Relativistic Center of Mass Shift

The following is an illustration of the analytic computation of the shift of the relativistic center of mass in a rotating hemispherical shell.





In the above, rc is the new center of mass that has been shifted by the relativistic mass increase effect, while ro is the original center of mass of the hemisphere when it is at rest. When the rotational speed is slow, which may be the case in a terrestrial experiment, the relativistic shift of the center of mass of a rotating hemisphere becomes



The gravitational dipole moment for this system in a slow rotational speed is given by the mass M times the above delta rc.

This aspect of center of mass shift is obvious in the sense that whoever has the knowledge of special relativity and a little bit of calculus background will be able to figure it out easily. The hardest part would have been to make the connection that it is a very abnormal mechanical system within the boundary of Newtonian mechanics, which may require extensive knowledge of general mechanics. Next hard step would have been to make the connection of this system with general relativity. If one misses any one of these steps or has insufficient knowledge on any of them, it would be next to impossible to get the solution of gravitational dipole moment.

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