[Paper Review] Monogenic functions in 5-dimensional spacetime used as first principle: gravitational dynamics, electromagnetism and quantum mechanics

José B. Almeida|ArXiv.org|Jan 11, 2006

Algebraic and Geometric Analysis12 references3 citations

TL;DR

This paper proposes that monogenic functions in 5-dimensional spacetime, defined by null vector derivatives in geometric algebra $G_{4,1}$, serve as a first principle unifying gravitational dynamics, electromagnetism, and quantum mechanics. By deriving two 4D spaces—one Euclidean and one Minkowski—the framework reproduces relativistic dynamics, Maxwell’s equations, and quantized electrodynamics from a single geometric foundation, offering a novel unification via monogenicity and null geodesics in 5D.

ABSTRACT

Monogenic functions are functions of null vector derivative and are here analysed in the geometric algebra of 5-dimensional spacetime, G(4,1), in order to derive several laws of fundamental physics. The paper introduces the working algebra and the definition of monogenic functions, showing that these generate two 4-dimensional spaces, one with Euclidean signature and the other one with Minkowski signature. The equivalence conditions between the two spaces are studied and relativistic dynamics, not entirely coincident with Einstein's general theory of relativity, is demonstrated. The monogenic condition is then shown to produce Maxwell's equations and electrodynamics both classical and quantized.

Motivation & Objective

• To establish monogenic functions in 5D spacetime as a foundational principle for fundamental physics.
• To demonstrate that monogenicity generates two 4D spaces: one with Euclidean signature and one with Minkowski signature.
• To show that the equivalence between these 4D spaces reproduces relativistic dynamics distinct from Einstein’s general relativity.
• To derive Maxwell’s equations and both classical and quantized electrodynamics from the monogenic condition.
• To unify gravitational, electromagnetic, and quantum phenomena under a single geometric algebra framework using 5D null geodesics.

Proposed method

• Utilizes geometric algebra $G_{4,1}$ in 5-dimensional spacetime with orthonormal basis vectors $oldsymbol{ au}_ u$ satisfying $(oldsymbol{ au}_0)^2 = -1$ and $(oldsymbol{ au}_i)^2 = 1$.
• Defines monogenic functions via the null vector derivative condition $Df = 0$, where $D = oldsymbol{ au}_ u rac{ar{ abla}}{ar{ abla}x^ u}$, ensuring the function is annihilated by the vector derivative.
• Constructs two 4D subspaces from the 5D algebra: one with Euclidean metric (4DO) and one with Minkowski metric (GTR), linked via refractive index coefficients.
• Derives the refractive index coefficients $n_4 = rac{1 + m/(2r)}{1 - m/(2r)}$ and $n_r = rac{(1 + m/(2r))^3}{1 - m/(2r)}$ from the monogenic condition, mapping Schwarzschild geometry to 4DO.
• Applies rotor formalism using exponentials of bivectors (e.g., $e^{-B/2}$) to represent rotations and boosts in 4D, with $B$ being a bivector whose norm defines rotation angle.
• Uses time derivatives along paths via $oldsymbol{ au} = rac{dx}{dt} = g_0 + g_i rac{dx^i}{dt}$, with $g_ u$ as frame vectors, to define $oldsymbol{ au} abla a = rac{da}{dt}$, enabling dynamic evolution.

Experimental results

Research questions

• RQ1 Can monogenic functions in 5D spacetime generate both Euclidean and Minkowski 4D spaces, and what are the conditions for their equivalence?
• RQ2 How does the monogenic condition in 5D spacetime reproduce relativistic dynamics distinct from Einstein’s general relativity?
• RQ3 Can Maxwell’s equations and classical electrodynamics be derived from the monogenic condition in 5D geometric algebra?
• RQ4 Can the monogenic framework also yield the quantized form of electrodynamics?
• RQ5 What is the geometric role of null geodesics in 5D spacetime in explaining the appearance of massive particles in 4D?

Key findings

• Monogenic functions in $G_{4,1}$ spacetime generate two 4D subspaces: one with Euclidean signature and one with Minkowski signature, linked by refractive index coefficients.
• The refractive index coefficients $n_4 = rac{1 + m/(2r)}{1 - m/(2r)}$ and $n_r = rac{(1 + m/(2r))^3}{1 - m/(2r)}$ are derived from the monogenic condition and map the Schwarzschild metric to 4D optics (4DO).
• Relativistic dynamics in 4D is derived from the monogenic condition, yielding a formulation not entirely coincident with Einstein’s general relativity.
• Maxwell’s equations and classical electrodynamics emerge directly from the monogenic condition in 5D spacetime.
• Quantized electrodynamics is also derived from the same monogenic framework, suggesting a unified origin for classical and quantum electrodynamics.
• The monogenic condition subsumes earlier null-geodesic approaches, with null paths in 5D generating massive particle trajectories in 4D, thus providing a deeper first principle.