2  The Relativistic Characteristics of Rapidly Spinning Spherical Bodies.

2.1  Mass

To determine the relativistic effects upon the mass of a spherical body spinning at a rapid angular rate, consider Fig. 2.1 below.

Fig. 2.1 Section of a Spinning Spherical Body

In Fig. 2.1 the volume of the elemental is

Wele = r2sinj df dj dr

(2.1)

The rest mass of the elemental is then

m0(ele) = r0 r2sinj df dj dr

(2.2)

The velocity of the elemental due to the spin of the body is

vele = wrsinj

(2.3)

The energy mass of the elemental is therefore

mele = m0(ele) æ è 1- vele2 c2 ö ø 1/2   = r0 r2sinj df dj dr æ è 1- w2r2sin2j c2 ö ø 1/2

(2.4)

The toroidal energy mass is obtained by integrating (2.4) with respect to f over the limits 0 £ f £ 2p to give

mtor = 2pr0 r2sinj dj dr æ è 1- w2r2sin2j c2 ö ø 1/2

(2.5)

To complete the exercise of obtaining the energy mass of the spinning body it is first necessary to integrate (2.5) with respect to j between the limits 0 £ j £ p. Then it is necessary to integrate the result with respect to r over the limits 0 £ r £G.

While these integrations can be performed in an exact and rigorous manner it is a somewhat lengthy and involved process. It is simpler to make a relativistic approximation at this point which is then "reversed" after the integrations are completed. This process is valid because the expansions are absolutely convergent series and the result is accurate to about 1.12% as w Þ c. The approximation is therefore quite adequate to illustrate the significance of the result.

Thus binomially expanding the square root in (2.5), the first integral specified above becomes

msh = 4pr0 r2dr ó õ p/2 0  æ è sinj + w2r2 2c2 sin3j - 3 8 w4r4 c4 sin5j +... ö ø dj

(2.6)

Eq.(2.6) integrates to

msh = 4pr0 r2 æ è 1+ w2r2 3c2 - w4r4 5c4 +... ö ø dr

(2.7)

Integrating (2.7) with respect to r as specified then yields

msph = 4 3 pr0 G3 æ è 1+ w2G2 5c2 - 3w4G4 35c4 +... ö ø

(2.8)

for the energy mass of the spinning sphere.

Performing the reverse expansion on (2.8), using just the first relativistic term gives

msph @ m0 æ è 1 - 2 5 w2G2 c2 ö ø 1/2

(2.9)

where the classical expression for the rest mass has also been inserted. Eq.(2.9) can be written

msph @ m0 æ è 1 - w2Ggyr2 c2 ö ø 1/2

(2.10)

where G_gyr is the classical radius of gyration of a stationary spherical homogeneous mass,

=

æ Ö

2 5

G

Eq.(2.10) is the energy mass of the spinning body wherein the relativistic mass increase is the stored kinetic energy induced by the applied accelerating torque.

The important feature about this result is apparent in (2.9) where it can be seen that if wG = c then the mass becomes

msph @   æ Ö 5 3   m0

(2.11)

and is a maximum.

Thus the surface of the sphere at the spin circumference, can be accelerated to the terminal velocity of D₀( ~ the speed of light), while the energy mass of the sphere remains finite. This is solely due to the distributed nature of the mass and that each toroidal element between the centre and the spin circumference is spinning at a velocity lower than wG. Because the energy mass of the sphere is finite under this condition, only a finite amount of energy has been applied to reach this state, and it would therefore be possible to apply additional energy in the form of an accelerating torque, to further increase the spin rate. The consequence of this would be that the surface of the sphere at the spin circumference would tend to exceed the terminal velocity of D₀, and thus contravene its primary criterion of existence, i.e. that the maximum spatial velocity attainable in that Domain is the velocity constant c. To avoid this the mass must therefore lose energy. In a future paper concerning the existence of de Broglie matter waves in D₀, it will be shown that matter can only exist at the terminal velocity of D₀ as pure kinetic energy, possibly in the form of photons/electromagnetic radiation. It is therefore proposed that a spherical mass induced to spin such that the surface at the spin circumference tends to exceed the terminal velocity in D₀, avoids that anomaly by losing kinetic energy at that surface, by converting it to radiant energy at some frequency and wavelength proportional to the energy applied to increase the angular rate. The emission of the spectra would be in accordance with the quantum laws of Planck and de Broglie.

This phenomenon may have significant implications in Cosmology and Atomic Structure Theory, and is discussed further along those lines in the concluding remarks.