Hilbert Book Model Project/Quaternionic Field Equations/Solutions

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Solutions

The names warp and clamp are new names for very old subjects that did not have a name

Here we consider the solutions of the homogeneous second order quaternionic partial differential equations. Two interesting second order quaternionic partial differential equations exist.

$\nabla ^{*}\varphi =\nabla ^{*}\nabla \psi =\nabla \nabla ^{*}\psi =(\nabla _{r}-{\vec {\nabla }})(\varphi _{r}+{\vec {\varphi }})=(\nabla _{r}+{\vec {\nabla }})(\nabla _{r}-{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi =0$

(1)

The inhomogeneous equation can be split into two first order partial wave equations $\chi =\nabla ^{*}\varphi$ and $\varphi =\nabla \psi$.

This equation does not offer waves as part of the solutions of the homogeneous equation.

$(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi =0$

(2)

This is the quaternionic equivalent of the wave equation. It offers waves as part of the solutions of the homogeneous equation.

Waves

Only the wave equation offers solutions in the form of waves.

$\nabla _{r}\nabla _{r}\psi =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi =\omega \,\psi \Rightarrow f=\exp(2\pi i\omega x\tau )$

(3)

Periodic harmonic actuators cause the appearance of waves,

Shock fronts

In odd numbers of participating dimensions, both homogeneous second order partial differential equations offer shock fronts as part of its set of solutions.

They consist of a retarded part and an advanced part. Single shot triggers cause the shock fronts.

The term "shock wave" is a misnomer. A shock front does not feature a frequency and it cannot be considered as a wave package. During travel and without a wave guide, wave packages disperse. Shock fronts don't disperse.

Warps

For one-time one-dimensional actuators, the wave equation $D\,\psi =0$ offers solutions in the form

$f(c\tau +x)+g(c\tau -x)$

(4)

For such actuators, equation $\boxdot \,\psi =0$ delivers solutions in the form

$f(c\tau +x{\vec {i}})+g(c\tau -x{\vec {i}})$

(5)

Warps are one-dimensional shock fronts. Warps show similarity with solitons. However, solitons are considered as wave packages.

In contrast to warps, wave packages disperse when they move.

Clamps

For one-time isotropic three-dimensional actuators, the wave equation $D\,\psi =0$ offers solutions in the form

$f(c\tau +r)/r+g(c\tau -r/r$

(6)

For such actuators, equation $\boxdot \,\psi =0$ delivers solutions in the form

$f(c\tau +r{\vec {i}})/r+g(c\tau -r{\vec {i}})/r$

(7)

Clamps are spherical shock fronts

After integration over a sufficient period, the clamps results in the Green’s function of the field under spherical conditions.

Gravitational waves are shock fronts. Clamps are miniature equivalents of the much larger gravitational shock fronts.

Plops

Two-dimensional single shot actuators cause plops as solutions of the second order partial differential equation.

The form looks like the response, when a stone falls vertically in a pool.

Waves, warps, and clamps

In contrast to waves and clamps, warps can travel huge distances through a nearly flat carrier field.

Waves

Solutions of the wave equation are known since centuries. Of these solutions waves are well-known. Waves require a periodic harmonic actuator.

The homogeneous second order partial differential equation that applies d'Alembert's operator accepts waves as solutions. For these solutions the equation can be converted in the Helmholtz equation.

$\nabla _{r}\nabla _{r}\psi =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi$

(8)

The Helmholtz equation considers the quaternionic function that defines the field separable.

$\psi (q_{r},{\vec {q}})=A({\vec {q}})T(q_{r})$

(9)

${\frac {\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \,A}{A}}={\frac {\nabla _{r}\nabla _{r}\,T}{T}}=-k^{2}$

(10)

$\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \,A=-k^{2}A$

(11)

$\nabla _{r}\nabla _{r}\,T=-k^{2}T$

(12)

For three-dimensional isotropic spherical conditions the solutions have the form

$A(r,\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left(a_{\ell m}j_{\ell }(kr)+b_{\ell m}y_{\ell }(kr)\right)Y_{\ell }^{m}({\theta ,\varphi })$

(13)

Here ${\textstyle j_{\ell }}$ and ${\textstyle y_{\ell }}$ are the spherical Bessel functions, and ${\textstyle Y_{\ell }^{m}}$ are the spherical harmonics. These solutions play a role in the spectra of atomic modules.

Shock fronts

For odd numbers of participating dimensions of a one-shot actuator, the second order partial differential equations deliver shock fronts. Warps and clamps form a category of super-tiny objects that cannot be observed as single objects. For that reason, these solutions went into oblivion.

In contrast to waves, these shock fronts do not feature a frequency.

Warps

Warps can travel huge distances through a flat continuum that acts as a carrier. During travel, warps keep their shape and their amplitude. Each warp carries a standard bit of energy. In reality, warps occur equidistant in strings. In that way, the string owns a frequency. To emulate a photon, the warp string must obey the Einstein-Planck relation $E=h\nu$. This restriction means that all warp strings must feature the same spatial length and the same emission duration.

In photons the vector ${\vec {i}}$ in solution $f(c\tau +r{\vec {i}})/r$ of equation $\boxdot \,f=0$ may appear to rotate as a function of the sequence index in the warp string.

Clamps

Isotropic single-shot triggers cause clamps. Clamps are volatile. During travel their amplitude diminishes as ${\frac {1}{r}}$ with distance $r$ from the trigger location. The clamp integrates into the Green’s function of the carrier field. In this way, the clamp temporary deforms its carrier. Consequently, each clamp carries a standard bit of mass. The volume of the Green’s function equals the extension of the carrier that causes the deformation. 

However, the deformation quickly fades away. Dense swarms of clamps that are recurrently regenerated can produce a persistent deformation of the embedding field.

Elementary particles are pointlike elementary modules that hop around in a stochastic hopping path. The hop landing locations form a coherent swarm. A location density distribution describes the swarm.

Each hop landing triggers a clamp, which integrates into the Green's function of the embedding field. This results in a persistent deformation of the field that equals the convolution of the Green's function and the location density distribution. Thus, the mass of the elementary particle is proportional to the number of elements of the hop landing location swarm.