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As you get going further and faster, this energy redshifts further and further making it less and less useful. Also there’s of course the problem of focusing a laser with that amount of energy tightly enough on an object increasingly far away.
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And crucially ensuring that beam continues to run for billions of years. a few decades may pass for you, but not for the laser that's pushing you
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Not sure if my interpretation here makes any sense but here it is. Due to relativity time slows down on the ship, the laser is still hitting the ship, how would the measurement of the energy received from the laser be perceived on the ship, from the ships perspective a 1000 years worth of energy arrives within a few decades. My head hurts thing about this. If you had a ship that could instantly jump to 0.5c travelling away from earth and we beamed a message using a laser at say 1Gb/s, how would the bandwidth appear to the ship?
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You have it backwards. The further and faster the ship goes, the less energy is received. The energy collected over a day will be spread over a week, a month, a year, a decade, etc.

GP has it backwards too. For 1g of acceleration, none of the beam that is produced after a year will ever reach the accelerating spacecraft. You only need to run it for a year.

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For 1g of acceleration, none of the light produced after 1 year will ever reach the spacecraft. You only have to operate the power source for a year, but you only get a year of source power spread across an infinity of time.
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So there is also the idea of a ramjet that uses interstellar hydrogen as fuel for fusion, which provides energy & propulsion: https://en.wikipedia.org/wiki/Bussard_ramjet

Some recent findings do give it lower feasibility though.

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Another option is to visit as many massive objects as possible on the way there, doing gravitational slingshots until you're half way, and then gravitational brakes on the approach. Of course now you're taking detours, but since certain amounts of fuel make the trip impossible, any workaround at all is an improvement. What's an extra few thousand years between friends?
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First, what “many objects along the way”? There are basically none. To do this requires many detours that significantly increase the distance you’re having to travel.

Second, the faster you’re going, the less useful a gravitational slingshot is because you spend less time in the object’s gravity well.

It’s a fun and useful trick within our solar system, but I don’t think the physics plays out well for interstellar travel.

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Stars is what I meant. Although upon further thought I think you're right. I guess the intervening stars are unlikely to be moving at helpful velocities for such a game of pinball, metric expansion being what it is.
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Slingshots require very specific conditions and configurations of orbiting bodies that won't really work in deep space - all the Kuiper Belt/Oort Cloud bodiees are far too small and too far away to be usable.

But assume for example that you don't care about how long it is going to take (you are an immortal cyborg, an upload living in the computer or a popsicle in hibernation & the AI is driving. Then you can just "island hop" between Kuiper Belt/Oort Cloud objects, never gaining any high % of the speed of light but always having the next body in your sights, not farther than a few decades or a couple hundred years of flight at most.

You reach the body, refuel, repair, maybe upgrade and then off you go, to the next one. And eventually, you might be at you destination, with a fully fueled and repaired ship. It might just take like 20000 years for start to finish, but hey, you can use it for something productive like learning to play Go or something. :)

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You're right, I overlooked some important slingshot requirements. As for the longevity, I figure the ship can build a facility for growing settlers when it arrives. With no humans on board it can sustain heavier accelerations.
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This is quite nicely described in the engenerator article on Orions Arm (including possible problems): https://www.orionsarm.com/eg-article/4a3d78dc77e6f

In this setting this was first done when a bunch of immortal vyborgs was on a colony ship, about hundred years from arrival but there was a probe with a matter printer already in the destination. So their got the idea to send the probe both blueprints for a humanoid body printer as well as their mind backups. That got their colony going quite a bit faster but might have resulted in some funny/awkward moments once they met in person later. :)

Presumably you could also run this withou a mind upload and let he individual develop normally.

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