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In discussing this question about propelling a spacecraft with photons and their relativistic momentum, the author asked that I restate my comment as another question.

If photons can really be used for propulsion, could two mirrors be affixed between two objects, focused well and continually adjusting their respective 'where they will be at the time of light arrival', and use the same photons bouncing back and forth between the two objects for perpetual thrust after the initial photon generation?

The initial question was on an assignment problem:

"A rocket of mass $m_0$ is propelled by a giant monochromatic laser mounted on the back of the rocket. The laser emits a beam with a power of $P_0$ watts and a frequency $f_0$, both measured in the rest frame of the rocket. When the beam is turned on, the rocket is driven in the opposite direction by the recoil."

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Ehryk
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  • Clearly the bounces and red-shifts will approach a limit because of conservation of energy. The momentum is equal and opposite and always ballances, but kenetic energy is scalar. All the energy of the photon can be given to the mirror's motion, and no more. – JDługosz Feb 01 '15 at 23:53

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The problem with your idea is that each time the light reflects off the mirror it transfers some of its energy to the mirror (to increase the mirror's kinetic energy) and is red shifted as a result. So the thrust would fade as the light red shifts away to nothing.

For obvious reasons the light can only transfer as much energy to the mirrors (in the form of kinetic energy) as it originally had. Otherwise you would have a perpetual motion device.

John Rennie
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  • More to the point, every rebound would rebound back, conserving the total momentum of the system, which would presumably be zero. – webb Mar 24 '14 at 20:00
  • @webb: I think the point is that the two mirrors are not connected, so they push each other apart. – John Rennie Mar 24 '14 at 20:02
  • I see, so energy would need to be added to keep the thrust constant. Does a laser get redshifted every time it bounces off something? I.E. if you pointed a green laser between two calibrated mirrors it would eventually be red? – Ehryk Mar 24 '14 at 20:04
  • @Ehryk: If the two mirrors are fixed in position, e.g. fastened to each other in a rigid frame, then the light wouldn't red shift because no energy is being transferred to the mirrors. In your question the red shift occurs because the mirrors aren't fixed in place so the light can transfer energy to them by pushing them apart. – John Rennie Mar 24 '14 at 20:06
  • So light bouncing off a mirror selectively transfers energy only if they aren't attached? – Ehryk Mar 24 '14 at 21:28
  • Well, you add kinetic energy to the mirror, which means you have to subtract the same kinetic energy from the photon, which means that if the mirror gets $\Delta E$, then the photon frequency has to drop to $\omega - \Delta E/\hbar$. – webb Mar 24 '14 at 21:45
  • @Ehryk Nothing so sinister is happening. A photon gets a net redshift off a stationary mirror (because it will transfer a bit of momentum to the mirror). If it strikes a moving mirror, there is a second Doppler term in the shift, which can be positive or negative. If you think about how the mirrors are moving in the attached/unattached cases, you should see that everything checks out OK. – Kyle Oman Mar 24 '14 at 23:42
  • @Ehryk you can model "attached mirror" as having very large mass. Then the photon hitting the mirror will lose very tiny amount of energy (by conservation of energy and momentum), and in the limit of mass $\to\infty$ the photon won't transfer any energy. – Ruslan Mar 25 '14 at 05:06
  • But the mass of the mirror is not infinite, and so with enough photons, you could move the mirror (as the other SO question alludes to), correct? – Ehryk Mar 25 '14 at 17:29
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I think that even if the mirrors are fixed, there would still be loss of momentum in the form of heat loss as the mirrors are made up of atoms and as such, are subject to thermodynamics.

Dan Park
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