Members of the NPA praise Dr. Lucas scientific work, reason enough to buy the first volume of his four volume work (The Universal Force). To orientate myself in any new work I usually read the opening and closing paragraphs of chapters and interspersed random bits that have caught my attention while scanning through the book. The typesetting and general layout was a bit confusing and inconsistent.

Nevertheless, the content is the essence, not the appearance. Lucas, in Chapter 4, makes his first important assertion, quote: Equation (4-44) is the derived version of the electrodynamic force F for an elastic finite-size charged particle to order v in Galilean transformation. The   \nicefrac{v }{c } factors typically identified with Special Relativity Theory are found to originate from finite-size electrical feedback effects, nonlinear effects, and conservation of energy and momentum. The last sentence is ambiguous: Is \nicefrac{v}{c} dependent on only one of the three cases under certain circumstance, or dependent on varying combination of the three cases?lucas92

Let’s derive Equation (4-44) on Page 92, the page can be viewed by clicking on the thumbnail to the left.  We start with the Equation (4-43) now renamed to (1) and (2) and we assume these to be correct.

(1)   \begin{equation*}  \vec {E}(\vec r,\vec v)  = \vec {E}_0(\vec r)\frac{1-\beta^2}{(1-\beta^2Sin^2\theta)^\frac32} \end{equation*}


(2)   \begin{equation*} \vec {B}_i(\vec r,\vec v) =\frac{\vec v}{c}\times\vec E(\vec r,\vec v) \end{equation*}

Lucas states: Now the total electromagnetic force F exerted by the moving charge distribution on a test charge q^\prime is using equation (4-26) for the Lorentz force and begins Equation (4-44) with

(3)   \begin{equation*}\vec {F}(\vec r,\vec v)=q\left\{\vec {E}(\vec r,\vec v)+ \frac{\vec v}{c}\times\vec {B}_i(\vec r,\vec v)    \right\}\end{equation*}

and the confusion and obfuscation begins. We forgive the typo,  the equation uses q and not q^\prime as stated in the preceding text. However, we cannot overlook Lucas’ gross mathematical mistakes. He develops (3) to

(4)   \begin{equation*}\vec F(\vec r,\vec v)= q \vec E_0(\vec r)\frac{1-\beta^2}{(1-\beta^2Sin^2\theta)^\frac32} \left[(\underbrace{1-\beta^2}_a+\underbrace{\beta^2\cos^2\theta)\hat{{r}}- (\vec\beta\cdot\hat{{r}})\hat{{r}}\times(\hat{{r}}\times\vec\beta)}_b \right]\end{equation*}

The 1 - \beta^2 term marked a should not be multiplied with the unit vecor \hat{{r}} and the remaining terms, marked b should be grouped with E_0 a scalar equal to the magnitude of \vec E_0(\vec r), that is a scalar is multiplied by a vector. The vector product \vec E_0 \hat{{r}} does not exist in mathematics! Furthermore, how the cos^2 \theta term is obtained remains a mystery to me.

The correct development of (3) is

(5)   \begin{equation*} \vec F(\vec r,\vec v)=q\left\{\vec E(\vec r,\vec v)+ \frac{\vec v}{c}\times\vec B_i(\vec r,\vec v)   \right\}\end{equation*}

using (2) to express \vec B_i in terms of \vec E

(6)   \begin{equation*} \vec F(\vec r,\vec v)=q\left\{\vec E(\vec r,\vec v)+ \frac{\vec v}{c}\times\left(\frac{\vec v}{c}\times\vec E(\vec r,\vec v)\right)    \right\}\end{equation*}

using (1) to express \vec E(\vec r,\vec v) in terms of \vec E_0(\vec r)

(7)   \begin{equation*} \vec F(\vec r,\vec v)=q \frac{1-\beta^2}{(1-\beta^2Sin^2\theta)^\frac32}\left\{\vec E_0(\vec r)+ \frac{\vec v}{c}\times\left( \frac{\vec v}{c}\times\vec E_0(\vec r)  \right)  \right\}\end{equation*}

using the identity \vec a \times (\vec b \times \vec c) = \vec b (\vec a \cdot \vec c) - \vec c (\vec a \cdot \vec b)

(8)   \begin{equation*} \vec F(\vec r,\vec v)=q \frac{1-\beta^2}{(1-\beta^2Sin^2\theta)^\frac32}\left\{\vec E_0(\vec r)+ \frac{\vec v}{c}\left(\frac{\vec v}{c}\cdot\vec E_0(\vec r)\right) - \vec E_0(\vec r)\left(\frac{\vec v}{c}\cdot\frac{\vec v}{c}\right)  \right\}\end{equation*}

evaluating the dot products i.e  \vec a \cdot \vec b = a b \cos \theta and \vec a \cdot \vec a= a^2, and grouping the vector quantities. Finally, also making the substitution \nicefrac{\vec v}c = \vec \beta to obtain

(9)   \begin{equation*} \vec F(\vec r,\vec v)=q \frac{1-\beta^2}{(1-\beta^2Sin^2\theta)^\frac32}\left\{\vec E_0(\vec r)\left(1-\beta^2\right)+ \vec \beta \Big(\beta E_0 \cos\theta \Big)   \right\}\end{equation*}

from which Lucas’ Equation (4-44),  that is Eq. (4) above, cannot be derived!

Four pages on, Lucas writes: The generalized potential energy U corresponding to equation (5-1) [which is the same as his (4-44)]  for the electrodynamic force that is accurate to order V in the Galillean transformation is

(10)   \begin{equation*}U(r,v)=\frac{qq^{\prime}}{r} \frac{(1-\beta^2)}{(1-\beta^2\sin^2\theta)^\frac12}= \frac{qq^\prime(1-\vec\beta^2)}{\left[\vec r^2 - \frac{\{\vec r \times (\vec r \times \vec \beta)\}^2}{\vec r^2} \right]^\frac12}\end{equation*}

At this point I closed the book since the work that followed was based on this flawed mathematics, there was no need for further review.  Mathematics is the final arbitrator; the Eq. (10), which is (5-4) on page 96, is also mathematically incorrect.  One can only square real and complex numbers, one cannot square, or take square roots, of vectors!

Lucas’ booked is filled with these and similar errors, therefor I cannot take this book seriously and consequently recommend others to dismiss it too.