Mass inflation exactly equals time dilation?

[HarryWertM]

Two spacecraft in inertial motion have relative velocity .5c. The ships have identical "grandfather" clocks very much like earthly grandfather clocks, except that the force of gravity is replaced by Coulomb repulsion.

[The clocks have charged plates on either side of a charged pendulum. Inertia drives the pendulum [penduli?] and Coulomb repulsion pushes the penduli back and forth. The clocks "run down" rapidly due to radiated energy, but for a short time they provide a weird highly predictable timing device.]

Each ship sees that the other ship has a heavier pendulum but equal [Coulomb] restoring force, so each ship sees that the other's pendulum swings back more slowly. My question is: If I simply plug in the appropriate higher relativistic mass into some equation [I know not what!] will I get exactly the usual "gamma" factor for the relative slowing of each ship's clock? I do not trust any equation I set up at all.

 

[bcrowell]

I think it's more complicated than that.

The period is proportional to m^2, not m.

Also, a field that is purely electric in one frame will be a combination of electric and magnetic in another frame. The exact result would depend on the orientation of the clock.

 

[pervect]

Things are definitely more complicated. For a traverse pendulum:http://en.wikipedia.org/w/index.php?title=Relativistic_electromagnetism&oldid=412215270

points out, in the section on "Uniform electric field — simple analysis", the electric field in the moving frame will be stronger by a factor of gamma.

But there will be a magnetic force as well as an electric force in the moving frame. The magnetic field will be proportional to velocity, and the magnetic force will be proportional to the magnetic field * velocity, or v^2.

Without working out the numbers in detail, I think that the "restoring force" would be equal to

gamma * E * (1-(v/c)^2) -> E / gamma

I.e, including the magnetic field, an apparent restoring force that's reduced by a factor of gamma.

Combine this with a "transverse mass" that's increased by a factor of gamma,

see http://en.wikipedia.org/w/index.php?title=Mass_in_special_relativity&oldid=420807342

the "transverse mass" is the right sort of mass to use for a transverse pendulum "longitudinal mass" would be needed for a parallel pendulum.

and note that the period is proportional to sqrt(E/m), and one gets the period increasing by a factor of gamma, as one expects.