The Generalized Holographic Model, Part III: The Electron and the Holographic Mass Solution

In the first part of this series, entitled The Generalized Holographic Model, Part I: The Holographic Principle, we introduced the holographic principle as developed by David Bohm, Gerard 't Hooft, Jacob Bekenstein and Stephen Hawking. This principle states that the information contained in the volume of a Black hole is holographically present in the boundary or event horizon of the black hole. Then, in the second part The generalized holographic model part II: Quantum Gravity and the Holographic Mass Solution, we introduced Nassim Haramein's generalization of such principle, where he includes the volume information or degrees of freedom in the volume as well. This generalization allows to define a holographic ratio that accounts for the surface-to-volume entropy or information potential transfer Φ, which is a steady state or thermodynamical equilibrium, equivalent to a kinetic rate constant. Through the generalized holographic approach, we had found that the mass of the proton emerges from the granular Planck-scale structure of spacetime in terms of a surface-to-volume information transfer potential Φ which decreases with the increasing radius of the particle.

This geometric solution Φ provides a steady-state calculation or thermodynamic equilibrium value for the vast number of little Planck oscillators or PSU spinning together (at the speed of light or very close to it) to form the vortex we call a proton. Therefore, this geometric solution also concerns dynamics and evolution in time, such as velocity, momentum, and acceleration, which are aspects that are subject to relativity theory. In this third section we will show how Dr. Amira Val Baker et al. unravel this dynamical aspect in the generalized holographic approach to explain the electron mass. This calculation was published in Physics Essays (2019) [1].

The rotational speed (also called angular speed) of the collection of PSUs creating the vortex we call a proton, would subject this system to special relativity and mass-dilation (changes in mass) which varies as a function of speed (as seen in Module 4, of our Unified Science Course, section 4.4.6 for the confining force).

Einstein’s special relativity theory explains that an object undergoes a mass-dilation with respect to an external observer (external reference frame of observation) when accelerated to the speed of light relative to the speed of the observer. Therefore, as an object moves away from the surface event horizon of the co-moving Planck units that make up the proton, Haramein reasoned that the velocity of the object would diminish very rapidly, and if it did, then the mass-dilation of the object would drop very rapidly too. If the mass dropped, so would the gravitational force of this object.

The mass of an object at rest is called rest mass (this is the standard mass m we are all used to, and it is a steady state value) and in this subsection, it will be denoted by m₀. Special relativity theory says that as the object moves and increases its speed, its mass increases following the equation m = ϒ m₀, where ϒ is a proportionality factor between m and m₀ (ϒ = m/m₀) that accounts for this dilation in mass (and time, too). Known as the Lorentz factor, ϒ depends on the ratio v/c of the speed of the object v to the speed of light c. We recall that c = 299.792.458 m/s (meters per second).

The figure above depicts the increase of the Lorentz factor ϒ (ϒ = m/m₀) for any object as the speed v of this object (plotted here in terms of the percentage of its speed v of the speed of light c) increases from zero movement (or v = 0) to the speed of light (or v = c).  We see this factor ϒ starts as value 1, meaning that the mass of the object is its rest mass, or m = m₀ when the object is at rest (v = 0), and as the object increases its speed, ϒ grows very slowly (ϒ > 1) and so does its mass, but then ϒ starts to increase very sharply when the speed v of the object reaches 90% of the speed of light c (v = 0.9c). Between 90% and the speed of light (0.9c < v <c), the Lorentz factor ϒ grows exponentially, indicating that mass increases too, becoming infinite. Therefore, it is often said that no object can reach the speed of light, and if it did, it would become a singularity (i.e., a point of infinite mass; it would curve spacetime like a black hole). In the same figure above, we see the complete formula for the Lorentz factor ϒ, which is often expressed as its inverse, ϒ^-1, such that 

An object is said to move at relativistic speeds when its speed is between 80% of the speed of light and c (0.8c < v < c). At the speeds we encounter in our daily lives, the relativistic effects like mass and time dilation are mostly unperceivable. Nevertheless, for our GPS to give accurate positions, these relativistic effects must be taken into account.

Using the analogy of a rubber ducky in a bathtub filled with water, if the rubber ducky is far from the drain, it will orbit slowly, but as it gets closer its rotation velocity will increase rapidly as it enters the region of the more coherent co-moving water particles that make up the vortex created by the gradient of the air-water exchange in the drain. This vortex is a perfect analogy of the one created by the singularity at the center of a black hole (it has recently been proved that Black holes admit vortex structure). Now imagine that the vortex is spinning at the speed of light, so that the rubber ducky sped up to almost the speed of light as it approached the drain; the ducky’s increase in mass would create an equivalent increase in the gravitational force that the rubber ducky has on its environment. If the ducky was moved away from the drain and its velocity reduced, the mass-dilation of the ducky would drop, as well.

In the context of our generalized Holographic model, a hydrogen atom consisting mainly of one proton in the nucleus and an electron orbiting around it can be pictured as follows: since the proton volume with a charge radius r_p is composed of these PSU rotating coherently (similar to the water molecules moving coherently to create the vortex in the drain in the bathtub example) at relativistic speeds, we first imagine the electron as a probe particle (the rubber ducky) propelled by this rotating field created by the proton. We picture the electron’s rotation speed increasing as it approaches the proton that is rotating at c in its center (equivalent to the rubber ducky approaching the drain), and therefore, the electron’s mass increases, as well. On the other hand, if a probe electron circulating this vortex moves away from the vortex, its speed will decrease, and therefore, its mass will, too.

If we evaluate the inverse of the Lorentz factor (ϒ^-1 = m₀/m) as a probe particle is moving away from a proton rotating at the speed of light c, and so is decreasing its speed, and plot ϒ^-1, we obtain the Figure below. On the left side of the plot, we see that mass m starts at the rest mass of the proton m = m_p= m₀, and as we move right along the horizontal axis, the speed decreases (and so the ratio v/c decreases) and the mass m decreases too, becoming the known mass of the electron m_e (m = m_e) when the speed has reached the value v = α c , which is the expected velocity of the electron in the first orbital of the Bohr hydrogen atom (v ≈ c/137 = 2.18 x 10⁶ m/s), α being the fine structure constant (α ≈ 1/137). Note that the plot below differs from the previous one - in which ϒ went to infinity - because it’s the inverse ϒ^-1 instead of ϒ, hence the speed v is decreasing (instead of increasing) as we move to the right along the horizontal axis.

The features of ϒ^-1 that we have just described are very telling. They tell us that there seems to be a continuous transition from proton mass to electron mass, expressed through their speeds. Therefore, instead of thinking about the electron as a separate system like a particle orbiting the nucleus, the electron could instead be thought of as an inherent feature of proton dynamics; a distribution or cloud of potential energy extending spatially out of the proton to the radius at which the volume encompasses the electron cloud of a hydrogen Bohr atom, since we saw from the Figure above that the electron reaches the mass m_e at the expected velocity of an electron in the first orbit of the Bohr model of the hydrogen atom. In Bohr’s model, the Bohr radius (a₀) is a constant that gives the most probable distance between the nucleus and the electron of a hydrogen atom in its ground state (ground state means that the electron is in the atom’s fundamental state: the first orbit or atomic level n=1), and its value is a₀ = 5.29177210903(80)×10⁻¹¹ m (the numbers inside the parenthesis indicate there is uncertainty in these last figures; it establishes the accuracy of the value). See the figure below for more clarity, keeping in mind that distances are not to scale; the radius of the proton r_p would be too small (two orders of magnitude smaller) with respect to a₀ to appear in this figure.

Therefore, we must consider the holographic ratio relationship as we extend the radius of the co-oscillating Planck PSUs beyond the charge radius of the proton (r > r_p) where r_p is the proton charge radius. It is thus reasonable to consider a velocity relationship in the holographic mass solution which becomes significant at speeds lower than c (v < c) which in this case would appear at r > r_p . Using the holographic approach we should expect that since the radius r is larger than r_p (i.e., for all r > r_p ) the mass m becomes smaller or decreases below the mass of the proton  m_p (m < m_p).

These considerations make us realize that the holographic equation already derived for the proton mass m_p (m_p = 2 Φ m_l presented in the second part of this series) could also be interpreted as a mass dilation (in this case, a decrease of mass since we are moving away from speed of light c) since m_p is much smaller than the Planck mass m_l, as if the velocity relationship or Lorentz factor was already taken into account by this factor 2 Φ, and all we have to do in order to extend this velocity relationship as the radius r increases beyond the proton radius r_p (and so does the holographic surface-to-volume ratio in terms of this variable radius r), is replace the factor 2 in the holographic solution for the proton, with a general and unknown geometric parameter β, obtaining a general equation m_r = β Φ_r,l m_l  that accounts for a decreased mass m_r as we move farther away from the proton radius r_p.

Therefore, the expression for a mass dilation now depends on a holographic surface-to-volume ratio Φ_r,l given in terms of PSU that tile a surface and fill in a volume (where both depend on a variable radius r) “scaled” by β, instead of depending on the former Lorentz factor ϒ which depends on the speed relationship v/c.

Outstandingly, when the radius r reaches the Bohr radius a₀ (i.e., when r = a₀) and using a geometric factor β = 1/(2α) authors find a mass in precise agreement with the experimental mass of the electron! This confirms the initial hypothesis of the system proton-electron as a continuous entity. The solution for the mass of the bounded electron can thus be given as:

where r_l is the Planck radius r_l = l / 2 , with l being the Planck length.

This solution supports the hypothesis that the geometric factor β = 1/(2α) in the expression for m_e is analogous to a Lorentz factor, but in terms of distances (radius) instead of speeds, and it extends from Planck mass directly to electron mass, without having to stop at the proton, though we know that the proton is implicitly included in this expression, since its radius is between Planck radius and Bohr radius (r_l < r_p < a₀). This also provides hints about the hierarchy problem, to be discussed in the next subsection.

With this solution, Dr. Val Baker et al. found a mass m_e = 9.10938(30)x10^-28 g with a precision of 10^-5, which is inside the precision of the experiment since it is accurate within 1σ compared to the CODATA 2018 value of 9.10938(37) x 10^-28 g. In comparison with the equation (1) for the electron mass described in our former RSF article entitled A Brief History of the Electron, where, as we can recall, m_e was calculated using the Rydberg constant and speed of light, c, among other experimental parameters. Our holographic solution is remarkable as it is derived from first principle geometric and velocity considerations alone!

In the example of the rubber ducky, two differences must be remarked. First, since the ducky is not an inherent part of the water in the bathtub, while the electron is inherently part of the dynamics of the proton in the hydrogen atom. Secondly, we know that the ducky would fall down the drain because the bathtub is not in a resonant condition, and it does not normally create resonant or rotational standing wave states. If the bathtub was in a condition to create this resonant state resulting from the whole dynamics of the water in a feedback loop, the ducky would rotate in a stable orbit, a stable orbit being what we call a bounded electron in the example of the hydrogen atom.

In this view, the bounded electron (i.e., the electron inside the hydrogen) is the first bounding condition (the first resonance or resonant state, standing wave) created by the spin and wave interference dynamics of the vacuum fluctuations inside the volume of the proton, and it happens at the Bohr radius a₀. Since it is the first stable boundary condition, it has been interpreted as an orbital, but in reality, this orbital is a cloud of potential energy, or more precisely, of charge that we call an electron, just as was predicted by quantum mechanics, except it’s not probabilistic in nature as it was interpreted by quantum theory, but real. It is a real standing wave distribution of charge (or polarization due to spin or angular momentum) around the proton.

^{Image courtesy of Dr. Val Baker.}

As Dr. Val Baker states:

This holographic solution for m_e is not only significantly accurate, it also gives insight into the physical and mechanical dynamics of the granular Planck scale vacuum structure of spacetime and its role as the source of angular momentum, mass, and charge. The definition clearly demonstrates that the differential angular velocities of the collective coherent behavior of Planck information bits (PSU) determines the specific scale boundary conditions and mass-energy relationships that we call proton, electron, etc.

These specific boundary conditions are defined by the general and complete geometric factor βΦ_r,l (analogous to a Lorentz factor for the generalized holographic model) in the expression for m_r that decreases from Planck mass to smaller mass particles as we move away from the Planck radius r_l (which is Planck length l divided by 2, (or r_l = l/2 ). The Universal scaling law that will be published in the upcoming Module 8 of our Unified Science Course is based on this principle of finding the appropriate, complete geometric factor to scale the masses and radii of all objects, from the Planck scale and below, up to the universe and beyond! Hence, it becomes a scaling or fractal holographic factor …

Additionally, this solution does not require the probabilistic interpretation of the electron, as the electron, proton and all masses emerge from a velocity gradient which creates a density gradient and coherent behavior of the vacuum fluctuations, which are real rotations of the quantum vacuum. Therefore, this solution challenges the probabilistic nature of the Copenhagen interpretation of quantum mechanics by reimprinting a classical or fluid mechanics (though relativistic, given the speeds involved) behavior onto the atomic realm. Furthermore, since the geometric part of the description implicit in the holographic solution is based on the geometry and discreteness of the PSU entities, it is quantized in nature, and so naturally bridges general relativity (implied in the β factor, which comes from relativistic considerations) and quantum physics (implied in the Planck mass m_l). And since these PSU are also bits of information, this solution relates both gravity and relativity with information theory (entropy).

_{Extension of the holographic solution for other atoms}

The solution for one electron can be extended to include other radii below the Bohr radius. This is done by extending the holographic mass solution for the electron in the n = 1 state to radii smaller than the Bohr radius (r < a₀), obtaining the following equation:

where Φ_r(r) is the holographic ratio, which is now a function that depends on the radius r for radii smaller than a₀  and where N is an integer.

The left hand side of the equation above, which expresses the holographic mass solution as a function of radius r, can be evaluated at each value for r, and can be plotted as shown in the Figure below, where we observe that the holographic mass solution increases with decreasing radius r (i.e., when going from right to left in the Figure below). We can also see that the increase in the holographic solution at specific integer amounts N of m_e each correspond sequentially with increasing N to the atoms after hydrogen (i.e., after m = m_e). Basically, this means that this solution gives the total electron mass for each atom in the periodic table as the complete electron potential at the ground state (n = 1) of each atom. The radius, which defines the extent of the vorticity, decreases with increasing number of electrons, suggesting that the vortex gets “tighter” as the electron potential increases.

_{Figure: Plot of the holographic mass solution as a function of radius. Note: the holographic mass is equal to me at corresponding radii of a0 /N. For example, the holographic mass equals the mass of one electron at a radius of the hydrogen atom in its n=1 state; equals the mass of two electrons at a radius of the helium atom in its n=1 state; equals the mass of three electrons at a radius of the Lithium atom in its n=1 state, and so on. Note: this relationship is only shown on the graph for the first three elements but continues for all known elements.}

More specifically, we can see that the holographic mass is equal to m_e at corresponding radii of a₀ / N. For example, the holographic mass equals the mass of one electron (m = m_e) at a radius of the hydrogen atom in its n = 1 state (written as r = r (H_n=1) in the figure above, meaning that r = a₀. It equals the mass of two electrons (m = 2m_e) at a radius of the helium element atom (He) in its n = 1 state (written as r = r (He_n=1) in the figure above). The holographic mass equals the mass of three electrons (m = 3m_e) at a radius of the Lithium atom (Li) in its n = 1 state (written as r = r (Li_n=1) in the figure above), and so on. This relationship is only shown on the graph for the first three elements, but it continues for all known elements.

With this generalized solution for all elements, we identify N as being the atomic number Z, where for progressively smaller fractions of a₀, we find an interesting proportional relationship between the holographic mass and the mass of the electron, as was seen in the Figure above.

_{Periodic table of elements. The atomic number Z (up to the left at each square) indicates the number of protons, which is equal to the number of electrons, in the atom. The atom is neutral (has equal positive and negative charge). Dmitri Mendeleev, via Wikimedia Commons/Jonathan Aprea}

This new derivation of the electron extends the holographic mass solution to the hydrogen Bohr atom and to all known elements in their fundamental state (n = 1 state), defining their atomic structure and charge through the electromagnetic fluctuation of the Planck scale. Furthermore, the atomic number Z emerges as a natural consequence of this geometric approach.

Regarding the dynamic description of the hydrogen atom, since Haramein’s holographic model depicts the proton as a black hole (as it has been explained the second part of this series), the entire atom (proton and its electron shells) could be considered as if it were a black hole, where the electron represents the ergosphere of the black hole proton. A similar parallelism between a Bohr atom and black holes has been suggested very recently by astrophysicist / theoretical physicist Christian Corda, Editor-in-Chief of the international journal “Journal of High Energy Physics, Gravitation and Cosmology” and “Theoretical Physics”. In Corda´s model [2] [3] a black hole is similar to the semiclassical model of Bohr's hydrogen atom, where the quantum normal modes (QNM) of vibration in the atom represent the electron jumps between orbitals, and the absolute value of the QNM electromagnetic frequencies triggered by the emission and absorption of particles (equivalent to Hawking radiation in Black holes) represent the energy shells of the gravitational “hydrogen atom” black hole.

_{... a little tiny bit of chemistry}

As we explained in a former article A Brief History of the Electron, the Bohr model was used to determine certain properties of the atomic spectra. Bohr's model considered concentric electron shells between which electrons jump, and these jumps were associated with the shapes seen in the emission spectra: s for sharp, p for principal, d for diffuse, and f for fundamental (this is why orbitals have these nomenclature s,p,d,f. For example, the n=1 state of all atoms is a spherical orbital also called orbital 1s). This regularity of line-spectra was linked to chemical property regularities, providing an argument for the X-ray spectra of the elements done by Henry Moseley in 1914 to establish an experimental basis in agreement with the periodic table of elements created decades before, in 1869, by the Russian chemistry professor Dmitri Mendeleev.

Since the atomic structure was unknown at the time, Mendeleev had organized the elements by measuring their atomic mass and this had proven to be an accurate criterion for predicting their chemical properties. It was after the discovery of the atomic nucleus by Ernest Rutherford in 1911 that the model of integer charge of the nucleus and its value was later confirmed experimentally by Moseley, who defined the nuclear charge as the number of protons in the nucleus and named it atomic number (Z). The atomic number creates an integer-based sequence, and it is the latest absolute definition of an element.  Since Z is an integer, this is also a feature of quantization.

There is a thermodynamic aspect to the holographic solution for the electron and its extension to all atoms, associated with the holographic ratio Φ, which is a volume-to-surface rate of information transfer, similar to the kinetic constant k(T) used in chemical reactions. A chemical reaction is basically a rearrangement of atoms inside a molecule or an exchange of atoms between molecules (an exchange of an integer number of atoms). For any of these to happen, there is a change in energy involved. If the chemical reaction is endothermic (endothermic means absorbing heat), one must supply a certain amount of energy for it to happen (usually by heating the receptacle containing the substances for the reaction). Hence, this reaction does not happen spontaneously. If the reaction is exothermic (releasing heat), it will happen spontaneously, liberating energy (usually in the form of heat, as well). This feature of chemical reactions is associated with k(T), a reaction rate constant which depends on temperature T and is associated with the Gibbs free energy of activation, a quantity that can be regarded as the energy needed to reach the transition state of a chemical reaction, converting reactants to products. The transition state theory (TST) of chemical reactions is used primarily to understand qualitatively how chemical reactions take place.

In the case of the holographic ratio Φ, it could be seen as a kinetic or reaction rate constant associated with the Gibbs free energy of the surface-to-volume information exchange, i.e., the energy exchange in the event horizon. In this sense, Φ represents a thermodynamic steady state calculation.

The Universal scaling law that will be published by Nassim Haramein and Olivier Alirol, entitled Scale invariant unification of forces, fields and particles in a Quantum Vacuum plasma (which once published,  will be addressed in detail in Module 8) is based on this principle of finding the appropriate, complete geometric factor to scale the masses and radii of all objects, from the Planck scale and below, up to the universe and beyond. Hence, it becomes a scaling or fractal holographic factor ...  It is a complete Unified Field Theory!

Note to the reader: This article is part of Module 7, section 7.1 of our unified Science Course, which is now available online, for free.  

References

[1] Val baker, A.K.F, Haramein, N. and Alirol, O. (2019). The Electron and the Holographic Mass Solution, Physics Essays, Vol 32, Pages 255-262.

[2] Corda, C.; Feleppa, F. Black Hole as Gravitational Hydrogen Atom by Rosen’s Quantization Approach. Preprints 2018, 2018100413 (doi: 10.20944/preprints201810.0413.v1).

[3] Quasi Normal modes: The "Electrons" of black holes as "Gravitational atoms"? Implications for the Black hole information puzzle. Advances in High Energy Physics, Volume 2015, Article ID 867601, 16

[4] Heilbron, John L. (1985). "Bohr's First Theories of the Atom". A. P.; Kennedy, P. J. (eds.). Niels Bohr: A Centenary Volume. Cambridge, Massachusetts: Harvard University Press. pp. 33–49