(1, 2, 3, 5)

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. $Δ V = V_{e} * ln (\frac{m_{f u l l}}{m_{e m p t y}})$ . $Δ V$ is your change in velocity. $V_{e}$ is your exhaust velocity. And $m_{f u l l}$ is the mass of the rocket full of propellant, with $m_{e m p t y}$ 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 antimatt...