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Gravitation

In the Hilbert Book Model, gravitation is the local deformation of the embedding continuum. The embedding continuum is defined by a continuous quaternionic function. Mass characterizes the strength of the local deformation. The deformation concerns the difference with respect to a flat equivalent of the embedding continuum. Often the at some distance of a rather central location the local deformation resembles, apart from a multiplication factor, the Green's function of the embedding field. The multiplication factor equals the mass of the object(s) that cause the deformation.

Interaction with point-like artifacts

One-dimensional one-shot triggers cause one-dimensional shock fronts. The time-integral over this disturbance stays zero.

Two-dimensional shock fronts don't exist.

Three-dimensional shock fronts integrate over time into the volume of the Green's function of the embedding continuum. In this case the front locally temporarily deforms the continuum, but it finally expands the whole continuum, The deformation corresponds to an injection of a small volume that is locally added to the carrier field. According to the dynamics of the spherical shock front, the added volume spreads over the field and in this way the deformation quickly fades away.

Requirements of the spherical pulse response

The conditions for a pulse to generate a spherical pulse response are very strickt.

POSTULATE: Only one kind of excitation exists that injects volume in a quaternionic field. That excitation is a spherical shock front and it is triggered by an isotropic point-like actuator in the form of a quaternion that represents a chiral symmetry breaking with the embedding field.

The product of a step function $\theta ()$ and a Dirac delta function $\delta (r)$ describes the spherical pulse response.

$u(R,\tau )=\theta (\tau )\delta (\tau -R/c)$

$\tau$ is proper time. R is the distance to the trigger location. c is the light speed. This pulse response integrates over time $\tau$ into the Green's function of the field. The Green's function has volume.

This means that the actuator must belong to a different version of the quaternionic number system that is used to formulate the quaternionic function, which describes the field.

The precise trigger requirement is considered to be the reason behind the color confinement phenomenon.

Electrons are applying quaternions that feature the proper isotropic symmetry breaking. Neutrinos do not break the symmetry, but instead they conflict with the handedness (chirality) . Quarks break the symmetry, but they don't do that in an isotropic way. Quarks must first conglomerate into hadrons before they can obtain mass.

Massive objects

Elementary modules

Elementary particles reside on a private platform that is implemented by a separable Hilbert space, and that platform provides them with a private version of the quaternionic number system that spans a private parameter space. A private stochastic process generates the hop landing locations of the particle on this parameter space. These locations form a hopping path and a hop landing location swarm. The process owns a characteristic function that ensures that the swarm is dense and coherent. It equals the Fourier transform of the location density distribution that describes the hop landing location swarm. The embedding process maps the swarm onto the embedding field. The parameter space of the platform of the particle floats over the parameter space of the function that describes the embedding field. Thus, the swarm moves as one unit over the embedding field. A displacement generator describes this movement and can be added in the form of a gauge factor to the characteristic function of the stochastic process. Together they represent the platform and the hop landing location swarm in Fourier space.

The local deformation of the embedding field by the swarm of overlapping spherical pulse responses equals the convolution of the Green’s function of the embedding field and the location density distribution of the swarm. If the location density distribution is close to a Gaussian distribution, then the deformation will look like ERF(r)/r. At a small distance from the swarm, it will already look like the usual 1/r shape function of the gravitation potential, but the deformation is a perfectly smooth function that does not feature the central singularity.

Several types of elementary particles exist. Each type corresponds to a type of version of the quaternionic number system. Important is the difference between the symmetry of the platform version and the symmetry of the background version. This difference results in the electric charge and the color charge of the platform. These charges characterize the platforms and not the swarms. The particle that corresponds to the swarm inherits the properties of the platform.

Embedding

The embedding continuum is the eigenspace of an operator that resides in a non-separable Hilbert space. It is the unique companion of an infinite dimensional separable Hilbert space that acts as the background platform and provides the background parameter space. This separable Hilbert space embeds fluently into the non-separable Hilbert space. They share the same parameter space, but the separable Hilbert space only uses the rational elements of the number system.

The floating platforms are represented by separable Hilbert spaces. The separable Hilbert space applies the selected version of the quaternionic Hilbert spaces for specifying its inner products, and indirectly this selects the kind of eigenvalues that operators support. A reference operator manages the private parameter space. Another operator manages the hop landing locations as a combination of a time stamp and a three-dimensional spatial location. The embedding of the floating platforms into the embedding continuum is an ongoing process in which for each platform step by step the content of the hop landing location storage bins is embedded in the embedding continuum.

The stochastic processes control the selection of the storage bins that will be embedded. The stochastic processes are combinations of a genuine Poisson process and a binomial process. A spatial Point Spread Function that equals the location density distribution of the hop landing location swarm implements the binomial process. The characteristic function of the stochastic process equals the Fourier transform of the Point Spread Function. It is the Optical Transfer Function of the stochastic process. It controls the coherence of the generated hop landing location swarm.

Modules

Elementary particles are elementary modules. Together the elementary particles constitute all modules that exist in the universe. Some of the modules constitute modular systems.

Composed modules own a stochastic process that governs their composition. This stochastic process also owns a characteristic function. That characteristic function equals a dynamic superposition of the characteristic functions of the stochastic processes that control the composing modules. The superposition coefficients can act as gauge factors that represent displacement generators. In this way the superposition coefficients control the internal locations of the components. It is a dynamic location. Since the internal movements stay within the module they are internal oscillations. The overall stochastic process controls the coherence of the footprints of the constituents. An extra gauge factor that combines with the overall characteristic function acts as a displacement generator for the whole module and ensures that it moves as a single object. With other words, the characteristic processes and the gauge factors control the binding of the components of the module.

The conclusion of this sketch is that superpositions occur in Fourier space and not in configuration space.

Dark matter

In isolation and in ensembles of spuriously distributed objects the shock fronts represent nature’s dark quanta. The spherical shock fronts can represent dark matter and the one-dimensional shock fronts may represent dark energy.

A halo of spurious spherical shock fronts that surrounds a series of coherent swarms of spherical shock fronts can implement gravitational lensing in a way that is like the swarms themselves can cause gravitational lensing. The Fourier transform of the location density distribution of the spherical pulse responses acts as an Optical Transfer Function that qualifies the imaging process. A similar phenomenon in optics is known as veiling glare.

Mass

Having mass is equivalent to having the capability to deform the carrier of the owner of the mass. It is also equivalent to having the capability to add volume to the carrier field. The volume spreads over the full carrier field.Thus in order to cause a persistent deformation the local extra volume must be recurrently resupplied. Each volume addition expands the carrier field.

Recurrent regeneration of mass

The origin of gravitation locates in the capability of spherical shock fronts to inject a bit of volume into the field that acts as our living space. On itself this mechanism does not cause a persistent deformation of our living space. The effect of each spherical shock front quickly fades away. Only the expansion of our living space stays. Special mechanisms must cause a recurrent regeneration of spherical shock fronts that overlap in time and in space, such that a persistent deformation is obtained.

The stochastic processes that result from the ongoing embedding of separable Hilbert spaces into a non-separable Hilbert space perform this job.

The requirement that the actuator of the spherical shock front must be isotropic represents the reason of existence of color confinement.

Black holes

Elementary particles are generated in "free" space and are represented by a swarm that can be described by a coherent location density distribution. The volume that is injected by the pulses that are caused by the hop landing locations will quickly spread over the full embedding field and the local deformation thus quickly fades away. In contrat a black hole is a region that is surrounded by a surface through which volume cannot depart. Neither the one-dimensional shock fronts, nor the three dimensional shock fronts can pass the border. Thus, for observers the border presents an event horizon. Other investigations of black holes show that the internals of the border represent the densest packaging of entropy. The black hole owns mass and when it is compared to a Greens'function, then this mass is proportional to the surface area of the black hole. When the extended Stokes theorem is applied to the black hole, then the area of its surface is proportional to the number of artifacts that this region contains, Further, the black hole acts as a black body. It emits one-dimensional shock fronts.

Huge gravitational vibrations

Very sensitive and rather large instruments like LIGO and Virgo can detect gravitational vibrations whose sources locate millions of light years away. The usual cause is a merger of two stars or black holes. The merger of stars will also produce visual effects. The duration of the final merger is very short. Thus, a big chance exists that this causes a spherical shock front. This shock front represents a significant amount of mass that travels with the font, but does not consist of matter. The front passes the detector with light speed.