In the quaternion electromagnetism formulation, the electromagnetic field quaternion denoted $\stackrel{\leftrightarrow }{F}=G-\stackrel{\to }{E}+i\stackrel{\to }{B}$ has a scalar part $G$, which is usually set equal to 0 due to L. Lorenz (see footnote 2). In Equation (12) below a differential equation for $G$ is derived. Although $G=0$ is a trivial solution more physically suggestive solutions exist. It will be suggested below that at least in the Classical Newtonian domain the $G$ could be proportional to the gravitational field. The time dependent “wave equation” for $G$ Equation (12) below suggests in addition to the static solution there is a wave that travels along with electromagnetic wave (which is related to the quanta of light known as the “photon”). It is proposed that the companion $G$ wave have a related quanta which should be called the “graviton”. It is proposed that the graviton-photon couple be named the Photo-graviton. Note that the graviton in this model has a spin 1 so that the photo-graviton should have a spin 2. The graviton is proposed to have negative mass extrapolating on the idea in reference [10] and that this mass exactly cancels the companion photon mass.

Whenever mass is created, even a small amount, an appropriate amount of gravity must be created also (with the common assumption that mass and gravity are related). According to an idea by Alan Guth [10] a gravitational field should have negative mass, extending this gravitons should have negative mass. When a “graviton” interacts with a positive mass, the positive mass, in order to conserve momentum, the positive mass must react by going in the direction of the graviton source. Since $E=m{c}^2$, Einstein’s idea that energy and mass are the same thing, when something as simple as a hydrogen atom absorbs an electromagnetic wave of an appropriate frequency, the hydrogen atom’s mass increases by + $\text{Δ}m=\frac{hf}{c^2}$. The hydrogen atom must then increase (actually decrease) the energy of its gravitational field by $-\text{Δ}m=\frac{-hf}{c^2}$. At first, this seems to be at odds with classical electromagnetism, however, it is not as will be shown below. If a negative mass graviton rides along with a positive mass photon it explains how a massive photon with $m=\frac{E}{c^2}$ can go at the speed of light, while other objects of finite mass are restricted by relativity from ever attaining this supposed universal speed limit. Negative mass gravitons in a vacuum are not in conflict with the curved-distorted Euclidean space of general relativity, in fact, they are why the space is distorted.

Figure 1 shows photo graviton absorption, and the completed Maxwell’s equation to the left, see subsection 6 below. It should be noted that the graviton wave has the “beat frequency” of the interference pattern of the two involved DeBroglie waves or Ψ functions. Figure 2 shows photo graviton spontaneous emission stimulated by graviton.

See Figure 1 , the photo graviton excites the lower state electron, to a higher more energetic state. The extra energy causes an increase in mass of the atom (from Einstein’s $E=m{c}^2$) that mass being; $\text{Δ}m=\frac{hf}{c^2}$. With a corresponding decrease in the gravitational field mass of $\text{Δ}{m}_{g-field}=\frac{-hf}{c^2}$. as shown by the emitted graviton.

The claim of Figure 2 is that spontaneous emission is triggered by free gravitons, the quanta of the field $G={\partial }_0{A}_0+\stackrel{\to }{\nabla }\cdot \stackrel{\to }{A}$. This claim is actually reinforced by standard physics graduate school text books such as reference [11] which give a simplified and not completely rigorous treatments of spontaneous emission.

Texts give an interaction Hamiltonian for the simplest spontaneous emission of $H_I=\stackrel{\to }{j}\cdot \stackrel{\to }{A}=-\frac{e\stackrel{\to }{p}\cdot \stackrel{\to }{A}}{mc}$ where $\stackrel{\to }{j}$ is the current of electron in upper state this is a gives transition matrix element $\langle \beta |H_I|\alpha \rangle$ going from the excited state $\langle \beta |$ to the lower state $|\alpha \rangle$ note that $\stackrel{\to }{p}$ is the electron momentum $\stackrel{\to }{A}$ is the vector potential. In quantum mechanics $\stackrel{\to }{p}$ is an operator $\stackrel{\to }{p}\to -ih\stackrel{\to }{\nabla }$ substituting this into $H_I=-\frac{e\stackrel{\to }{p}\cdot \stackrel{\to }{A}}{mc}$ yields $H_I=\frac{-ieh\left(\stackrel{\to }{\nabla }\cdot \stackrel{\to }{A}\right)}{mc}$ note that the interaction Hamiltonian for spontaneous emission contains ( $\stackrel{\to }{\nabla }\cdot \stackrel{\to }{A}$), which is the vector potential part of what is claimed to be proportional to the gravitational potential $G=\left({\partial }_0{A}_0+\stackrel{\to }{\nabla }\cdot \stackrel{\to }{A}\right)$ therefore bolstering the claim that gravitational fluctuations or gravitons cause spontaneous emission!

Devices such as lasers depend on the stimulated emission of light. It should involve a dragging ambient graviton into the process of light amplification however, this topic deserves a better treatment than can be presented here.

For two “real” Pauli quaternions $\stackrel{\leftrightarrow }{A}=w_A+x_A{\hat{\sigma }}_1+y_A{\hat{\sigma }}_2+z_A{\hat{\sigma }}_3=\left(w_A+{\stackrel{\to }{r}}_A\right)$ and $\stackrel{\leftrightarrow }{B}=w_B+x_B{\hat{\sigma }}_1+y_B{\hat{\sigma }}_2+z_B{\hat{\sigma }}_3=\left(w_B+{\stackrel{\to }{r}}_B\right)$

$\stackrel{\leftrightarrow }{A}\ast \stackrel{\leftrightarrow }{B}=\left(w_A+{\stackrel{\to }{r}}_A\right)\left(w_B+{\stackrel{\to }{r}}_B\right)=\left(w_A{w}_B+\left({\stackrel{\to }{r}}_A\cdot {\stackrel{\to }{r}}_B\right)+w_A{\stackrel{\to }{r}}_B+w_B{\stackrel{\to }{r}}_A+i\left({\stackrel{\to }{r}}_A\times {\stackrel{\to }{r}}_B\right)\right)$(1)

Note that the dot product has a positive sign instead of the negative sign of real quaternions and the cross product becomes imaginary and has the opposite parity of the Hamilton representation (i.e. right-hand rule vs left hand rule).

A “Complex” Pauli quaternion takes the form;

$\begin{array}{c}{\stackrel{\leftrightarrow }{P}}_C=\left(w_R+x_R{\hat{\sigma }}_1+y_R{\hat{\sigma }}_2+z_R{\hat{\sigma }}_3\right)+i\left(w_I+x_I{\hat{\sigma }}_1+y_I{\hat{\sigma }}_2+z_I{\hat{\sigma }}_3\right)\\ =\left(w_R+{\stackrel{\to }{r}}_R\right)+i\left(w_I+{\stackrel{\to }{r}}_I\right)\end{array}$(2)

the Pauli quaternion has a quasi real (on the left denoted with subscript R) and quasi imaginary part (on the right denoted with subscript I).

This quaternion in terms of the Pauli matrices, is actually more convenient for the way that human physics has evolved; being usable for Maxwell’s equations, relativity and relativistic quantum mechanics. Although they could be presented in Hamilton Biquaternion form. For now rather than rewrite Physics text books of the last hundred years it is convenient to go along with the Pauli quaternion version of Physics or even the more awkward Pauli Algebra version where matrices are used instead of scalars and vectors.

The full Pauli quaternion product is given by;

${\stackrel{\leftrightarrow }{P}}_A{{\stackrel{\leftrightarrow }{P}}^{\prime }}_B=\left({\stackrel{\leftrightarrow }{P}}_R+i{\stackrel{\leftrightarrow }{P}}_I\right)\left({{\stackrel{\leftrightarrow }{P}}^{\prime }}_R+i{{\stackrel{\leftrightarrow }{P}}^{\prime }}_I\right)=\left({\stackrel{\leftrightarrow }{P}}_R{{\stackrel{\leftrightarrow }{P}}^{\prime }}_R-{\stackrel{\leftrightarrow }{P}}_I{{\stackrel{\leftrightarrow }{P}}^{\prime }}_I\right)+i\left({\stackrel{\leftrightarrow }{P}}_R{{\stackrel{\leftrightarrow }{P}}^{\prime }}_I+{\stackrel{\leftrightarrow }{P}}_I{{\stackrel{\leftrightarrow }{P}}^{\prime }}_R\right)$(3)

Pauli quaternions can be used for Maxwell’s Equations, Relativity and Relativistic quantum mechanics.

The del operator in Pauli quaternion form is

$\stackrel{\to }{\nabla }\equiv {\hat{\sigma }}_1\frac{\partial }{\partial x}+{\hat{\sigma }}_2\frac{\partial }{\partial y}+{\hat{\sigma }}_3\frac{\partial }{\partial z}$(4)

the quaternion differential operator.

(Silberstein’s Operator [7] [8] ) Pauli quaternion version is

$\stackrel{\leftrightarrow }{D}=\frac{\partial }{c\partial t}+{\hat{\sigma }}_1\frac{\partial }{\partial x}+{\hat{\sigma }}_2\frac{\partial }{\partial y}+{\hat{\sigma }}_3\frac{\partial }{\partial z}=\left({\partial }_0+\stackrel{\to }{\nabla }\right)$(5)

Its parity conjugate is

${\stackrel{\leftrightarrow }{D}}^{\text{p}}=\frac{\partial }{c\partial t}-{\hat{\sigma }}_1\frac{\partial }{\partial x}-{\hat{\sigma }}_2\frac{\partial }{\partial y}-{\hat{\sigma }}_3\frac{\partial }{\partial z}=\left({\partial }_0-\stackrel{\to }{\nabla }\right)$

operation on the “real” part of a Pauli quaternion

$\stackrel{\leftrightarrow }{D}\left(A_0+\stackrel{\to }{A}\right)=\left({\partial }_0+\stackrel{\to }{\nabla }\right)\left(A_0+\stackrel{\to }{A}\right)=\left({\partial }_0{A}_0+\stackrel{\to }{\nabla }\cdot \stackrel{\to }{A}\right)+\left({\partial }_0\stackrel{\to }{A}+\stackrel{\to }{\nabla }{A}_0+i\stackrel{\to }{\nabla }\times \stackrel{\to }{A}\right)$(6)

note

$\left({\partial }_0-\stackrel{\to }{\nabla }\right)\left({\partial }_0+\stackrel{\to }{\nabla }\right)=\left({\partial }_0^2-{\nabla }^2\right)$(7)

This is the d’Alembert operator also known as the d’Alembertian it is a scalar operator used in a number of Partial differential Equations.

Pauli algebra is basically Pauli quaternions with 2 × 2 matrices rather than unit vectors. The standard Pauli matrix representations (although there are an infinite collection of the matrix representations) are:

$I=\left[\begin{array}{ll}1\hfill & 0\hfill \\ 0\hfill & 1\hfill \end{array}\right]{\sigma }_1=\left[\begin{array}{ll}0\hfill & 1\hfill \\ 1\hfill & 0\hfill \end{array}\right]{\sigma }_2=\left[\begin{array}{ll}0\hfill & -i\hfill \\ i\hfill & 0\hfill \end{array}\right]{\sigma }_3=\left[\begin{array}{ll}1\hfill & 0\hfill \\ 0\hfill & -1\hfill \end{array}\right]$(8)

Maxwell’s four Equations can be written as one Pauli Quaternion Equation as follows:

$\left({\partial }_0-\stackrel{\to }{\nabla }\right)\left(G-\stackrel{\to }{E}+i\stackrel{\to }{B}\right)=4\pi \left(\rho +\frac{\stackrel{\to }{J}}{c}\right)$(9)

$G=\left({\partial }_0{A}_0+\stackrel{\to }{\nabla }\cdot \stackrel{\to }{A}\right),\stackrel{\to }{E}=\left(-{\partial }_0\stackrel{\to }{A}-\stackrel{\to }{\nabla }{A}_0\right),i\stackrel{\to }{B}=\left(i\stackrel{\to }{\nabla }\times \stackrel{\to }{A}\right)$

$G$ = Lorenz condition field, $\stackrel{\to }{E}$ = Electric field, $\stackrel{\to }{B}$ = Magnetic Induction field.

$A_0$ = Scalar Potential, $\stackrel{\to }{A}$ = Vector Potential.

Conversely all four Maxwell’s equations can be extracted from Eq. (9).

$\stackrel{\to }{\nabla }\cdot \stackrel{\to }{E}=4\pi \rho -{\partial }_{0}G$ is the real scalar part Coulomb’s Law (modified) (9A)

$\stackrel{\to }{\nabla }\cdot \stackrel{\to }{B}=0$ is the imaginary scalar part No Magnetic monopoles (9B)

$\stackrel{\to }{\nabla }\times \stackrel{\to }{E}+{\partial }_0\stackrel{\to }{B}-\stackrel{\to }{\nabla }G=0$ is the real vector part Faraday’s Law (modified) (9C)

$\stackrel{\to }{\nabla }\times \stackrel{\to }{B}-{\partial }_0\stackrel{\to }{E}=4\pi \frac{\stackrel{\to }{J}}{c}$ is the imaginary vector part Ampere’s Law (9D)

$\stackrel{\to }{E}$ is the electric field and $\stackrel{\to }{B}$ is the magnetic flux density or magnetic induction $\rho$ is the charge density and $\stackrel{\to }{j}$ is the current density.

$G$ is the Lorenz condition field, typically it is set equal to zero however if ${\partial }_{0}G\ll 1$ and $\stackrel{\to }{\nabla }G\ll 1$ then the standard Maxwell’s equations are obtained. The physical interpretation for $G$ which is more satisfying than the “gauge ambiguity” is that G is proportional to the Newtonian gravitational potential this will be demonstrated in the following:

$\left({\partial }_0+\stackrel{\to }{\nabla }\right)\left({\partial }_0-\stackrel{\to }{\nabla }\right)\left(G-\stackrel{\to }{E}+i\stackrel{\to }{B}\right)=4\pi \left({\partial }_0+\stackrel{\to }{\nabla }\right)\left(\rho +\frac{\stackrel{\to }{J}}{c}\right)$(10)

$\left({\partial }_0^2-{\nabla }^2\right)\left(G-\stackrel{\to }{E}+i\stackrel{\to }{B}\right)=4\pi \left[\left({\partial }_0\rho +\stackrel{\to }{\nabla }\cdot \frac{\stackrel{\to }{J}}{c}\right)+\left(\nabla \rho +{\partial }_0\frac{\stackrel{\to }{J}}{c}\right)+\left(i\stackrel{\to }{\nabla }\times \frac{\stackrel{\to }{J}}{c}\right)\right]$(11)

from the continuity equation, the scalar part of the “Electromagnetic quaternion” wave equation is:

$\left({\partial }_0^2-{\nabla }^2\right)G=4\pi \left({\partial }_0\rho +\stackrel{\to }{\nabla }\cdot \frac{\stackrel{\to }{J}}{c}\right)=0$(12)

The following is to demonstrate the plausibility of claiming that the scalar part of the electromagnetic field quaternion could be associated with the gravitational potential.

There are a number of Solutions to Equation (12) the simplest is $G=0$ which the Lorenz condition “gauge” assuming the continuity equation ${\partial }_0\rho +\stackrel{\to }{\nabla }\cdot \frac{\stackrel{\to }{J}}{c}=0$ is true, while time dependence and angular dependence are not present, Equation (12) becomes Laplace’s equation, ${\nabla }^{2}G=0$, so that the simplest non-zero solution (ignoring added integration constants) for the static case in spherical coordinates for the Laplace equation is:

$G=\frac{K}{r}$(13)

where $K$ is an unspecified integration constant. Choosing $K=-\kappa {G^{\prime }}_{Newton}Mm$ $\kappa =\frac{1}{q^{\prime }d}$ so that $\kappa$ is an inverse dipole moment from dimensional analysis which yields a form of Newton’s Gravity potential equation.

$G=\frac{1}{q^{\prime }d}\frac{-{G^{\prime }}_{Newton}Mm}{r}$(14)

where ${G^{\prime }}_{Newton}$ is Newton’s gravitational constant. In the typical model with the charge $q$ on a mass $m$, since the model is so simple, a charge $-q$ should appear on $M$ and $q^{\prime }d\to -qr$.

A Gravitational Traveling wave solution for Equation (12) in rectangular coordinates is

$G=G_0{\text{e}}^{i\left(\stackrel{\to }{k}\cdot \stackrel{\to }{r}-wt+\varphi \right)}$(15)

Maxwell’s Equations in terms of Potentials are:

$\begin{array}{c}\left({\partial }_0^2-{\nabla }^2\right)\left(A_0+\stackrel{\to }{A}\right)=\left({\partial }_0-\stackrel{\to }{\nabla }\right)\left(\left({\partial }_0{A}_0+\stackrel{\to }{\nabla }\cdot \stackrel{\to }{A}\right)+\left({\partial }_0\stackrel{\to }{A}+\stackrel{\to }{\nabla }\text{Φ}\right)+\left(i\stackrel{\to }{\nabla }\times \stackrel{\to }{A}\right)\right)\\ =\left({\partial }_0-\stackrel{\to }{\nabla }\right)\left(G-\stackrel{\to }{E}+i\stackrel{\to }{B}\right)=4\pi \left(\rho +\frac{\stackrel{\to }{J}}{c}\right)\end{array}$(16)

Relativity is an obvious application for quaternions. In addition to discovering the equivalent of Equation (9) without the $G$, the Lorentz Transformation in quaternion form was discovered by both Ludwik Silberstein [7] [8] [13] and Arthur Conway [6] [14] independently, a minor dispute followed. See Figure 3 .

$\left(c{t}^{\prime }+{\stackrel{\to }{r}}^{\prime }\right)=\gamma \sqrt{1+\frac{\stackrel{\to }{v}}{c}}\left(ct+\stackrel{\to }{r}\right)\sqrt{1+\frac{\stackrel{\to }{v}}{c}}$(17)

where $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$.

Recent experiments [15] have shown that antimatter i.e. anti-Hydrogen reacted to gravity the same as “normal” matter i.e. Hydrogen. That is, contrary to some speculation, antimatter doesn’t fall up! Alan Guth had previously suggested [10] that the gravitational field has negative mass. Agreeing with Guth, this presentation suggests that the gravitational potential field which is claimed to be proportional to the Lorenz condition field, has negative mass! This report also goes a step further, to say that gravitons exist, have negative mass, and curve space, which is not observed in the classical Newtonian region.

Since the full Maxwell’s equations yield a wave equation for $G-\stackrel{\to }{E}+\stackrel{\to }{B}$ the electromagnetic-gravitational fields, assigning a composite quanta, a “photo-graviton” to it seems appropriate. The photo graviton should have a net mass of zero and a spin of 2.

The zero mass photo-graviton explains the unresolved observations in the so-called Crookes radiometer. The standard explanation was disputed by Maxwell and later Einstein⁴ The model presented in Figure 4 and Figure 5 . suggests that the reflected part of the Crookes Radiometer does not respond to the zero mass photo-graviton while the absorbing side, absorbs only the positive-mass electromagnetic part of the photo-graviton while the negative mass part of the photo-graviton becomes part of the gravitational field; causing the radiometer to spin in the observed direction. There have been some experiments that have changed the geometry of the Crookes radiometer, but the side that absorbs the most light still is pushed more, an experiment with an extreme vacuum should settle the question of the “radiometer” only working in a “partial vacuum”. So called reflective solar sails in miniature made to fit into a Crookes radiometer might clarify the issue, since they supposedly work by reflection, pitted against the absorbing “dark” side of the radiometer should get the same results that the radiometer gives now.