A classical Unified Field Theory using Riemanian Quarternions
- Here is the Abstract:
The equations governing gravity and electromagnetism show both profound
similarities and unambiguous differences. Albert Einstein worked to unify
gravity and electromagnetism, mainly by trying to generalize Riemannian
geometry. Hamilton's quaternions are a 4-dimensional topological algebraic
field related to the real and complex numbers equipped with a static
Euclidean 4-basis. Riemannian quaternions as defined herein explicitly allow
for dynamic changes in the basis vectors. The equivalence principle of
general relativity which applies only to mass is generalized because for any
Riemannian quaternion differential equation, the chain rule means a change
could be caused by the potential and/or the basis vectors. The Maxwell
equations are generated using a quaternion potential and operators.
Unfortunately, the algebra is complicated. The unified force field proposed
is modeled on a simplification of the electromagnetic field strength tensor,
being formed by a quaternion differential operator acting on a potential,
Box* A* . This generates an even, antisymmetric-matrix force field for
electricity and an odd, antisymmetric-matrix force field for magnetism,
where the even field conserves its sign if the order of the differential and
the potential are reversed unlike the odd field. Gauge symmetry is broken
for massive particles by the even, symmetric-matrix term, which is
interpreted as being due to gravity. In tensor analysis, a differential
operator acting on the field strength tensor creates the Maxwell equations.
The unified field equations for an isolated source are generated by acting
on the unified force field with an additional differential operator, Box*
Box* A* = 4 pi J*. This contains a quaternion representation of the Maxwell
equations, a classical link to the quantum Aharonov-Bohm effect, and dynamic
field equations for gravity. Vacuum solutions to the unified field equations
are discussed. The field equations conserve both electric charge density and
mass density. Under a Lorentz transformation, the gravitational and
electromagnetic fields are Lorentz invariant and Lorentz covariant
respectively, but there are residual terms whose meaning is not clear
presently. An additional constraint is required for gauge transformations of
a massive field.
Einstein's vision using quaternions
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