Pseudo-Euclidean Gravity

A new theory of a (flat) spacetime gravitational interaction is presented. This theory follows almost effortlessly from a new Lagrangian formulation of Maxwell's theory for photons and electrons (and positrons) whose associated Euler Lagrange equations imply the conventional Maxwell equations, but which possesses new \textbf{\emph{bosonic}} degrees of freedom that may be associated with a fundamental gravitational interaction. The precise character of this gravitational interaction with photons is explicitly defined in terms of a local U(1)-invariant Lagrangian in Eq.[\ref{Lagrangian3}]. The new formulation of Maxwell's theory is cast on the real, eight dimensional pseudo-Euclidean vector space defined by the split octonion algebra, regarded as a vector space over $\mathbb{R}$, and denoted $\mathbb{R}^{4,4} \cong M_{3,1} \oplus {}^*M_{3,1}$. (Here $M_{3,1}$ denotes real four-dimensional Minkowski space-time and ${}^*M_{3,1}$ denotes its dual; $\mathbb{R}^{4,4}$ resembles the phase space of a single relativistic particle.) This gravitational interaction is carried by a field that defines an algebraically distinguished element of the split octonion algebra, namely, the multiplicative unit element. We call this interaction the "unit" interaction, since any equivalence with Newton-Einstein gravity has yet to be established.