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Longitudinal Displacement - The Separation Restoring Force


In order to derive the aetherial force equations it is useful to have a picture of the aether in mind. Figure 7 is therefore a useful construct. In Figure 7, the squares represent the attached negative aether, while the circles represent the attached positive aether. Recall that it is postulated above that each aether species is attached to itself and to its counterpart.



Now consider Figure 8, which depicts the situation wherein an infinite thin sheet of detached, positive aether is introduced into the attached aether. In such a case, the attached positive aether is pushed outward by the introduced detached aether. In this case, the circles of positive aether now have their centers upon the boundaries of the squares of negative aether. This leaves a condition where the displacement is P = (1/2)i for x > 1/2 and P = -(1/2)i for x < -1/2. (Here x is the horizontal coordinate of Figure 8, i is a unit vector in the x direction, and the unit for length is arbitrary.) In the case shown in Figure 8 there will be no net force on the detached aether, since the situation is completely symmetric. (Recall that it is postulated that the aether is incompressible in the presence of electromagnetic effects. Hence, when a sheet of detached positive aether is introduced into the attached positive aether it must push aside the attached positive aether in order to retain the constant overall aetherial density.) Next, Figure 9 depicts the situation wherein two infinite sheets of detached, positive aether are introduced into the attached aether. In this case, the outward forces on the attached positive aether add for the attached positive aether outside of both sheets, and the forces cancel for the attached positive aether between the sheets. Hence, on the outside of each sheet we have attached aether that is displaced from its negative counterpart, while between the sheets there is no such displacement. With a linear restoring force, the energy of each displacement is proportional to the square of the displacement, $$E = (1/2)k_1x^2$$, which is the familiar equation for the energy stored in a spring. It is therefore useful to think of the situation as one of a field of springs. The springs between the sheets are in their relaxed, non-extended, state and they have no extra energy. The springs outside of the sheets are extended and thus have have extra internal stored energy.



Since the detached aether is free to move, the situation shown in Figure 9 is one that will lead to motion of the detached aether. This is because, should the detached aether move outward, the outward springs will relax to their un-extended lengths, releasing their energy. That released energy will go into the sheets of detached aether as kinetic energy. The force on the sheets comes from the flow of the attached aether toward the position occupied by the detached aether, as the in-rushing attached aether pushes the detached aether outward. Hence it is seen that two detached aether sheets will repel each other.

As a next step toward the more general case, the density of detached aether within the sheets can be assumed to be less than the attached aetherial density, and the densities of the two sheets might be different from one another. Defining $$\rho_R$$ to be the amount of detached “red” aether per unit volume ($$\rho_R$$ is the “red” aetherial density) and defining the thickness of the sheet as $$\delta x$$, the displacement (the length of the red arrows) will become

$$x_R$$ = $$\rho_R \delta x$$/$$\rho_0$$

where $$\rho_0$$ is the density of the undisturbed attached positive aether. Similarly,

$$x_G$$ = $$\rho_G \delta x$$/$$\rho_0$$, where $$\rho_G$$ is the “green” aetherial density. (The displacements will be reduced from what is shown in Figure 9 if the density of the detached aether is less than $$\rho_0$$, since there will be less detached aether pushing upon the neighboring attached aether. The reduction in the displacement will be proportional to the reduction in the density of the detached aether within the sheet.)

If $$x_R$$ is not equal to $$x_G$$ the displacement of the attached aether would no longer be zero between the sheets. Rather, the displacement would be $$x_R - x_G$$ between the sheets, where $$x_R$$ is the displacement caused by the detached aether on the left, and $$x_G$$ is the displacement caused by the detached aether on the right. In this unbalanced case, the displacement of the attached aether outside of the sheets will be $$x_R + x_G$$. In this unbalanced case, the energy of the “springs” between the sheets is $$E_1 = (1/2)k_1(x_R - x_G)^2$$, while the energy of the “springs” outside of the sheets is $$E_2 = (1/2)k_1(x_R + x_G)^2$$. Hence, in this unbalanced case, the energy liberated from the “springs” as the sheets move is given by $$\Delta E = E_2 - E_1 = (1/2)k_1(x_R + x_G)^2 - (1/2)k_1(x_R - x_G)^2 = (1/2)k_1(x_R^2 + 2x_Rx_G + x_G^2) - (1/2)k_1(x_R^2 - 2x_Rx_G + x_G^2) = 2k_1x_Rx_G$$. (Note that in the balanced case, $$x_R = x_G = x/2$$ where is the total stretching of the spring and hence in that case the energy release is $$(1/2)k_1x^2$$ as in the balanced case discussed above.)

As the detached aether moves outward, this will decrease the energy stored in the springs in the layers it passes as those springs become relaxed. The energy that was once in the springs will transfer to the detached aether. Hence, the energy gained by the sheet is proportional to the distance it moves, and the force per unit area will be:

where $$\Delta E$$ is the energy released by all springs in the volume $$A\Delta x$$ and C is some constant. (As an area of the sheet moves, it passes through a volume $$A\Delta x$$. The number of springs in the volume will be proportional to the size of the volume. Hence, the energy liberated from an area A passing through a distance $$\Delta x$$ is $$\Delta E = CA\Delta x$$, and since energy is force times distance, Eq. (1) results.)

From the above discussion, we also know that the energy gained by the sheet as it passes a single spring is $$\Delta E_{spring} = 2k_1 x_R x_G$$. So, if $$A\Delta x$$ contains n springs, then the energy will be $$\Delta E = 2nk_1 x_R x_G$$ and C would be $$2nk_1 x_R x_G/\Delta x$$, leaving $$F/A = k_2 x_R x_G$$, where $$k_2 = 2nk_1/\Delta x$$.

So far, the analysis (and the Figures) involve an atomic aether, and while it is possible that the aether is atomic, it is also possible that it is a continuum. Either way, the aether is approximated by a continuum if the number of springs (n) becomes very large, and each spring becomes very small, in our region of analysis $$\Delta x$$. If the aether is a continuum, the analysis lets the spring size go to zero and n go to infinity, and the analysis will now switch over to the consideration of just such a continuum.

More generally, it is also possible to introduce detached negative aether into the analysis. If, instead of positive detached aether, the “red” aether depicted in Figure 3 is negative detached aether, then it would be the negative attached aether that would be displaced by the red detached aether. The red arrows remain the same, but now they represent the distance that the negative attached aether has moved. Since it is the difference between the positive and negative attached aether that results in the stretching of the springs, for the case where $$\rho_R = \rho_G = \rho_0$$ (and with the green aether still positive detached aether) it is the springs between the sheets that are stretched, while it is the springs outside of the sheets that remain relaxed. Hence, the energy inside the sheets is now $$\Delta E_1 = (1/2)k_1(w_R + w_G)^2$$, while the energy outside of the sheets is $$E_2 = (1/2)k_1(w_R - w_G)^2$$, and thus $$\Delta E_{spring} = E_2- E_1 = (1/2)k_1(w_R - w_G)^2 - (1/2)k_1(w_R + w_G)^2 = (1/2)k_1(w_R^2 - 2w_Rw_G + w_G^2)^2 - (1/2)k_1(w_R^2 + 2w_Rw_G + w_G^2)^2 = -2k_1w_Rw_G$$. Using the same analysis as above, the force becomes $$F/A = -k_1w_Rw_G$$.

Equation Examples
$$\mathbf{F} = m\mathbf{a} = \mu_P \rho_0 \Delta x \Delta y \Delta z \frac{\partial^2 \mathbf{P}} {\partial t^2} = \mathbf{F_T} + \mathbf{F_F} $$

$$ = \Delta x \Delta y \Delta z T_0 \nabla^2\mathbf{P} + k_2 \Delta x \Delta y \Delta z [\rho_{dp} \mathbf{v_{dp}} - \rho_{dn} \mathbf{v_{dn}} - \rho_0(\frac{\partial \mathbf{N}} {\partial t} - \frac{\partial \mathbf{P}} {\partial t})] $$

$$\nabla^2\mathbf{P} - \frac{\mu_P \rho_0} {T_0} \frac{\partial^2 \mathbf{P}} {\partial t^2} = \frac{k_2} {T_0}[-\rho_{dp} \mathbf{v_{dp}} + \rho_{dn} \mathbf{v_{dn}} + \rho_0(\frac{\partial \mathbf{N}} {\partial t} - \frac{\partial \mathbf{P}} {\partial t})] $$

$$\frac{1} {c^2} = \frac{\mu_P \rho_0} {T_0}$$

$$\frac{k_2} {T_0} = \frac{4 \pi a^2} {c}$$

$$\nabla^2\mathbf{N} - \frac{\mu_N \rho_0} {T'_0} \frac{\partial^2 \mathbf{N}} {\partial t^2} = \frac{k_2} {T'_0}[-\rho_{dn} \mathbf{v_{dn}} + \rho_{dp} \mathbf{v_{dp}} + \rho_0(\frac{\partial \mathbf{P}} {\partial t} - \frac{\partial \mathbf{N}} {\partial t})] $$

$$\frac{1} {c^2} = \frac{\mu_N \rho_0} {T'_0}$$

$$\frac{k_2} {T'_0} = -\frac{4 \pi a^2} {c}$$

$$\Psi_P - \Psi_N = -\phi/4\pi\rho_0$$