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John Rennie's Q&A How long would it take me to travel to a distant star? discusses about interstellar travel taking into consideration. There was a case that discussed about constant acceleration-deceleration journey where the traveller would have clocked only 12 years. But from Earth's perspective the travel took 491 years. Also the traveller experienced space contraction.

My question is should a ship be fuelled from the traveller's frame or from the earth's frame? Or will a relativistic math yield that both are same quantity?

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karthikeyan
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You can think of the fuel being stored in barrels, and observers in all frames will agree on how many unused barrels are left when the ship reaches its destination, since discrete quantities aren't affected by the Lorentz boost.

If you're wondering how this makes sense with the time dilation, imagine that on board the ship, it appears that the barrels are being used up at a constant rate. However, from the Earth it appears that, as the ship accelerates, it uses fuel at an ever-slower pace and correspondingly accelerates at an ever-slower pace. This is related to the energy equation in lurscher's answer, since an ever-increasing amount of the work of the engines goes to increasing the energy rather than the velocity.

ragnar
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To being able to reach a high fraction of the speed of light, you need also a significant fraction of your equivalent payload mass as mass-energy. According to the formula for energy (rest and kinetic)

$$ E = \gamma m c^2$$

so, to reach $0.1c$, that means your gamma is $\gamma = (1-\beta^2)^{-1/2} = 0.99^{-1/2} = 1.01$, so that means that for each kilogram of payload you want to accelerate to $0.1c$ (c being the speed of light) you need 0.01 kilograms as energy.

Accelerating a rocket with even still nonexistent fusion engines to speeds higher than $0.1c$ becomes exponentially harder (i.e. your rocket gets exponentially bigger)

That's why the best possibility for interstellar ships is laser-pushed sails, as you leave all the fuel near the sun at home, and just beam the energy toward a very large sail

As Rod Vance comments, if you want to consider constant acceleration instead of reaching a given (relativistic) delta-V, then the energy requirements go up substantially. Look at the John Baez FAQ on relativistic rockets for more info

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lurscher
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    The laser sail is an enormous waste of energy, though, until the ship is near the speed of light. Below that you end up using hundreds, if not thousands of times more energy than you actually need to accelerate. The simple answer to near-lightspeed travel is: DON'T! – CuriousOne Jun 19 '15 at 00:18
  • the energy might seem go to waste, but not its momentum (the sail gets roughly twice if reflective), and that's what matters for propulsion – lurscher Jun 19 '15 at 00:40
  • if you are worried that a more energy-efficient interstellar propulsion should be used, you can wait until we reach Kardashev II and start manufacturing micro black holes, they make nearly perfect energy storage systems, optimal for relativistic propulsion – lurscher Jun 19 '15 at 00:45
  • Please write to your electricity company and ask them to bill you for momentum rather than energy use. :-) Good luck with the mini black hole production program, you are going to make Elon Musk look like a baby. Or not. – CuriousOne Jun 19 '15 at 03:58
  • You might like to add a link to John Baez's page where he calculates the fuel usage using the analysis found in Misner-Thorne-Wheeler (nicely illustrates your point). To reach the center of our galaxy, accelerating at 1g all the way there, do a flyby and decelerate to come back, you'll need to take 124 tonnes of fuel for each kilogram of payload (assuming perfect conversion of matter) for a 40 year round trip. If you want to stop at the center of the galaxy, that'll cost you nearly a million tonnes of fuel for each kg payload. See http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html – Selene Routley Jun 19 '15 at 06:03
  • @WetSavannaAnimalakaRodVance that's correct, that is for constant acceleration along the whole way. In this case the consideration was to reach a certain deltaV and coast for the rest of the trip – lurscher Jun 19 '15 at 15:34
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The spaceship and the Earth start out in the same frame, so the amount of fuel agrees in both frames.

Zo the Relativist
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