Moving Frames of Reference.

In these articles, an inertial frame of reference (FoR) is one where no external energy is being supplied to change its speed.
A “moving” FoR is one that has been accelerated to a constant speed relative to the home frame by externally applied energy.
A FoR attached to an object in free fall in a gravitational field does not qualify as a moving frame for relativistic purposes as its total energy is not changed, but it does qualify as an inertial frame.
A FoR attached to a particle on a disc rotating at a constant speed is both inertial and moving.

First, a derivation of the equivalence of mass and energy is given, as it is accepted to be true in some of the articles on this website.
Consider a photon emitted by a light source. It is moving at light speed c relative to its source and has energy E, but no mass and therefore no momentum, yet, when it strikes mass it exerts a force with the attributes of momentum. It was well known in the late 1800’s that momentum has a quantitative value of its energy divided by its speed. Therefore its apparent (virtual) momentum, when it strikes some mass, is E/c, and this has to be equivalent to the momentum of a mass M travelling at the speed of light, i.e. Mc.
Therefore, E/c = Mc, or E=Mc².
This means that mass is concentrated energy, and in appropriate circumstances mass and energy can be converted from one to the other.

For this discussion we will assume that a material object A is accelerated to some speed relative to an observer B, by applied energy.
There are two basic places for observation by an observer, one is where the observer travels with the object A, and he is then said to be in the object's "own" frame of reference (FoR), and the other is in any other FoR, called an "external" FoR herein, one of which becomes the chosen frame (often referred to as the stationary frame) from which the observer B observes the moving object A.

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 a speed by the application of energy, the energy transferred to the object takes the form of kinetic energy (KE) in external FoRs, but, in its own FoR the object always has zero velocity so it cannot have kinetic energy. Therefore, in its own FoR, as it cannot be created nor destroyed, the applied energy (E) must be stored in the object (often called potential energy) as mass. From the mass/energy equation, the stored mass is E/c², where E is also equal to the external KE and the new mass M is equal to the original "rest" mass Mo + KE/c².
Thus, observer B measures the speed (V) of A and calculates its kinetic energy from the classic definition, KE = MoV²/2. This assumes that all the applied energy is utilised to obtain speed, so that none of it is diverted to increasing the mass in external FoRs. 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 = Mo(1 + V²/2c²) in home frame units (not the same as Einstein's equation, where the maximum speed was restricted to that of light), and the mass of A as observed from external frames remains at Mo.

No problem so far? Some people do have a problem with it and assume the mass gain occurs in all FoRs, but what follows is where some, even highly qualified physicists (!), can get it wrong. When changing this example so that an observer at A observes an object in B to be apparently moving, a 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 occurred as no energy has been applied to object B. This apparent anomaly is solved for this situation as follows.
Firstly, consider that the total energy of an object (mass plus any kinetic energy plus any other form of energy associated with it) cannot be different by merely observing it from different frames. Therefore the Total Energy of an object must be the same in all FoRs.
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 Moc² = mc² + mV²/2, or Mo/m = 1 + V²/2c², and is the inverse of M/Mo, as one should expect from relativity for this circumstance.
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. Thus, the Doppler Effect for light does not qualify as a relativistic effect, it does not change any property of the source (mass, frequency), it is only an optical effect of the relative velocity of the observer.

To end, just a few words on relativistic momentum (MV); all text books assume M is the relativistic increased mass at velovity V. 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 MoV is valid for unrestricted motion. It is shown in Natural Relativity (Section III.B) that mass can increase in external frames if motion is restricted (with no dissipation of energy), but is only ever equal to M when V is zero, and it is then the Static case, such as when energy is added to an object to lift it against the force of gravity and there is no resulting motion (V=0).

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