2.2  Moment of Inertia and Radius of Gyration.

Proceeding as in the last Section the moment of inertia of the toroid is determined by integration of (2.12) with respect to f between the limits 0 £ f £ 2p to give

Again proceeding as in the previous Section and binomially expanding the square root in (2.13) gives

Now, integrating (2.16) with respect to r between the limits 0 £ r £ G gives the moment of inertia of the spinning sphere as

Reversing the earlier expansion process, using just the first relativistic term, finally gives

To determine the relativistically corrected radius of gyration, note that according to classical mechanics the relativistically corrected moment of inertia would be given by

G_gyr^* is the relativistically corrected radius of gyration.

Inserting (2.9) for the energy mass, equating the result to (2.18) and solving for the relativistic radius of gyration yields

Thus both the moment of inertia and the radius of gyration are relativistically variable parameters. The reason is because as the spin rate of the spherical body increases, the outer part of the body, which contains the greater part of the mass, also experiences a higher spin velocity so absorbing the greater proportion of kinetic energy. This results in that part of the body gaining the greater part of the relativistic mass increase, so causing an increase in the radius of gyration and thereby the moment of inertia.

2.3  Angular Momentum.

Eq. (2.25) is the maximum angular momentum that can be obtained by any spinning homogeneous spherical mass within Pseudo-Euclidean Space-Time, D₀.

2.4  Spin Energy.

The kinetic energy gained from the accelerating torque by the spinning mass is given by the classical equation

Ek = w2Isph 2

(2.26)

and from (2.18) this becomes

Ek @ w2I0 2 æ è 1- 4 7 w2G2 c2 ö ø 1/2

(2.27)

The maximum is again given when wG = c, viz.

Ek @   æ Ö 7 75   m0 c2

(2.28)

This is a little over 30% of the rest energy, quite a low value for a relativistic energy maximum. The application of further accelerating torque does not add kinetic energy to (2.28) but, as proposed in Section 2.1, is radiated away by that part of the surface experiencing the terminal velocity.

2.5  Volume, Surface Area and Average Matter Density.

Using the results of the previous Sections, it is easy to show that the volume of a spinning sphere, measured in units associated with stationary axes in D₀ is

Wsph @ W0 æ è 1+ 2 5 w2G2 c2 ö ø 1/2   @ W0 æ è 1+ w2Ggyr2 c2 ö ø 1/2

(2.29)

and the surface area

L sph @ L 0 æ è 1+ 2 3 w2G2 c2 ö ø 1/2

(2.30)

It is interesting to note that these last two parameters, volume and surface area reduce as spin rate increases in contrast to the other parameters studied here which increase. This is caused by the Lorentz/Fitzgerald contraction of the circumference as is shown in Section 2.6 below.

From (2.9) and (2.29) it is clear that the average density of a spinning sphere is also a relativistic variable given by

r @ r0 æ è 1+ 2 5 w2G2 c2 ö ø 1/2   æ è 1- 2 5 w2G2 c2 ö ø 1/2   @ r0 æ è 1- 2 5 w2G2 c2 ö ø @ r0 æ è 1- w2Ggyr2 c2 ö ø

(2.31)

and when wG = c

rmax @ 5 3 r0

(2.32)

Eq.(2.32) is the maximum average density attainable by a spinning spherical homogeneous mass.

2.6  The Circumference in the Spin Plane.

Consider the elemental in Fig. 2.2 below

Fig. 2.2 Rotating Elemental.

In Fig. 2.2 rdf is the width of the elemental at rest. The effect of Lorentz/Fitzgerald contraction is to reduce dimensions in the direction of motion. Thus the width of the above elemental rotating at a constant angular rate of w would, in stationary axes in D₀, become

w = rdf æ è 1- w2r2 c2 ö ø 1/2

(2.33)

If now the elemental in Fig. 2.2 is extended to become a rotating toroid, i.e. integrating df over the limits 0 £ f £ 2p, then (2.33) becomes

wcir = 2p r æ è 1- w2r2 c2 ö ø 1/2

(2.34)

An alternative way of representing this effect is as follows.

The axes attached to a body in motion are sometimes referred to as having rotated into the temporal dimension, [1], [3]. To depict the effect proposed here, if the spin circumference of the rotating toroid is represented using Fig. 2.3 below

Fig. 2.3 Representation of Circumference in Relativistic Axes

In Fig. 2.3

C is an axis stationary in D₀.

C^/ is an axis moving with the velocity of the toroidal circumference, i.e. wr. This axis is therefore effectively attached to the rotating toroid.

q is the spatial axis rotation angle into the temporal dimension and where sinq = wr/c

If the circumference of the toroid is laid out on the two above axes, then the distance 0 ® B₁ on the C axis is the circumference of the toroid at rest. i.e. 2pr. The distance 0 ® B₂ is the circumference of the toroid on the moving axes measured in the units of the moving axis. This therefore also measures the circumference as at rest. Consequently,

The distance 0 ® B₃ is the circumference of the spinning toroid measured in the stationary C axis in D₀. Thus

This result leads to the apparent anomaly that in the limit, when the outer surface of the toroid reaches the terminal velocity of D₀, i.e. wr = c, the circumference of this outer surface becomes zero for a finite radius. In the axis attached to the spinning toroid the circumference remains unchanged at 2pr. This is however, no different from the reduction to zero of length in a linear Lorentz/Fitzgerald contraction.

2.7  Time Dilatation and its Radial Gradient

This is similar to the gravitational case where the term on the left is shown in [2] Eq.(3.17) and [2] Eq. (4.4) et seq. to represent the gravitational Acceleration Potential. In the case shown here it represents the centripetal acceleration of a spinning body in D₀, which is therefore seen to be the result of the radial temporal dilatation gradient due to the spin motion. The similarity is however, only a mathematical one because the Acceleration Potential of D₁, i.e. gravity, is the result of the time dilatation gradient produced by the gravitational source, i.e. there is no external force applied to produce this effect. Centripetal acceleration on the other hand is caused by the change in direction of the rotating body as it spins about the centre of rotation. An external force is exerted on it by whatever means it is connected to the centre of rotation. Eq. (2.42) is consequently only an alternative mathematical expression for centripetal acceleration.

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