Upon reading the riddle again, it’s rather easier than I thought.
He’s right, the two ships would continue to accelerate from one another since the laser would be supplying a small, constant force (not an infinite force due to the fading effect between the mirrors as described above). However, this acceleration would be tiny due to the large mass of the ships (a=F/m) and would take centuries to accelerate the ships to a useful speed.
Well yes. The momentum transfer would go on forever (assuming no absorption via dust) but the initial momentum of the photon is finite (E/c) so that alone is transferable to the ships
Umm Stars are in orbit around the center of the Galaxy…
okay, this sounds pretty damn cool. Any SuperString Theory for Dummies[sub]TM[sub] Books you could recommend?
qts
It makes “I want to visit another solar system” not be a valid reason to get on a ship.
Grey
Each gain in momentum in one ship is balanced by a gain in the other in the opposite. So the total amount of momentum in the whole system is constant.
While the total momentum does, I think, end up being finite, I don’t think that it is limited to the momentum of the photon. When the first ship shoots the photon, it (the ship) experiences a recoil equal to the momentum of the photon. When the photon comes back to the first ship and bounces off, it imparts a finite amount of momentum. So clearly the total momentum is greater than that of the photon. In fact, suppose the two ship both have mass M. It’s probably a reasonable simplification that the limit of their velocities are almost equal, so the total energy is twice Mvv/2, or just Mvv. This must be equal to the energy of the photon, so hn=Mvv (where n=nu). V would therefore be sqrt (hn/M). The momentum of the ships is Mv, and that of the original photon is hn/c. So if p= photon momentum, P= ship momentum, then p=hn/c, P=M sqrt(hn/M)= sqrt (hnM). So say n=1Mhz M=10^9 kg. p=2.210^-36 kgm/s. P=8.14^-10 kgm/s. So if we had a beam of one mole of photons, p=1.3^10^-12 kgm/s; P=4.9*10^14 kgm/s for a final velocity of ~500km/s. So actually, if we could solve the attenuation and focusing problems, we could get two really large ships going rather fast while exchanging only a microscopic amount of momentum at a time.
BioHazard
Yes, but I did say currently. And currently even .1c is way beyond our means.
Scr4
That’s what I was thinking. The attenuation and diffraction answers are technically correct, but rely on practical considerations rather than physical laws, so theoretically we could design a system that gets around those problems, while redshift is inherent to the process.
SentientMeat
No, that doesn’t explain it because no matter how small the force is, a finite force over an infinite time period would result in an infinite amount of acceleration.
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THE RYAN:
“…no matter how small the force is, a finite force over an infinite time period would result in an infinite amount of acceleration…”
Depends on EXACTLY what you mean.
The rate of acceleration will at no point be infinite. What you will have is a DECLINING rate of acceleration that never quite falls to zero. In other words, zero is the limit of the curve.
Depending on the details–I can’t work them out–you will either have have a certain maximum finite velocity as the limit, or the velocity graph will progressively pass through all successive velocities on its way to (but obviously never “reaching”) the infinite case. (But in this universe the “infinite case” is actually C.)
It’s one of those amazing-but-true things that some kinds of infinitely-iterated incrementation series fail to go beyond a definite finite limit.
Go figure.
Ryan
Your syntax is a little inexact - is an infinite amount of acceleration identical to an infinite acceleration? If so, you are mistaken. A permanent force will eventually accelerate the ships to c (light speed in vacuo). The closer one gets to c the less effect the already tiny acceleration will have (think of “the mass increasing”, not strictly true but a useful simplification). However, considering the insignificance of the force it would seem unlikely that the ships would get anywhere near c within the lifetime of the universe.
No, the total available momentum is that of the photon. If it were a pool ball, what you are positing is the ball striking another heavier ball and both travelling off in opposite directions with no change of speed of the first ball. This would not be the case (even if photons did behave like pool balls), since the momentum transfered by the small ball to the bigger one would result in a slower speed of the first. Upon each collision, more momentum transfer would occur until the ball’s speed tended to zero.
In reality, there is merely a probability upon each reflection that the photon is absorbed by a surface atom, which increases towards 100% with more reflections.
This explanation can be tested by hanging a bathroom mirror around your neck whilst looking in another mirror, shining a laser pointer into one mirror and singularly failing to be blown across the room by the violation of momentum conservation.
I think a lot of people in this thread are relying on the magic science genies to fix everything up for them.
The thing is, we are pretty sure that we got at least some concepts reasonably well understood, stuff like energy, speed etc.
If we stick to the conventional stuff and assume for the moment that Warp drives are just wishful thinking, then we can start to pin some solid numbers down.
How fast do we wan’t a ship to go? How long are we willing to wait? How heavy is it going to be?
As a previous poster has pointed out, to even GET to 0.5c is going to require serious amounts of energy. We are pretty sure that matter-anti-matter is about the best way we can store energy. We are alos pretty sure that all our calculations are going to be roughly correct. In that case, it would take ungodly amounts of energy just to do anything.
The rate of acceleration won’t be infinite, but the acceleration will be. And by “a force”, I meant a specific force of a specific, and unchanging magnitude.
SentientMeat
You are correct. I omitted the fact that I was using the word “acceleration” to refer to the change in the relativistic parameter, where the velocity is equal to the hyperbolic tangent of the relativistic parameter. I thought that including this would confuse more than it elucidated.
No, because each time it bounces back its momentum is “charged up” again. Think about a pail of water. Is the amount of water I can pour onto a fire limited by the volume of the pail? No, because I can go back and refill the pail as many times as I want to.
Yes, the smaller one would have its speed slightly reduced. But one plus something slightly less than one is more than one, and the momentum of the beginning photon plus the the momentum of the returning momentum is going to be greater than the momentum of just the beginning photon.
I thought that I had already explained that there is no violation of momentum conservation; as long as there are two ships, moving it opposite directions, and acclerating at the same rate (in absolute value), the change in momentum cancels out.
i dont know how to solve the energy problem, but we could build a spaceship big enough to live on and have the people on the ship live their lives like they would on earth - except they would use hydroponic farms and other substitutes for things that they cant normally get on the ship. After many generations pass by, the ship reaches its destination.
You know, when the vectors confuse the hell out of you go for the scalars…
Eo = 1/2 MshipV1o+E/c+1/2MshipV2o
Now initially V1o and V2o are going to be close enough to 0 to not really matter. So
Eo’=E/c
Now eventually the photons will be red shifted enough that their wavelength tends to oo so the Ephotonfinal ---->0
Ef = 1/2MshipV1f+1/MshipV2f
Eo’=Ef right? So 1/2MshipV1f+1/2MshipV2f=E/c
Looks damn finite to me.
As for
1 + 1/10 + 1/100 + 1/1000 + …1/10^n tends to 10/9 which is finite despite the infinite iterations.
Well, you seem to confusing momentum with energy (if you’re talking about energy, the velocity ters should be squared, and if you’re talking about momentum, you shouldn’t be dividing by two).
But you are correct that the momentum ends up being limited for any particular photon momentum and ship mass. The point that I’m making is that although the momentum is limited, it is not limited solely by the initial momentum of the photon. The initial momentum of the photon is just one factor among many that goes towards determing the final momentum. I can keep the photon momentum constant, and by varying the ship mass make the final momentum as large as I want it.
The “bouncing light between two lightsails” discussion is getting out of hand, but it’s not that difficult. Forget about momentum - the problem is about energy. Remember, reflection of a photon is a perfectly elastic collision so energy is conserved. (Momentum too, but that’s always conserved.) Say you have two lightsail ships at rest facing away from each other, and you release a single photon onto one sail. Assuming perfect focusing and reflectivity, the photon will bounce between them repeatedly and the ship starts to move away from each other. Because the mirrors are now moving away, the photon loses energy on each bounce due to redshift. Eventually all the photon’s kinetic energy will be transferred to the kinetic energy of the ships, and the photon will disappear. (Maybe not completely, but its energy would asymptotically apprach zero.)
So, the energy of each photon (and no more) is eventually shared out between the ships, and these photons keep being “replenished” by the laser.
Agreed.
Big deal.
Are we agreed that the total force is finite (and tiny), and so therefore the acceleration (a=F/m) is tiny also (unless you can make a spaceship out of an atom)?
Sorry to harp on about it, but perpetual motion machines are a hobby of mine.
Are talking about light sails in general? The force is of course finite. The impulse from a single photon is very small. But it’s not zer so if you have enough photons the impulse adds up, and you can generate as much acceleration as you want. It’s just a matter of building a powerful enough laser to generate the photons. Using a 1-square kilometer sail to capture sunlight results in about 9 Newtons of thrust. If you had a laser capable of illuminating that entire sail at 1000 times the brightness of the sun, you can accelerate a 100 ton spacecraft at 0.01 G. If you keep it up for 10 years you reach a very respectable speed of 0.1c. Robert Forward proposed something like this in his novel Rocheworld. IIRC the scheme involved a solar powered laser array in orbit around Mercury, and a focusing lens for the laser constructed in the outer solar system.
Now we’ve cleared up that little bone of contention, whatever you like.
I’d think the laser very difficult to aim after a few days. Not to mention that I think I’m correct in saying thet the beam would have to spread out due to Heisenberg’s Uncertainty Principle.
The Ryan yeah I meant energy. There’s a reason I should never write equations after 10pm. But basically I’m saying you start with a finite energy and at the end its will be contained in the kinetic energy of the ships and what ever is left in the photons.
Now I did a little number crunching on engine types and getting to .1c, just how big a ship do you need? You’re not going to like it. 
Fission, fusion and antimatter all require initial masses that dwarf the delivered ship. That’s not even taking into account that we’d want to slow down once we get there.
Propellant ships don’t seem to be the way to go, if you want to get there quick.
So say we send an 800 t ship to .1c how much energy?
Turns out it’s about 3.6x10^20 J which seems big but if we build solar panels at 1 AU (1400W/m^2) with an efficiency of 10% and operate for 1 year we can harvest 3.6x10^20 J with an array only (he said trying not to laugh) 280 km per side.
Way back at the beginning of this thread, Scylla stated the following:
OK, that makes a certain amount of sense, but how do you come to a stop once you have reached your destination? If I remember my basic physics correctly, it takes just as much energy to go from top speed to zero as it does to go from zero to that top speed. If you accelerate by chucking nuclear bombs out the back and set them off, you will need to set off just as many bombs to decelerate.
The only problem is that instead of setting them off behind you, you now have to set them off in front of you. I can’t imagine shielding good enough to withstand the effect of detonating multiple hydrogen bombs directly in your flight path. At least during acceleration you are moving away from the bomb blast and not directly into it.
Maybe the problem is simply with my lack of sufficient imagination, but it really seems that there is a huge difference between the two situations.
Regards,
Barry