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The Generalized Holographic Model, Part II: Quantum Gravity and the Holographic Mass Solution

Image credit: Shutterstock

By Dr. Inés Urdaneta, Physicist at Resonance Science Foundation

In the former article 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. We then introduced the generalization of such principle by Nassim Haramein, 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, which is a steady state or thermodynamical equilibrium, equivalent to a kinetic rate constant.  

In this second part we will see why Haramein’s generalized holographic approach gives a quantized solution to mass and gravity in terms of the quanta of space. To achieve these remarkable results, Haramein defined a spherical volume unit or voxel for space that he named Planck Spherical Unit (PSU) which is the fundamental unit of energy. The PSU represents a quantum of electromagnetic oscillation, and it also represents a bit of information.

Therefore, the seemingly smooth and continuous empty space has a granularity composed of these units (just as the water molecules in the example of liquid water) and they pack in to fill any volume in the flower of life pattern in 3D as shown below, to the left. This is what a volume filled with packed PSU looks like:

When projecting the former 3D pattern in any direction, we obtain the very well-known 2D flower of life pattern shown above, to the right. This is what a flat surface tiled by PSU would look like.

As seen in the previous  article, each fundamental spherical unit volume voxelating all space, is an oscillating Planck Spherical Unit (PSU) defined geometrically by [1]:

Where the Planck radius is rl = l/2 and l is the Planck length. Remember that each PSU also represents a unit of entropy or bit of information.

As we explained in the former article, the holographic principle only considered the information contained in the surface of a black hole taken as a spherical system but where the unit of tilling is a square of Planck size l2. Exploring this holographic principle further along with the maximal entropy of a black hole [2] addressed there, Haramein proposes a generalized holographic approach in terms of BOTH the surface AND volume entropies of a spherical system, using a sphere as a first order approximation for the system under consideration (such as a PSU, a proton, an electron, or the Universe). The results he obtained have proven this geometry to be a very good assumption.

This surface-to-volume entropy of a spherical system gives the holographic ratio ɸ, is obtained by tiling the surface and filling the volume of such a spherical system with these PSU, like the figure below shows:

Figure: schematic to illustrate the Planck Spherical Units (PSU) packed within a spherical volume. These PSUs or Planck voxels tile along the area of a spherical surface and fill in the volume of the spherical system as seen in the figure.

This approach calculates the density of information/entropy of a spherical surface η in equatorial disc units  and thus defines the surface information/entropy in terms of PSUs, such that

where the Planck area is the equatorial disk of a Planck spherical unit PSU taken as one unit of information/entropy and A = 4π r2 is the surface area of a spherical system. This video offers an amazing depiction of the relationship between the two areas, a feature that is closely related to the generalized holographic solution. And this second video offers an amazing explanation of the generalized holographic solution.


Note that this geometric calculation of surface entropy η gives a very similar formula to Beckenstein-Hawking’s formula for entropy from the previous article S = A/4l2 [3], though they differ in that in Haramein’s approach, the information bit is defined as the equatorial disk area of a PSU with radius rl instead of a squared area of size l2. For this reason, the ¼ factor is no longer present in the expression for η, and instead, there is 1/π.


Following this definition for surface information η, the density of information/entropy within a volume V of a sphere with radius r is similarly defined in terms of PSU as,

With these very simple densities that we have named surface entropy η and volume entropy R, we obtain the fundamental holographic ratio Φ = η / R, which is a ratio that expresses the surface-to-volume entropy and represents the information potential transfer between the volume and the surface of the spherical system.

Image courtesy of Dr Amira Val Baker.

This holographic ratio Φ is the main concept of the generalized holographic theory. Basically, it means that we are counting the units of information stored in the surface and comparing it to the amount of information stored in the volume. And this count is being performed in a geometrical way, meaning that the shape of each unit (given by its geometry) and its size (or scale) are what define how many of these units can be stored in a certain space. If the size and geometry of this unit were different, the number of units that fit into a region (i.e., the density) would change. This is why choosing the correct geometry is critical to achieving this solution to quantum gravity.

Therefore, having the correct geometry provides the correct shape and having the correct size provides the correct scale of the unit. Haramein achieves this correct unit when defining it as a Planck (referring to the size of the unit) Spherical (referring to the shape of the unit) Unit (referring to a real, physical unit). Since we are counting an integer number of these units, the system or spacetime is quantized. Here, the feature of quantization appears, without resorting to quantum theory. The real origin of quantization, which is not explained anywhere in quantum theory (apart from the fact that it must be related to the Planck constant h), is therefore explained in the generalized holographic model. This particular pattern of division of space was the missing piece of the puzzle. Haramein’s initial intuition from more than 25 years ago that we should look at the pattern of division of space instead of trying to identify the tiniest fundamental particle was correct!

First Televised interview to Nassim Haramein (1995).
In this interview, Nassim Haramein explains that in order to have any field, there must be a singularity, and this implies an infinite division of space. See the interesting visualizations in the video showing that there can be infinite information in a bounded system.


In this view, quantization has more to do with the fact that we can count, and therefore, exchange information. And it just so happens that this counting can be traced back to the Planck scale. This means that spacetime at all scales must be quantized with the very small granular structure of the Planck scale, just as water appears smooth but when closely studied, it is found to be made of water molecules and particles that quantize it. General relativity as defined by Einstein assumes a smooth spacetime (like water) and it is applied at the large scale of planets, stars, and galaxies. What we are discovering is that spacetime at the quantum level is not smooth but granular like the molecules and atoms that make up water.  Since all big things are made of little things, Haramein discovered that this quantization defines the masses and dynamics not only of atomic and subatomic particles, but of the structure of the universe as a whole, as well.

The ratio of the number of PSU in the surface to the number of these units in the volume (or what we call surface-to-volume information content, or entropy, Φ) and its inverse, 1/Φ, represent the rate of information exchange between surface and volume, and vice versa, respectively. The generalized holographic solutions prove that it is this ratio that explains the emergence of features such as mass and gravity.

As we will see, the volume-to-surface entropy, given by the inverse of the holographic ratio 1/Φ, when multiplied by the Planck mass, gives the gravitational mass of a black hole (also called holographic mass, or mass-energy), while the surface-to-volume entropy (given by the holographic ratio Φ), when multiplied by the Planck mass, gives the rest mass of the system.


Note that this first principle calculation of the generalized holographic model is related to the original holographic principle from Beckenstein et al. from the former section, though it differs from Beckenstein’s in that Haramein considers not only the surface, but also the volume information content. Additionally, the geometry of the bit is related to a sphere, and not a square or a cube.


Mass of a Black hole (or holographic mass)

The holographic relationship between the energy transfer potential of the volume information and the surface information (in other words, the inverse of the fundamental ratio, or 1/Φ, which is R/η or volume entropy R divided by surface entropy η, also known as the volume-to-surface entropy) gives the gravitational mass ms (that we also call holographic mass) for any black hole with this equation:

where ml is the Planck mass, η is the number of PSU on the horizon of the black hole assumed to be a spherical surface, and R is the number of PSU within the spherical volume. This equation renders the same numerical value for mass than the one obtained using the Schwarzschild equation.The Schwarzschild radius rS solution of Einstein’s field equation for a non-rotating and uncharged black hole is given by:

where G is the gravitational constant, c is the speed of light, and m is the mass of a non-rotating and uncharged black hole.

When calculating the mass of the black hole using this last equation for the Schwarzschild radius, we obtain the same numerical value as the holographic mass ms. Therefore, the holographic mass obtained in terms of a discrete granular structure of spacetime at the Planck scale is equivalent to the Schwarzschild solution from general relativity (and hence, achieving quantum gravity). This has been verified for Cygnus-X black hole [3].

If we input the Planck mass (m = ml) in the above equation for rs , we obtain rsl = 2l . Since l is the Planck length, then 2l equals 4 rl being rl the Planck radius.

This rsl = 2l is the solution for the Schwarzschild radius of a black hole with Planck mass (that we will refer to as rSl to emphasize that it is a Planck-sized black hole). This means that the radius of a Planck mass black hole rSl is 4 times larger than the Planck radius rl and this fact will be important when we look at the possibility of the PSU as a wormhole termination.

The factor 2 in the solution above for rSl is the link between quantum theory and general relativity; it is the relationship between mass and radius. And it is also the key to unlocking the complete scaling law from the Planck scale down and up to the Universe and beyond!

This geometry is linked to relativity through the radius since it expresses the curvature of spacetime, while mass is an aspect of the quantum vacuum, since it emerges from the information dynamics implicit in the surface-to-volume holographic ratio Φ. This is to say that mass emerges from the information dynamics at the quantum or Planck scale. Another way of understanding this relationship between relativity and quantum physics is to recall that Einstein’s field equations predict a singularity of Planck-scale size.

This radius  rsl = 2l = 4 rl  obtained with a relativistic equation, also corresponds at the other side of the scale (quantum), to the radius at which all the volume information is encoded on the surface, that is, R = η.  At this radius, the holographic ratio Φ (surface-to-volume ratio) is equal to its inverse 1/Φ (volume-to-surface ratio) such that Φ = 1/Φ = 1. When calculating the surface entropy and the volume entropy we obtain ηl = Rl = 64, which results in a holographic ratio Φ = ηl / Rl = 1 yielding ml = (ηl / Rl) ml such that a state of equilibrium of the surface-to-volume information transfer potential is achieved. This supports the conjecture that due to its ultimate stability, the Planck is the fundamental unit of spacetime at the very fine scale of the quantum vacuum. This also means that the Planck mass is the quantum or unit of mass; it is the reference.

The result, 64, is very important: it confirms that the 64 tetrahedron grid representation of space is correct! We recall this grid in the figure below, where we can see how it is related to the flower of life when protected from top to bottom (in the figure on the left) and to the tree of life from the kabbalah Jewish tradition (see the figure on the right).

Figure taken from Article by Arlenz Kim


Such a coincidence is remarkable, and it is now supported by scientific models!

Additionally, the scale size of the Planck length, being so small (34 orders of magnitude smaller than a centimeter), is associated with the quantum scale, while the relatively large Planck mass ml (of the order of 10-5 g), which is 19 orders of magnitude bigger than the mass of a proton, places the Planck mass at the macroscopic scale, and is therefore associated with the relativistic scale. This crossover between radius and mass, considering their association with either relativity or quantum theory, is the manifestation of a unification of the relativity and quantum theories that occur at this Planck mass ml or equivalently, at this 2l radius, inside which 64 PSU are coherently rotating, creating a vortex that is curving spacetime. And this brings us again to the time when Haramein, inspired by Buckminster Fuller’s work, proposed the 64 tetrahedron grid as the real geometry of space.

The factor 2 between the Planck length l and the Schwarzschild radius  of a Planck mass black hole could be the reason why the factor 2 appears in many fundamental equations related to geometric considerations of motion, particle physics and cosmology, and occurs often in the most fundamental equations of physics [5]. In Haramein´s work, the origin of this factor 2 could be explained as the holographic surface-to-volume consideration of the fundamental geometric clustering of the structure of spacetime at the Planck scale, where one Planck mini black hole is an aggregation of Planck spherical vacuum oscillators [6].

The energy transfer between the surface information and the volume information, where R > η for all r > 2l, suggests that gravitational curvature (simply put, the curvature or bending of spacetime due to any mass) is the result of an asymmetry in the information storage of spacetime. The volume information is not only the result of the information/entropy surface bound of the local environment, but may also be non-local, due to wormhole interactions like those proposed by a conjecture (known as ER=EPR conjecture) in which black hole interiors are connected to each other through micro wormhole interactions [4].


Mass of a proton

Inversely, at the scale of the proton, the granular structure clustering of PSUs leads to precise values of the rest mass mp and charge radius rp of a proton, given by the equations:

where η is the ratio between the surface area of the proton and the equatorial area of a PSU, and gives the number of Planck equatorial areas that tile the surface of the proton, R is the amount of PSUs that fill in the volume of the proton (i.e., the volume of the proton divided by the volume of a PSU), Φ is the fundamental holographic ratio η / R and ml is the Planck mass (the mass of a PSU). We see that the factor 2 appears in this equation for m, which, as we explained before, comes from the asymmetry in the information storage of spacetime.

Significantly, this value for the proton radius is within 1σ agreement with the latest muonic and electronic measurements of the charge radius of the proton [3] [7], compared to a 7σ variance in the standard approach at that time [8].

Haramein’s generalized holographic solution recalls Bohm’s holographic principle; the enfolded structure corresponds to the volume-to-surface ratio (holographic mass), while the unfolded structure corresponds to the surface-to-volume ratio (rest mass). Reality results from the dynamic between the information enfolded or confined in the volume of a system and the information that it can effectively exchange with its surroundings, and therefore, unfolds expressing as mass. Mass, in this context, is the unfolded portion of all the information that is enfolded within such a bounded system. We could say that mass is the equilibrium state of the information transfer inertia in a bounded volume, and this inertia represents the degree of possibility for the surface to express all the information enfolded within this volume.

Similar approaches to that of Haramein, such as Erik Verlinde’s idea of entropic gravity, attempt to explain the phenomena in the frame of information theory and entropy, where gravity is an emergent property of information dynamics, similar to the generalized holographic model solution. Verlinde's calculations head in the same direction, though they are still far from achieving the same results.

The CODATA recommends an updated value for the proton radius from 2018, which is before they confirmed that the 2013 muonic hydrogen measurements yield consistently the same results as the latest 2019 electronic hydrogen measurements. This mystery, which is part of the so-called Proton Puzzle, has been addressed in a previous article, were we explain the history of the proton radius measurements and calculations, showing that Haramein's results are the most accurate theoretical prediction and are the result of first principle calculations!

And not only has this model predicted the features of protons within the certitude of experiments ... it predicts as well the mass of the electron, form first principles! Stay tuned for more ...

Note to the reader: This article is part of Module 7, section 7.1.3 of the free Unified Science course, which you can access at this Resonance Academy link



[1] N. Haramein, Phys. Rev. Res. Int. 3, 270 (2013).

[2] ’t Hooft G. The Holographic Principle. arXiv:hep-th/0003004v2. 2000;1-15

[3] J D. Bekenstein, Nuovo Cim. Lett. 4, 737 (1972).

[4] J. Maldacena and L. Susskind, e-print arXiv:1306.0533 (2013).

[5] P. Rowlands, e-print arXiv:physics/0110069 (2001).

[6] A. Val Baker et al. Physics Essays 32, 2 (2019)

[7] N. Haramein, e-print (2013).

[8] A. Antognini, F. Nez, K. Schuhmann, F. D. Amaro, F. Biraben, J. M. R. Cardoso, D. S. Covita, A.  Dax, S. Dhawan, M. Diepold, L. M. P. Fernandes, A. Giesen, A. L. Gouvea, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C-Y. Kao, P. Knowles, F. Kottmann, E-O. Le Bigot, Y-.W Liu, J. A. M. Lopes, L. Ludhova, C. M. B. Monteiro, F. Mulhauser, T. Nebel, P. Rabinowitz, J. M. F. dos Santos, L. A. Schaller, C. Schwob, D. Taqqu, J. F. C. A. Veloso, J. Vogelsang, and R. Pohl, Science 339, 417 (2013).




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