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3.5.8 Force

In the energy constraint equation, \(E=A\ c\ \kappa\), the energy content has a scalar value, and since \(A\), too, is a scalar quantity, the product \(c\ \kappa\) must also be scalar. It is therefore necessary that the imaginary part of the product of vectors, \(c=v+Jc_o\), and \(\kappa=\gamma + J\kappa_o\), is zero. That is,
\begin{align}
(v+Jc_o)(\gamma+J\kappa_o) &= v\gamma - c_o\kappa_o + J(v\kappa_o + c_o \gamma)\nonumber \\
&= v\gamma -c_o\kappa_o +J0 \nonumber \\
\ \nonumber \\
Hence, \ \ (v\kappa_o + c_o\gamma) = 0 \nonumber
\end{align}
This establishes the following relationship between $v, c_o, \gamma, and\ \kappa_o$
\begin{align}
\frac{v}{c_o} &=- \frac{\gamma}{\kappa_o}\ \ \ \ \ so\ that\ v=- \gamma \frac{c_o}{\kappa_o} \nonumber
\end{align}

That is, the velocity of the energy element is proportional to \(\gamma\) the excess curvature within the element. Since the energy element occupies only a small volume of the continuum, we can regard \(\gamma\) as a 'blip' in the overall curvature of the continuum.

To find acceleration, we differential the above equation with respect to time, so that
\begin{align}
acceleration\ = \frac{\partial v}{\partial t} &=-\frac{\partial \gamma}{\partial t}\ \frac{c_o}{\kappa_o} \nonumber
\end{align}

To make this equation meaningful we multiply each side by the mass of the energy element, \(m=E_o/c_o^2\). This gives
\begin{align}
m \frac{\partial v}{\partial t} &=-\frac{\partial \gamma}{\partial t}\ m\ \frac{c_o}{\kappa_o} \nonumber \\
&= -\frac{\partial \gamma}{\partial t}\ \frac{E_o}{c_o^2}\ \frac{c_o}{\kappa_o} \nonumber \\
&= -\frac{\partial \gamma}{\partial t}\ \frac{E_o}{c_o \kappa_o} \nonumber
\end{align}
Using the energy constraint relationship that \(E_o=Ac_o\kappa_o\), we find that the equation becomes
\begin{align}
m \frac{\partial v}{\partial t} &=-\frac{\partial \gamma}{\partial t}\ A \nonumber
\end{align}
The term on the right hand side had the dimensions of force and is proportional to rate of change of curvature \(\gamma\) within the volume element. The equation now reads, 'mass times rate of change of velocity equal force', which is Newton's second law of motion.

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