The Schwarzschild-de Broglie Modification of Special Relativity for Massive Field Bosons (SBM)

Table of Contents:

Abstract

1.0 Introduction

2.0 The Schwarzschild – de Broglie modification of SRT for massive field bosons (SBM)

3.0 Spontaneous symmetry breaking with the formation of a phase boundary

4.0 Higgs mechanism from the SBM model perspective

5.0 Discussion and Conclusion

6.0 Acknowledgement

7.0 Literature

The Schwarzschild – de Broglie Modification of Special Relativity for Massive Field Bosons (SBM)

A study about dark matter and dark energy from the SBM model perspective

Author: Siegfried Gantert

s.gantert@email.de

Abstract

This work is a presentation of a modified form of special relativity for field-bosons – in short SBM. Field bosons, for the purposes of this work, are synonymous with the condensates from spin 0-particles. The starting point is the hypothesis that a minimum size of uncertainty ( =Schwarzschild radius) becomes effective with relativistic velocities, from which different limit velocities are derived, depending on the size of the field bosons. In accordance with the SBM model, field bosons under a defined phase limit become massive through spontaneous symmetry breaking. Field bosons can melt into larger condensates through the effects of gravity, whereby their effective mass is reduced, thus also reducing their large-scale gravitative coherence.

1.0 Introduction

According to the latest background radiation measurements (CMB) of the ESA’s Planck satellite, about 68 percent of the universe is composed of a mysterious dark energy and 27 percent of dark matter, with the more familiar visible matter only making up about 5 percent of the universe [1].

F. Zwicky was one of the first to point out the possible existence of dark matter, after investigations into the proper motion of galaxies in galaxy clusters showed great compatibility with this idea [2, 3]. Similar observations were made in the rotation curve of spiral galaxies [4-9]. In these cases, the amount of dark matter increases with distance from the center, while it is comparatively low in the center of the galaxy.

However, the result of an investigation into the distribution of mass in the matter around the sun [10] showed that this mass is distributed almost exactly as it appears visibly. There were thus no indications for dark matter found.

Due to the difficulties of reconciling these sometimes-contradictory observations with known physical concepts, many concepts of dark matter and dark energy have been proposed. Some models that have attempted to describe the enigmatic behavior are string theory [11], loop quantum gravity [12], the quintessence model [13], the axion [14], phantom energy [15], and the MOND theory [16].

If one takes the cosmological constant L as a basis, the background radiation data [17, 18] regarding the distribution of dark matter and dark energy can be well reconciled with the predictions of the cosmological standard model (L-CDM model). However, the fundamental mechanisms that could explain a dark energy that works against gravity remain largely unexplained at this point.

The predictions of the L-CDM model also lead to inconsistencies in order of magnitude scales of a galaxy [19]. In terms of the model’s projections, the center of a galaxy should rotate faster than the observed measurements. One would also expect to observe a greater density of cold dark matter towards the center of a galaxy. The discrepancy between the measurements and the predictions of the L-CDM model were referred to as a “cold dark matter catastrophe” in astrophysics literature [20].

Mayer et al. ascribed the lack of dark matter in the center region of a galaxy to supernova explosions [21]. A research group led by Benoit Famaey came to the conclusion, on the basis of their investigations [22, 23], that there must be a close connection between the distribution of visible matter and dark matter. Their observations correspond more with the predictions of the MOND theory (Modified Newtonian Dynamics) of Mordehai Milgrom [24].

Models interpreting dark matter as a condensate of a scalar boson field have recently received increased attention [25-38].

At the elementary particle level, hypothetical particles such as the axion or WIMPs (weakly interacting massive particles) have been considered as candidates for dark matter. In the SUSY theory(-ies), it is the neutralino, the lightest (LSP) of a series of hypothetical particles, which is seen as a possible candidate for cold dark matter (CDM) [39].

2.0 The Schwarzschild-de Broglie modification of SRT for massive field bosons (SBM)

The work presented here aims to utilize astrophysical data [1-10, 50] in order to check the results of the SBM model for consistency. It is motivated by the possibility of better understanding the physical aspects of dark energy and dark matter.

The work is structured as follows: the underlying SBM model will first be presented along with the different limit velocities of field bosons with masses of various scales of magnitude derived from the model. Subsequently, the phase limit will be determined at which the relativistic field bosons receive their effective mass through spontaneous symmetry breaking. In the following section, a mechanism similar to Higgs will be proposed, on the basis of the SBM model, in order to place the total energy of the field bosons involved into a quantitative relationship with their mass-giving effect. In conclusion, the results of the findings will be employed to discuss the usefulness of the proposed SBM model, using examples from some astrophysical problem areas.

Within the concept of the Planck scale, the Planck length is often taken as a limit for the validity of currently known physical laws [40]. The Planck length (3) is determined through a comparison of the size of the Schwarzschild radius (1) with the Compton wavelength, equation (2). According to a concept from Max Planck [41, 42], the description of physical phenomena over distances smaller than the Planck length (1.616E – 35m), equation (3), is impossible with our current level of scientific understanding, and can only be formulated with a quantum theory of gravitation.

In principle, the Compton wavelength can be determined by the increase in wavelength of the scattered radiation at a right angle that results from an elastic impact of a photon with a particle at rest. After the collision, this normally leads to the suppression of the coherent properties of quantum mechanical states, which is related to the loss of the interference capability of the particle. As photons do not collide with dark matter, having only a gravitational interaction, it is probable that a stable and coherent system exists. It thus seems worthwhile to examine the appearance of dark matter in terms of a coherent interaction of bosons, especially the coherent interaction of Higgs bosons.

Due to their integral spins, Higgs bosons can arrange themselves into condensates, given stable quantum mechanical conditions. It can be shown that appropriate potentials develop with a certain modification of the special relativity theory, and that these could contribute to the development of such condensates. In the context of this work, the term “field boson” will usually be used in place of condensate in order to highlight the particulate character of such an assembly. Because Higgs bosons, as the constituents of such assemblies, are subject to Bose-Einstein statistics due to their integral spins, a respective field boson (condensate) can be described as a single particle wave function and can be understood quantum mechanically as an independent particle.

Therefore, the Compton wavelength, as a collision result, will not be taken as the central point of consideration in this work, but rather the de Broglie wavelength (4) of field bosons of differing sizes.

The SBM model presented here differs not only from the SRT of Albert Einstein in that it differentiates between fermionic and bosonic particles; it also differs from the double special relativity theory [43-45], which takes the approach that Planck length (or Planck energy) should be taken into account, in addition to the speed of light, as a further Lorentz invariant quantity.

First, possible consequences of coherent conditions of Higgs bosons with regard to relativistic speed behavior of field bosons will be discussed.

As it is impossible to make any remarks about the strength of the particles’ interconnections a priori, the scale mass of a field boson is defined as the sum of the masses of the (Higgs) bosons of which it is composed, equation (5).

The scale mass should mainly serve as a reference scale to show the effective sizes of the field bosons.

For further approaches, the special relativity theory (SRT) was used, with the reservation that a description of a relativistic particle of the field particle type described above is impossible in the SRT. As a result, the laws of SRT were only used to lead towards the point where a “collapse” in wave function was expected. In a second step, a specific relativistic limit velocity that prevented a collapse in wave function, or the loss of coherence was searched for every field boson. A "l in the index of limit velocities stands for “limit,” while “n” designates the number of Higgs bosons constituting a specific field boson. If there are no attractive or repellant interactions between the Higgs bosons, equation (6) applies to the de Broglie wavelength of a (free) field boson with mass center , where stands for the impulse of the field boson.

If the field boson moves very quickly from the point of view of an observer at rest (inertial system), then the relativistic impulse in equation (7) must be taken into account.

From the point of view of an observer at rest () the matter wave of a coherent field particle is compressed in the direction of motion at relativistic speeds . It is now assumed that, for an observer at rest, the material wave around the limit speed of a field boson passes over the stage of a flattened rotation ellipsoid into a toroidal limit form; it is simplified into a horn torus, as shown in Fig. 1. For the uncertainty of such a field particle close to the speed limit, there is a minimum size, with relation (8), of which the value is larger than the double of the Schwarzschild radius.

The validity of the relativistic length contraction per (9) close to the mass center can actually be sustained with requirement (8), without it leading to a collapse of the matter wave or the creation of a punctiform singularity, see Fig. 1.

From the point of view of an observer at rest, the “length” of a field boson under limiting condition (8) approaches 0 at relativistic energies at the mass center, as shown in Fig. 1. The de Broglie wavelength relationship (10) applies to this limiting condition:

The dotted lines at the right in Fig. 1 schematically show the outer Schwarzschild solution as a sectional drawing through a Flamm's paraboloid in two different position with the distance Dx. In the SBM model, this thus results in differences with the SRT, for which a break in known physical laws is predicted near the Planck length [41].

illustration not visible in this excerpt

Figure 1: Toroidal symmetry of the limit configuration (horn torus) with the de Broglie wavelength for determining the relativistic limit velocity of a field boson (sectional drawing at right).

For the explicit calculation of the limit velocity of a field boson of scale mass , relationship (10) will be used to establish the conditional equation (11) for , taking into account the Lorentz factor

Solving the equation (11) for gives (12) as a solution, whereby only positive values should be taken into account before the radical term.

The graphical presentation of the limit velocities (12) contrasted with the scale mass (Fig. 2) shows that below a scale mass of about 1E-10 kg there are hardly any noticeable deviations of the SBM from the special relativity theory (SRT), as applies to field bosons with a low scale mass.

However, above 1E-10 kg on the mass scale, there are stronger deviations from the SRT. The calculated limit velocities gradually approach 0 as scale masses increase, see Fig. 2.

According to equation (12), the limit velocities are only dependent on the scale mass (3) of a field boson and the natural constants G, ħ and c. In the case of a bosonic particle without resting mass, a photon for example, applies , as one can easily verify from equation (12) for As will be shown in more detail later, contrary to the SRT, the SBM predicts the dispersion of limit velocities of field bosons with non-zero resting masses. Bosons without resting mass, for example photons (of differing energies), should therefore not display any velocity dispersion. This conforms well to run time measurements of photons from GBR events [46, 47] (GBR gamma ray burst).

illustration not visible in this excerpt

Figure 2: Calculated limit velocities according to the SBM model on a mass scale from 0 to about 1.6E-7 kg.

If one retains relativistic principles and substitutes the newly acquired limit velocities for the speed of light c in the energy impulse equation of the SRT (13), or in equation (15) for total energy, there is a strong deviation from the SBM model, in contrast to the SRT, see Fig. 3 below.

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