Action and reaction when fast object is pushed

[jartsa]

Let's say Bob floats in space. Bob's identical twin Joe moves past Bob at relativistic speed. As Joe is passing by, Bob gives him more speed by pushing him with his hand.

Somehow I just happen to know that Joe will feel a smaller force pushing him than what Bob feels. So my question is: Why is the proper force felt by Joe smaller than the proper force felt by Bob?

(Bob and Joe are like superman. They are not breaking any laws of physics though)

"Somehow I just happen to know" is perhaps somewhat obnoxious. So let's say that in a little bit different scenario Bob "pushes" Joe by shooting him with a laser gun, Bob feels the normal recoil of the gun, Joe feels an arbitrarily small force, depending on how much redshift there is.

 

[Mister T]

The twins will undergo equal but opposite changes in momentum. Note that in Newtonian physics conservation of momentum is equivalent to the Third Law. In relativistic physics the Third Law is valid only as an approximation in the low speed limit, but conservation of momentum is valid at all speeds.

 

[Dale]

let's say that in a little bit different scenario Bob "pushes" Joe by shooting him with a laser gun, Bob feels the normal recoil of the gun, Joe feels an arbitrarily small force, depending on how much redshift there is.

This is actually a very good way to look at it. Any exchange of energy and momentum between Bob and Joe can be decomposed into an exchange of two light pulses of different energies pointing in opposite directions. In Joes frame one will be redshifted and one will be blue shifted. The amount of momentum transfer depends on whether the redshifted or blue shifted pulse is more energetic in Joe's frame.

 

[A.T.]

So let's say that in a little bit different scenario Bob "pushes" Joe by shooting him with a laser gun, Bob feels the normal recoil of the gun, Joe feels an arbitrarily small force, depending on how much redshift there is.

The force will be different, but so will be the duration. The total momentum change will be the same if the light is absorbed.

 

[jartsa]

The force will be different, but so will be the duration. The total momentum change will be the same if the light is absorbed.

The duration will increase as the wavelength. The force must decrease as square of the inverse of the wavelength, so that the energy will decrease as inverse of the wavelength, as it should.

Maybe it's not worth the effort to try to make sense of the above. My point is: The proper force decreases surprisingly fast. Faster than the proper duration increases.

 

[Dale]

Yes. And insofar as a "why" question can even be answered it can be understood in terms of the Doppler shift.