User:HansVanLeunen/sandbox

= Tensor Differential Calculus =

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Parameter spaces
We restrict to quaternionic parameter spaces, thus to 3+1 D parameter spaces.

Parameter spaces can differ in the way Cartesian and polar coordinate systems can sequence them and in the way the scalar part relates to the spatial part.

Quaternionic fields are defined by quaternionic functions that have values, which depend on the selected parameter space. Quaternionic fields exist in scalar fields, vector fields and as combined scalar and vector fields.

Quaternionic fields exist as continuum eigenspaces of normal operators that reside in quaternionic non-separable Hilbert spaces. Quaternionic functions of quaternionic parameter spaces can represent these combined fields. However, the same field can also interpret as the eigenspaces of the Hermitian and anti-Hermitian parts of the normal operator. A normal quaternionic reference operator that features a flat continuum eigenspace can represent the quaternionic parameter space. This reference operator can also split into a Hermitian and an anti-Hermitian part.

Due to the four dimensions of quaternions, the quaternionic number systems exist in 24 versions that differ in their Cartesian ordering. This number requires that the axes of the Cartesian coordinates systems are parallel.

If the approach pursues spherical ordering, then for each Cartesian start ordering two extra polar orderings are possible. All these choices correspond to different parameter spaces.

Base model
The Hilbert Book Model contains a base model consisting of a series of separable Hilbert spaces that each use a private parameter space that applies the same version of the quaternionic number system for specifying the values of pairs of vectors of the underlying vector space. The separable Hilbert spaces share that vector space. The model archives all dynamic geometric data that the model uses in a read-only repository that comprises all participating separable Hilbert spaces. All corresponding parameter spaces float on a background parameter space. The parameter spaces of the separable Hilbert spaces float over the background parameter space. The differences in sequencing of the parameter spaces by their coordinate systems with the sequencing of the background parameter space by its coordinate systems determine the symmetry flavor of the floating platforms. The symmetry flavors determine the symmetry related charges of the platforms. These charges act as sources or drains of a symmetry related field. The charges locate at the geometric centers of the platforms. On every participating platform resides an elementary module. At every subsequent progression instant, a private stochastic process generates a new location. Consequently the elementary module hops around in a stochastic hopping path. The stochastic process owns a characteristic function that ensures that the hop landing locations form a dense and coherent swarm. The characteristic function contains a gauge factor that acts as a displacement generator. Therefore, at first approximation, the swarm moves as a single unit. The geometric center of the swarm stays at the geometric center of the platform. Thus, the displacement generator controls the displacement of the platform.

Each separable Hilbert space own a reference operator that provides the private parameter space of the separable Hilbert space as its eigenspace. However, the eigenspace only contains rational values.

A non-separable Hilbert space embeds all separable Hilbert spaces. A selected separable Hilbert space applies the version of the quaternionic number system, which forms the background parameter space to specify the values of its inner product. The non-separable Hilbert space also embeds the symmetry flavor of this separable Hilbert space. Thus the normal reference operator that supplies the background parameter space in the selected separable Hilbert space has a direct equivalent in the non-separable Hilbert space.

Fields
The existence of the reference operator enables the specification of a series of define operators that apply a quaternionic function, the eigenspace of the reference operator and shares the eigenvectors of the reference operator to define the eigenspace as the target space of the quaternionic function for the values that the parameter space contains. In this way it is possible to define operators that correspond to the field that embeds the hop landing locations and gets deformed by the interaction.It is also possible to define the operator that represents the symmetry related fields.

These two quite different fields play an important role. One field is the superposition of all symmetry related fields. The other is the field that embeds all hop landing locations and gets deformed by these triggers. Both fields couple at the geometric centers of the floating platforms.

Basic quanta
Super-tiny shock fronts present nature's basic quanta. Spherical shock fronts temporary carry a standard bit of mass. During travel, one-dimensional shock fronts keep the shape and the amplitude of their front. They carry a standard bit of energy. In separation these quanta cannot be perceived. In huge assemblies these quanta become noticeable.

Format conversion
A read-only repository in the form of the combination of a quaternionic infinite dimensional separable Hilbert space and its non-separable companion stores the dynamic geometric data that constitute the observed event in a Euclidean format in the form of combinations of a timestamp and a three-dimensional spatial location. Quaternions act as storage containers. A private timestamp and a spatial location characterize the observer. The observer can only access storage locations whose timestamp predates his own timestamp. A continuum transfers this information to the observer. The speed of information transfer of the continuum is fixed. Therefore, the information transfer affects the format of the information that the observer perceives. A non-zero speed difference between observed event and observer will contract observed lengths will dilate durations. The Lorentz transform is a hyperbolic transform that describes the format conversion.

Quaternionic differential calculus describes the interaction between discrete objects and the continuum at the location where events occur. Converting the results of this calculus by the Lorentz transform will describe the information that the observers perceive. Observers perceive in spacetime format. This format features a Minkowski signature. The Lorentz transform converts from the Euclidean storage format at the situation of the observed event to the perceived spacetime format.

The path along which the field transfers the information deforms with the field. The elementary module follows a stochastic hopping path. The geometric center of the swarm of hop landing locations follows a much smoother path. If the field deforms, then the average path gets extra characteristics, such as polarization and magnetization. Further, the path implements translations ans rotations of traveling information.

General format conversion
The metric tensor defines a general format conversion

.$$g_{\mu\nu} = \begin{bmatrix} g_{00} & g_{01} & g_{02} & g_{03} \\ g_{10} & g_{11} & g_{12} & g_{13} \\ g_{20} & g_{21} & g_{22} & g_{23} \\g_{30} & g_{31} & g_{32} & g_{33}\end{bmatrix}$$

The coordinate transformations $$dx^\nu \Rightarrow dX^\nu $$ define the elements $$g_{\mu\nu}$$ as

$$g_{\mu\nu}={d X^\mu \over d x_\nu}$$

Christoffel symbols
Christoffel symbols of the second kind $Γ^{k}_{ij}$ are defined as the unique coefficients such that the equation
 * $$\nabla_i \mathrm{e}_j = \Gamma^k{}_{ij}\mathrm{e}_k$$

holds, where $∇_{i}$ is the Levi-Civita connection on $M$ taken in the coordinate direction $e_{i}$.

The Christoffel symbols can be derived from the vanishing of the covariant derivative of the metric tensor $g_{ik}$:

0 = \nabla_l g_{ik} = \frac{\partial g_{ik}}{\partial x^l} - g_{mk}\Gamma^m{}_{il} - g_{im}\Gamma^m{}_{kl} = \frac{\partial g_{ik}}{\partial x^l} - 2g_{m(k}\Gamma^m{}_{i)l}. $$ Using that the symbols are symmetric in the lower two indices, one can solve explicitly for the Christoffel symbols as a function of the metric tensor by permuting the indices and resumming:
 * $$\Gamma^i{}_{kl}=\tfrac12g^{im} \left(\frac{\partial g_{mk}}{\partial x^l} + \frac{\partial g_{ml}}{\partial x^k} - \frac{\partial g_{kl}}{\partial x^m} \right) $$

where $(g^{jk})$ is the inverse of the matrix $(g_{jk})$, defined as $g^{ji}g_{ik} = δ&thinsp;^{j}_{k}$.

Although the Christoffel symbols are written in the same notation as tensors with index notation, they are not tensors, since they do not transform like tensors under a change of coordinates.

Curvature
The metric $$g$$ completely determines the curvature of spacetime.

According to the fundamental theorem of Riemannian geometry, there is a unique connection ∇ on any semi-Riemannian manifold that is compatible with the metric and torsion-free. This connection is called the Levi-Civita connection.

The curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the Levi-Civita connection ∇. In local coordinates this tensor is given by:
 * $${R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho {}_{\nu\sigma}

- \partial_\nu\Gamma^\rho {}_{\mu\sigma} + \Gamma^\rho {}_{\mu\lambda}\Gamma^\lambda {}_{\nu\sigma} - \Gamma^\rho {}_{\nu\lambda}\Gamma^\lambda {}_{\mu\sigma}.$$ The curvature is then expressible purely in terms of the metric $$g$$ and its derivatives.