Here we see a paradox, since theoretically relativistic mass
and energy of the particle can approach infinity.
In fact, in the reference frame S the mass at rest [M.sub.0]([infinity]), which is equal to the value [[mu].sub.0], moves with linear momentum p, or, equivalently, with constant velocity 1/[square root of (1 + [[mu].sup.2.sub.0]/[p.sup.2])], and becomes the relativistic mass
[M.sub.p]([infinity]), which is equal to the value [square root of ([[mu].sup.2.sub.0] + [p.sup.2])].
The relativistic mass
increase of the oscillating electrons leads to a decrease in the plasma frequency and thus a local increase in the refractive index.
You see that the Doppler effect causes the correct de Broglie wavelength and relativistic mass
increase to appear in the observed waves, as a function of the relative velocity.
B is the magnetic field, c is the speed of light, E is the electric field, [P.sub.[perpendicular to]] is the momentum perpendicular to the magnetic field, t is the time, m is the rest mass of the charged particle, [gamma]m is the relativistic mass
of the charged particle, and q is the charge of the charged particle.
The concept of relativistic mass
has been a part of special relativistic physics since it was first introduced by Einstein [1,2] and explored by the early relativists (see for example [3,4]).
The relativistic mass
and energy are equal in all the calculations.
The MSR was specifically designed to carry out muon (g - 2) precession experiments (g is the Lande [gamma]-factor) with muons of momentum 3.094 GeV/c corresponding to a [gamma]-factor of 29.3 (effective relativistic mass
), so that the electrons emitted from muon decay in the lab frame were very nearly parallel to the muon momentum.
Real mass-bearing particles (the rest-mass [m.sub.0] [not equal to] 0, the relativistic mass
m = [m.sub.0]/[square root of (1 - [V.sup.2]/[c.sup.2])] is real) move along the non-isotropic lines (ds [not equal to] 0) at sub-luminal velocities V < c; imaginary mass-bearing particles or hypothetical tachyons (the rest-mass [m.sub.0] [not equal to]] 0, the relativistic mass
m = i[m.sub.0]/[square root of (1 - [V.sup.2]/[c.sup.2])] is imaginary) move along non-isotropic lines (ds [not equal to] 0) at superluminal velocities V > c; massless particles (the rest-mass [m.sub.0] = 0, the relativistic mass
m [not equal to] 0) move along isotropic lines (ds = 0) at light velocity V = c.
Advantage of Periodic relativity (PR) over general relativity can be seen in its use of revised principle of equivalence which states that the gravitational mass is equal to the relativistic mass
. Application of this principle gives a very simple derivation for the orbital period derivative of the binary star .
By solving the scalar geodesic equation for a mass-bearing particle ("stone-like objects"), we shall obtain that the relativistic mass
of the object changes according to the remoteness to the observer in the particular space.
The projective unprimed intrinsic affine coordinates [phi][??] and [phi]c[phi][??] that constitute the observer's intrinsic frame, containing one-dimensional intrinsic relativistic mass
[phi]m of the particle in [phi][??], are then made manifest outwardly in the unprimed affine spacetime coordinates [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the observer's fame on flat four-dimensional spacetime, containing the three-dimensional relativistic mass
, m = [gamma][m.sub.0], of the particle in affine 3-space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the observer's frame.