Unprovability of first Maxwell’s equation in light of EPR’s completeness condition: a computational approach from logico-linguistic perspective

Abstract

Maxwell’s verbal statement of Coulomb’s experimental verification of his hypothesis, concerning force between two electrified bodies, is suggestive of a modification of the respective computable expression on logical grounds. This modification is in tandem with the completeness condition for a physical theory, that was stated by Einstein, Podolsky and Rosen in their seminal work. Working with such a modification, I show that the first Maxwell’s equation, symbolically identifiable as $\mathbf {\nabla }\cdot \textbf{E}=\rho /\epsilon _0$ from the standard literature, is unprovable. This renders Poynting’s theorem to be unprovable as well. Therefore, the explanation of ‘light’ as ‘propagation of electromagnetic energy’ comes into question on theoretical grounds.

Appendices

Appendix A: An estimate of $s, s'$: Computational content of Maxwell’s statement of Coulomb’s hypothesis

Maxwell’s statement of Coulomb’s hypothesis, in light of ECC, calls for an explanation of the length scales s and $s'$, that characterise the electrified bodies, in (2) which must appear only as an approximate expression along with correction terms in powers of $s/r, s'/r$, etc. such that these terms become negligible as $r\ggg s, s'$ in tandem with the experimental data that verify Coulomb’s hypothesis. It should appear as if the electrified bodies themselves are the most natural uncertainties, characterised by $s, s'$, in the experimental demonstration of the Coulomb’s hypothesis. Such an analysis indeed has been penned for gravitational interaction between two bodies in ref. [31], which respects experimental observations. While I plan to report the details of such analysis for the interaction between two electrified/charged bodies on a different occasion, here I shall provide a glimpse of the relevant outcome of the analysis which will show an estimate of $s, s'$ in (2). In what follows, to make it familiar in accord with the modern notations available in the textbooks, I shall write $q_1, q_2$ in stead of $e, e'$. Also, for similarity of notation, I shall write $\xi _1, \xi _2$ instead of $s, s'$.

The analysis that has been presented in ref. [31], for the case of gravity, takes into account the extension of the dot that is needed to demonstrate the notion of ‘a point’ and hence its smallness can only be understood in relation to what we may call ‘a line’. Adopting such a philosophy of incorporating the natural uncertainties that are necessary for the demonstration of numbers by making cuts on a line, and following the steps of reasoning presented in ref. [31], the relevant expressions for the present discussion, from which the analysis shall begin, can be written as follows:

$$\begin{aligned}{} & {} \frac{F}{F_C}=\frac{\xi _1\xi _2}{d^2}~~:~F_C=\frac{m^2c^3}{h},~\xi _i=\zeta \frac{q_i}{m}~~\forall i\in [1,2],~\nonumber \\{} & {} \zeta =\left( \frac{h}{4\pi \epsilon _0c^3}\right) ^{\frac{1}{2}}. \end{aligned}$$

(42)

In comparison to the notations of ref. [31], and in order to manifest the difference between the quantities involved, I have written $F_C, \xi _1, \xi _2$ here in place of $F_0, s_1, s_2$ from ref. [31], respectively. Here,

$$\begin{aligned}{} & {} \epsilon _0= 8.854\times 10^{-12}~\text {C}^2\,\text {kg}^{-1}\,\text {m}^{-3}\,\text {s}^2 \end{aligned}$$

(43)

$$\begin{aligned}{} & {} h=6.626\times 10^{-34}~\text {kg}\,\text {m}^2\,\text {s}^{-1}\end{aligned}$$

(44)

$$\begin{aligned}{} & {} c=3\times 10^8~\text {m}\,\text {s}^{-1}\end{aligned}$$

(45)

$$\begin{aligned}{} & {} \therefore \zeta = \left( \frac{h}{4\pi \epsilon _0c^3}\right) ^{\frac{1}{2}}=4.696\times 10^{-25}~\text {C}^{-1}\,\text {kg}\,\text {m}. \end{aligned}$$

(46)

For $m=1$ kg and $q_i=1$ C, $\xi _i=4.696\times 10^{-25}$ m.

By construction, $d=r_i+\xi _i\forall i\in [1,2]$. The condition $d>2\xi _i \equiv r_i>\xi _i$ is called large distance with respect to $\xi _i$ and the condition $d<2\xi _i \equiv r_i<\xi _i$ is called small distance with respect to $\xi _i$.

For large distance analysis with respect to $\xi _1$:

$$\begin{aligned}{} & {} \frac{F}{F_C}=\frac{\xi _2\xi _1}{r_1^2}\left[ 1-\frac{2\xi _1}{r_1}+\frac{3\xi _1^2}{r_1^2}-\cdots \right] ~:~d>2\xi _1,\nonumber \\{} & {} \quad ~r_1>\xi _1 \end{aligned}$$

(47)

$$\begin{aligned}{} & {} \quad \equiv F=\frac{1}{4\pi \epsilon _0}\frac{q_2q_1}{r_1^2}\left[ 1-2\zeta \frac{q_1}{r_1}+3\zeta ^2\frac{q_1^2}{r_1^2}-\cdots \right] ~:~r_1\nonumber \\{} & {} \quad >\zeta \frac{q_1}{m}\qquad \hbox {[using (42)]}. \end{aligned}$$

(48)

For large distance analysis with respect to $\xi _2$:

$$\begin{aligned}{} & {} \frac{F}{F_C}=\frac{\xi _1\xi _2}{r_2^2}\left[ 1-\frac{2\xi _2}{r_2}+\frac{3\xi _2^2}{r_2^2}-\cdots \right] ~:~d\nonumber \\{} & {} \quad>2\xi _2, ~r_2>\xi _2 \end{aligned}$$

(49)

$$\begin{aligned}{} & {} \quad \equiv F=\frac{1}{4\pi \epsilon _0}\frac{q_1q_2}{r_2^2}\left[ 1-2\zeta \frac{q_2}{r_2}+3\zeta ^2\frac{q_2^2}{r_2^2}-\cdots \right] ~:~\nonumber \\{} & {} \quad r_2>\zeta \frac{q_2}{m}\qquad \hbox {[using (42)]}. \end{aligned}$$

(50)

The following may be called the point charge limit with respect to both $\xi _1$ and $\xi _2$:

$$\begin{aligned}{} & {} \frac{F}{F_C}\simeq \frac{\xi _1\xi _2}{r^2}: d=r_i+\xi _i\simeq r_i\simeq r ~(\text {say}) ~\text {when}~ \nonumber \\{} & {} \quad r_i\ggg \xi _i~\forall i\in [1,2],\end{aligned}$$

(51)

$$\begin{aligned}{} & {} \quad \equiv {F}\simeq \frac{1}{4\pi \epsilon _0} \frac{q_1q_2}{r^2}~:~ r\simeq r_i\ggg \xi _i~\forall i \in [1,2]\nonumber \\{} & {} \quad \quad \hbox {(using (42))}. \end{aligned}$$

(52)

This provides a justification of Maxwell’s statement of Coulomb’s hypothesis along with an estimate of the length scales $\xi _1,\xi _2$ which characterise the two charged bodies. The essence of the associated condition, and why this can be called the point charge limit, can be understood by calling a point as that is extremely small compared to what can be called a line – a refinement of the first axiom of geometry by considering the concreteness of a dot that is necessary for the purpose of demonstration of what a point is [31].

In passing, it can be noted that the small distance analysis, with respect to both $\xi _1$ and $\xi _2$, is well behaved as follows:

$$\begin{aligned}{} & {} F=F_C\left[ 1-2\frac{r}{\xi }+3\frac{r^2}{\xi ^2}-\cdots \right] ~:~\xi _1=\xi _2=\xi \nonumber \\{} & {} \quad >r=r_1=r_2, \end{aligned}$$

(53)

$$\begin{aligned}{} & {} \quad \equiv F=\frac{m^2c^3}{h}\left[ 1-2\frac{r}{\zeta q}+3\frac{r^2}{\zeta ^2q^2}-\cdots \right] ~:~r\nonumber \\{} & {} \quad <\zeta \frac{q}{m}\qquad \hbox {(using (42))}. \end{aligned}$$

(54)

All the above expressions and the corresponding analysis can be cast into a form that manifests the role of fine structure constant. However, such discussions are not necessary for the present purpose and will be reported in future on a different occasion.

Appendix B: A decision problem as the basis of the continuity equation

The continuity equation, the so-called local conservation law, forms the basis of our general understanding of the flow of quantities like charge, mass, etc. in physics (see refs [2, 3, 49,50,51]). Here, I point out how the continuity equation is founded on a decision problem that allows for making suitable choices concerning two apparently contradictory propositions. For the present discussion it is convenient to consider the continuity equation involving charge and current [3]. To understand the issue at hand, it is sufficient and convenient to discuss the flow of charge along a wire that can be written as follows:

$$\begin{aligned} \mathbf {\nabla }\cdot \textbf{J}+\frac{\partial \lambda }{\partial t}=0, \end{aligned}$$

(55)

where $\textbf{J}=\lambda \textbf{v}$, $\textbf{v}=\textrm{d}\mathbf {\ell }/\textrm{d}t$, $\textrm{d}\mathbf {\ell }={\textrm{d}x}~{\hat{i}}+{\textrm{d}y}~{\hat{j}}+{\textrm{d}z}~{\hat{k}}$ and $\textbf{v}$ satisfies the condition $\mathbf {\nabla }\cdot \textbf{v}=0$ [49,50,51]; $\lambda [x(t), y(t), z(t), t]$ is the linear charge density (charge per unit length) at some instant t, $\textbf{J}[x(t), y(t), z(t), t]$ is the line current density and [x(t), y(t), z(t)] denote the spatial coordinate of any point on the line of flow at some time t. The construction of the relation $\textbf{J}=\lambda \textbf{v}$ is based on two scenarios of interpreting the charge flow that can be demonstrated through figure 6, which can be generalised for surface and volume flows (see ref. [3]).

• Scenario 1 (S1): Consider that a point charge $\textrm{d}q$ is displaced through $\mathbf {\textrm{d}l}$ in time $\textrm{d}t$ such that we can write $\mathbf {\textrm{d}l}=\textbf{v}\textrm{d}t$. Then, we can write $|\mathbf {\textrm{d}l}|=|\textbf{v}|\textrm{d}t$ and hence, $\textrm{d}q=\lambda |\textbf{v}|\textrm{d}t$. In this case, the charge $\textrm{d}q$ is at the point A at some time t and then it is at the point B at some time $t+\textrm{d}t$, so that the displacement in time duration $\textrm{d}t$ is $\textbf{AB}=\mathbf {\textrm{d}l}$. In such a scenario the following assertion holds true:

  $$\begin{aligned} M \equiv \textrm{d}q\text { is a point charge}. \end{aligned}$$

  (56)

• Scenario 2 (S2): At any instant t of the passage of a line current, there is a line charge density $\lambda$ (charge per unit length) such that the charge $\textrm{d}q$ distributed over the length $\textrm{d}\ell$ is written as $\textrm{d}q=\lambda \textrm{d}x$. In this case, the charge $\textrm{d}q$ is neither at point A, nor at point B, but it is spread over the line AB. Therefore, the following proposition holds true for such explanation:

  $$\begin{aligned} \lnot M \equiv \textrm{d}q\text { is NOT a point charge}. \end{aligned}$$

  (57)

Therefore, the relation $\textbf{J}= \textrm{d}q/\textrm{d}t=\lambda \textbf{v}$ holds true if and only if M and $\lnot M$ in one and the same process of reasoning that leads to such a construction. It may be written as

$$\begin{aligned} \textbf{J}=\lambda \textbf{v}~: M\wedge \lnot M. \end{aligned}$$

(58)

The situation can be viewed as a decision problem where it cannot be decided whether $\textrm{d}q$ is a point charge or not [14]. This, however, allows for suitable choices to be made according to context, i.e. $\textrm{d}q$ is considered as a point charge in S1 and it is not considered as a point charge in S2.