Argand plane as asymmetric imaginary time: James M. Chappell, et. al., non-commuting relativity algebra

 

 https://archive.org/details/geometryofmeanin00arth/page/54/mode/1up

AI says: 

An Argand diagram is not directly used for relativity, which is visualized using

Minkowski diagrams (also called spacetime diagrams). However, the mathematical relationship between the two fields is significant: a Wick rotation is a technique used in special relativity where a real-time axis is rotated onto an imaginary axis in a four-dimensional spacetime plane, similar to using an Argand diagram. This rotation is achieved by replacing the real time variable with the imaginary unit

$i$

multiplied by a real time parameter (

$\tau$

), which can simplify calculations by making the spacetime interval resemble a four-dimensional version of the Pythagorean theorem

 

 So the process of asymmetric time is key to life itself....as imaginary time....

How the Argand diagram is relevant to relativity 

• Wick Rotation: This is a mathematical transformation where the real time axis (
  

  $t$
  ) is replaced with an imaginary one (
  

  $it$
  ). This is equivalent to rotating the time axis by
  

  ${90}^{\circ }$
  in an Argand-like plane.
• Simplifying calculations: This rotation transforms the negative sign in the spacetime interval formula, making it look like a Euclidean metric:
  

  $d{s}^2=c^{2}d{t}^2-d{x}^2$
  becomes
  

  $d{s}^2=d{x}^2+c^{2}d{\tau }^2$
  where
  

  $\tau =it$
  .
• Relating to Euclidean Geometry: This transformation allows for mathematical tools developed for Euclidean geometry to be applied to spacetime, making it easier to solve certain problems in relativity.
• Conceptual tool: The idea of rotating to an imaginary axis is a way to map the unique properties of spacetime, especially in extreme conditions, to a more familiar geometric space, even though real time is not truly imaginary
•  

 This professor in Australia - James M. Chappell - has about six papers on how noncommutativity is the truth of reality. His latest paper shows how this truth enables a better understanding of Einstein's equivalence principle, due to an asymmetric time dynamic....thereby predicting an earlier formation of galaxies as now proven by the James Webb telescope! The "increase in dark matter" is actually the change in the gravitational potential energy as trapped light in a box (meaning frequency as inverse to time proportional to momentum inverse to wavelength as first realized by Louis de Broglie).

 https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0051756

 the plane that also squares to minus one, but which can be included without the addition of an extra dimension,

 This also leads to momentum and energy being represented as components of a multivector and we give a new efficient derivation of Compton’s scattering formula, and a simple formulation of Dirac’s and Maxwell’s equations. Based on the mathematical structure of the multivector, we produce a semi-classical model of massive particles, which can then be viewed as the origin of the Minkowski spacetime structure and thus a deeper explanation for relativistic effects. We also find a new perspective on the nature of time, which is now given a precise mathematical definition as the bivector of the plane.

  Chappell JM, Iqbal A, Iannella N, Abbott D (2012) Revisiting Special Relativity: A Natural Algebraic Alternative to Minkowski Spacetime. PLoS ONE 7(12): e51756. https://doi.org/10.1371/journal.pone.0051756

 Clifford’s geometric algebra of two-dimensions can be adopted as a suitable algebraic framework to describe special relativity, because the Lorentz transforms act separately on the parallel and perpendicular components of vectors relative to a boost direction thereby defining a two-dimensional space.
Clifford algebra has been used previously to describe spacetime [18–21], however these approaches follow Minkowski in describing a four-dimensional spacetime framework with an associated mixed metric, such as the STA of Hestenes [18] which uses the
four algebraic non-commuting basis elements

 the bivector, defining an associative non-commuting algebra.

https://www.eleceng.adelaide.edu.au/personal/dabbott/wiki/images/2/29/3D_Space.pdf 

 We conclude however, that Clifford’s geometric algebra (GA), provides the most elegant description of space

 In two dimensions this formula reduces to the single sided operator v′ = eiwθv due to the anticommuting nature of i = e12 over vectors. The rotation formula in two dimension now analogous to the conventional formula for the rotation of vectors in the Argand plane.

 we suggest the notation I = e1234 and J = e12345, where the capitalization
indicates that they have a positive square, but capital I like lower case i is anticommut-
ing and J is commuting similar to j. As we continue to higher dimension this patterns of
positive or negative squares, and commuting versus anticommmuting pseudoscalars repeats
with a period of four.
The development of GA is now expanding rapidly, with benefits being found in research
into black holes11, quantum field theory12, quantum tunneling13, quantum computing14,
general relativity and cosmology15, beam dynamics and buckling16, computer vision17 and EPR-Bell experiments18.
Many commentators believe that Cliffords mathematical system ‘should have gone on to
dominate mathematical physics’19, but, Clifford died young, at the age of just 33 and vector calculus was heavily promoted by Gibbs and rapidly became popular, eclipsing Clifford’s work, which in comparison appeared strange with its non-commuting variables. In hindsight, non-commuting reflects the non-commutivity of rotations in three-space, and hence is exactly what is required for these variables.

https://arxiv.org/pdf/1509.00501 

 There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. With such a range of vector formalisms in use, it thus appears that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one of the primary goals of nineteenth century science was to suitably describe vectors in three-dimensional space. This situation has also had the unfortunate consequence of fragmenting knowledge across many disciplines, and requiring a significant amount of time and effort in learning the various formalisms. We thus historically review the development of our various vector systems and conclude that Clifford's multivectors best fulfills the goal of describing vectorial quantities in three dimensions and providing a unified vector system for science.

 The Cartesian coordinate system proposed by Descartes, appears to become confused,
however, with the later development of the Argand diagram, which, while isomorphic to
the Cartesian plane, consists of one real and one imaginary axis, and so not rotationally
symmetric. To add to the confusion, Hamilton in 1843 generalized the complex numbers
to three space, defining the algebra of the quaternions using the basis elements i, j, k that
can also be used as a substitute for three dimensional Cartesian coordinates. This confused
state of affairs, on exactly how to represent three-space coordinates and rotations, was fi-
nally resolved by William Clifford in 1873. Clifford adopted the Cartesian coordinate system
of Descartes, but then also integrated the algebra of complex numbers and quaternions as
the rotation operators within this space. Clifford also achieved a fulfillment of Descartes’
original vision of a vector being able to be manipulated in the same way as normal num-
bers, by deriving a multiplication and division operation for vectors that allowed them to
be treated as algebraic variables.

 

 a vector quaternion squares to the negative Pythagorean length. Indeed, Maxwell commented on this
unusual fact, noting that the kinetic energy, which involves the
square of the velocity vector, would therefore be negative [9].
Maxwell, despite these reservations, formulated the equations
of electromagnetism in quaternionic form. Maxwell, however,
backed away from a complete endorsement of the quaternions,
recommending in his treatise on electricity and magnetism ‘the
introduction of the ideas, as distinguished from the operations
and methods of Quaternions’ [10]

 The Gibbs’ side of the debate though argued that the non-commutativity of the quaternions

added many difficulties to the algebra compared with the much
simpler three-vector formalism in which the dot and cross
products were each transparently displayed separately. Ulti-
mately, with the success of the Gibbs formalism in describing
electromagnetic theory, exemplified by the developments of
Heaviside [11], and with an apparently more straightforward
formalism, the Gibbs system was adopted as the standard
vector formalism to be used in engineering and physics. This
outcome to the debate is perhaps surprising in hindsight as in
comparison with quaternions, the Gibbs vectors do not have
a division operation and two new multiplication operations
are required beyond elementary algebra and so therefore did
not satisfy the basic principles of Descartes or Hankel. The
trend of adopting the Gibbs vector system, however, continued
in 1908 when Minkowski rejected the quaternions

   the square of vector quaternions being negative—
the same problem that was identified earlier by Maxwell

 The Clifford geometric algebra Cℓ(ℜ3), being an
eight-dimensional linear space, is indeed able to subsume
the quaternion algebra and the Gibbs vectors into a single
formalism, as required. We now therefore describe the Clifford
geometric algebra in three dimensions.

 In order to restore isotropy to complex numbers and the
Argand plane, we can introduce the Clifford geometric algebra
Cℓ(ℜ2) where the complex numbers are isomorphic to the
even subalgebra. 

 Hamilton’s non-commutivity was one of the things
that counted against his vector system at the time it actually
is exactly what is needed in a vector system in three di-
mensions, as three-dimensional rotations are intrinsically non-
commuting

https://arxiv.org/pdf/1708.09725

https://arxiv.org/pdf/1611.02564 

https://arxiv.org/pdf/1509.06707 

 we distinguish a
second type of time-like quantity that we here refer to as rotational time represented by....It was noted that the bivectors were also time-like and so could be included as a description of time. Thus time can become quaternionic combining scalar and bivector components. 

 https://arxiv.org/pdf/1708.09725

 as the true time asymmetry of accelerating frames in special relativity, we can assert that velocity
dependent effects are not simply coordinate artifacts of general relativity. The result of
Hilbert also appears to indicate ‘gravitational repulsion’,

 

Front. Phys., 16 November 2016

Sec. Statistical and Computational Physics

Volume 4 - 2016 | https://doi.org/10.3389/fphy.2016.00044

Time As a Geometric Property of Space

 The proper description of time remains a key unsolved problem in science. Newton conceived of time as absolute and universal which "flows equably without relation to anything external." In the nineteenth century, the four-dimensional algebraic structure of the quaternions developed by Hamilton, inspired him to suggest that he could provide a unified representation of space and time. With the publishing of Einstein's theory of special relativity these ideas then lead to the generally accepted Minkowski spacetime formulation of 1908. Minkowski, though, rejected the formalism of quaternions suggested by Hamilton and adopted an approach using four-vectors. The Minkowski framework is indeed found to provide a versatile formalism for describing the relationship between space and time in accordance with Einstein's relativistic principles, but nevertheless fails to provide more fundamental insights into the nature of time itself. In order to answer this question we begin by exploring the geometric properties of three-dimensional space that we model using Clifford geometric algebra, which is found to contain sufficient complexity to provide a natural description of spacetime. This description using Clifford algebra is found to provide a natural alternative to the Minkowski formulation as well as providing new insights into the nature of time. Our main result is that time is the scalar component of a Clifford space and can be viewed as an intrinsic geometric property of three-dimensional space without the need for the specific addition of a fourth dimension.

 https://www.mdpi.com/2218-1997/11/6/190

 a clock at the
periphery p′, in the frame K′, experiencing a radial force, and running asymmetrically
slower in time with respect to a clock, also in K′, at the centre z′, is deemed completely
equivalent to one lying fixed and deeper in a static gravitational field. Of course, the equiv-
alence principle relates an accelerated (and in particular a rotating) frame on Minkowski
spacetime to a gravitational field only if restricted to a spacetime neighbourhood that is
small enough so that the acceleration in the rotating frame can be considered as constant,
which means that in the equivalent situation, the gravitational field is homogeneous.

 s in the case of gravitational time dilation, where a true asymmetry exists between
the stationary clock rates in the field, there is also a true asymmetry in the internal energy
of the stationary masses in a gravitational field.

  Hence, the mass increase, m, changes in the same ratio as the frequency and length, dl, while inversely changing in comparison to time dilation.

 The unseen matter, coupled to the mass increase, has characteristics similar to the
behaviour of ‘dark matter’. That is, the observed larger-than-expected lensing effects
imply an increased mass, and Equation (21) implies less visible ‘redshifted mass’. Current
estimates of dark matter are approximately a factor of 5.5 times the observed baryonic
matter [ 21]. Gravitational lensing and extreme redshift data can thus be used to directly
confirm mass discrepancies in a given galaxy or a galaxy cluster.

 It also posits that a frequency decrease in radiation emitted from source masses in a gravitational field is accompanied by a mass–energy increase, according to Equation (21), showing a general link between time dilation and inertial mass.