The Imaginary Universe (on the Three Complementary Sets of Measurement Units Defining Three Dark Electrons)

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The Imaginary Universe (on the Three Complementary Sets of Measurement Units Defining Three Dark Electrons)

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Szymon Łukaszyk^*

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Submitted:

27 August 2024

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28 August 2024

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Abstract

Three Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene establish three complementary fine-structure constants, two of which are negative. Each introduces its own specific set of Planck units. Hence, two sets of basic Planck units are real and two are imaginary. The elementary charge is the same in all those sets of Planck units, establishing equality between the products of each fine-structure constant and the speed of light it is associated with, and defining the dark electron in each of these three complementary systems. All fine-structure constants are related to each other through the constant of pi, which indicates that they do not vary over time. The negative complementary fine-structure constant established by the graphene reflectance is dual to the fine-structure constant. The assumption of universality of the black hole entropy formula to the remaining two stellar objects emitting perfect black-body radiation less dense than a black hole (neutron stars and white dwarfs) renders their energies exceeding their mass-energy equivalence. To accommodate this unphysical result, we introduced an imaginary mass and defined three complex energies in terms of real and imaginary Planck units, storing the surplus energy in their imaginary parts. It follows that black holes are fundamentally uncharged and have a vanishing imaginary mass. We have derived the lower bound on the mass of a charged black-body \textit{object}, the upper bound on a white dwarf radius, and the equilibrium density of all three complex energies. The complex force between real masses and imaginary charges leads to the complex black-body \textit{object's} surface gravity and generalized Hawking radiation complex temperature. Furthermore, based on the Bohr model for the hydrogen atom, we show that complex conjugates of this force represent atoms and antiatoms. The proposed model considers the value(s) of the fine-structure constant(s), which is(are) otherwise neglected in general relativity, and explains the registered (GWOSC) high masses of neutron star mergers and the associated fast radio bursts (CHIME) without resorting to any hypothetical types of exotic stellar objects.

Keywords:

Subject: Physical Sciences - Mathematical Physics

1. Introduction

The universe began with the Big Bang, a prevailing scientific opinion. However, this Big Bang was not an explosion of 4-dimensional spacetime, which is also a current prevailing scientific opinion, but an explosion of dimensions that started in

(- 1)

-dimensional void. The only exception to the general formula for the integral of a volume unit

a^{n}

, where a is the length of an edge in any dimension n is

n = - 1

, where a jump discontinuity occurs and

\int a^{- 1} d a = ln (a)

. A first 0-dimensional point appeared in the

(- 1)

-dimensional void, inducing the appearance of other points that were indistinguishable from the first. The breach made by the first operation of the dimensional successor function of the Peano axioms inevitably continued leading to the formation of 1-dimensional, real and imaginary lines, allowing for an ordering of points using multipliers of real units (ones) or imaginary units (

a \in R \Leftrightarrow a = 1 b

, and

a \in I \Leftrightarrow a = i b

, where

b \in R

). Then, out of the two lines of each kind, crossing each other only at one initial point

(0, 0)

, the dimensional successor function formed 2-dimensional

R^{2}

I^{2}

, and

R \times I

Euclidean planes, with

I^{2}

being a mirror reflection of

R^{2}

. Thus, forming n-dimensional Euclidean spaces

R^{a} \times I^{b}

with

a \in N

real and

b \in N

imaginary lines,

n = a + i b

, and the scalar product defined by

x \cdot y = (x_{1}, \dots, x_{a}, i x_{1}^{'}, \dots, i x_{b}^{'}) (y_{1}, \dots, y_{a}, i y_{1}^{'}, \dots, i y_{b}^{'}) : = \sum_{k = 1}^{a} x_{k} y_{k} + \sum_{l = 1}^{b} x_{l}^{'} \bar{y_{l}^{'}},

(1)

where

x, y \in R^{a} \times I^{b}

. With the appearance of the first 0-dimensional point, information has begun to evolve [1,2,3,4,5,6,7,8,9], initially using undirected exploration in a selectionless [8] and a timeless [9] assembly process.

Mathematics is theoretically infinite, so determining why some math corresponds to our observable universe and the rest does not is a highly nontrivial problem [10]. Fortunately, the mathematical properties of particular dimensions are not the same. For regular convex n-polytopes, for example, there are countably infinitely many regular convex polygons, five regular convex polyhedra (Platonic solids), six regular convex 4-polytopes and only three regular convex n-polytopes if

| n | > 4

and

| n | < 0

[11]. In particular, a 4-dimensional Euclidean space is endowed with a peculiar property known as exotic

R^{4}

[12], absent in other dimensionalities. Owing to this property,

R^{3} \times I

Euclidean space provides a continuum of homeomorphic but non-diffeomorphic differentiable structures and this is necessary for biological evolution, as it allows the variation of phenotypic traits within populations of individuals [13]: each piece of individually memorized information is homeomorphic to the corresponding piece of individually perceived information but remains non-diffeomorphic (non-smooth). Hence, selection [8] and time [9] emerged and the evolution of information had to inevitably exploit directed exploration provided by biological evolution. Exotic

R^{4}

solves the problem of extra dimensions of nature, and perception requires a natural number of (thus independent) dimensions to form perceived space [14]. Each biological cell and each biological agent perceives an emergent space of three real dimensions and one imaginary (time) observer-dependently [15] and at present, when

i 0 = 0

is real, through a spherical Planck triangle corresponding to one bit of information in units of

- c^{2}

, where c is the speed of light in vacuum. This is the principle of emergent dimensionality (ED) [5,6,9,13,16,17].

Human perception involves measuring and measuring requires measurement units. In 1899 Max Planck derived the natural units of measure as "independent of special bodies or substances, thereby necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and nonhuman ones" [18]. Planck units utilize the Planck constant h introduced in his black-body radiation formula. Earlier, in 1881, George Stoney derived a system of natural units [19] based on the elementary charge e (Planck’s constant was unknown at this time). The ratio of Stoney units to Planck units is

\sqrt{α}

, where

α

is the fine-structure constant. This study derives three complementary sets of Planck units based on the three complementary fine-structure constants established by the Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene (MLG) and hints at certain areas of their applicability. We chose Planck units over other natural unit systems not only because they incorporate the fine-structure constant

α

and the Planck constant h. Other systems of natural units (except for Stoney units) also incorporate them. This is because only the Planck area defines one bit of information on a patternless black hole surface given by the Bekenstein bound (52) and the binary entropy variation [5,6].

The paper is organized as follows. Section 2 shows that the Fresnel coefficients for the normal incidence of electromagnetic radiation on MLG include three complementary fine-structure constants, including two negative ones and hinting that the negative complementary fine-structure constant

α_{2}

established by the graphene reflectance is dual to the fine-structure constant

α

. Section 3 presents complementary sets of

α_{*}

-Planck units established by these three fine-structure constants. Section 4 introduces the concept of a black-body object in thermodynamic equilibrium that emits perfect black-body radiation and reviews its necessary properties. Section 5 introduces complex energies expressed in terms of real and imaginary

α_{-}

-Planck units and applies them to black-body objects to show that in this model

α

and

α_{2}

are indeed dual to each other. Section 6 defines complex forces based on the products of the complex energies. The complex force between real masses and imaginary charges is applied in Section 7 to the Bohr model of the hydrogen atom and in Section 8 to derive a black-body object surface gravity and the generalized Hawking radiation temperature. Section 9 considers the observed mergers of black-body objects to show that the observed data can be explained without the need to introduce hypothetical exotic stellar objects. Section 10 discusses fluctuations of black-body objects. Section 11 summarizes the findings of this study.

2. Three Complementary Fine-Structure Constants

Numerous publications provide Fresnel coefficients for the normal incidence of electromagnetic radiation (EMR) on monolayer graphene (MLG), which are remarkably defined only by

π

and the fine-structure constant

α

having the inverse

α^{- 1} = {(\frac{q_{P}}{e})}^{2} = \frac{4 π ϵ_{0} ℏ c}{e^{2}} \approx 137.036,

(2)

where

q_{P}

is the Planck charge, ℏ is the reduced Planck constant,

ϵ_{0} \approx 8.8542 \times 10^{- 12} [{kg}^{- 1} \cdot m^{- 3} \cdot s^{2} \cdot C^{2}]

is the vacuum permittivity (electric constant), and e is the elementary charge. We choose this set of units over [F

\cdot m^{- 1}

] for

ϵ_{0}

, since the mass, length, time, and charge units can express all the electrical units and together with temperature, amount of substance, and luminous intensity, these are the base units of the International System of Quantities (ISQ). Furthermore, in this notation we see that

ϵ_{0}

is dependent on the unit of time. Transmittance (T) of MLG

T = \frac{1}{{(1 + \frac{π α}{2})}^{2}} \approx 0.9775,

(3)

for normal EMR incidence was derived from the Fresnel equation in the thin-film limit [20] (Eq. 3), whereas a spectrally flat absorptance (A)

A \approx π α \approx 2.3 %

has been reported [21,22] for photon energies between approximately

0.8011 \times 10^{- 19}

and

4.0054 \times 10^{- 19}

[J]. T is related to reflectance (R) [23] (Eq. 53) as

R = π^{2} α^{2} T / 4

, i.e,

R = \frac{\frac{1}{4} π^{2} α^{2}}{{(1 + \frac{π α}{2})}^{2}} \approx 1.2843 \times 10^{- 4},

(4)

The above equations for T and R, as well as the equation for the absorptance

A = \frac{π α}{{(1 + \frac{π α}{2})}^{2}} \approx 0.0224,

(5)

were also derived [24] (Eqs. 29-31) based on the thin-film model (setting

n_{s} = 1

for the substrate). The sum of the transmittance (3) and reflectance (4) at normal EMR incidence on the MLG was derived [25] (Eq. 4a) as

T + R = 1 - \frac{4 σ η}{4 + 4 σ η + σ^{2} η^{2} + k^{2} χ^{2}} = \frac{1 + \frac{1}{4} π^{2} α^{2}}{{(1 + \frac{π α}{2})}^{2}} \approx 0.9776,

(6)

where

η \approx 376.73

[

kg \cdot m^{2} \cdot s \cdot C^{- 2}

] is the vacuum impedance,

σ = e^{2} / (4 ℏ) = π α / η \approx 6.0853 \times 10^{- 5}

[

{kg}^{- 1} \cdot m^{- 2} \cdot s^{- 1} \cdot C^{2}

] is the MLG conductivity [26], k is the wave vector of light in vacuum, and

χ = 0

is the electric susceptibility of vacuum. Therefore, these coefficients are well established both theoretically and experimentally [20,21,22,25,27,28].

As a consequence of the conservation of energy

R + (T + A) = 1 .

(7)

In other words, the transmittance in the Fresnel equation describing the reflection and transmission of EMR at normal incidence on a boundary between different optical media is, in the case of the 2-dimensional (boundary) MLG, modified to include its absorption (cf. Appendix C). Furthermore,

\sqrt{R} + \sqrt{T} = 1

and

A = 2 \sqrt{R T}

. Notably for Fresnel coefficients (A12) (cf. Appendix C),

\sqrt{R_{F}} + \sqrt{T_{F}} = 1

solves to

n_{1} = n_{2}

which implies no refraction. We note that each conservation law is associated with a certain symmetry, as asserted by Noether’s theorem. In this case, the symmetry involves the fine-structure constant

α

and

π

2.1. Reflectance

The reflectance

R = 0.013 %

(4) of the MLG can be expressed as a quadratic equation of

α

R {(1 + \frac{π α}{2})}^{2} - \frac{1}{4} π^{2} α^{2} = 0, \frac{1}{4} (R - 1) π^{2} α^{2} + R π α + R = 0,

(8)

which can be expressed in terms of the reciprocal of

α

, defining

β 1 / α

R β^{2} + R π β + \frac{1}{4} (R - 1) π^{2} = R {(β + \frac{π}{2})}^{2} - \frac{π^{2}}{4} = 0 .

(9)

The quadratic equation (9) has two roots

β = α^{- 1} = \frac{- π R + π \sqrt{R}}{2 R} \approx 137.036, and

(10)

β_{2} = α_{2}^{- 1} = \frac{- π R - π \sqrt{R}}{2 R} \approx - 140.178 .

(11)

Therefore, equation (8) includes the second negative fine-structure constant

α_{2}

. It turns out that the sum of the reciprocals of these fine-structure constants (10) and (11)

\begin{matrix} α^{- 1} + α_{2}^{- 1} & = \frac{- π R + π \sqrt{R} - π R - π \sqrt{R}}{2 R} = \frac{- 2 π}{2} = - π, \forall R \neq 0, \end{matrix}

(12)

is a transcendental number independent of the value of the reflectance R. Furthermore, the minimum of parabola (9) amounts

- π^{2} / 4 \approx - 2.4674

and occurs at

- π / 2 \approx - 1.5708

, as shown in Figure 1. Also, these values are independent of the reflectance.

These results are also intriguing in the context of a peculiar expression of the fine-structure constant [29] as a transcendental number

α^{- 1} = 4 π^{3} + π^{2} + π \approx 137.036303776

(13)

that contains a free

π

term and is very close to the physical definition (2) of

α^{- 1}

, which according to the CODATA 2022 value is

137.035999177

. We note that CODATA values are computed by averaging the measurements.

Using equations (12) and (13), we can express the negative reciprocal of the 2nd fine-structure constant

α_{2}^{- 1}

that emerged in the quadratic equation (8) also as a function of

π

only:

α_{2}^{- 1} = - π - α_{1}^{- 1} = - 4 π^{3} - π^{2} - 2 π \approx - 140.177896429 .

(14)

2.2. Transmittance and Absorptance

The transmittance (3) of MLG can also be expressed as a quadratic equation of

α

\frac{1}{4} T π^{2} α^{2} + T π α + (T - 1) = 0,

(15)

having two roots

α = \frac{- T + \sqrt{T}}{\frac{1}{2} π T} \approx {(137.036)}^{- 1} and α_{3} = \frac{- T - \sqrt{T}}{\frac{1}{2} π T} \approx {(- 0.7809)}^{- 1} \approx - 1.2805,

(16)

where

α_{3}

is the third negative fine-structure constant. Their sum hints taxicab geometry (where

π = 4

)

α + α_{3} = - \frac{4}{π} \forall T \neq 0,

(17)

and is also independent of the value of the transmittance T. The third complementary fine-structure constant expressed by

π

using relations (13) and (17) is

α_{3} = - \frac{16 π^{2} + 4 π + 5}{4 π^{3} + π^{2} + π} .

(18)

Finally, the absorptance (5) of MLG can be expressed as a quadratic equation of

α

\frac{1}{4} A π^{2} α^{2} + (A - 1) π α + A = 0,

(19)

having two roots

α = \frac{2 ((1 - A) - \sqrt{1 - 2 A})}{π A} \approx {(137.036)}^{- 1} and α_{4} = \frac{2 ((1 - A) + \sqrt{1 - 2 A})}{π A} \approx {(0.0180)}^{- 1} \approx 55.5387,

(20)

where

α_{4}

is the fourth positive fine-structure constant. Their product

α α_{4} = \frac{4}{π^{2}} \forall A \neq 0,

(21)

is also a transcendental number independent of the value of the reflectance A. The minimum of parabola (19) amounts

(2 A - 1) / A \approx - 42.6257

and occurs at

2 (1 - A) / (π A) \approx 27.7730

. The fourth complementary fine-structure constant expressed by

π

using relations (13) and (21) is

α_{4} = \frac{4}{π^{2}} (4 π^{3} + π^{2} + π) .

(22)

2.3. Summary

The form of the relations (13), (14) is the same (contains the same like-terms with respect to

π

) as the relation (126) describing the information capacity of a black body object after absorption or emission of a particle. Furthermore, the relation (12) corresponds to the following identity

\frac{α + α_{2}}{α α_{2}} = - π,

(23)

between the roots (10) and (11), which is also derivable from the Fresnel equations and the corresponding Euclid formula (cf. Appendix C). Because the fine-structure constants

α_{*}

are expressible by

π

only, we conjecture that they do not vary over time. They could not vary in the first undirected [8] and timeless [9] part of the evolution of information in the universe. Also, combining relations (23) with (17) and (21) yield

\frac{α + α_{2}}{α α_{2}} = \frac{α + α_{3}}{α α_{4}} = - π .

(24)

Furthermore, we can interpret

β_{*}

as a phase of a complex number, as shown in Figure 2. Using relations (13) and (14) along with angle sum and difference trigonometric identities we obtain

\begin{matrix} z_{1} & = e^{i / α} = cos (4 π^{3} + π^{2} + π) + i sin (4 π^{3} + π^{2} + π) = - cos (4 π^{3} + π^{2}) - i sin (4 π^{3} + π^{2}) \approx 0.3682 - 0.9298 i, \\ z_{2} & = e^{i / α_{2}} = cos (- 4 π^{3} - π^{2} - 2 π) + i sin (- 4 π^{3} - π^{2} - 2 π) = cos (4 π^{3} + π^{2}) - i sin (4 π^{3} + π^{2}) \approx - 0.3682 - 0.9298 i, \\ z_{3} & = e^{i / α_{3}} \approx 0.7103 - 0.7039 i, \\ z_{4} & = e^{i / α_{4}} \approx 0.9998 + 0.0180 i . \end{matrix}

(25)

Finally,

α_{3} < α_{2} < 0 < α < α_{4}

. Therefore, we conclude that only

α

and

α_{2}

are dual to each other and we introduce

α_{★}

for

α

α_{2}

Using relations (13), (14), (18), and (22) the MLG coefficients (3)-(5) can be expressed simply by

π

(cf. Appendix A) and introduce three pairs of

π

-like constants for two surfaces with positive and negative Gaussian curvatures (cf. Appendix B). Interestingly,

T (α_{4}) = R (α)

and

R (α_{4}) = T (α)

In the following section, we derive the complementary sets of Planck units corresponding to the complementary fine-structure constants

α_{*}

. We shall use the subscript "*" as a placeholder for all four fine-structure constants, and the subscripts "+", "−" to describe respectively positive

α

α_{4}

and negative

α_{2}

α_{3}

and quantities associated with them. Occasionally we use the subscript 1 for quantities associated with

α_{1} α

3. Complementary Sets of Planck Units

Natural units can be derived from numerous starting points [6,30] (cf. Appendices Appendix E). The central assumption in all systems of natural units is that the quotient of the unit of length

ℓ_{*}

and time

t_{*}

is a unit of speed; we call it

c = ℓ_{*} / t_{*}

. It is the speed of light in vacuum c in all systems of natural units, except for Hartree and Schrödinger units, where it is

c α

, and Rydberg units, where it is

c α / 2

. Hence,

α

is coupled with c in Hartree, Schrödinger, and Rydberg measurement units.

The fine-structure constant can be defined as the quotient (2) of the squared (and thus positive) elementary charge e and the squared Planck charge

α = e^{2} / q_{P}^{2}

. To accommodate the negativity of the fine-structure constants

α_{-}

discovered in the preceding section, we must introduce the imaginary Planck charge such that its square would yield a negative value.

For example, in the case of

α_{2}

q_{P}^{2} = \frac{e^{2}}{α} \neq q_{P 2}^{2} = \frac{e^{2}}{α_{2}} \Rightarrow q_{P 2} = \pm a e, a = \frac{1}{\sqrt{α_{2}}} \in I .

(26)

Therefore

e^{2} = q_{P *}^{2} α_{*} .

(27)

We note that an imaginary

q_{P -}

, which must have a physical definition analogous to

q_{P}

, requires either a real, negative speed of light or some complementary real, negative electric constant (h is positive). Let us call them

c_{-}

and

\tilde{ϵ_{0}}

q_{P}^{2} = 4 π ϵ_{0} ℏ c > 0, q_{P -}^{2} = 4 π \tilde{ϵ_{0}} ℏ c_{-} < 0 .

(28)

From this equation, we find that

\tilde{ϵ_{0}} c_{-} < 0

, because the values of the other constants are known.

We assume the universality of the Planck constant h and the gravitational constant G. However, this assumption may be too far-reaching, given that we do not need to know the gravitational constant G or Planck constant h to find the product of the Planck length and c,

ℓ_{P} c = \sqrt{ℏ G / c}

[31], i.e., we know how to express three physical constants

{ℏ, G, c}

by two

{ℓ_{P}, c}

only. This resembles a qubit, which in general requires three real numbers to be described, whereas if it encodes only one of its states, then just two real numbers suffice.

The fine-structure constant can also be defined as the ratio of Coulomb’s law for two elementary charges to Newton’s law of gravity for two Planck masses separated by the same distance. Constructing the same ratio for the remaining

α_{*}

leads to

\frac{1}{4 π R^{2}} \frac{e^{2}}{ϵ_{0}} = α_{*} G \frac{m_{P *}^{2}}{R^{2}},

(29)

where the area of a 3-ball (

4 π R^{2}

) in the denominator of the Coulomb force requires further investigation. Hence, the quotient of the squared Planck charge and mass must be the same for all sets of Planck units. Therefore

\frac{q_{P *}^{2}}{m_{P *}^{2}} = 4 π ϵ_{0} G,

(30)

is independent of the unit of time (even though

ϵ_{0}

and G are) and introduces the imaginary Planck masses

m_{P -}

. The masses

m_{P *}

can be calculated from equation (30) by determining the value of the Planck charge

q_{P *}

from equation (27). From (30) we also conclude that

\tilde{ϵ_{0}} = ϵ_{0} > 0

and then by (31) that

μ_{2} > 0

and

c_{2} < 0

. Next, we assume that the solution of Maxwell’s equations in vacuum is valid for other values of the constants involved. Let us call the unknown magnetic constant, corresponding to

c_{*}

μ_{*}

. Therefore,

μ_{0} ϵ_{0} c^{2} = μ_{*} ϵ_{0} c_{*}^{2} = 1,

(31)

from which the bivalued

c_{*} = \pm 1 / \sqrt{μ_{*} ϵ_{0}}

can be obtained. Unlike the electric constant

ϵ_{0}

, magnetic constants

μ_{*}

are independent of the unit of time. Furthermore, Maxwell’s equations in vacuum are not directly dependent on the fine-structure constant(s), which is included in the magnetic constant

μ_{0}

. Finally, combining relations (27) and (28) yields

e^{2} = 4 π ϵ_{0} ℏ c_{*} α_{*},

(32)

which establishes the universality of the elementary charge e that defines both matter and antimatter and leads to the following important relation between the speeds of light in vacuum

c_{*}

, and the fine-structure constants

α_{*}

c α = c_{2} α_{2} = c_{3} α_{3} = c_{4} α_{4} = \frac{e^{2}}{4 π ϵ_{0} ℏ} .

(33)

c α

is also the velocity of the electron in the first circular orbit in the Bohr hydrogen atom model to which we shall return in Section 7. Because each

c_{*}

derivable from the final Maxwell equation (31) is bivalued, all sets of

α_{*}

-Planck units have four forms equal in modulus: real positive, real negative, imaginary positive and imaginary negative. However, due to the relation (33), we consider mostly real, positive

α_{+}

-Planck units and imaginary, positive

α_{-}

-Planck units. We note that switching the signs of

c_{*}

in the relation (33) would require the imaginary elementary charge

e_{i} = i e

. Therefore, we consider real Planck units associated with the positive speed of light

c_{2 +}

associated with

α_{2}

in the Appendix D only.

Complementary speeds of light

c_{*}

(33) introduce complementary sets of Planck units, wherein basic

α_{-}

-Planck units are imaginary since they are defined as square roots containing

c_{-}

raised to odd powers (1, 3, 5). Furthermore, the speed of electromagnetic radiation is the product of its wavelength and frequency, and these quantities would be imaginary in terms of imaginary Planck units; a negative speed of light is necessary to accommodate this.

α_{2}

-Planck units that can be expressed, using the relation (33), in terms of base Planck units

q_{P}

ℓ_{P}

m_{P}

t_{P}

, and

T_{P}

are

q_{P 2} = \sqrt{4 π ϵ_{0} ℏ c_{2}} = q_{P} \sqrt{\frac{α}{α_{2}}} \approx i 1.8969 \times 10^{- 18} [C] (| q_{P 2} | > | q_{P} |),

(34)

ℓ_{P 2} = \sqrt{\frac{ℏ G}{c_{2}^{3}}} = ℓ_{P} \sqrt{\frac{α_{2}^{3}}{α^{3}}} \approx i 1.5622 \times 10^{- 35} [m] (| ℓ_{P 2} | < | ℓ_{P} |),

(35)

m_{P 2} = \sqrt{\frac{ℏ c_{2}}{G}} = m_{P} \sqrt{\frac{α}{α_{2}}} \approx i 2.2012 \times 10^{- 8} [kg] (| m_{P 2} | > | m_{P} |),

(36)

t_{P 2} = \sqrt{\frac{ℏ G}{c_{2}^{5}}} = t_{P} \sqrt{\frac{α_{2}^{5}}{α^{5}}} \approx i 5.0942 \times 10^{- 44} [s] (| t_{P 2} | < | t_{P} |),

(37)

T_{P 2} = \sqrt{\frac{ℏ c_{2}^{5}}{G k_{B}^{2}}} = T_{P} \sqrt{\frac{α^{5}}{α_{2}^{5}}} \approx i 1.4994 \times 10^{32} [K] (| T_{P 2} | > | T_{P} |) .

(38)

Most Planck units derived from the

α_{2}

-Planck base units (34)-(38) are also imaginary. These include the

α_{2}

Planck volume

ℓ_{P 2}^{3} = {(\frac{ℏ G}{c_{2}^{3}})}^{3 / 2} = ℓ_{P}^{3} \sqrt{\frac{α_{2}^{9}}{α^{9}}} \approx i 3.8127 \times 10^{- 105} [m^{3}] (| ℓ_{P 2}^{3} | < | ℓ_{P}^{3} |),

(39)

the

α_{2}

Planck momentum

p_{P 2} = m_{P 2} c_{2} = \sqrt{\frac{ℏ c_{2}^{3}}{G}} = m_{P} c \sqrt{\frac{α^{3}}{α_{2}^{3}}} \approx i 6.7506 [kg m / s] (| m_{P 2} c_{2} | > | m_{P} c |),

(40)

the

α_{2}

Planck energy

\begin{matrix} E_{P 2} = m_{P 2} c_{2}^{2} = \sqrt{\frac{ℏ c_{2}^{5}}{G}} = E_{P} \sqrt{\frac{α^{5}}{α_{2}^{5}}} \approx i 2.0701 \times 10^{9} [J] (| E_{P 2} | > | E_{P} |), \end{matrix}

(41)

and the

α_{2}

Planck acceleration

a_{P 2} = \frac{c_{2}}{t_{P 2}} = \sqrt{\frac{c_{2}^{7}}{ℏ G}} = a_{P} \sqrt{\frac{α^{7}}{α_{2}^{7}}} \approx \pm i 6.0198 \times 10^{51} [m / s^{2}] (| a_{P 2} | > | a_{P} |) .

(42)

However, the

α_{2}

-Planck density

ρ_{P 2} = \frac{m_{P 2}}{ℓ_{P 2}^{3}} = \frac{c_{2}^{5}}{ℏ G^{2}} = ρ_{P} \frac{α^{5}}{α_{2}^{5}} \approx - 5.7735 \times 10^{96} [kg / m^{3}] (| ρ_{P 2} | > | ρ_{P} |),

(43)

and the

α_{2}

-Planck area

ℓ_{P 2}^{2} = \frac{ℏ G}{c_{2}^{3}} = ℓ_{P}^{2} \frac{α_{2}^{3}}{α^{3}} \approx - 2.4406 \times 10^{- 70} [m^{2}] (| ℓ_{P 2}^{2} | < | ℓ_{P}^{2} |),

(44)

are real and negative. Interestingly, both Planck forces

F_{P}

and

F_{P 2} = \frac{c_{2}^{4}}{G} = \frac{c^{4}}{G} \frac{α^{4}}{α_{2}^{4}} = F_{P} \frac{α^{4}}{α_{2}^{4}} \approx 1.3251 \times 10^{44} [N] (F_{P 2} > F_{P}),

(45)

are strictly positive. The remaining sets of Planck units are listed in Table 1.

Contrary to the elementary charge e (27), there is no physically meaningful elementary mass

M_{e} = \pm 1.8592 \times 10^{- 9} [kg]

satisfying the relation (36)

M_{e}^{2} = α_{*} m_{P *}^{2} .

(46)

There is no physically meaningful elementary length

L_{e} \approx \pm 2.5927 \times 10^{- 32} [m]

satisfying the relation (35)

L_{e}^{2} = \frac{ℓ_{P *}^{2}}{α_{*}^{3}},

(47)

or an elementary temperature

T_{e} \approx \pm 6.4450 \times 10^{26} [K]

satisfying the relation (38)

T_{e}^{2} = α_{*}^{5} T_{P *}^{2},

(48)

and close to the Hagedorn temperature of grand unified string models, or an elementary period

t_{e} = \pm 1.1851 \times 10^{- 38}

satisfying the relation (37)

t_{e}^{2} = \frac{t_{P *}^{2}}{α_{*}^{5}} .

(49)

However, the relations between periods (49) and temperatures (48) are inverted. Hence, the energy-time version of Heisenberg’s uncertainty principle (HUP) is saturated using energy from the equipartition theorem for one bit of information [5,6,32] both by Planck temperatures and times and elementary temperatures and periods (49), (48)

\frac{1}{2} k_{B} T_{P *} t_{P *} = \frac{1}{2} k_{B} T_{e} t_{e} = \frac{ℏ}{2} .

(50)

The Planck charge relation (27) and charge conservation principle imply that the elementary charge e, unlike mass

M_{e}

is the same in all systems of Planck units, even though the same forms of the relations (27) and (46) reflect the same forms of Coulomb’s law and Newton’s law of gravity, which are inverse-square laws.

Condensed-matter systems having two pairs of sublattices located at half-translation positions and related by multiple glide-reflection symmetries, such as palladium diselenide, make their relative quantum phases polarized into only four kinds, three of which become dark due to double destructive interference [33]. This is akin to two qubits with one fixed quantum phase. The dark quantum state does not absorb or emit photons and, therefore, is undetectable by spectroscopic means. Considering this result and the equality of the elementary charge e in all systems of Planck units that we derived based on a condensed-matter system having only one pair of sublattices, and thus analogous to only one qubit, we note the elementary charge e in these three complementary systems defines the dark electron of these systems. Thus

(0) 000

(blue) dark state disclosed in [33] corresponds to

α

-electron,

(0) 0 π π

(red) state corresponds to

α_{2}

-electron, and

(0) π π 0

(green) and

(0) π 0 π

(yellow) states correspond to

α_{3}

- and

α_{4}

-electrons.

In the following, where deemed appropriate, we express the physical quantities in Planck units:

\begin{matrix} M_{+} : = m_{+} m_{P +}, & M_{-} : = m_{-} m_{P -}, m_{+}, m_{-} \in R \\ E_{+} : = m_{+} E_{P +} & E_{-} : = m_{-} E_{P -}, \\ Q_{+} : = q e, & Q_{-} : = i Q_{+} = i q e, q \in Z, \\ λ_{+} : = l ℓ_{P}, & λ_{-} : = l_{-} ℓ_{P -}, l_{+} = \frac{2 π}{m_{+}}, l_{-} = \frac{2 π}{m_{-}}, \\ {R_{+}, D_{+}} : = {r_{+}, d_{+}} ℓ_{P +}, & {R_{-}, D_{-}} : = {r_{i}, d_{i}} ℓ_{P -}, r_{+}, d_{+}, r_{-}, d_{-} \in R, \end{matrix}

(51)

where uppercase letters M, E, Q,

λ

, R, and D denote masses, energies, charges, Compton wavelengths, radii, and diameters (or lengths), lowercase letters m, l, etc. denote multipliers of the real

α_{+}

and imaginary

α_{-}

Planck units, respectively, and q is an integer multiplier of the elementary charge e. The latter assumption is most likely too far-reaching, considering the fractional charges of quasiparticles, particularly in the open research problem of the fractional quantum Hall effect (cf. Appendix G) and energy-dependent fractional charges in electron pairing [34].

4. Black Body Objects

There are only three observable objects in nature that emit perfect black-body radiation: unsupported black holes (BHs, the densest), neutron stars (NSs), supported, as believed, by neutron degeneracy pressure, and white dwarfs (WDs), supported, as believed, by electron degeneracy pressure (the least dense). We collectively refer to these black-body objects (BBs). The spectral density in sonoluminescence, that is light emission by sound-induced collapsing gas bubbles in fluids, was also shown to have the same frequency dependence as black-body radiation [35,36]. Thus, sonoluminescence, particularly shrimpoluminescence [37], is probably emitted by collapsing micro-BBs. Micro-BH induced in glycerin by modulating acoustic waves has also been reported [38].

The term black-body object is not used in general relativity (GR) and standard cosmology, but standard cosmology scrunches under embarrassingly significant failings, not just tensions as is sometimes described, as if to somehow imply that a resolution will eventually be found [39]. In addition, James Webb Space Telescope data show multiple galaxies that grew too massive too soon after the Big Bang, which is a strong discrepancy with the

Λ

cold dark matter model (

Λ

CDM) expectations of how galaxies formed at early times at both redshifts, even when considering observational uncertainties [40]. For example, the supermassive BH of J1120+0641 quasar with mass

M_{BH} = 1.52 \pm 0.17 \times 10^{9} M_{⊙}

assembled in less than 0.77 billion years after the Big Bang [41,42]. This is an important unresolved issue, indicating that fundamental changes to the reigning

Λ

CDM model of cosmology are required [40]. In particular, it is well known that entropic gravity [32] explains the galaxy rotation curves without resorting to dark matter (dark matter is not required to explain the rotation curves of certain galaxies, such as the massive relic galaxy NGC 1277 [43]), has been experimentally confirmed [44], and is decoherence-free [45].

The term object as a collection of matter is a misnomer because it neglects the (quantum) nonlocality [9,46] that is independent of the entanglement among particles [47], as well as the Kochen-Specker contextuality [48], and increases as the number of particles increases [49,50]. Macro-realistic theories are false [51]. Thus, we use emphasis for (perceivably indistinguishable) particle and (perceivably distinguishable) object, as well as matter and distance. The ugly duckling theorem [52,53] asserts that every two objects we perceive are equally similar (or equally dissimilar). These terms do not have an absolute meaning in the ED. In particular, given the observation of quasiparticles in classical systems [54]. Within the ED framework, no object is enclosed in space. The interiors of the BBs are inaccessible to an exterior observer [55], which makes them similar to interior-less mathematical points representing real numbers on a number line. Thus, the term object is a particularly staring misnomer if applied to BBs.

It has been experimentally confirmed that (so-called) accretion instability is a fundamental physical process [56]. We conjecture that this process, which has already been recreated under laboratory conditions [57], is common for all BBs. As black-body radiation is a radiation of global thermodynamic equilibrium, it is patternless [58] (thermal noise) radiation that depends only on one parameter. In the case of BHs, this is known as Hawking [59] radiation, and this parameter is the BH temperature

T_{BH} = T_{P} / (2 π d_{BH})

corresponding to the BH diameter [6]

D_{BH} = d_{BH} ℓ_{P}

. Furthermore, BHs absorb patternless information [6,60]. Therefore, because Hawking radiation depends only on the diameter of a BH, it is the same for a given BH, even though it fluctuates (cf. Section 10).

As black-body radiation is patternless, triangulated [6] BBs contain a balanced number of Planck area triangles, each having binary potential

δ φ_{k} = - c^{2} \cdot {0, 1}

, as has been shown for BHs [6], based on the Bekenstein-Hawking (BH) entropy [55]

S_{BH} = k_{B} N_{BH} / 4

, where

N_{BH} 4 π R_{BH}^{2} / ℓ_{P}^{2} = π d_{BH}^{2}

is the information capacity of the BH surface, i.e., the

⌊N_{BH}⌋ \in N_{0}

Planck triangles corresponding to bits of information [5,6,32,55,61], and the fractional part triangle(s) having the area

{N_{BH}} ℓ_{P}^{2} = (N_{BH} - ⌊N_{BH}⌋) ℓ_{P}^{2}

too small to carry a single bit of information [5,6], where "

⌊x⌋

" is the floor function that yields the greatest integer less than or equal to its argument x.

BH entropy can be derived from the Bekenstein bound

S \leq \frac{2 π k_{B} R E}{ℏ c} = π k_{B} m_{1} d_{1},

(52)

which defines an upper limit on the thermodynamic entropy S that can be contained within a sphere of radius R and energy E. Substituting BH (Schwarzschild) radius

R_{BH} = 2 G M_{BH} / c^{2}

and mass-energy equivalence

E_{BH} = M_{BH} c^{2}

, where

M_{BH}

is the BH mass, into the bound (52), it reduces to the BH entropy. In other words, the BH entropy saturates the Bekenstein bound (52). Furthermore, the Bekenstein bound can be derived from the BH entropy

S_{BH} = \frac{1}{4} k_{B} N_{BH} = \frac{1}{4} k_{B} \frac{4 π R^{2}}{ℓ_{P}^{2}} = k_{B} \frac{π R R}{ℓ_{P}^{2}} \leq k_{B} π R \frac{2 G E}{c^{4}} \frac{c^{3}}{ℏ G} = \frac{2 π k_{B} R E}{ℏ c},

(53)

where we used

M \leq \frac{R c^{2}}{2 G}

and

E = M c^{2}

The patternless nature of perfect black-body radiation was derived [6] by comparing the BH entropy with the binary entropy variation

δ S = k_{B} N_{1} / 2

([6] Eq. (55)), which is valid for any HS, where

N_{1} \in N

denotes the number of active Planck triangles with a binary potential

δ φ_{k} = - c^{2}

. Thus, the entropy of all BBs is

S_{BB} = \frac{1}{4} k_{B} N_{BB} .

(54)

Furthermore,

N_{1} = N_{BB} / 2

confirms the patternless thermodynamic equilibrium of BBs by maximizing Shannon entropy [6]. In complex Euclidean

R^{a} \times I^{b}

space, a n-ball (

n = a + b i \in C

) is spherical only for

b = 0

, i.e., when perceived. Not only BBs are perfectly spherical when perceived; their mergers (cf. Section 9) are also perfectly spherical. Furthermore, the trigonometric member of its volume and surface formulas vanishes for the radius

r = 1 / \sqrt{π}

[5,17]. This is an important result since

r = 1 / \sqrt{π}

is a dimensionless mathematical value like

π

α_{*}

, while

R = r ℓ_{P} = ℓ_{P} / \sqrt{π}

, providing the information capacity

N = 4

and hence one unit of BB entropy (54) is a physical one. Anyway, this corroborates the universality and applicability of BH entropy (54) to all BBs. Furthermore, some binary strings of of length

N_{BB} \geq 4

can be assembled in less than

N_{BB} - 1

steps. There is no disorder or uncertainty in a binary string of length

N_{BB} \leq 3

[9].

We shall define the generalized radius of a BB (this definition applies to all HSs) having mass

M_{BB}

as a function of

G M_{BB} / c^{2}

multiplier

k \in R, k \geq 2

\begin{matrix} R_{BB} : = k \frac{G M_{BB}}{c^{2}}, d_{BB} = 2 k m_{BB}, \end{matrix}

(55)

and the generalized BB energy

E_{BB}

(this definition also applies to all HSs) as a function of

M_{BB} c^{2}

multiplier

a \in R

E_{BB} : = a M_{BB} c^{2}, E_{BB} = a m_{BB} E_{P} .

(56)

Substituting

M_{BB}

from the generalized radius definition (55) into the generalized BB energy definition (56) and the latter into the Bekenstein bound (52), we obtain

S \leq \frac{1}{2} k_{B} \frac{a}{k} N_{BB},

(57)

which equals BB entropy (54) if

\frac{a}{2 k} = \frac{1}{4} \Leftrightarrow a = \frac{k}{2}

. Thus, the energy (56) of all BBs with a generalized radius (55) turns into

E_{BB} = \frac{k}{2} M_{BB} c^{2} = \frac{k}{2} m_{BB} E_{P} = \frac{d_{BB}}{4} E_{P},

(58)

with

k = 2

in the case of the BHs, setting the lower bound for the other BBs. We further call coefficient k the size-to-mass ratio (STM). This is similar to the specific volume (reciprocal of density) of BB. We derive the upper STM bound in Section 5. The energy (58) of BBs other than BHs (i.e., for

k > 2

) exceeds the mass-energy equivalence

E = M c^{2}

, which is the limit of the maximum real energy. Therefore, we consider this surplus energy that exceeds

M_{BB} c^{2}

as related to charge and we shall model it as imaginary and thus unmeasurable.

According to the no-hair theorem, all BHs general relativity (GR) solutions are characterized by only three parameters: mass, electric charge, and angular momentum. However, BHs are fundamentally uncharged, because the parameters of any conceivable BH, in particular, charged (Reissner–Nordström) and charged-rotating (Kerr–Newman) BH, can be arbitrarily altered, provided that the BH area does not decrease [62] using Penrose processes [63,64] to extract the electrostatic and/or rotational energy of BH [65]. Thus, any BH is defined by only one real parameter: its diameter, mass, temperature, energy, etc., each differing from the other by a multiplicative constant. A BH can embrace the real number that defines it as a curvature of a spherical triangle corresponding to one bit of classical information. The area of a spherical triangle is larger than that of a flat triangle defined by the same vertices and depends on its curvature.

However, it is accepted that in the case of NSs, electrons combine with protons to form neutrons, such that NSs are composed almost entirely of neutrons. However, it is never the case that all electrons and protons of an NS become neutrons. WDs are charged by definition because they are believed to be mostly composed of electron-degenerate matter. But how can a charged BB store both its curvature corresponding to its mass and an additional parameter corresponding to its charge? Fortunately, the relation (27) ensures that the elementary charge is the same in all systems of Planck units. Therefore, the charge of a spherical Planck triangle of a BB surface can link the perceivable Euclidean space

R_{α}^{3} \times I_{α}

parameterized by Planck units with the parameterization provided by one of the remaining negative fine-structure constants

α_{-}

. This can be considered in a polyspherical coordinate system, in which gravitation/acceleration acts in a radial direction (with the entropic gravitation acting inwardly and acceleration acting in both radial directions) [6], while electrostatic acts in tangential directions. Contrary to the no-hair theorem, we characterize BBs only by mass and charge, neglecting the angular momentum because the latter introduces the notion of time, which we find redundant in the BB description of a patternless thermodynamical equilibrium. Time is required for directed exploration only [8,9].

5. Complex Energies

A complex energy formula

E_{R} : = E_{M_{R}} + i E_{Q_{R}} = M_{R} c^{2} + \frac{i Q_{R}}{2 \sqrt{π ϵ_{0} G}} c^{2},

(59)

where

E_{M_{R}}

and

i E_{Q_{R}}

represent the real and imaginary energies of an object having mass

M_{R}

and charge

Q_{R}

was proposed in ref. [66]. Equation (59) considers the real masses

M_{R}

and charges

Q_{R}

which in ref. [66] are defined in CGS units. We adopted SI units and modified this formula to a form involving real physical expressed in terms of real

α_{+}

-Planck units and imaginary quantities expressed in terms of the imaginary

α_{-}

-Planck units using relations (33), (36), (41), (51), and (32) as

\begin{matrix} \frac{e}{2 \sqrt{π ϵ_{0}}} = \sqrt{α_{+} c_{+} ℏ} = \sqrt{α_{-} c_{-} ℏ} . \end{matrix}

(60)

To this end, we defined the following three complex energies linking mass, imaginary mass, and charge: the complex energy of real mass and imaginary charge

E_{M Q_{i}} : = E_{M_{+}} + E_{Q_{-}} = M_{+} c_{+}^{2} + \frac{Q_{-}}{2 \sqrt{π ϵ_{0} G}} c_{+}^{2} = (m_{+} m_{P +} + i q \sqrt{α_{+}} m_{P +}) c_{+}^{2} = (m_{+} + i q \sqrt{α_{+}}) E_{P +},

(61)

of real charge and imaginary mass

\begin{matrix} E_{Q M_{i}} : = E_{Q_{+}} + E_{M_{-}} = \frac{Q_{+}}{2 \sqrt{π ϵ_{0} G}} c_{-}^{2} + M_{-} c_{-}^{2} = (q \sqrt{α_{-}} m_{P -} + m_{-} m_{P -}) c_{-}^{2} = \frac{α_{+}^{2}}{α_{-}^{2}} (q \sqrt{α_{+}} + \sqrt{\frac{α_{+}}{α_{-}}} m_{-}) E_{P +}, \end{matrix}

(62)

and of real mass and imaginary mass

\begin{matrix} E_{M M_{i}} : = & M_{+} c_{+}^{2} + M_{-} c_{-}^{2} = (m_{+} + \sqrt{\frac{α_{+}^{5}}{α_{-}^{5}}} m_{-}) E_{P +} . \end{matrix}

(63)

We neglected the energy of real and imaginary charges

E_{Q Q_{i}}

, because by equation (27), the elementary charge is the same in all systems of measurement units, and hence we use the same elementary charge multiplier q in (51). Furthermore, the mass-energy equivalence relates the mass

M_{+}

M_{-}

to the speed of light

c_{+}

c_{-}

, which subsequently parameterizes both parts of the energies

E_{M Q_{i}}

and

E_{Q M_{i}}

(cf. Appendix F). We express all energies (61)-(63) in terms of the same Planck energy

E_{P +}

in order to be able to compare them, as we assume that they are fundamental to any object and particle. We also note that the complex conjugates

E_{M Q_{i}}

and

\bar{E_{M Q_{i}}}

of the energy (61) represent respectively particle and antiparticle. We note that antimatter obeys gravity [67], which is consistent with the definition (61) and the findings of this study.

In the remainder of this section we shall analyze the energies (61)-(63) for different pairs of

{α_{+}, α_{-}}

in order

{α_{1}, α_{2}}

{α_{4}, α_{2}}

{α_{1}, α_{3}}

, and

{α_{4}, α_{3}}

. Our aim is to determine which pair is the most plausible physically.

Energies (61) and (62) yield two different charge energies corresponding to the elementary charge, imaginary quantum

E_{Q_{-}} (q = \pm 1) = \pm i \sqrt{α_{+}} E_{P +} \approx \pm {1.6710 \times 10^{8}, 2.8848} i [J],

(64)

and the - larger in modulus - real quantum

E_{Q_{+}} (q = \pm 1) = \pm \sqrt{α_{-}} E_{P -} \approx \pm {1.7684 \times 10^{8}, 5.4265 \times 10^{3}} [J] .

(65)

Furthermore,

\forall q

α_{+}^{2} E_{Q_{-}} = i α_{-}^{2} E_{Q_{+}}

. The universal character of the charges is additionally emphasized by the real

\sqrt{α}

multiplied by i in the imaginary charge energy (61) and imaginary

\sqrt{α_{2}}

in the real charge energy (62).

Theorem 1

The complex energies (61)-(63) cannot be all balanced complex quantities.

Proof.

The complex energies

E_{M Q_{i}}

and

E_{Q M_{i}}

are balanced if their real and imaginary parts are equal in modulus. This holds for

q^{2} α_{+} = m_{+}^{2} = - \frac{α_{+}}{α_{-}} m_{-}^{2} .

(66)

However, if balanced with each other, they cannot be balanced with the energy

E_{M M_{i}}

, which is balanced for

m_{+}^{2} = - \frac{α_{+}^{5}}{α_{-}^{5}} m_{-}^{2} \neq - \frac{α_{+}}{α_{-}} m_{-}^{2},

(67)

defining the energy balance of uncharged masses. Similarly, the complex energies

E_{Q M_{i}}

and

E_{M M_{i}}

can be balanced with each other but not with

E_{M Q_{i}}

since

\frac{α_{+}^{4}}{α_{-}^{4}} q^{2} α_{+} = - \frac{α_{+}^{5}}{α_{-}^{5}} m_{-}^{2} = m_{+}^{2} \neq q^{2} α_{+}

(68)

and likewise for

E_{M Q_{i}}

and

E_{M M_{i}}

. □

The squared moduli of the complex energies (61)-(63), expressed in terms of the Planck energy, are

| E_{M Q_{i}} |^{2} = (M_{+}^{2} + q^{2} α_{+} m_{P +}^{2}) c_{+}^{4} = (m_{+}^{2} + q^{2} α_{+}) E_{P +}^{2},

(69)

| E_{Q M_{i}} |^{2} = \frac{α_{+}^{4}}{α_{-}^{4}} (q^{2} α_{+} m_{P +}^{2} - M_{-}^{2}) c_{+}^{4} = \frac{α_{+}^{4}}{α_{-}^{4}} (q^{2} α_{+} - \frac{α_{+}}{α_{-}} m_{-}^{2}) E_{P +}^{2},

(70)

| E_{M M_{i}} |^{2} = (M_{+}^{2} - \frac{α_{+}^{4}}{α_{-}^{4}} M_{-}^{2}) c_{+}^{4} = (m_{+}^{2} - \frac{α_{+}^{5}}{α_{-}^{5}} m_{-}^{2}) E_{P +}^{2},

(71)

ad shown in Figure 3.

Equalities of the squared moduli of the complex energies lead to the following results

| E_{M Q_{i}} |^{2} = {| E_{Q M_{i}} |}^{2} \Leftrightarrow m_{-} = \pm \sqrt{\frac{α_{-}}{α_{+}} [q^{2} α_{+} (1 - \frac{α_{-}^{4}}{α_{+}^{4}}) - \frac{α_{-}^{4}}{α_{+}^{4}} m_{+}^{2}]},

(72)

| E_{M Q_{i}} |^{2} = {| E_{M M_{i}} |}^{2} \Leftrightarrow m_{-} = \pm q \sqrt{- \frac{α_{-}^{5}}{α_{+}^{4}}} = \pm q {0.0807, 1.3935 \times 10^{- 9}, 3.4846 \times 10^{4}, 6.0157 \times 10^{- 4}},

(73)

| E_{Q M_{i}} |^{2} = {| E_{M M_{i}} |}^{2} \Leftrightarrow m_{+} = \pm q \sqrt{\frac{α_{+}^{5}}{α_{-}^{4}}} = \pm q {0.0894, 4.5170 \times 10^{8}, 2.7741 \times 10^{- 6}, 1.4019 \times 10^{4}},

(74)

Because the mass multiplier

m_{-} \in R

, the square root argument must be nonnegative in equation (72), which leads to (the sign of this inequality changes, as

α_{-} < 0

)

\begin{matrix} m_{+} \geq | q | \sqrt{α_{+} (\frac{α_{+}^{4}}{α_{-}^{4}} - 1)} \approx | q | {0.0263, 4.5170 \times 10^{8}, 0.0854 i, 1.4019 \times 10^{4}}, \end{matrix}

(75)

which is imaginary for

\sqrt{α (α^{4} / α_{3}^{4} - 1)}

ruling out the third pair of alphas

{α_{1}, α_{3}}

. Otherwise, we would have to exclude objects having the same energies

E_{M Q_{i}}

and

E_{Q M_{i}}

from our model. Furthermore, for

q = 0

equation (72) yields

m_{-} = \pm m_{+} \sqrt{- \frac{α_{-}^{5}}{α_{+}^{5}}} \approx \pm m_{+} \{0.9449, 1.8699 \times 10^{- 10}, 4.0791 \times 10^{5}, 8.0722 \times 10^{- 5}\},

(76)

which corresponds to equation (67).

Furthermore, the relation (75) means that the masses of the uncharged micro-BHs (

q = 0

) can be arbitrary (

m \geq 0

). However, micro NSs and micro WDs, which are also in thermodynamic equilibrium, are charged. Thus, even a single elementary charge (

q = 1

) of a white dwarf renders its minimum allowable mass (75)

M_{WD} \geq 5.7275 \times 10^{- 10} [kg]

comparable to the mass of a sand grain.

Theorem 2

Moduli of the complex energies (69)-(71) are equal

\begin{matrix} | E_{M Q_{i}} |^{2} = | E_{Q M_{i}} |^{2} = {| E_{M M_{i}} |}^{2} = (1 + \frac{α_{2}^{4}}{α^{4}}) m^{2} E_{P}^{2} = (1 + \frac{α^{4}}{α_{2}^{4}}) q^{2} α E_{P}^{2} = - (1 + \frac{α_{2}^{4}}{α^{4}}) \frac{α^{9}}{α_{2}^{9}} m_{i}^{2} E_{P}^{2} A^{2} E_{P}^{2}, A \in R, \end{matrix}

(77)

for

q^{2} α = - \frac{α^{5}}{α_{2}^{5}} m_{i}^{2} = \frac{α_{2}^{4}}{α^{4}} m^{2}, m_{i}^{2} = - \frac{α_{2}^{9}}{α^{9}} m^{2} \approx 0.8155 m^{2} .

(78)

Proof.

If the squared moduli (69)-(71) are equal to some constant energy then

q^{2} α

drops out by subtracting

| E_{M Q_{i}} |^{2} - \frac{α_{2}^{4}}{α^{4}} {| E_{Q M_{i}} |}^{2}

, yielding

m^{2} + \frac{α}{α_{2}} m_{i}^{2} = A^{2} (1 - \frac{α_{2}^{4}}{α^{4}}) .

(79)

Subtracting this from

| E_{M M_{i}} |^{2}

yields

m_{i}^{2} = - A^{2} \frac{α_{2}^{9}}{α^{5} (α^{4} + α_{2}^{4})},

(80)

which substituted into the relation (79) yields

m^{2} = A^{2} \frac{α^{4}}{α^{4} + α_{2}^{4}} .

(81)

Finally, substituting

m^{2}

(81) into equation

| E_{M Q_{i}} |^{2}

(69) yields

q^{2} α = A^{2} \frac{α_{2}^{4}}{α^{4} + α_{2}^{4}} .

(82)

□

We can interpret the squared generalized energy of the BBs (58) as the squared modulus of each of the complex energies (69)-(71). That is, for

E_{M Q_{i}}

(61) and for

E_{M M_{i}}

(63), we assume that the real mass energy

E_{BB} = M_{BB} c^{2}

of the BB is the observable real part of this complex energy. For

| E_{M Q_{i}} |^{2}

(69) we have

\frac{k^{2}}{4} m_{BB}^{2} = m_{BB}^{2} + q_{BB}^{2} α_{+} \Leftrightarrow q_{BB} \sqrt{α_{+}} = \pm m_{BB} \sqrt{\frac{k^{2}}{4} - 1},

(83)

where

q_{BB}^{2} α_{+}

represents the charge surplus energy exceeding

M_{BB} c^{2}

. We note that

k = 2

implies

q_{BB} = 0

, confirming the vanishing charge of BHs. For

| E_{M M_{i}} |^{2}

(71) we have

\frac{k^{2}}{4} m_{BB}^{2} = m_{BB}^{2} - \frac{α_{+}^{5}}{α_{-}^{5}} m_{i BB}^{2} \Leftrightarrow m_{i BB} = \pm m_{BB} \sqrt{\frac{α_{-}^{5}}{α_{+}^{5}} (1 - \frac{k^{2}}{4})},

(84)

where the square root argument must be nonnegative, which implies (along with (83)), as expected,

k \geq 2

, where

k = 2

implies

m_{i BB} = 0

. Finally, for

| E_{Q M_{i}} |^{2}

(70) we have

\frac{k^{2}}{4} m_{BB}^{2} = \frac{α_{+}^{4}}{α_{-}^{4}} (q_{BB}^{2} α_{+} - \frac{α_{+}}{α_{-}} m_{i BB}^{2}) \Leftrightarrow m_{i BB} = \pm m_{BB} \sqrt{\frac{α_{-}}{α_{+}} [\frac{k^{2}}{4} (1 - \frac{α_{-}^{4}}{α_{+}^{4}}) - 1]},

(85)

where we substituted

q_{BB}^{2} α_{+}

as a function of

m_{BB}^{2}

from equation (83). Comparison of

m_{i BB}

given by (85) with

m_{i}

given by (72) also leads to

k \geq 2

, and for

k = 2

q = 0

corresponds to the balance of uncharged masses (67), (76) which are unrelated to any assumptions regarding BB energy and independent of STM. Comparing equations (72), as a function of charge q, and (85), as a function of the STM k, leads, as expected, to the relation (83).

Furthermore, the square root argument in equation (85) must be nonnegative, because

m_{BB}, m_{i BB} \in R

. This leads to the maximum STM-bound

k \leq \frac{2}{\sqrt{1 - \frac{α_{-}^{4}}{α_{+}^{4}}}} = k_{\max} \approx {\underset{̲}{6.7933}, 2.0000, 6.4949 \times 10^{- 5} i, 2.0000},

(86)

where again third pair of examined alphas

{α_{1}, α_{3}}

must be ruled out, while pairs

{α_{2}, α_{4}}

and

{α_{3}, α_{4}}

do not allow to extend our model above the STM of BH. The relations (83) and (85) are shown in Figure 4 for

α

and

α_{2}

The maximum STM-bound

k_{\max}

(86) sets the bounds on BB energy (58), mass, and radius (55)

R_{BH} = \frac{2 G M_{BB}}{c^{2}} \leq R_{BB} \leq \frac{k_{\max} G M_{BB}}{c^{2}} .

(87)

In particular, using the relations (51),

2 m_{BB} \leq r_{BB} \leq k_{\max} m_{BB}

r_{BB} / k_{\max} \leq m_{BB} \leq r_{BB} / 2

From the relation (78) we see that uncharged BHs (

q = 0

) cannot remain in a nonvanishing equilibrium of complex energies (77), even though they are in thermodynamic equilibrium. It was shown [5] based on the Mandelstam-Tamm [68], Margolus–Levitin [69], and Levitin-Toffoli [70] theorems on the quantum orthogonalization interval that a BH is represented by a qubit

| ψ_{BH} = \frac{1}{\sqrt{2}} (| 0 + | E_{BH})

in an equal superposition of the eigenstates corresponding to the BH energy

E_{BH} = M_{BH} c^{2}

and vanishing ground state energy. However, such a nonvanishing equilibrium of complex energies (77) is possible for charged BBs. We obtain it by substituting squared generalized energy of the BBs (58) into equation (77) as

A^{2} = \frac{1}{4} k^{2} m_{BB}^{2}

and solving for

(1 + α_{-}^{4} / α_{+}^{4}) m_{BB}^{2}

. This yields

k_{eq} = 2 \sqrt{1 + \frac{α_{-}^{4}}{α_{+}^{4}}} \approx {\underset{̲}{2.7665}, 2.0000, 6.1586 \times 10^{4}, 2.0000}

(88)

where all three energies are equal. The equilibrium

k_{eq}

(88) and maximum

k_{\max}

(86) STMa satisfy

k_{eq}^{2} + 16 / k_{\max}^{2} = 8

(which resolves to

2 \leq | k_{eq} | < 2 \sqrt{2}, | k_{\max} | \geq \sqrt{2}, \forall k_{eq}, k_{\max} \neq 0 \in R

The relations (75), (86), and (88) show that the negative fine-structure constant

α_{-}

corresponding to

α

α_{2}

not

α_{3}

. As we have seen in Section 2.3,

α

and

α_{2}

are dual to each other. The BB having the STM

k_{eq} \approx 2.7665

(88) and the elementary charge (

q^{2} = 1

) would have mass

M_{BB} (k_{eq}) \approx \pm 1.9455 \times 10^{- 9} [kg]

, imaginary mass

M_{i BB} (k_{eq}) \approx \pm i 1.7768 \times 10^{- 9} [kg]

, corresponding to the Compton wavelength

λ_{BB} (k_{eq}) \approx \pm 1.1361 \times 10^{- 33} [m]

, and the imaginary Compton wavelength

λ_{i BB} (k_{eq}) \approx \pm i 1.2160 \times 10^{- 33} [m]

. On the other hand, equation (83) provides the BB charge in equilibrium (77) as

q_{BB} (k_{eq}) \approx 11.1874 m_{BB}

and the limit of the BB charge

q_{BB} (k_{\max}) \approx 37.9995 m_{BB}

We note that BBs with STMs

2 \leq k \leq 3

are referred to as ultracompact [71], where

k_{ps} = 3

is a photon sphere radius, at which, according to an accepted photon sphere definition, the strength of gravity forces photons to travel in orbits. The author wonders why the photons would not travel in orbits at a radius

R = G M / c^{2}

corresponding to the orbital velocity

v_{orb}^{2} = G M / R

of mass M. Obviously, according to the ED principle, photons do not travel. Any object that undergoes complete gravitational collapse passes through an ultracompact stage [72], where

k < 3

. Collapse can be approached by gradual accretion, increasing the mass to the maximum stable value, or by the loss of angular momentum [72]. During the loss of angular momentum, the star passes through a sequence of increasingly compact configurations until it finally collapses and becomes a BH. It was also pointed out [73] that for a neutron star of constant density, the pressure at the center would become infinite if

k = 2.25

, which is the radius of the maximal sustainable density for gravitating spherical matter given by Buchdahl’s theorem. It was shown [74] that this limit applies to any well-behaved spherical star, where the density increases monotonically with the radius. Furthermore, some observers would measure a locally negative energy density if

k < 2.6 (6)

thus breaking the dominant energy condition, although this may be allowed [75]. As the surface gravity increases, photons from further behind the NS become visible. At

k \approx 3.52

the entire NS surface becomes visible [76]. The relative increase in brightness between the maximum and minimum of a light curve is greater for

k < 3

than for

k > 3

[76]. Furthermore,

k_{c} = \sqrt{27} \approx 5.1962

defining a photon capture radius [77], the effective radius for capturing photons that approach the black hole from infinity is about

76 %

k_{\max}

(86). Therefore, the equilibrium and maximum STM ratios (88), (86) satisfying

k_{eq} < k_{ps} < k_{c} < k_{\max}

are well within the range of radii of ultracompact objects researched in the state-of-the-art within the GR framework.

However, aside from the Schwarzschild radius, derivable from the escape velocity

v_{e s c}^{2} = 2 G M / R

of mass M by setting

v_{e s c}^{2} = c^{2}

, and discovered in 1783 by John Michell [78], all the remaining significant radii of GR are only approximations. GR neglects the value of the Planck constant and the fine-structure constants

α

and

α_{2}

, which, similar to

π

or the base of the natural logarithm, are fundamental constants of nature. Constructive criticism of GR can be found, for example, in [79,80,81,82,83,84,85].

Therefore, we conjecture that

k_{eq}

is the correct value of the photon sphere radius and

k_{\max}

is the correct value of a photon capture radius. These radii are indirectly measured by the Event Horizon Telescope (EHT), a telescope array consisting of a global network of radio telescopes, each associated with an atomic clock. The signals collected by each telescope are stored along with time stamps and subsequently shipped to one location to be synchronized and processed. The EHT observational targets are the Milky Way’s BH, Sagittarius A* (Sgr A*) and the M87* BH at the center of the Messier 87 galaxy. The EHT data processing model assumes a Kerr BH (uncharged, spinning) [77,86] and is suitable for prograde accretion disks [86] (spinning in the same direction as a BH). The first assumption is congruent with our results; in this and in the preceding sections we have shown that BHs are fundamentally uncharged.

The processed Sgr A* and M87* signals revealed ring-like structures surrounding these BHs, which were compared with a large suite of general-relativistic magnetohydrodynamics (GRMHD) simulations [77,86]. However, the GRMHD simulations turned out to be more variable than the observed signals and only a few configurations could satisfy the full set of observational constraints apart from variability, hinting that more work is needed to fully explore the physical parameter space and to understand this variability, as these variations challenge standard approaches to interferometric analysis. Subsequently, measurements of the first in an infinite series of photon rings around M87* were reported [87] based on simultaneous modeling and imaging of the EHT signals. However, the photon ring calculated by this method turned out to be much brighter than expected, while it should be emitting only around 20 percent of the light in the image, as general relativity predicts [88]. Therefore, it may be just picking out an unrelated structure in the image [89] and M87* and Sgr A* EHT observations have yet to experimentally resolve any photon ring [90,91,92,93]. This ambiguity requires further research of these ring-like structures revealed by the EHT within the ED framework, considering the equilibrium and maximum STM ratios

k_{eq}

(88) and

k_{\max}

(86) that may accurately describe them.

6. Complex Forces

The complex energies (61)-(63) define the complex forces (similarly to the complex energy of real masses and charges (59); cf. ref. [66] Eq. (7)). Using the relations (51), we obtain the following three complex products of energies

E_{12 m q_{i}} : = E_{1 M Q_{i}} E_{2 M Q_{i}} / E_{P}^{2} = m_{1} m_{2} - q_{1} q_{2} α + i \sqrt{α} (m_{1} q_{2} + m_{2} q_{1}),

(89)

E_{12 q m_{i}} : = E_{1 Q M_{i}} E_{2 Q M_{i}} / E_{P}^{2} = \frac{α^{4}}{α_{2}^{4}} (α q_{1} q_{2} + \frac{α}{α_{2}} m_{i 1} m_{i 2} + \sqrt{\frac{α}{α_{2}}} \sqrt{α} (m_{i 1} q_{2} + m_{i 2} q_{1})),

(90)

E_{12 m m_{i}} : = E_{1 M M_{i}} E_{2 M M_{i}} / E_{P}^{2} = m_{1} m_{2} + \frac{α^{5}}{α_{2}^{5}} m_{i 1} m_{i 2} + \sqrt{\frac{α^{5}}{α_{2}^{5}}} (m_{1} m_{i 2} + m_{2} m_{i 1}),

(91)

where

E_{1 A B_{i}}

is the energy of object 1 and

E_{2 A B_{i}}

is the energy of object 2. Their squared moduli are

\begin{matrix} | E_{12 m q_{i}} |^{2} & = m_{1}^{2} m_{2}^{2} + α^{2} q_{1}^{2} q_{2}^{2} + α (m_{1}^{2} q_{2}^{2} + m_{2}^{2} q_{1}^{2}), \\ | E_{12 q m_{i}} |^{2} & = \frac{α_{+}^{10}}{α_{-}^{10}} (m_{i 1}^{2} m_{i 2}^{2} + α_{2}^{2} q_{1}^{2} q_{2}^{2} - α_{2} (m_{i 1}^{2} q_{2}^{2} + m_{i 2}^{2} q_{2}^{2})), \\ | E_{12 m m_{i}} |^{2} & = m_{1}^{2} m_{2}^{2} + \frac{α^{10}}{α_{2}^{10}} m_{i 1}^{2} m_{i 2}^{2} - \frac{α^{5}}{α_{2}^{5}} (m_{1}^{2} m_{i 2}^{2} + m_{2}^{2} m_{i 1}^{2}) . \end{matrix}

(92)

Products of energies (89)-(91) define three complex forces acting over a real distance

R_{+}

F_{A B_{i}} = \frac{G}{c^{4} R_{+}^{2}} E_{1 A B_{i}} E_{2 A B_{i}} = \frac{F_{P}}{r_{+}^{2}} E_{12 a b_{i}},

(93)

and three complex forces acting over an imaginary distance

R_{-}

{\tilde{F}}_{A B_{i}} = \frac{G}{c_{2}^{4} R_{-}^{2}} E_{1 A B_{i}} E_{2 A B_{i}} = \frac{α_{2}}{α} \frac{F_{P}}{r_{-}^{2}} E_{12 a b_{i}},

(94)

where

A, B \in {M, Q}

and

a, b \in {m, q}

. We express the forces (93) and (94) in terms of the Planck force

F_{P}

in order to compare them, which yields

α_{2} r_{+}^{2} F_{A B_{i}} = α r_{-}^{2} {\tilde{F}}_{A B_{i}} .

(95)

In the case of two charged masses

m_{1}

and

m_{2}

in a balanced state, such as a nucleus and an electron cloud of an atom in its non-ionized state,

q_{1} = - q_{2}

and switching the signs of charges (

q_{1} \to - q_{2}

q_{2} \to - q_{1}

) provides complex conjugates of the energies (89) and (90), which - after such switch - represent antimatter.

7. Extended Bohr Model

The Bohr model of the hydrogen atom is based on three assumptions that can be conveniently expressed in terms of Planck units using relations (51). The first assumption of a natural number of electron wavelengths

λ_{e}

that fits along the circumference of the electron’s orbit of radius R becomes:

n λ_{e} = 2 π R \Leftrightarrow n l_{e} = 2 π r, n \in N .

(96)

The second assumption, de Broglie’s relation between electron mass

M_{e}

, velocity

V_{e}

and wavelength becomes

λ_{e} = \frac{h}{M_{e} V_{e}} = \frac{2 π ℏ}{M_{e} V_{e}} \Leftrightarrow l_{e} = \frac{2 π}{m_{e} v_{e}}, V_{e} v_{e} c, v_{e} \in R .

(97)

Finally, the third assumption postulated equality between the centripetal force exerted on the electron orbiting around the proton (assuming an infinite mass of the latter) and the Coulomb force between the electron and proton becomes:

\frac{M_{e} V_{e}^{2}}{R} = \frac{1}{4 π ϵ_{0}} \frac{e^{2}}{R^{2}} \Leftrightarrow m_{e} v_{e}^{2} r = \frac{e^{2}}{4 π ϵ_{0} ℏ c} = α .

(98)

It is remarkable that such a simple postulate alone, expressed in terms of Planck units, introduces the fine-structure constant

α

. Joining relations (96) and (97) yields

m_{e} v_{e} r = n,

(99)

which combined with (98) and using the relation (33) yields

V_{e} = v_{e} c = \frac{1}{n} α c \Leftrightarrow v_{e} = \frac{1}{n} α,

(100)

Thus, in the first circular orbit (

n = 1

) of this model, the electron velocity factor

v_{e} = α

. In the Bohr model of atoms other than hydrogen this equality of forces is extended to a point-like set of Z electrons orbiting around a nucleus, where Z is the atomic number.

Furthermore, since the proton and the electron have different signs of the elementary charge e, the Coulomb force should be considered negative in this model.

To quantify charges, we assume that the centripetal force acting on the electron is equal to the complex force

F_{M Q_{i}}

(93) with the product of real mass and imaginary charge energies (89) and, motivated by the short multiplication formula

(a + b) (a - b)

and the two-body problem we describe, use the reduced mass of the proton-electron system. This yields

\frac{m_{e} m_{p}}{m_{e} + m_{p}} \frac{v_{e}^{2}}{r} = \frac{m_{e} m_{p} + α + i \sqrt{α} (m_{e} - m_{p})}{r^{2}},

(101)

where we set

q_{e} = - 1

and

q_{p} = 1

as the electron and proton charges, respectively,

m_{e}, m_{p} \in R

from the electron mass

M_{e} = m_{e} m_{P} = 9.1094 \times 10^{- 31}

[kg] and the proton mass

M_{p} = m_{p} m_{P} = 1.6726 \times 10^{- 27}

[kg]. The equation (101) yields complex velocity

v_{e} = \sqrt{\frac{m_{e} + m_{p}}{r} (1 + \frac{α}{m_{e} m_{p}}) + i \frac{\sqrt{α}}{r} \frac{m_{e}^{2} - m_{p}^{2}}{m_{e} m_{p}}} \approx 7.2993 \times 10^{- 3} - i 3.2816 \times 10^{- 21} \approx α

(102)

assuming that R is equal to the Bohr radius

a_{0} = 5.2918 \times 10^{- 11} [m]

or the complex radius

R = (\frac{m_{e} + m_{p}}{v_{e}^{2}} (1 + \frac{α}{m_{e} m_{p}}) + i \frac{\sqrt{α}}{v_{e}^{2}} \frac{m_{e}^{2} - m_{p}^{2}}{m_{e} m_{p}}) ℓ_{P} \approx (5.2946 \times 10^{- 11} - i 4.7607 \times 10^{- 29}) [m] \approx a_{0}

(103)

assuming that the Bohr model gives the velocity of the electron, that is,

v_{e} = α

. However, neglecting their insignificant imaginary parts, they correspond to their Bohr model counterparts. This is because the gravitational attraction between the proton and the electron in hydrogen atom (

10^{- 47}

N) is negligible compared to the Coulomb force between them (

10^{- 8}

N).

We further note that switching the signs of charges (

q_{e} = 1

q_{p} = - 1

) provides complex conjugates of the relation (101), which in this case describes the antihydrogen. Thus, we conjecture that the energy generated by a hydrogen-antihydrogen collision, predicted by this extended Bohr model, is

E_{H M Q_{i}} + E_{\bar{H} M Q_{i}} = 2 (m_{e} m_{p} + α) E_{P} \approx 2.8549 \times 10^{7} [J] .

(104)

Expressed in terms of Planck units and the reduced mass of the proton-electron system, the Rydberg constant for hydrogen is

R_{H} = \frac{m_{e} m_{p}}{m_{e} + m_{p}} \frac{α_{*}^{2}}{4 π ℓ_{P}} \approx 1.0968 \times 10^{7} [1 / m],

(105)

The inverse of the corresponding Rydberg formula for hydrogen can be expressed using the relation (13) for

α

\frac{m_{e} m_{p}}{m_{e} + m_{p}} l = \frac{4 π}{α^{2}} \frac{n_{1}^{2} n_{2}^{2}}{n_{2}^{2} - n_{1}^{2}} = \frac{n_{1}^{2} n_{2}^{2}}{n_{2}^{2} - n_{1}^{2}} 4 π^{3} (16 π^{4} + 8 π^{3} + 9 π^{2} + 2 π + 1),

(106)

using the wavelength relation (51) and

α

expression (13). The coefficients

{16, 8, 9, 2, 1}

form part of the OEIS sequence A158565 for

n = 10

8. BB Complex Gravity and Temperature

We can use the complex force

F_{M Q_{i}}

(93) with the product (89) (i.e., complex Newton’s law of universal gravitation) to calculate the BB surface gravity

g_{BB}

, assuming an uncharged (

q_{2} = 0

) test mass

m_{2}

and comparing this force with Newton’s 2nd law of motion

\frac{F_{P}}{r_{BB}^{2}} (m_{BB} m_{2} + i \sqrt{α_{+}} m_{2} q_{BB}) = M_{2} g_{BB} = m_{2} m_{P} {\hat{g}}_{BB} a_{P}, {\hat{g}}_{BB} = \frac{1}{r_{BB}^{2}} (m_{BB} + i \sqrt{α_{+}} q_{BB}),

(107)

where

g_{BB} = {\hat{g}}_{BB} a_{P}

{\hat{g}}_{BB} \in R

. Substituting

q_{BB} \sqrt{α_{+}}

from the BB energy relation (83) and the mass taken from the generalized BB radius (55)

r_{BB} = k m_{BB}

into the relation (107) yields

{\hat{g}}_{BB} = \frac{1}{k r_{BB}} (1 \pm i \sqrt{\frac{k^{2}}{4} - 1}),

(108)

which reduces to BH surface gravity for

k = 2

and in modulus

{\hat{g}}_{BB}^{2} = \frac{1}{k^{2} r_{BB}^{2}} (1 + i \sqrt{\frac{k^{2}}{4} - 1}) (1 - i \sqrt{\frac{k^{2}}{4} - 1}) = \frac{1}{4 r_{BB}^{2}} .

(109)

for all k and is remarkably independent of

α_{*}

. In particular,

\begin{matrix} g_{BB} (k_{\max}) = \pm \frac{a_{P}}{d_{BB}} (0.2944 \pm 0.9557 i), \end{matrix}

(110)

\begin{matrix} g_{BB} (k_{eq}) = \pm \frac{a_{P}}{d_{BB}} (0.7229 \pm 0.6909 i) . \end{matrix}

(111)

The BB surface gravity (108) leads to the generalized complex Hawking blackbody radiation equation:

\begin{matrix} T_{BB} = \frac{ℏ}{2 π c k_{B}} g_{BB} = \frac{T_{P}}{k π d_{BB}} (1 \pm i \sqrt{\frac{k^{2}}{4} - 1}), \end{matrix}

(112)

describing the BB temperature by including its charge in the imaginary part, which also for

k = 2

and in modulus reduces to the BH temperature for all k. In a commonly used, equivalent form it is

T_{BB} = \frac{ℏ c^{3}}{2 k^{2} π G M_{BB} k_{B}} (1 \pm i \sqrt{\frac{k^{2}}{4} - 1}) .

(113)

In particular,

\begin{matrix} T_{BB} (k_{\max}) & = \frac{T_{P}}{2 π d_{BB}} (\frac{\sqrt{α_{+}^{4} - α_{-}^{4}}}{α_{+}^{2}} \pm i \frac{α_{-}^{2}}{α_{+}^{2}}) \end{matrix}

(114)

T_{BB} (k_{eq}) = \frac{T_{P}}{2 π d_{BB}} \frac{α_{+}^{2} \pm i α_{-}^{2}}{\sqrt{α_{+}^{4} + α_{-}^{4}}}

(115)

reduce to the BH temperature for

α_{-} = 0

. Therefore, a universe without the negative fine structure constants

α_{-}

(i.e., with

α_{-} = 0

) would be a black hole disallowing the directed exploration [8,9] of the evolution of information [1,2,3,4,6,7]. This kind of exploration requires imaginary time [6]. And we cannot zero

α_{2}

as we would have to neglect the existence of graphene, estimated to form about

1.9 %

of total interstellar carbon [94] and recently discovered on the Moon [95].

9. BB Mergers

As the entropy (Boltzmann, Gibbs, Shannon, von Neumann) of independent systems is additive, a merger of BB₁ and BB₂ having entropies (54)

S_{1} = \frac{1}{4} k_{B} π d_{1}^{2}

and

S_{2} = \frac{1}{4} k_{B} N_{2}

, produces a BB_C with entropy (we drop the HS subscripts in this section for clarity)

S_{1} + S_{2} = S_{C} \Leftrightarrow d_{1}^{2} + d_{2}^{2} = d_{C}^{2},

(116)

which shows that the resultant information capacity is the sum of the information capacities of the merging components. Thus, a merger of two primordial BHs, each with the Planck length diameter, the reduced Planck temperature

\frac{T_{P}}{2 π}

(the largest physically significant temperature [5]) produces a BH having

d_{BH} = \pm \sqrt{2}

which represents the minimum BH diameter allowing for the notion of time [5]. In comparison, a collision of the latter two BHs produces a BH with

d_{BH} = \pm 2

and the triangulation defining only one precise diameter between its poles (cf. [6] Figure 3(b)), which is also recovered from HUP (cf. Appendix E).

Substituting the generalized diameter (55) into the entropy relation (116) establishes a Pythagorean relation between the generalized energies (58) of the merging components and the merger

\frac{k_{C}^{2}}{4} m_{C}^{2} = \frac{k_{1}^{2}}{4} m_{1}^{2} + \frac{k_{2}^{2}}{4} m_{2}^{2}, \forall m_{j} \in {R, I} .

(117)

It is accepted that gravitational events observations alone allow measuring the masses of the merging components, setting a lower limit on their compactness, but it does not exclude mergers that are more compact than neutron stars, such as quark stars, BHs, or more exotic objects [96]. We note in passing that describing the registered gravitational events as waves is misleading: normal modulation of the gravitational potential, registered by LIGO and Virgo interferometers, and caused by rotating (in the merger case, inspiral) objects is wrongly interpreted as a gravitational wave understood as a carrier of gravity [97]. It has also been shown that from a mathematical point of view, the quadrupolar waves in a quantum spin nematic are in one-to-one correspondence with quantized gravitational waves in a flat, 4-dimensional spacetime [98].

The accepted value of the Chandrasekhar WD mass limit, which prevents its collapse into a denser form, is

M_{Ch} \approx 1.4 M_{⊙}

[99] and the accepted value of the analogous Tolman-Oppenheimer-Volkoff NS mass limit is

M_{TOV} \approx 2.9 M_{⊙}

[100,101]. There is no accepted value for the BH mass limit. The conjectured value is

5 \times 10 M_{⊙} \approx 9.95 \times 10^{40} kg

. We note in passing that a BH with a surface gravity equal to the Earth’s surface gravity (9.81 m/s²) would require a diameter of

D_{B H} \approx 9.16 \times 10^{15} m

(slightly less than one light year) [6] and mass

M_{B H} \approx 3.08 \times 10^{42} kg

exceeding the conjectured limit. The masses of most registered merging components go well beyond

M_{TOV}

. From those that do not, most of the total or final masses exceed this limit. Therefore, these mergers (

BH ⇆ BH

BH ⇆ NS

BH ⇆ WD

NS ⇆ NS

NS ⇆ WD

WD ⇆ WD

) are classified as BH mergers. Only a few were classified otherwise, including GW170817, GW190425, GW200105, and GW200115, as listed in Table 2.

Equation (117) explains the measurements of large masses of BB mergers with at least one charged merging component without resorting to any hypothetical types of exotic stellar objects such as quark stars. Interferometric data, available online at the Gravitational Wave Open Science Center (GWOSC) portal1, indicates that the total mass of a merger is the sum of the masses of the merging components. Thus

m_{C} = m_{1} + m_{2} \Leftrightarrow m_{C}^{2} = m_{1}^{2} + m_{2}^{2} + 2 m_{1} m_{2} \Leftrightarrow \{\begin{matrix} m_{C}^{2} \geq m_{1}^{2} + m_{2}^{2} & if m_{1} m_{2} \geq 0 \\ m_{C}^{2} \leq m_{1}^{2} + m_{2}^{2} & if m_{1} m_{2} \leq 0 \end{matrix} .

(118)

We can use the squared moduli

| E_{M Q_{i}} |^{2}

| E_{Q M_{i}} |^{2}

, and

| E_{M M_{i}} |^{2}

(83)-(84) and the relation (117) to derive some information about the merger from equation (117). We shall initially assume

m_{j} \geq 0 \Rightarrow m_{1} m_{2} \geq 0

, since negative masses, similar to negative lengths, and their products with positive ones, are (in general [102]) inaccessible for direct observation, unlike charges.

| E_{M Q_{i}} |^{2}

(83) with the first inequality (118) yields:

| E_{M Q_{i}} |_{C}^{2} = | E_{M Q_{i}} |_{1}^{2} + {| E_{M Q_{i}} |}_{2}^{2}, m_{C}^{2} = m_{1}^{2} + m_{2}^{2} + (q_{1}^{2} + q_{2}^{2}) α - q_{C}^{2} α \geq m_{1}^{2} + m_{2}^{2}, q_{C}^{2} \leq q_{1}^{2} + q_{2}^{2},

(119)

On the other hand,

| E_{Q M_{i}} |^{2}

(85) with inequality (119) leads to (

α_{-} < 0

, and thus the direction of the inequality is reversed):

q_{C}^{2} \leq q_{1}^{2} + q_{2}^{2} \Rightarrow m_{i C}^{2} \geq m_{i 1}^{2} + m_{i 2}^{2} .

(120)

But

| E_{M M_{i}} |^{2}

(84) with the first inequality (118) leads to:

m_{C}^{2} \geq m_{1}^{2} + m_{2}^{2} \Rightarrow m_{i C}^{2} \leq m_{i 1}^{2} + m_{i 2}^{2},

(121)

contradicting inequality (120) (

α_{-}^{5} < 0

), while

| E_{M M_{i}} |^{2}

with inequality (120) leads to:

m_{i C}^{2} \geq m_{i 1}^{2} + m_{i 2}^{2} \Rightarrow m_{C}^{2} \leq m_{1}^{2} + m_{2}^{2},

(122)

contradicting the first inequality (118) and is consistent with the second inequality (118) introducing the product of positive and negative masses.

| E_{Q M_{i}} |^{2}

with inequality (121) yields:

m_{i C}^{2} \leq m_{i 1}^{2} + m_{i 2}^{2} \Rightarrow q_{C}^{2} \geq q_{1}^{2} + q_{2}^{2},

(123)

contradicting the inequality (120) and so on.

The additivity of the entropy (116) of statistically independent merging BBs, both in global thermodynamic equilibrium, defined by their generalized radii (55), introduces the energy relation (117). This relation, universality of the charges (27), and the BB complex energies (69)-(71) establish imaginary, negative, and mixed masses during the merger. The BB merger spreads as a gravitational event associated with a fast radio burst (FRB) event, as reported [103] based on the gravitational event GW1904251 and FRB 20190425A event2. Furthermore, IXPE3 observations show that the polarized X-rays detected from 4U 0142+61 pulsar exhibit a

90^{\circ}

linear polarization swing from low to high photon energies [104]. In addition, direct evidence for a magnetic field strength reversal based on the observed sign change and extreme variation of FRB 20190520B’s rotation measure, which changed from

\sim 10000 [rad \cdot m^{- 2}]

\sim - 16000 [rad \cdot m^{- 2}]

between June 2021 and January 2022, has been reported [105], and such extreme rotation measure reversal has never been observed before in any FRB or any astronomical object. It has been suggested that outside the GR, merging BHs may differ from their GR counterparts [106]. Furthermore, it was experimentally confirmed [107], based on the registered gravitational event GW170817, that BB mergers are perfectly spherical. It is concluded [107] that an additional process seems necessary to make the merger distribution uniform. However, one can hardly expect the collision of two perfectly spherical, patternless thermal noises to produce an aspherical pattern instead of another perfectly spherical patternless noise. Where would information about this pattern come from at the moment of collision? From the point of impact? No point of impact can be considered unique on the patternless surface.

During the merger, the STM ratio

k_{C}

decreases, making the BB_C denser until it becomes a BH for

k_{C} = 2

and no further charge reduction is possible (see Figure 4). From equation (117) and the first inequality (118), we see that this holds for

k_{C}^{2} (m_{1}^{2} + m_{2}^{2}) \leq k_{1}^{2} m_{1}^{2} + k_{2}^{2} m_{2}^{2} .

(124)

For two merging BHs

k_{1} = k_{2} = 2

and the relation (124) yields

k_{C}^{2} \leq 4 \Rightarrow k_{C} = 2 = k_{{BH}_{C}}

. On the other hand, if

m_{1} = m_{2}

then

k_{C} \leq \sqrt{(k_{1}^{2} + k_{2}^{2}) / 2}

. The tendency to decrease the STM, given by relation (124), is reflected in the merger statistics registered by LIGO and Virgo interferometers: the registered fraction of BH mergers is much higher than might be expected by chance.

Table 2 lists the STM ratios

k_{{BB}_{C}}

calculated according to equation (117), which provide the measured mass

M_{{BB}_{C}}

of the merger and satisfy inequality (124). The STM ratios

k_{{BB}_{1}}

and

k_{{BB}_{2}}

of the merging components were arbitrarily selected based on their masses, considering the limit of mass

M_{TOV}

of the NS.

10. BB Fluctuations

A relation [108] (p.160) describing a BH information capacity, having an initial information capacity4

N_{j} = 4 π R_{j}^{2} / ℓ_{P}^{2}

, after absorption of a particle having the Compton wavelength equal to the BH radius

R_{j}

N_{j + 1}^{A} = 64 π^{3} \frac{ℓ_{P}^{2}}{R_{j}^{2}} + 32 π^{2} + 4 π \frac{R_{j}^{2}}{ℓ_{P}^{2}},

(125)

was subsequently generalized [6] (Eq. (18)) to all Compton wavelengths

λ = l ℓ_{P} = \frac{2 π}{m} ℓ_{P}

(or frequencies

ν = c / λ = 1 / (l t_{P})

) and thus to all radiated Compton energies

E = m E_{P}

m \in R

absorbed (+) or emitted (−) by a BH as

N_{j + 1}^{A / E} (m) = 64 π^{3} \frac{1}{l^{2}} \pm 16 π^{2} \frac{d}{l} + π d^{2} = 16 π m^{2} \pm 8 π d m + π d^{2} .

(126)

Equation (126) can be further generalized, using the generalized diameter

d = 2 k \hat{m}

(55), to all BBs as follows

Δ N^{A / E} N_{j + 1}^{A / E} (k, m) - N_{j} = 16 π m (m \pm k \hat{m}),

(127)

where

\hat{m}

represents the BB mass, and its roots are

\begin{matrix} m^{A / E} = {0, \mp k \hat{m}} = \{0, \mp \frac{d}{2}\} = {0, \mp r}, \end{matrix}

(128)

where it vanishes. Thus, in general, BB changes its information capacity by:

Δ N^{A} \{\begin{matrix} > 0 & m \in (- \infty, - k \hat{m}) \cap (0, \infty) \\ = 0 & m = {- k \hat{m}, 0} \\ < 0 & m \in (- k \hat{m}, 0) \end{matrix}, Δ N^{E} \{\begin{matrix} > 0 & m \in (- \infty, 0) \cap (k \hat{m}, \infty) \\ = 0 & m = {0, k \hat{m}} \\ < 0 & m \in (0, k \hat{m}) \end{matrix},

(129)

absorbing or emitting energy m with

min (Δ N) = - 4 π k^{2} {\hat{m}}^{2}

m = \pm k \hat{m} / 2

, as shown in Figure 5. Equation (129) shows that, depending on its mass

\hat{m}

, a BB can expand or contract by emitting or absorbing energy m [6]. However, expansion by emission (

Δ N^{E} > 0

), for example, requires energy

m > k \hat{m}

exceeding the mass-energy equivalence of BB for

k > 2

, which is consistent with the results presented in Section 5. We note that the same form of the relation (127), expressed as

N_{j + 1}^{A / E} (\hat{m}, l) = 64 π^{3} \frac{1}{l^{2}} \pm 32 π^{2} \frac{k \hat{m}}{l} + 4 π k^{2} {\hat{m}}^{2}

(130)

contains the same like terms with respect to

π

as the transcendental expressions for the fine-structure constants (13) and (14), which requires further investigation. The forms of the fine-structure constants (18) and (22) are different.

11. Discussion

Complex, imaginary, and negative physical quantities are the subject of research. In particular, the subject of scientific research is the thermodynamics in the complex plane. For example, Lee–Yang zeros [109,110] and photon-photon thermodynamic processes under negative optical temperature conditions [111] have been experimentally observed. Furthermore, the rendering of synthetic dimensions through space modulations has recently been suggested because it does not require any active materials or other external mechanisms to break time-reversal symmetry [112]. Complexified geodesics are investigated [113] and it was shown that from a geometric point of view the unitary symmetries U(1) and SU(2) stem fundamentally from Schwarzschild and Reissner-Nordström metrics through spacetime complexification if a new Euclidean metric on a complex Hermitian manifold is provided [114]. In Lorentz signature, a Hermitian structure must necessarily be complex-valued, so its integrability properties are more subtle than in the Euclidean case [115].

Physical quantities accessible for direct everyday observation are mostly real and positive with the negativity of distances, velocities, accelerations, etc., induced by the assumed orientation of space. Quantum measurement results, for example, are the necessarily real eigenvalues of Hermitian operators. Unlike charges, negative, real masses are also generally inaccessible for direct observations. However, dissipative coupling between excitons and photons in an optical microcavity leads to the formation of exciton polaritons with negative masses [102]. In Section 9, we show that negative masses also result from the merging of BBs.

The notion of atemporality, related to the prebiotic, timeless [9] part of the evolution of information in the universe, was also investigated [116] on a black hole (BH) surface as the dynamical mechanism responsible for the transition from a regime with a real-valued interval to an imaginary interval via the Wick rotation. The Wick rotation between real and imaginary intervals was also analyzed in the context of kinematics on holographic spheres [6] and quantum orthogonalization intervals of BHs [5].

By Noether’s theorem, the conservation of energy (7) between three Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene is associated with a symmetry between the fine-structure constant

α

and

π

. This symmetry establishes three complementary fine-structure constants, of which two are negative, wherein all fine-structure constants are related to each other through the constant of

π

, which indicates that they do not vary over time. The complementary fine-structure constants are associated with their own sets of Planck units that allow for different parameterization of the perceivable space

R_{α}^{3} \times I_{α}

Euclidean space parameterized by the fine-structure constant

α

. Since the elementary charge and the products

c_{*} α_{*}

are the same in all these four parameterizations, the complementary electrons correspond to dark states recently discovered in condensed-matter systems having two pairs of sublattices.

As the two sets of basic Planck units are real and two are imaginary, we have applied four complex pairs of masses and charges defined by

{α_{+}, α_{-}}

to the complex energy formula [66] defining three complex energies (61) and (62). The generalized energy (58) of all perfect black-body objects (black holes, neutron stars, and white dwarfs) with a generalized radius

R_{BB} = k R_{BH} / 2

, where

R_{BH}

is the Schwarzschild radius, exceeds the mass-energy equivalence if

k > 2

. However, the complex energies (61)-(63) allow storage of this excess energy in their imaginary parts. Further analysis of this model showed that the negative complementary fine-structure constant established by the graphene reflectance is dual to the fine-structure constant and Appendix D presents some arguments to support the claim that

α

parameterization sets favorable conditions for biological evolution to emerge. The lower bound on the mass of a charged black-body object is

M_{BB} > 5.7275 \times 10^{- 10} [kg]

and the upper bound on a white dwarf radius is

R_{WD} ≲ 3.3967 R_{BH}

, where

R_{BH}

is the Schwarzschild radius of the white dwarf mass. A charged black-body object is in the equilibrium of complex energies if its radius

R_{eq} \approx 1.3833 R_{BH}

, which is close to the photon sphere radius

R_{ps} = 1.5 R_{BH}

, and is marginally greater than the locally negative energy density bound of

4 / 3 R_{BH}

. The maximum radius of the black-body object is

R_{\max} \approx 3.3967 R_{BH}

, which is close to the photon capture radius

R_{c} = \sqrt{27} R_{BH} / 2 \approx 2.5981 R_{BH}

. The complex force between real masses and imaginary charges leads to the complex black-body object’s surface gravity and generalized Hawking radiation complex temperature. Furthermore, on the basis of the Bohr model for the hydrogen atom, we show that complex conjugates of this force represent atoms and antiatoms. The proposed model considers the value(s) of the fine-structure constant(s), which is(are) otherwise neglected in general relativity, and explains the registered (GWOSC) high masses of neutron star mergers and the associated fast radio bursts (CHIME) without resorting to any hypothetical types of exotic stellar objects.

MLG is a truly 2-dimensional material with no thickness. Although its thickness is reported [117] as 0.37 [nm] with other reported values up to 1.7 [nm], these results are not credible, considering that 0.335 [nm] is the established interlayer distance and consequently the thickness of bilayer graphene: the thickness of bilayer graphene is not

2 \times 0.37 + 0.335 = 1.075

[nm]. In the context of the results of this study, graphene is a keyhole to ED. The history of graphene is also instructive. Discovered in 1947 [118], graphene was long considered an academic material until it was eventually pulled from graphite in 2004 [119] using ordinary Scotch tape (introduced into the market in 1932). These fifty-seven years, along with twenty-nine years (1935-1964) between the condemnation of quantum theory as incomplete [120] and Bell’s mathematical theorem [121] asserting that it is not true, and the fifty-eight years (1964-2022) between the formulation of this theorem and the 2022 Nobel Prize in Physics for its experimental loophole-free confirmation, should remind us that Max Planck, the genius who discovered Planck units, has also discovered Planck’s principle.

Acknowledgments

The author truly thanks his wife, Magdalena Bartocha, for her support since this research [122,123] began; Wawrzyniec Bieniawski for inspiring discussions, constructive ideas concerning the layout of this paper, and his feedback concerning the sections on BB mergers and BB fluctuations; Piotr Masierak for constructive and insightful remarks; Mariola Bala for handling these esoteric vapors under control; and Andrzej Tomski for defining the scalar product for the real/imaginary Euclidean spaces

R^{a} \times I^{b}

(1).

Abbreviations

The following abbreviations are used in this paper:

ED	emergent dimensionality
EMR	electromagnetic radiation
MLG	monolayer graphene
T	transmittance
R	reflectance
A	absorptance
$α_{*}$	any of the fine-structure constants
$α_{★}$	$α$ or $α_{2}$
$α_{+}$	$α$ or $α_{4}$
$α_{-}$	$α_{2}$ or $α_{3}$
BH	black hole
NS	neutron star
WD	white dwarf
BB	black-body object
HS	holographic sphere
STM	size-to-mass ratio
GR	general relativity
HUP	Heisenberg’s uncertainty principle

Appendix A. Mlg Transmittance, Absorptance, and Reflectance as Functions of π Only

With definitions of

α

(13),

α_{2}

(14),

α_{3}

(18), and (22) MLG Fresnel coefficients (3), (4), and (5) can be expressed simply by

π

. For

α^{- 1} = 4 π^{3} + π^{2} + π

(13) and for

α_{4}

(22) they are

T (α) = R (α_{4}) = \frac{4 {(4 π^{2} + π + 1)}^{2}}{{(8 π^{2} + 2 π + 3)}^{2}} \approx 0.9775,

(A1)

A (α) = A (α_{4}) = \frac{4 (4 π^{2} + π + 1)}{{(8 π^{2} + 2 π + 3)}^{2}} \approx 0.0224,

(A2)

R (α) = T (α_{4}) = \frac{1}{{(8 π^{2} + 2 π + 3)}^{2}} \approx 1.2843 e - 4 .

(A3)

For

α_{2}^{- 1} = - 4 π^{3} - π^{2} - 2 π

(14) they are

T (α_{2}) = \frac{4 {(4 π^{2} + π + 2)}^{2}}{{(8 π^{2} + 2 π + 3)}^{2}} \approx 1.0228,

(A4)

A (α_{2}) = - \frac{4 (4 π^{2} + π + 2)}{{(8 π^{2} + 2 π + 3)}^{2}} \approx - 0.0229,

(A5)

with

R (α_{2}) = R (α)

and for

α_{3}

(18) they are

A (α_{3}) = \frac{48 π^{2} + 12 π + 16}{{(8 π^{2} + 2 π + 3)}^{2}} - 4 \approx - 3.9323,

(A6)

R (α_{3}) = \frac{{(16 π^{2} + 4 π + 5)}^{2}}{{(8 π^{2} + 2 π + 3)}^{2}} \approx 3.9548,

(A7)

with

T (α_{3}) = T (α)

A (α) > 0

and

A (α_{2}) < 0

imply a sink and source, respectively, whereas the opposite holds for T, as illustrated schematically in Figure A1. We conjecture that the negative A and T values exceeding 100% for

α_{2}

(11) and (14) could be explained in terms of graphene spontaneous EMR emission.

Figure A1. Illustration of the concepts of negative absorptance and excessive transmittance of EMR under normal incidence on MLG for

{α, α_{2}}

Figure A1. Illustration of the concepts of negative absorptance and excessive transmittance of EMR under normal incidence on MLG for

{α, α_{2}}

Appendix B. π-like Constants

The quadratic equation (8) can also be solved for

π

, which yields two roots.

π {(R, α_{★})}_{1} = \frac{2 \sqrt{R}}{α_{★} (1 - \sqrt{R})} and π {(R, α_{★})}_{2} = \frac{- 2 \sqrt{R}}{α_{★} (1 + \sqrt{R})},

(A8)

dependent on R and

α_{★}

, where

α_{★}

indicates

α

α_{2}

. This can be further evaluated using the MLG reflectance R (4) or (A3), yielding four, yet only three, distinct possibilities,

π_{1} = π {(α)}_{1} = - π \frac{4 π^{2} + π + 1}{4 π^{2} + π + 2} = π \frac{α_{2}}{α} \approx - 3.0712,

(A9)

π {(α)}_{2} = π {(α_{2})}_{1} = π \approx 3.1416, and

(A10)

π_{2} = π {(α_{2})}_{2} = - π \frac{4 π^{2} + π + 2}{4 π^{2} + π + 1} = π \frac{α}{α_{2}} \approx - 3.2136 .

(A11)

The modulus of

π_{1}

(A9) corresponds to a convex surface with a positive Gaussian curvature, whereas the modulus of

π_{2}

(A11) corresponds to a negative Gaussian curvature. The product

π_{1} π_{2} = π^{2}

is independent of

α_{★}

, the quotient

π_{1} / π_{2} = α_{2}^{2} / α^{2}

is not directly dependent on

π

, and

|π_{1} - π| \neq |π - π_{2}|

. It remains to be determined whether each of these

π

-like constants describes the ratio of the circumference of a circle drawn on the respective surface to its diameter (

π_{c}

) or the ratio of the area of this circle to the square of its radius (

π_{a}

). These definitions produce different results for curved surfaces, whereas

π_{a} > π_{c}

on convex surfaces and

π_{a} < π_{c}

on saddle surfaces [124]. The remaining

π

-like constants corresponding to

α_{3}

and

α_{4}

are listed in Table 1.

Appendix C. Mlg Fresnel Equation and Euclid’s Formula

The Fresnel equation for the normal incidence of EMR at the boundary of two media with refractive indices

n_{1}

and

n_{2}

R_{F} + T_{F} = \frac{{(n_{1} - n_{2})}^{2}}{{(n_{1} + n_{2})}^{2}} + \frac{{(2 \sqrt{n_{1} n_{2}})}^{2}}{{(n_{1} + n_{2})}^{2}} = 1 .

(A12)

Substituting MLG reflectance (4) and the sum of transmittance (3) and absorptance (5) into the Fresnel equation (A12) yields

\frac{{(n_{1} - n_{2})}^{2}}{{(n_{1} + n_{2})}^{2}} = \frac{\frac{1}{4} π^{2} α^{2}}{{(1 + \frac{π α}{2})}^{2}}, \frac{4 n_{1} n_{2}}{{(n_{1} + n_{2})}^{2}} = \frac{1 + π α}{{(1 + \frac{π α}{2})}^{2}},

(A13)

which resolves to

n_{1} = 1

independent on

α

and two roots for

n_{2}

n_{2} (α) = \{1 + π α, \frac{1}{1 + π α}\} = {1.0229, 0.9776},

(A14)

satisfying

1 + π α = \frac{1}{1 + π α_{2}},

(A15)

which corresponds to the identity (23). The refractive index

n_{2} \approx 1.0229

is close to that of liquid helium

n \approx 1.025

at 3 K. The refractive index

n_{2} \approx 0.9776

is close to the refractive index of water

n = 0.99999974 = 1 - 2.6 \times 10^{- 7}

for X-ray radiation at a photon wavelength of

0.04

nm. We note that these results are different from the complex refractive index of MLG (

{\tilde{n}}_{g} = 2.4 - 1.0 i

at 532 nm to

{\tilde{n}}_{g} = 3.0 - 1.4 i

at 633 nm at room temperature). However, because

n_{1} = 1

, the equation (A14) relates to the absolute (

n = c / V

) refractive indices; it models MLG as a boundary between vacuum and some other bulk medium. Refractive indices (A14) correspond to the phase velocities

V (- \frac{α_{2}}{α}) = - c \frac{α}{α_{2}} = - c_{2}, V (- \frac{α}{α_{2}}) = - c \frac{α_{2}}{α} = - \frac{c^{2}}{c_{2}} \approx 2.9307 \times 10^{8} [m / s]

(A16)

using the relation (33). We note that the phase velocity of light

- c_{2} > c

does not carry information and thus can be faster than the speed of light in vacuum c.

On the other hand, the Fresnel equation (A12) has the same form as the Euclid’s formula for generating Pythagorean triples

a = k^{2} - l^{2}

b = 2 k l

c = k^{2} + l^{2}

\frac{{(k^{2} - l^{2})}^{2}}{{(k^{2} + l^{2})}^{2}} + \frac{{(2 k l)}^{2}}{{(k^{2} + l^{2})}^{2}} = 1,

(A17)

with

k^{2} = n_{1}

and

l^{2} = n_{2}

. Substituting MLG reflectance (4) and the sum of transmittance (3) and absorptance (5) into the Euclid formula (A17) yields

\begin{matrix} k & = \{\sqrt{π α + 1}, - \sqrt{π α + 1}, \sqrt{\frac{1}{π α + 1}}, - \sqrt{\frac{1}{π α + 1}},\} \approx \{\pm 1.0114, \pm 0.9887\}, \\ l & = \{1, 1, 1, 1\}, \end{matrix}

(A18)

generating four right triangles with edges

\begin{matrix} a (α) & = \{π α, π α, \frac{- π α}{π α + 1}, \frac{- π α}{π α + 1}\} \approx {0.0229 x 2, - 0.0224 x 2}, \\ b (α) & = \{2 \sqrt{π α + 1}, - 2 \sqrt{π α + 1}, \frac{2}{\sqrt{π α + 1}}, \frac{- 2}{\sqrt{π α + 1}}\} \approx {\pm 2.0228, \pm 1.9775}, \\ c (α) & = \{π α + 2, π α + 2, \frac{π α + 2}{π α + 1}, \frac{π α + 2}{π α + 1}\} \approx {2.0229 x 2, 1.9776 x 2}, \end{matrix}

(A19)

and

\begin{matrix} a (α_{2}) & \approx {- 0.0224 x 2, 0.0229 x 2}, \\ b (α_{2}) & \approx {\pm 1.9775, \pm 2.0228}, \\ c (α_{2}) & \approx {1.9776 x 2, 2.0229 x 2} . \end{matrix}

(A20)

In each case, different edge lengths satisfy

π α = - π α_{2} / (π α_{2} + 1)

, which also corresponds to the identity (23). Furthermore

c (α_{★}) - a (α_{★}) = 2, b {(α_{★})}^{2} = 4 \sqrt{a (α_{★}) + 1} .

(A21)

We further note that

a (α_{★}) \approx - A (α_{★})

, (A2), (A5) and

| b (α_{★}) | \approx T (α_{★}) + 1

, (A1), (A4).

Appendix D. Why α Is Better for Biological Evolution than α 2 ?

The probability that two nuclear particles a and b will undergo nuclear fusion by overcoming their electrostatic barriers is given by Gamow–Sommerfeld factor

p (E) = e^{- \sqrt{\frac{E_{G}}{E}}},

(A22)

where

E_{G} : = 2 \frac{m_{a} m_{b}}{m_{a} + m_{b}} E_{P} {(π α Z_{a} Z_{b})}^{2}

(A23)

is the Gamow energy,

m_{a}

m_{b}

are the masses of those particles in terms of

α

α_{2}

Planck units (51) and

Z_{a}

Z_{b}

are their respective atomic numbers.

{(π α)}^{2} \approx 5.2557 \times 10^{- 4}

is larger than

{(π α_{2})}^{2} \approx 5.0227 \times 10^{- 4}

, the probability (A22) is higher for the same dimensionless parameters

m_{*}

Z_{*}

. Therefore, perceivable

(R_{α}^{3} \times I_{α})

-space provides more favorable conditions for the evolution of information (by nuclear fusion) than nonperceivable

(R_{c_{2 +}}^{3} \times I_{c_{2 +}})

-space parameterized with negative

α_{2}

, positive

c_{2 +}

and with imaginary charge due to the relation (33).

Furthermore, the

c_{2 +}

-Planck energy

E_{P 2 +}

and temperature

T_{P 2 +}

are higher than the Planck energy

E_{P}

and temperature

T_{P}

. Therefore, the perceivable

(R_{α}^{3} \times I_{α})

-space provides more favorable conditions for the evolution of information than would be provided by nonperceivable

(R_{c_{2 +}}^{3} \times I_{c_{2 +}})

-space, also owing to the minimum energy principle.

Appendix E. Planck Units and HUP

Perhaps the simplest derivation of the squared Planck length is based on HUP

δ P_{HUP} δ R_{HUP} \geq \frac{ℏ}{2} or δ E_{HUP} δ t_{HUP} \geq \frac{ℏ}{2},

(A24)

where

δ P_{HUP}

δ R_{HUP}

δ E_{HUP}

, and

δ t_{HUP}

denote momentum, position, energy, and time uncertainties, respectively. Replacing energy uncertainty

δ E_{HUP} = δ M_{HUP} c^{2}

with mass uncertainty using mass-energy equivalence, and time uncertainty with position uncertainty using

δ t_{HUP} = δ R_{HUP} / c

[30] yields

δ M_{HUP} δ R_{HUP} \geq \frac{ℏ}{2 c} .

(A25)

Interpreting

δ M_{HUP} = δ R_{HUP} c^{2} / (2 G)

as the BH mass in (A25) we derive the Planck length as

δ R_{HUP}^{2} = ℓ_{P}^{2} \Rightarrow δ D_{HUP} = \pm 2 ℓ_{P}

and recover [6] the BH diameter

d_{BH} = \pm 2

However, using the same procedure but inserting the BH radius instead of the BH mass into the uncertainty principle (A25) leads to

δ M_{HUP}^{2} = \frac{1}{4} ℏ c / G = \frac{1}{4} m_{P}^{2}

. In general, using the generalized radius (55) in both procedures, we obtain

δ M_{HUP}^{2} = \frac{1}{2 k} m_{P}^{2} and δ R_{HUP}^{2} = \frac{k}{2} ℓ_{P}^{2} .

(A26)

Thus, if k increases, the mass

δ M_{HUP}

decreases, and

δ R_{HUP}

increases and the factor is the same for

k = 1

i.e., for the orbital speed radius

δ R = G δ M / c^{2}

or orbital speed mass

δ M = δ R c^{2} / G

Appendix F. Other Definitions of Complex Energies

Let us define the mass/charge energies (61), (62) with different speeds of light, i.e., the charge part of the energy

E_{M Q_{i}}

with

c_{-}

and the charge part of the energy

E_{Q M_{i}}

with

c_{+}

\begin{matrix} {\tilde{E}}_{M Q_{i}} : = M_{+} c_{+}^{2} + \frac{Q_{i}}{2 \sqrt{π ϵ_{0} G}} c_{-}^{2} = m_{+} m_{P +} c_{+}^{2} + i q \sqrt{α_{+}} m_{P +} \frac{α_{+}^{2}}{α_{-}^{2}} c_{+}^{2} = (m_{+} + i q \sqrt{α_{+}} \frac{α_{+}^{2}}{α_{-}^{2}}) E_{P +}, \\ {\tilde{E}}_{Q M_{i}} : = \frac{Q}{2 \sqrt{π ϵ_{0} G}} c_{+}^{2} + M_{-} c_{-}^{2} = q \sqrt{α_{+}} m_{P +} c_{+}^{2} + m_{-} m_{P +} \sqrt{\frac{α_{+}}{α_{-}}} \frac{α_{+}^{2}}{α_{-}^{2}} c_{+}^{2} = (q \sqrt{α_{+}} + m_{-} \sqrt{\frac{α_{+}^{5}}{α_{-}^{5}}}) E_{P +}, \end{matrix}

(A27)

If we assume that their squared moduli are equal then

m_{-} = \pm \sqrt{\frac{α_{-}^{5}}{α_{+}^{5}} [q^{2} α_{+} (1 - \frac{α_{+}^{4}}{α_{-}^{4}}) - m_{+}^{2}]},

(A28)

leads to a type mismatch since

m_{+} \in R

and

| q | \sqrt{α_{+} (1 - α_{+}^{4} / α_{-}^{4})} \in I

. The same category error would be obtained if

E_{M Q_{i}}

was parameterized with

c_{-}

and

E_{Q M_{i}}

with

c_{+}

. Therefore, both parts of the complex energy

E_{M Q_{i}}

(61) must be expressed by

c_{+}

, while both parts of the complex energy

E_{Q M_{i}}

(62) - by

c_{-}

Appendix G. Hall Effect

The fractional quantum Hall (FQHE) effect shows a stepwise dependence of the conductance on the magnetic field (as compared to the linear dependence of the Hall effect) with steps quantized as

R = \frac{h}{ν e^{2}} = \frac{1}{2 ν ϵ_{0} α_{*} c_{*}},

(A29)

where

ν

is an integer or fraction (for example, for

ν = 5 / 2

R = 1 / (5 ϵ_{0} α c)

). The relations (A29) and (33) suggest that the 2D FQHE links real and imaginary Planck units, similar to 2D graphene, establishing the complementary fine-structure constants.

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1	https://www.gw-openscience.org/eventapi/html/allevents
2	Data available online at the Canadian Hydrogen Intensity Mapping Experiment (CHIME) portal (https://www.chime-frb.ca/catalog).
3	X-ray Polarimetry Explorer (https://ixpe.msfc.nasa.gov).
4	We drop the HS subscripts in this section for clarity.

Figure 1. MLG reflectance as a function of

β 1 / α

Figure 1. MLG reflectance as a function of

β 1 / α

Figure 2. Four fine-structure constants as phases of a complex number

z_{*} e^{i / α_{*}}

in order

z_{2}

z_{1}

z_{3}

z_{4}

from left to right.

Figure 2. Four fine-structure constants as phases of a complex number

z_{*} e^{i / α_{*}}

in order

z_{2}

z_{1}

z_{3}

z_{4}

from left to right.

Figure 3. Squared moduli of three complex energies linking mass

m_{+}

, imaginary mass

m_{-}

, and charge q through pairs of positive and negative fine-structure constants.

Figure 3. Squared moduli of three complex energies linking mass

m_{+}

, imaginary mass

m_{-}

, and charge q through pairs of positive and negative fine-structure constants.

Figure 4. Ratios of imaginary mass

M_{i BB}

to real mass

M_{BB}

(green) and real charge

q_{BB} m_{P} \sqrt{α}

M_{BB}

(red) of a BB as a function of the size-to-mass ratio

0 \leq k \leq 10

for

{α, α_{2}}

. The mass

M_{i BB}

is imaginary for

k ⪅ 6.79

. The charge

q_{BB}

is real for

k \geq 2

Figure 4. Ratios of imaginary mass

M_{i BB}

to real mass

M_{BB}

(green) and real charge

q_{BB} m_{P} \sqrt{α}

M_{BB}

(red) of a BB as a function of the size-to-mass ratio

0 \leq k \leq 10

for

{α, α_{2}}

. The mass

M_{i BB}

is imaginary for

k ⪅ 6.79

. The charge

q_{BB}

is real for

k \geq 2

Figure 5. BB information capacity variations

Δ N

after absorption (red) or emission (green) of energy m (

k = 2

\hat{m} = 1

Figure 5. BB information capacity variations

Δ N

after absorption (red) or emission (green) of energy m (

k = 2

\hat{m} = 1

Table 1. Complementary fine-structure constants

α_{2}

α_{4}

and the associated physical quantities and sets of Planck units.

Table 1. Complementary fine-structure constants

α_{2}

α_{4}

and the associated physical quantities and sets of Planck units.

	$α$	$α_{2}$	$α_{3}$	$α_{4}$
$π$ -form	$\frac{1}{4 π^{3} + π^{2} + π}$	$\frac{- 1}{4 π^{3} + π^{2} + 2 π}$	$- \frac{16 π^{2} + 4 π + 5}{4 π^{3} + π^{2} + π}$	$\frac{4}{π^{2}} (4 π^{3} + π^{2} + π)$
$α_{*}$	$0.0073$	$- 0.0071$	$- 1.2805$	$55.5387$
$α_{*}^{- 1}$	$137.0363$	$- 140.1779$	$- 0.7809$	$0.0180$
$c_{*}$ [m/s]	299792458	$- 3.0667 \times 10^{8}$	$- 1.7084 \times 10^{6}$	$3.9390 \times 10^{4}$
$q_{P *}$ [C]	$1.8755 \times 10^{- 18}$	$1.8969 \times 10^{- 18} i$	$1.4158 \times 10^{- 19} i$	$2.1499 \times 10^{- 20}$
$ℓ_{P *}$ [m]	$1.6162 \times 10^{- 35}$	$1.5622 \times 10^{- 35} i$	$3.7570 \times 10^{- 32} i$	$1.0731 \times 10^{- 29}$
$m_{P *}$ [kg]	$2.1765 \times 10^{- 8}$	$2.2013 \times 10^{- 8} i$	$1.6430 \times 10^{- 9} i$	$2.4948 \times 10^{- 10}$
$t_{P *}$ [s]	$5.3911 \times 10^{- 44}$	$5.0941 \times 10^{- 44} i$	$2.1991 \times 10^{- 38} i$	$2.7243 \times 10^{- 34}$
$T_{P *}$ [K]	$1.4168 \times 10^{+ 32}$	$1.4994 \times 10^{+ 32} i$	$3.4733 \times 10^{+ 26} i$	$2.8037 \times 10^{+ 22}$
$ℓ_{P *}^{3}$ [m³]	$4.2218 \times 10^{- 105}$	$3.8124 \times 10^{- 105} i$	$5.3030 \times 10^{- 95} i$	$1.2358 \times 10^{- 87}$
$p_{P *}$ [kg m/s]	$6.5249$	$6.7506 i$	$0.0028 i$	$9.8272 \times 10^{- 6}$
$E_{P *}$ [J]	$1.9561 \times 10^{9}$	$2.0702 \times 10^{9} i$	$4.7954 \times 10^{3} i$	$0.3871$
$a_{P *}$ [m/s²]	$5.5609 \times 10^{+ 51}$	$6.0200 \times 10^{+ 51} i$	$7.7686 \times 10^{+ 43} i$	$1.4459 \times 10^{+ 38}$
$ρ_{P *}$ [kg/m³]	$5.1553 \times 10^{+ 96}$	$- 5.7740 \times 10^{+ 96}$	$- 3.0982 \times 10^{+ 85}$	$2.0188 \times 10^{+ 77}$
$ℓ_{P *}^{2}$ [m²]	$2.6122 \times 10^{- 70}$	$- 2.4404 \times 10^{- 70}$	$- 1.4115 \times 10^{- 63}$	$1.1516 \times 10^{- 58}$
$F_{P *}$ [N]	$1.2103 \times 10^{+ 44}$	$1.3252 \times 10^{+ 44}$	$1.2764 \times 10^{+ 35}$	$3.6072 \times 10^{+ 28}$
$μ_{*}$ [kg m C⁻²]	1.2569e-6	1.2012e-6	0.0387	72.8061
T	$T (α) \approx 0.9775$	$T (α_{2}) \approx 1.0228$	$T (α_{3}) = T (α)$	$T (α_{4}) = R (α)$
A	$A (α) \approx 0.0224$	$A (α_{2}) \approx - 0.0229$	$A (α_{3}) \approx - 3.9323$	$A (α_{4}) = A (α)$
R	$R (α) \approx 1.2843 \times 10^{- 4}$	$R (α_{2}) = R (α)$	$R (α_{3}) \approx 3.9548$	$R (α_{4}) = T (α)$
$π_{1} = π α_{*} / α$		-3.0712	-551.2868	$2.3910 \times 10^{4}$
$π_{2} = π α / α_{*}$		-3.2136	-0.0179	$4.1278 \times 10^{- 4}$

Table 2. Selected BB mergers discovered with LIGO and Virgo. Masses in

M_{⊙}

Table 2. Selected BB mergers discovered with LIGO and Virgo. Masses in

M_{⊙}

Event	$M_{1}$	$M_{2}$	$M_{C}$	$k_{1}$	$k_{2}$	$k_{C}$
GW170817	$1 . 46_{- 0.10}^{0.12}$	$1 . 27_{- 0.09}^{0.09}$	$2.8$	4.39	4.39	3.03
GW190425	$2 . 00_{- 0.2}^{0.6}$	$1 . 4_{- 0.3}^{0.3}$	$3 . 4_{- 0.1}^{0.3}$	4.39	4.39	3.15
GW200105	$8 . 9_{- 1.5}^{+ 1.2}$	$1 . 9_{- 0.2}^{0.3}$	$10 . 9_{- 1.2}^{+ 1.1}$	2.76	4.39	2.38
GW200115	$5 . 7_{- 2.1}^{+ 1.8}$	$1 . 5_{- 0.3}^{0.7}$	$7 . 1_{- 1.4}^{+ 1.5}$	3	4.39	2.64

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The Imaginary Universe (on the Three Complementary Sets of Measurement Units Defining Three Dark Electrons)

Abstract

1. Introduction

2. Three Complementary Fine-Structure Constants

2.1. Reflectance

2.2. Transmittance and Absorptance

2.3. Summary

3. Complementary Sets of Planck Units

4. Black Body Objects

5. Complex Energies

6. Complex Forces

7. Extended Bohr Model

8. BB Complex Gravity and Temperature

9. BB Mergers

10. BB Fluctuations

11. Discussion

Acknowledgments

Abbreviations

Appendix A. Mlg Transmittance, Absorptance, and Reflectance as Functions of π Only

Appendix B. π-like Constants

Appendix C. Mlg Fresnel Equation and Euclid’s Formula

Appendix D. Why α Is Better for Biological Evolution than α 2 ?

Appendix E. Planck Units and HUP

Appendix F. Other Definitions of Complex Energies

Appendix G. Hall Effect

References

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