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Consider the following scenario :

From here on I would like the word "laser" to be treated as just a source of monochromatic beam that emits light . You can take any other light source(if you do not want to get caught up in unimportant details) the arguments should not differ much(at least in my opinion).

A laser is kept on a frictionless plane and switched on. The laser is emitting light in a particular direction and we know light has momentum. Now as the net force on system is zero, conservation of momentum states that the laser must also acquire a momentum in direction opposite to light.

So laser also starts moving, but where does the required kinetic energy come from? One simple answer is to say the from the source powering laser, which seems right.

Now if we take a spherically symmetrical source of light which employs same mechanism of light production as our laser above, it should be reasonable to assume it has same efficiency to light conversion as our laser.

Except now the spherical source of light moves nowhere( using conservation of momentum) and thus now it has zero kinetic energy.

But if the efficiency of light production is same where does that extra energy(that was supposed to be converted to kinetic energy) go?

Question :

Why do we have this contradiction? What am I arguing wrong here?

Edit: I have changed the title to suit the question more.

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  • $\begingroup$ Do you think the energy carried by the photons in both the cases will be the same? $\endgroup$
    – user258881
    Commented Apr 3, 2020 at 13:00
  • $\begingroup$ As you stated, because of energy conservation. The photons energy (and frequency) will increase because of this. $\endgroup$
    – user258881
    Commented Apr 3, 2020 at 13:10
  • $\begingroup$ @FakeMod would it not? Why? As far as I can see spherical source can be considered as a limiting case of joining infinitely many lasers symmetrically opposite to each other, each with same mechanism to produce light and thus each having some unaccounted energy. Please explain if I am wrong. $\endgroup$
    – Hrishabh Nayal
    Commented Apr 3, 2020 at 13:14
  • $\begingroup$ @FakeMod how can we say that the frequency will increase if the mechanism producing it is same. Can this be verified experimentally?(I do not think it can as the energies involved should be too low to measure) $\endgroup$
    – Hrishabh Nayal
    Commented Apr 3, 2020 at 13:16
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    $\begingroup$ IMO You could simplify this question. Start by removing all discussion of "efficiency." You seem to assume that a laser should be "totally efficient," but that is very far from the truth. For example, all lasers convert a substantial fraction of the input energy to heat. I think all you need to do is talk about a laser that outputs a beam of so-many milliWatts, and then compare that to a rigid assembly of two identical lasers that point in opposite directions from each other. $\endgroup$
    – Solomon Slow
    Commented Apr 3, 2020 at 13:34

1 Answer 1

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Consider a laser with photon energy $E_\gamma$. This means that the photons emitted by it have an energy of $E_\gamma$ in the reference frame of the laser emitter. Since the laser emitter is on a frictionless plane, by conservation of momentum, it will move in the opposite direction of the beam (albeit very slowly, as the force applied by the beam is very small), which means that the photons emitted will have less energy in the reference frame of the plane, being redshifted according to the relativistic Doppler effect; this is where the kinetic energy of the laser emitter comes from. If we now look at the case of two identical, attached laser emitters, pointing in opposite directions, momentum is conserved when the laser assembly remains at rest, so none of the power goes toward the kinetic energy of the emitters, so the beam does not become redshifted.

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  • $\begingroup$ Thanks! I get that answer. $\endgroup$
    – Hrishabh Nayal
    Commented Apr 4, 2020 at 2:35

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