Okay, this occurred to me as I was falling asleep a couple of days ago.
If a solar sail absorbs a photon, it gains momentum from that photon. If it reflects a photon, it gains twice the momentum, right?
But, suppose there is another solar sail set up directly behind in the opposite direction. It also reflects that photon back, gaining momentum. Can these two solar sails arbitrarily reflect light between them, gaining momentum all the while?
If so, where does the energy come from? Diminishing returns obviously applies as they begin to speed away from each other, but it seems that the photon (or photons) can be exchanged an arbitrary number of times, transferring momentum to both ships for free.
I’m undoubtedly missing something, or got something wrong, so let me know what it is.
Sounds to me like you did something right. I don’t know anything about photons, etc, but I can see that if you have two things going in opposite directions then the average wouldn’t change- You’d have conservation.
It like when then swinging balls in that desk toy collide. Pull one up on one side and two on the other and drop at once. You get two balls pop out the first side and one the second because the total momentum has to be the same before and after collision.
Momentum is a vector quantity not a scalar quantity. As long as the vector total is conserved, the momentum of either ship can be as large as you please. As the distance increased, you would have trouble making sure the photon hit the other ship. It might take a long time to get much speed using a single photon, I’d suggest you use a few. Or better yet, use bowling balls and trampolines.
I think that the OP realizes that momentum is not a problem here. The question is energy, which is a scalar quantity, and hence does not cancel. The key here is Doppler shift. If we look for a moment at just a single photon making one collision with a single sail, then in the zero-momentum frame (also known as the center of mass frame), the photon moves in from the left and the sail moves in from the right with equal and opposite momentum, collide, and then each rebounds in the opposite direction with the same momentum.
Now let’sput in our other sail, and put ourselves in the frame where the center of mass of the two sails is equal. Now, in each collision, the incoming photon is slightly blueshifted, and the outgoing one is slightly redshifted, in every collision. This means that the photon lost a little energy, which works out to exactly the amount of energy gained by the sails.