You wouldn't need a sudden jump. If you had a rocket that accelerated at a pleasant 1G forever, you could reach and stop at the center of the milky way in about 20 (your time) years, and you could reach and stop at the Andromeda galaxy in about 28 years. Play around with some of the online space travel relativity calculators--it's wild!
Of course building and fueling such a rocket is what's totally out of reach.
No, it would direct the photons out the back of the space craft, producing a beam intense enough to vaporize any solids or liquids close by. This beam would ideally be able to provide perhaps 1G or so of continuous acceleration.
Is there drag in space? I.e. would you need increasing energy to accelerate at a constant rate as the speed goes up?
With a traditional rocket, I believe you'd need decreasing energy to maintain the same acceleration as the flight progressed, since you are carrying along with you and burning the fuel, and so the total mass (payload + fuel) that needs to be accelerated is constantly decreasing.
Of course there's the pesky problem that for every N kg of mass you want to accelerate at 1G for that kind of a trip, you're probably going to need somewhere on the order of N billion kg of fuel to burn.
> Of course building and fueling such a rocket is what's totally out of reach.
We'd need a device that could efficiently transform several kg of matter to photons.