gwyn@Brl-Bmd.ARPA (01/30/83)
From: Doug Gwyn <gwyn@Brl-Bmd.ARPA> Since I am having trouble with net mailing (brought about by the TCP/IP switch on the ARPAnet and deficient UUCP mailers everywhere), I am posting a summary of this argument to the mailing list. I welcome constructive comments and hope to write up an "eqn" document with more detail someday. (I submitted it for publication years ago, but it was reviewed by a person with years of research wasted, according to this theory.) The key to the argument is the observation that one cannot interpret space-time coordinates physically without reference to the metric tensor to decode their meaning. A local Lorentz metric diag(1,1,1,-1) permits one to identify the coords as a local cartesian intertial frame, with the fourth coord physically timelike. The simple coord transformation (x,y,z,t) -> (t,y,z,x) has a corresponding metric diag(-1,1,1,1) which must of course be unscrambled to be properly interpreted. No one would dream of calling the new x' == t a "space" coordinate, etc. The derivation of the Lorentz transformation relating two relatively- moving inertial frames makes no special reference to the speed of motion being less than light (1 in the units used here); therefore the same formulae are used to relate a supposedly faster-than-light system to a "stationary" reference system. By some fairly simple algebraic manipulation, one can verify the following: The same state of motion between two inertial frames is obtained by either: (a) Postulating motion in the x direction with speed v and swapping (x,t) coordinates, or (b) Postulating motion in the x direction with speed V = 1 / v (V = c^2 / v if c != 1). Of course, unless v = V = 1 one of these speeds will be super- luminal and the other sub-luminal. One interpretation of this result is "Any faster-than-light motion is indistinguishable from a (definite) slower-than-light motion in the same direction but with the coordinate axes perversely misnamed." Note that this interpretation explains in a particularly simple way why experimenters have not been able to observe free "tachyons". Once this basic correspondence is on hand, it can be tested in many ways. My favorite is to take the "relativistic velocity-addition" law v1 "plus" v2 = (v1 + v2) / (1 + v1*v2) and examine the result of replacing v1, v2, and/or v1 "plus" v2 by their reciprocal speeds 1/v1, etc. There are other details to be checked, such as motion in non-collinear directions, but it is apparent (to me at least) that nothing new will result from these other considerations.