100 meters down in the earth, which is why many sensitive particle physics occur in deep mines, and pions are about equally penetrating due to a similar mass. Now, these pions and muons are much lower-energy than cosmic ray muons, so the situation isn't quite that bad, but they still have a tendency to require an awful lot of shielding. And a couple percent of the energy is radiated as kaons, which have similar issues, and can be charged or uncharged, the latter of which is unaffected by magnetic fields.
Amusingly enough, kaons contain a strange quark, which marks the only time the strange quark is actually relevant to a practical engineering design.
I'll get into more details later, but you'll require a pretty healthy weight of shielding mass unless you want to lose a bunch of your antimatter energy to space and hose every starship part in the vicinity with enough gamma radiation to give a person an instant-coma radiation dose in a few seconds. Yes, there are no people, just silicon chips, but radiation hardening isn't that advanced. (Yet)
So, in the limit of large amounts of energy, antimatter is definitely the best. But for smaller amounts, antimatter power's total mass is dominated by shielding mass, fusion's mass is dominated by the mass of whatever the most-compact fusion device of a given wattage the future can come up with (and neutron shielding, if they go for that type of fusion power), and fission... has a bunch of weight by itself, but you also need your nuclear reactor and the associated shielding.
My design has about 160 g of antimatter on board, which is both quite manageable to produce relative to the absurd 150,000 tons an antimatter starship needs (Edit: see above, maybe not), and in the realm where it's kind of unclear which power source does best. I picked antimatter over "ultra-compact fusion reactor" mainly because it's sexy and more fun to speculate about. I used about 3 tons of shielding, so maybe fusion would be better if the future can make a fusion reactor that produces 10 megawatts and weighs under 3 tons. Or maybe a fission reactor could make it work, although the fuel alone (with a very efficient 20% burnup) would weigh a ton, and this neglects the rest of the reactor and neutron shielding.
Part 5d: You've Gotta Have Radiators
Vacuum is a great insulator! This is why vacuum-layer windows are awesome for insulation, because the only way heat can leave is by radiating away. This is a big problem in space travel, though. If you're cranking out a gigawatt of heat energy, your spacecraft will heat up until it's radiating a gigawatt in thermal radiation and glowing bright orange, toasting anything onboard that requires temperatures lower than molten iron to function.
So most of a practical spaceship's visual space is composed of radiators. Ordinary chemical rockets drop much of their energy in the form of hot escaping propellant, but fusion, fission, and antimatter rockets are very efficient with their propellant, so this avenue isn't available.
You'll need some way to deal with this if you want to do any space mission with a fission, fusion, or antimatter power source of any appreciable magnitude. Remember the Frisbee Antimatter Starship I mentioned earlier? 500 of the 700 km of length is just a gigantic radiator to dissipate the heat being absorbed by the gamma-radiation shield of the antimatter engine. I got my radiator for the antimatter reactor down to a paltry 1/4 of a kilometer, and I feel pretty proud about that.
An especially cool technology for this is the liquid drop radiator, which uses some sort of molten metal, and sprays it out as a sheet of fine droplets which has massive area, which is then collected and recirculated. It's unsuitable for really long missions because of very slow metal evaporation into space, but pretty nifty.
Due to the unsuitability of these for really long missions, the part in my design where there's a 10,000-year burn of a dusty-plasma-fission rocket, (The antimatter beam core is also acceptable, and probably has more manageable radiation shielding issues, but I wanted to highlight an obscure design that shows that fission can be surprisingly effective) for steering to a good-looking star, cranking out 3.5 gigawatts of heat the whole way, required something a bit more... solid. Diamond is the best heat conductor, and I'm assuming it's available by nanotech, so the giant cylindrical plug of graphite is also going to have extensible diamond radiator fins that will glow bright orange on approach.
Part 5e: More Notes On Antimatter Shielding
I think magnetic fields can confine the pions and muons and charged kaons to a finite region until they interact with something, dissipating their energy, and then you just need gamma ray shielding. Also, beams of charged particles can have energy extracted from them in a much more efficient way than dissipating heat. This would probably be used in a practical design, but I was being stubborn and wanted to capture the neutral kaons too, and figured "hey, if we're shielding gamma rays, is it practical to shield everything and drop the mass of the magnet and energy-extraction subsystem and have a vanilla turbine operating off the heat from the shield?" Basically, it'd just be a solid ball of shielding, and you shoot the antimatter into the center, where ~all of the radiation is absorbed, and the ball can be cooled down by a coat of liquid metal being pumped over it.
Now, the muons only show up later, and if you can stop the pions, the kinetic energy of the decay muons is low enough that they actually aren't that penetrating. So the task is to stop a flux of high energy gamma rays and pions and kaons. The dominant energy loss mechanism at these energies is inelastic collisions, where the pion or kaon smacks an atomic nucleus directly, blasting it to bits, which smack into other nuclei, and the energy level and penetratingness of the radiation drastically falls as energy drops, until the entire cascade is contained. For gamma rays, they smack an atomic nucleus directly, and turn into an electron-antielectron pair, which does a smaller cascade and is less penetrating. Still, even a more sensible design with charged particle energy extraction is going to need the gamma ray shielding (or just incredible radiation resistance) and weigh a decent amount.
Crunching the numbers, I discovered something hilarious. There's a number that is basically "what thickness of material gets half of your beam to interact", and for very dense elements, this gets thin enough to counteract the increased density of your ball. Lead is used in conventional gamma-shielding because it's cheap and pretty dense. But, as starship design is a very important priority for a civilization, they'd probably splurge on whatever material is optimal.
The optimal material turned out to be osmium (although iridium and platinum would be about as good). Yes, in starship design, where every gram counts, I found a perfectly legitimate engineering reason to stick a 3-ton ball of osmium in the middle, as the antimatter reactor core. As a bonus, antimatter reactions tend to split heavy nuclei, so there's an energy boost from induced fission in the osmium, and osmium is really hard to melt so it can definitely accommodate the reaction.