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Force and special relativity.

  1. Aug 12, 2013 #1


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    Does the magnitude of the force change when the frame moves with a high speed with respect to another frame at rest? If yes, How?

    Thanks for any replies.
  2. jcsd
  3. Aug 12, 2013 #2


    Staff: Mentor

    Are you familiar with four-vectors? If so, the easiest way to understand force in relativity is to look at how the four-force transforms:


    If not, let me know and I will try to find a different reference, but it won't be as easy to understand.
  4. Aug 12, 2013 #3


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    Actually I am sorry, I am not familiar with four-vectors; however, I am not looking for a detailed derivation, I am just looking for a simple equation that allows me to calculate the magnitude of a force in an inertial frame if the magnitude is known in an other inertial frame - something like Lorentz transformation.
  5. Aug 12, 2013 #4


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    In Special Relativity the force acting on a particle is not always parallel to its acceleration. It is parallel only when the acceleration is either parallel or perpendicular to the velocity. Also, force can be defined in more than one way. If you define it as the change of the particle's momentum with respect to coordinate time, then

    f = γ ma + γ3 mv dv/dt v/c2

    On the other hand, if you want to define it as the change of momentum with respect to the particle's proper time, multiply the above by γ:

    F = γ2 ma + γ4 mv dv/dt v/c2
    Last edited: Aug 12, 2013
  6. Aug 12, 2013 #5


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    First what do you mean by "the force"? What force are you talking about? We know, since Galileo, that force is proportional to acceleration. As long as one frame is moving at a constant speed, with respect to another, acceleration, and so "force" should remain the same.
  7. Aug 12, 2013 #6


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    Well, the magnitude of 4-acceleration is invariant, thus also magnitude of 4-force. However, 3-acceleration (either coordinate or by proper time) is not a vector under the Lorenz transform, and its magnitude is not a scalar invariant.

    Trivial example: a world line with constant coordinate acceleration in one IRF transforms to a world line with time varying coordinate acceleration in another IRF.
    Last edited: Aug 12, 2013
  8. Aug 13, 2013 #7
    I take that you mean if the magnitude of a force is measured differently with systems in relative motion. It depends on the direction of the force: along the direction of motion the force is measured the same.
    Fx'= Fx

    However, perpendicularly it is not the same. For an object that is at rest in the moving system:
    Fy'= γ Fy

    It's more complex if the object is also moving in the moving system.

    In order to avoid confusion (in view of the different answers here), it may be useful if you give an example of what you have in mind exactly.
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