It is well known that spacetime metric is connected with the Dirac algebra. However, the question of the connection of space-time curve metric with algebra has not yet received wide coverage.
If you are curious about the origin of the pseudo-Riemannian metric, then look to the chapter "On some applications of vector field algebra" of the book "Mathematical Notes...", where it is argued that the reason for the curvature of space-time is the curvature of the vacuum vector field of accelerations of particles of matter moving along the surface of a 7-dimensional sphere and perceived as a local algebra of vector fields.
Research Proposal MATHEMATICAL NOTES ON THE NATURE OF THINGS
Dear Igor. In a Lorentzian spacetime, which all relativists agree describes spacetime, there exists a line element field (X,-X) in 4-D provided the Euler-Poincare characteristic vanishes. There also exists a unit vector u collinear to the line element vector; x=fu where f is the magnitude of X. These vectors are time-like so u^au_a=-1 and X^aX_a=-f^2. The Lorentzian metric g can be then written as g=g^+-2uu where g^+ is a Riemannian metric which always exists. The line element vectors are regular, so they never vanish.
The spin-1 Klein-Gordon equation in curved spacetime can be derived directly from the line element field. For further discussion on how Modified general relativity links to quantum theory, please read carefully, arXiv:1904.10803v4 [gr-qc] and
arXiv:1909.03845v1 [gr-qc] .
Gary Nash , I’ll definitely look at your article, but for now let me note that a reduced pseudo-Riemannian metric can be induced by both a scalar and a vector field defined in Minkowski space, but I’m interested in what induces an unreduced metric.
However, perhaps you have two independent vector fields. Then how do you interpret them?
PS As for my version, in my case the pseudo-Riemannian metric is induced by a 4-dimensional complex vector field.
It is clear, that is, you use the basic elements of the frame to construct the metric. And how do you interpret them from a metaphysical point of view?
The line element field is a dynamical object independent of the Riemannian metric. There is no physical requirement that either objects exist, they just do in a Lorentzian spacetime. The physics comes out from what the combination of those objects explains. For example, general relativity is explainable from the Riemannian metric as long as it is Minkowskian in free fall. The line element fields in MGR provide for the description of dark energy and dark matter that GR cannot explain by itself. It must be emphasized that Einstein's equation is the same in both GR and MGR, but the latter contains a tensor that describes the e-m of the gravitational field. Please carefully go through the papers so that we can get on the same wavelength.
This way:
Preprint Lerch's Φ and the Polylogarithm at the Positive Integers
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