by Otto E. Rössler, Faculty of Science, University of Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
Abstract: An unfamiliar result in special relativity is presented: non-conservation of rest mass. It implies as a corollary a resolution of the Ehrenfest paradox. The new result is inherited by general relativity. It changes the properties of black holes. (June 21, 2012)
Rest mass is conserved in special relativity in the absence of acceleration. Under this condition, the well-known relativistic increase of total mass with speed is entirely due to the momentum part of the total-mass formula, so rest mass stays invariant as is well known. However, the presence of acceleration changes the picture. Two cases in point are the constant-acceleration rocketship of Einstein’s equivalence principle of 1907, and the rotating disk of Einstein’s friend Ehrenfest 5 years later.
First the Einstein rocket:
If light emitted from a point close to the tip of the constantly accelerating rocketship arrives with its finite speed at the bottom, it is blueshifted there because the bottom has in the meantime picked up a constant upwards speed. This at first sight absurd implication of special relativity was spotted by Einstein in 1907 in a famous mental tour de force. The arriving photons possessed their higher frequency from the beginning. Since they were at equilibrium with the local masses at their point of origin (think of positronium-annihilation generated photons being used), all masses at their height of origin are increased by the pertinent blueshift factor with respect to the same masses residing at the bottom. The converse argument holds true in the other direction for the redshift of photons from the bottom arriving at the tip, and for the correspondingly lower relative rest mass of all stationary particles at the bottom.
Second the Ehrenfest disk:
If light emitted from a more peripheral point of the constantly rotating disk arrives at the motionless center, it is redshifted by the transverse Doppler-shift factor discovered by Einstein in 1905. Much as in the previous case, the emitted photons are locally inter-transformable with solid rest mass. The implied local decrease in rest mass entails a proportional size increase via the Bohr radius formula of quantum mechanics (the parallel size change went unmentioned in the preceding case). But this is not the end of the story: Simultaneously, Lorentz contraction holds true at the light-emitting point on the rotating disk. The two local size change factors – that of the transverse Doppler shift and that of Lorentz contraction – happen to be each other’s inverses. Since they thus cancel out (the ratio is unity), the rotating disk remains perfectly flat. This prediction, deduced from special relativity with acceleration included, solves the Ehrenfest paradox.
Rest mass is not conserved in “special relativity with acceleration included.” Rest mass decreases more downstairs (or outwards, respectively) in proportion to the so-called gravitational (or rotational, respectively) redshift factor. This proposed new result in special relativity is bound to carry over to general relativity. Indeed the gravitational-redshift proportional reduction of rest mass has been described in general relativity by Richard J. Cook (in his 2009 arXiv paper “Gravitational space dilation”). The non-constancy of rest mass despite the fact that it appears locally un-changed has a tangible consequence: it affects the properties of black holes. The implications are incisive enough to let a currently running attempt at producing black holes on earth appear contraindicated from the point of view of planetary survival. This fact makes it desirable to find a flaw in the above chain of reasoning. (For J.O.R.)