Relativistic centripetal force question

[teve]

The last post in the thread https://www.physicsforums.com/showthread.php?t=466305 seems to say that the relativistic force on a mass moving on a circular path is the centripetal acceleration, v*v/r, times rest mass times gamma squared.

But the articles http://en.wikipedia.org/wiki/Force#Special_relativity and http://en.wikipedia.org/wiki/Special_theory_of_relativity#Force seem to suggest the force is the centripetal acceleration time rest mass times just gamma.

What am I missing? Are the situations different? If so, how?

 

[bcrowell]

I think this is because the spacelike part of the four-force differs from the force three-vector by a factor of gamma: http://en.wikipedia.org/wiki/Four-force

 

[teve]

So then what is the magnitude of the (relativistic) centripetal force on a particle with rest mass m traveling at velocity v in a circle of radius r as observed from a frame at rest with the center of the circle? Is it (mvv/r) times gamma, or gamma squared? It is still not clear to me. It seems this should be a simple question and should have one answer.

 

[bcrowell]

Gamma to the first power. The spacelike part of the force four-vector simply isn't the force.

 

[teve]

OK. Then in the LHC with gamma=7500, r=4300m (C=27000m), 2808 bunches of 1.15e11 protons each (proton m=1.67e-27kg), the LHC dipole magnets exert (on the average) a radial inward force per meter of very nearly (7500*(1.67e-27kg*2808*1.15e11)(c^2)/4300m)/27000m=3.131 N/m. 6.262 N/m for the two beams. Is this a correct application of the formula?

 

[Dale]

What am I missing? Are the situations different? If so, how?

The magnitude of the four-force (gamma squared) is the force felt by the accelerating particle, it is a relativistic invariant. The magnitude of the force three-vector (gamma) is the force measured in the "lab" frame, it is frame variant.

It seems this should be a simple question and should have one answer.

Things rarely have a single answer in relativity unless you pay very careful attention to the details.

 

[MikeLizzi]

The magnitude of the four-force (gamma squared) is the force felt by the accelerating particle, it is a relativistic invariant. The magnitude of the force three-vector (gamma) is the force measured in the "lab" frame, it is frame variant.

Things rarely have a single answer in relativity unless you pay very careful attention to the details.

Thanks for that simple explanation. Now I can check that off on my list of things to figure out.