Moving Frames of Reference.                                                        Copyright ©2004-2008 David V Connell.

      For this discussion we will assume two objects, A and B, which are identical when at rest, and then assume A is accelerated to some speed relative to B, by applied energy.
      There are two basic places for observing (frames of reference) for an observer, one is when the observer travels with the object A, and he is then said to be in the objects "own" frame of reference (FoR), and the other is that of an observer in any other FoR, called "external" FoRs herein (often referred to as the stationary frame), observing the moving object.
      The principle of Conservation of Total Energy (which includes mass since it has been shown to be a concentrated form of energy) indicates that when mass is accelerated to some speed by the application of force, the energy transferred to the object takes the form of kinetic energy (KE) in external FoRs, but, in its own FoR, since the object always has zero velocity it cannot have kinetic energy there, so the applied energy (E) must be stored in the object (as it cannot be destroyed, or created), and is therefore stored as mass (often called potential energy). From E = Mc², the stored mass is E/c², where E is equal to the KE and the new mass M is the original "rest" mass M_o + KE/c².
      Thus, observer B measures the speed (V) of A and calculates its kinetic energy from the classic definition, KE = M_oV²/2. This assumes that all the applied energy is utilised to obtain speed, that none of it is diverted to increasing the mass in external FoRs, as could happen if the speed was restricted.That is, the maximum unrestricted speed is obtained from the applied energy. From above, the new mass M of object A in its own FoR, is given by M = M_o(1 + V²/2c²).
      No problem so far (?), but what follows is where some people, including highly qualified physicists (!), get a little confused. It may be noted that total energy (mass plus any kinetic energy plus any other form of energy associated with it) cannot be different by merely observing it from different frames at the same instant. Therefore it must be the same in all FoRs, and when energy is added to an object it is added in all FoRs.
      When, in this example, the observer at A observes B to be moving, his simple calculation of the total energy of B, being its original rest mass plus its KE, offends the Conservation principle, as it seems to have increased by the amount of the KE. This cannot have ocurred as no energy has been applied to object B.
      From the discussion above, it is known that the mass of object A has been increased by the adsorbed energy, therefore the mass of B is reduced relative to the mass of A, so the reduced mass must be used in the calculation of the total energy of B as measured by A and is therefore mc² + mV²/2, where m is the apparent (reduced) mass.
      This must be equal to its original total energy, so M_oc² = mc² + mV²/2, giving, M_o/m = 1 + V²/2c², which is identical to M/M_o, as one might expect from relativity.
      Thus, only the mass in an object's own FoR can be real mass, so that any relativistic changes (which are dependent on a change in mass) are independent of the FoR of the observer. The Doppler Effect for light does not qualify as a relativistic effect, it does not change the emitted frequency, it is only an optical effect of relative velocity.
      A word on momentum (MV), all text books assume M is the relativistic increased mass. But, for unrestricted motion, M is only increased in its own FoR, and V belongs only to external FoRs, so MV is a mixture of FoRs and cannot be correct. Therefore, only M_oV is valid for unrestricted motion.

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