The Imaginary Universe

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The Imaginary Universe

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Szymon Łukaszyk^*

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Social Change and Transition

Submitted:

29 May 2024

Posted:

30 May 2024

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Abstract

Maxwell's equations in vacuum provide the negative speed of light -c, which leads to imaginary Planck units. However, the second, negative fine-structure constant $\alpha_2^{-1} \approx -140.178$, present in the Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene, establishes the different, negative speed of light in vacuum $c_2 \approx -3.06 \times 10^8~\text{[m/s]}$, which introduces imaginary Planck units different in magnitude from those parametrized with $c$. Furthermore, algebraic relations between the fine-structure constant hint that the fine-structure constant does not vary over time. It follows that electric charges are the same in real and imaginary dimensions. We model neutron stars and white dwarfs, emitting perfect black-body radiation, as \textit{objects} having energy exceeding their mass-energy equivalence ratios. We define complex energies in terms of real and imaginary natural units. Their imaginary parts, inaccessible for direct observation, store the excess of these energies. It is conjectured that the maximum atomic number $Z=238$. A black-body \textit{object} is in the equilibrium of complex energies if its radius $R_\text{eq} \approx 1.3833~R_{\text{BH}}$, which is close to the photon sphere radius $R_{\text{ps}}=1.5~R_{\text{BH}}$, and marginally greater than a locally negative energy density bound of $4/3~R_{\text{BH}}$. The complex force between real masses and imaginary charges leads to the black-body object's surface gravity and generalized Hawking radiation temperature, which includes its charge. Furthermore, this force agrees with the physical parameters of the hydrogen atom. The proposed model takes into account 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 stars' mergers and the associated fast radio bursts (CHIME) without resorting to any hypothetical types of exotic stellar \textit{objects}.

Keywords:

Subject: Physical Sciences - Mathematical Physics

I. 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. More precisely, in the

- 1

-dimensional void, a 0-dimensional point appeared, inducing the appearance of other points that were indistinguishable from the first one. 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

1, 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 + b

, and the scalar product defined by

x \cdot y = (x_{1}, t o 1 e m . . ., x_{a}, i x_{1}^{'}, t o 1 e m . . ., i x_{b}^{'}) (y_{1}, t o 1 e m . . ., y_{a}, i y_{1}^{'}, t o 1 e m . . ., 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 selection-less [8] and a time-less [9] assembly process.

However, dimensional properties are not uniform. For regular convex n-polytopes in natural dimensions, 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 > 3

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

R^{4}

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

R^{3} \times I

space provides a continuum of homeomorphic but non-diffeomorphic differentiable structures. Each piece of individually memorized information is homeomorphic to the corresponding piece of individually perceived information but remains non-diffeomorphic (non-smooth). This allows the variation of phenotypic traits within populations of individuals [12]. Hence, selection [8] and time [9] emerged and the evolution of information entered into directed exploration provided biological evolution. Exotic

R^{4}

solves the problem of extra dimensions of nature, and perceived space requires a natural number of dimensions [13]. Each biological cell perceives an emergent space of three real dimensions and one imaginary (time) observer-dependently [14] 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 called the emergent dimensionality (ED) [5,6,9,12,15,16]. Appendix E presents some arguments to support the claim that perceived dimensionality sets favorable conditions for biological evolution to emerge.

Each dimension requires certain measurement units. In real dimensions, Max Planck in 1899 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" [17]. Planck units utilize the Planck constant h introduced in his black-body radiation formula. However, in 1881, George Stoney derived a system of natural units [18] based on the elementary charge e (Planck’s constant was unknown at this time). The ratio of the Stoney units to the Planck units is

\sqrt{α}

, where

α

is the fine-structure constant. This study derives a complementary set of natural units applicable to imaginary dimensions, including imaginary units, based on the discovered negative fine-structure constant,

α_{2}

Imaginary and negative physical quantities are the subject of ongoing research. In particular, the subject of scientific research is the thermodynamics in the complex plane. For example, Lee–Yang zeros [19,20] and photon-photon thermodynamic processes under negative optical temperature conditions [21] 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 [22]. However, 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 real eigenvalues of Hermitian operators. Unlike charges, negative, real masses are 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 [23]. In Section VI, we show that negative masses also result from the merging of black-body objects.

Furthermore, the study introduces a model for storing the excess energy of neutron stars and white dwarfs that exceeds their mass–energy equivalences in imaginary dimensions. The model results in an upper bound on the size-to-mass ratio of their cores, where the Schwarzschild radius sets the lower bound.

The remainder of this paper is organized as follows. Section II shows that the Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene include the second negative fine-structure constant

α_{2}

as a fundamental constant of nature. Section III shows that this second fine-structure constant endows us with the

α_{2}

-natural units. Section IV introduces the concept of a black-body object in thermodynamic equilibrium that emits perfect black-body radiation and reviews its necessary properties. Section V introduces the complex mass and charge energies expressed in terms of real and imaginary

α_{2}

-Planck units introduced in Section III and applies them to black-body objects. Section VI 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 VII discusses fluctuations of black-body objects. Section VIII defines complex forces. The complex force between real masses and imaginary charges is used in Section IX to derive a black-body object surface gravity and the generalized Hawking radiation temperature. Section X discusses complex forces in the context of the Bohr model of the hydrogen atom. Section XI summarizes the findings of this study. Certain prospects for further research are provided in the Appendices.

II. The Second Fine-Structure Constant

Numerous publications have 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

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. 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 [24] (Equation. 3), whereas a spectrally flat absorptance (A)

A \approx π α \approx 2.3 %

has been reported [25,26] for photon energies between approximately

0.5

and

2.5

[eV]. T is related to reflectance (R) [27] (Equation 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 [28] (Equations 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 [29] (Equation 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 [Ω]

is the vacuum impedance,

σ = e^{2} / (4 ℏ) = π α / η \approx 6.0853 \times 10^{- 5} [Ω^{- 1}]

is the MLG conductivity [30], 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 [24,25,26,29,31,32].

As a consequence of the conservation of energy

(T + A) + R = 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.

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, \end{matrix}

(12)

is remarkably 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 (4) value, and the same results can (only) be obtained for

T + A

(cf. Appendix A).

We further note that the relation (12) corresponds to the following identity

\frac{α + α_{2}}{α α_{2}} = - π,

(13)

between the roots (10) and (11), which is also present in the MLG Fresnel equations and the corresponding Euclid formula (cf. Appendix C).

Owing to these dependencies on

π

only between the fine-structure constants

α

and

α_{2}

they do not vary over time. They could not vary in the first undirected [8] and time-less [9] phase of the evolution of information in the universe.

These results are also intriguing in the context of a peculiar algebraic expression for the fine-structure constant [33]

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

(14)

that contains a free

π

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

α^{- 1}

, which according to the CODATA 2018 value is

137.035999084

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

Using equations (12) and (14), we can express the negative reciprocal of the 2^nd 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 .

(15)

Using equations (14) and (15), T (3), R (4), and A (5) of MLG for the normal incidence of EMR can be expressed simply by

π

(cf. Appendix B). Moreover, equation (8) includes two

π

-like constants for two surfaces with positive and negative Gaussian curvatures (see Appendix D).

III. Set of $α_{2}$ -Planck Units

In this section, we derive the complementary Planck units based on the second fine-structure constant

α_{2}

. We further call these

α_{2}

-Planck units. Natural units can be derived from numerous starting points [6,34] (cf. Appendices Appendix F and Appendix G). 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 the Rydberg units, where it is

c α / 2

2. Furthermore, c as the velocity of the electromagnetic wave is derivable from Maxwell’s equations in vacuum

\nabla^{2} E = μ_{0} ϵ_{0} \frac{\partial^{2} E}{\partial t^{2}}, \frac{\partial^{2} E}{\partial x^{2}} = μ_{0} ϵ_{0} \frac{\partial^{2} E}{\partial t^{2}},

(16)

where

E

is the electric field, and

μ_{0}

is the vacuum permeability (magnetic constant). Without postulating any solution to this equation but by simple substitution

\partial x ℓ_{*}

and

\partial t t_{*}

\partial^{2} E E_{*}

factors out, and we obtain the well-known

1 = μ_{0} ϵ_{0} c^{2},

(17)

symmetric in its electric and magnetic parts [35] from which the bivalued

c = \pm 1 / \sqrt{μ_{0} ϵ_{0}}

can be obtained, knowing the values of

μ_{0}

and

ϵ_{0}

. We note that it is

c^{2}

, not c, present in mass-energy equivalence, the Lorentz factor, the black hole (BH) potential, etc. 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}

In the following, we assume the universality of the real elementary electric charge e that defines both matter and antimatter, Planck constant h, uncertainty principle parameter, and gravitational constant G (i.e., we assume that there are no counterparts to these physical constants in other physical dimensions in our model and that these dimensional constants are positive). The last two assumptions are probably 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

ℓ_{P}

and the speed of light in vacuum [36]. We note that antimatter obeys gravity [37], which is consistent with the findings of this study.

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}

. 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 (47) and the binary entropy variation [5,6].

To accommodate the negativity of the fine-structure constant discovered in the preceding section, we must introduce the imaginary Planck charge

q_{P i}

such that its square would yield a negative value of

α_{2}

\begin{matrix} q_{P}^{2} = \frac{e^{2}}{α} & \neq q_{P i}^{2} = \frac{e^{2}}{α_{2}} \Rightarrow q_{P i} = a e, a \in I, \\ e^{2} = q_{P}^{2} α & = q_{P i}^{2} α_{2} . \end{matrix}

(18)

Next, we note that an imaginary

q_{P i}

, 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 (we assume that h is positive). Let us call them

c_{2}

and

\tilde{ϵ_{0}}

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

(19)

From this equation, we find that

\tilde{ϵ_{0}} c_{2} < 0

, because the values of the other constants are known. Next, we assume that the solution (17) of Maxwell’s equations in vacuum is valid for other values of the constants involved. Let us call the unknown magnetic constant

μ_{2}

. Therefore,

μ_{0} ϵ_{0} c^{2} = μ_{2} \tilde{ϵ_{0}} c_{2}^{2} = 1 .

(20)

From this and from

\tilde{ϵ_{0}} c_{2} < 0

, we conclude that the product

μ_{2} c_{2} < 0

. The quotient of the squared Planck charge and mass introduces the imaginary Planck mass

m_{P i}

\frac{q_{P}^{2}}{m_{P}^{2}} = \frac{q_{P i}^{2}}{m_{P i}^{2}} = 4 π ϵ_{0} G .

(21)

The value of the imaginary Planck mass

m_{P i}

can be calculated from the equation (21) by determining the value of the imaginary Planck charge

q_{P i}

from the equation (18). From (21) we also conclude that

\tilde{ϵ_{0}} = ϵ_{0} > 0

and then by (20) that

μ_{2} > 0

and

c_{2} < 0

. Knowing

m_{P i}

we can determine the value of the negative non-principal square root of

c_{2} = \pm 1 / \sqrt{μ_{2} ϵ_{0}}

of the relation (20) as

c_{2} = \frac{q_{P i}^{2}}{4 π ϵ_{0} ℏ} \approx - 3.066653 e 8 [m / s],

(22)

which is greater than the speed of light in vacuum c in modulus.

The mass, length, time, and charge units can express all the electrical units. Therefore, along with temperature, amount of substance, and luminous intensity, these are the base units of the International System of Quantities (ISQ). We further conclude that the magnetic constant

μ_{2}

is lower than

μ_{0}

μ_{0} = \frac{4 π ℏ α}{c e^{2}} \approx 1.2569 \times 10^{- 6} [kg \cdot m \cdot C^{- 2}], μ_{2} = \frac{4 π ℏ α_{2}}{c_{2} e^{2}} \approx 1.2012 \times 10^{- 6} [kg \cdot m \cdot C^{- 2}] .

(23)

Unlike the electric constant

ϵ_{0}

, both magnetic constants

μ

are independent of the unit of time. Furthermore, negative

α_{2}

and

c_{2}

lead to a second, time-dependent but negative vacuum impedance

η_{2} = - \frac{4 π α_{2} ℏ}{e^{2}} = - \frac{1}{ϵ_{0} c_{2}} \approx - 368.29 [kg \cdot m^{2} \cdot s^{- 1} \cdot C^{- 2}] (| η_{2} | < | η |) .

(24)

Finally, combining relations (18) and (19) yields

e^{2} = 4 π ϵ_{0} ℏ c α = 4 π ϵ_{0} ℏ c_{2} α_{2},

(25)

which leads to the following important relation between the speeds of light in vacuum c and

c_{2}

, and the fine-structure constants

α

and

α_{2}

c α = c_{2} α_{2},

(26)

valid for both the principal and non-principal square roots of the equation (20).

c α

is also the electron’s velocity at the first circular orbit in the Bohr hydrogen atom model3 to which we shall return in Section X. Furthermore, the relation (26) introduces an interesting interplay between

α

vs.

α_{2}

and c vs.

c_{2}

that, as we conjecture, should be able to explain

ν = 5 / 2

state in the fractional quantum Hall effect in the 2D system of electrons, as well as other fractional states with an even denominator [38] (cf. Appendix H). The equation (26) is not the only

α

α_{2}

relation. Along with the two

π

-like constants

π_{1}

π_{2}

(relations (D.3) and (D.5), cf. Appendix D)

\frac{α_{2}}{α} = \frac{c}{c_{2}} = \frac{π_{1}}{π} = \frac{π}{π_{2}} = \frac{m_{P}^{2}}{m_{P i}^{2}} = \frac{q_{P}^{2}}{q_{P i}^{2}} \approx - 0.9776 .

(27)

Therefore, the non-principal square root of

c = \pm 1 / \sqrt{μ_{0} ϵ_{0}}

and principal square root of

c_{2} = \pm 1 / \sqrt{μ_{2} ϵ_{0}}

in (20) also introduce imaginary (

- c

)-Planck units and real (

- c_{2}

)-Planck units, respectively. In particular, the imaginary (

- c

)-Planck time parameterizes holographic sphere (HS) time relations [5,6]. We conjecture that

α_{2}

-Planck units are appropriate for expressing physical quantities of

I^{3} \times R

Euclidean space rather than

R^{3} \times I

Euclidean space that we perceive because of the minimum energy principle (cf. Appendix E). 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.

The negative speed of light

c_{2}

(22) leads to the complementary Planck charge

q_{P i}

, length

ℓ_{P i}

, mass

m_{P i}

, time

t_{P i}

, and temperature

T_{P i}

that redefined by square roots containing

c_{2}

raised to odd powers (1, 3, 5) become bivalued and real-imaginary because c and

c_{2}

are bivalued. In other words, both the Planck and

α_{2}

-Planck units have four forms equal in modulus: real positive, real negative, imaginary positive, and imaginary negative. However, here, we consider mostly real, positive

α

-Planck units and imaginary, positive

α_{2}

-Planck units (hence the subscript i).

Principal square roots of the base

α_{2}

-Planck units (for negative

c_{2}

) that can be expressed, using the relation (26), in terms of base Planck units

q_{P}

ℓ_{P}

m_{P}

t_{P}

, and

T_{P}

are

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

(28)

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

(29)

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

(30)

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

(31)

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

(32)

Most Planck units derived from the

α_{2}

-Planck base units (28)-(32) are also imaginary. These include the

α_{2}

Planck volume

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

(33)

the

α_{2}

Planck momentum

p_{P i} = m_{P i} c_{n} = \sqrt{\frac{ℏ c_{n}^{3}}{G}} = m_{P} c \sqrt{\frac{α^{3}}{α_{2}^{3}}} \approx i 6.7504 [kg m / s] (| m_{P i} c_{n} | > | m_{P} c |),

(34)

the

α_{2}

Planck energy

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

(35)

and the

α_{2}

Planck acceleration

a_{P i} = \frac{c_{n}}{t_{P i}} = \sqrt{\frac{c_{n}^{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 i} | > | a_{P} |) .

(36)

However, the

α_{2}

-Planck density

ρ_{P 2} = \frac{m_{P i}}{ℓ_{P i}^{3}} = \frac{c_{n}^{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} |),

(37)

and the

α_{2}

-Planck area

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

(38)

are real and bivalued, similar to Planck density

ρ_{P}

and area

ℓ_{P}^{2}

. 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}),

(39)

are strictly positive.

Coulomb’s law for elementary charges and Newton’s law of gravity for Planck masses define the fine-structure constants

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

(40)

where

R_{*}

is a real or imaginary distance and

m_{P i}

is imaginary. The area of a sphere in the denominator of the Coulomb force requires further investigation.

The relations between time (31) and temperature (32)

α_{2}

-Planck units are inverted,

α^{5} t_{P i}^{2} = α_{2}^{5} t_{P}^{2}

α_{2}^{5} T_{P i}^{2} = α^{5} T_{P}^{2}

, and the energy-time version of Heisenberg’s uncertainty principle (HUP) is saturated using energy from the equipartition theorem for one bit of information [5,6,39]

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

(41)

Furthermore, eliminating

α

and

α_{2}

from the relations (28)-(30), yields

\begin{matrix} ℓ_{P} m_{P}^{3} = ℓ_{P i} m_{P i}^{3} and ℓ_{P} q_{P}^{3} = ℓ_{P i} q_{P i}^{3} . \end{matrix}

(42)

Contrary to the elementary charge e (18), there is no physically meaningful elementary mass

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

that would satisfy the relation (30)

M_{e}^{2} = α m_{P}^{2} = α_{2} m_{P i}^{2} .

(43)

Furthermore, there is no physically meaningful elementary (and imaginary) length

L_{e} \approx \pm i 9.7382 \times 10^{- 39} [m]

satisfying the equation (38)

L_{e}^{2} = α^{3} ℓ_{P i}^{2} = α_{2}^{3} ℓ_{P}^{2},

(44)

(which in modulus is almost 1660 times smaller than the Planck length), or an elementary temperature

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

abiding by (32)

T_{e}^{2} = α^{5} T_{P}^{2} = α_{2}^{5} T_{P i}^{2},

(45)

and close to the Hagedorn temperature of grand unified string models.

The Planck charge relation (18) and charge conservation principle imply that the elementary charge e is the quantum of charge in real and imaginary dimensions, while masses, lengths, temperatures, and other derived quantities that can vary with time are not similarly quantized. The universal character of the charges is additionally emphasized by the real

\sqrt{α}

multiplied by i in the imaginary charge energy (58) and imaginary

\sqrt{α_{2}}

in the real charge energy (59). Furthermore, the same forms of the relations (18) and (43) reflect the same forms of Coulomb’s law and Newton’s law of gravity, which are inverse-square laws.

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

\begin{matrix} M m m_{P}, & M_{i} m_{i} m_{P i}, m, m_{i} \in R \\ E m E_{P} & E_{i} m_{i} E_{P i}, \\ Q q e, & Q_{i} i Q = i q e, q \in Z, \\ λ l ℓ_{P}, & λ_{i} l_{i} ℓ_{P i}, l = \frac{2 π}{m}, l_{i} = \frac{2 π}{m_{i}}, \\ {R, D} {r, d} ℓ_{P}, & {R_{i}, D_{i}} {r_{i}, d_{i}} ℓ_{P i}, r, d, r_{i}, d_{i} \in R, \end{matrix}

(46)

where uppercase letters M, E, Q,

λ

, R, and D denote masses, energies, charges, Compton wavelengths, radii, and diameters (or lengths), lowercase letters denote multipliers of the positive real Planck units and imaginary

α_{2}

-Planck units, respectively, and subscript i refers to the multiplication of imaginary quantities. We note that the discretization of charges by integer multipliers q of the elementary charge e seems too far-reaching, considering the fractional charges of quasiparticles, particularly in the open research problem of the fractional quantum Hall effect (cf. Appendix H) and energy-dependent fractional charges in electron pairing [40].

IV. 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 [41,42]. Thus, sonoluminescence, particularly shrimpoluminescence [43], is probably emitted by collapsing micro-BBs. Micro-BH induced in glycerin by modulating acoustic waves has also been reported [44].

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 [45]. In adition, 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 [46]. This is an important unresolved issue, indicating that fundamental changes to the reigning

Λ

CDM model of cosmology are required [46]. The term object as a collection of matter is a misnomer because it neglects the (quantum) nonlocality [9,47] that is independent of the entanglement among particles [48], as well as the Kochen-Specker contextuality [49], and increases as the number of particles increases [50,51]. 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), however ridiculous and contrary to common sense4 that may sound. 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.

Entropic gravity [39] explains galaxy rotation curves without resorting to dark matter (which is not required to explain the rotation curves of certain galaxies, such as the massive relic galaxy NGC 1277 [55]), has been experimentally confirmed [56], and is decoherence-free [57]. It has been experimentally confirmed that (so-called) accretion instability is a fundamental physical process [58]. We conjecture that this process, which has already been recreated under laboratory conditions [59], is common for all BBs. As black-body radiation is a radiation of global thermodynamic equilibrium, it is patternless [60] (thermal noise) radiation that depends only on one parameter. In the case of BHs, this is known as Hawking [61] 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}

, where

d_{BH} \in R

. Furthermore, BHs absorb patternless information [6,62]. Therefore, because Hawking radiation depends only on the diameter of a BH, it is the same for a given BH, even though it is momentary as the BH fluctuates (cf. Section VII).

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 [63]

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 triangles5 corresponding to bits of information [5,6,39,63,64], 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].

BH entropy can be derived from the Bekenstein bound

S \leq \frac{2 π k_{B} R E}{ℏ c} = π k_{B} m d,

(47)

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 (47), it reduces to the BH entropy. In other words, the BH entropy saturates the Bekenstein bound (47)6.

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] Equation (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} .

(48)

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

N_{BB} \leq 3

[9]. Furthermore,

N_{1} = N_{BB} / 2

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

We 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}

(49)

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} .

(50)

Substituting

M_{BB}

from definition (49) into definition (50) and the latter into the Bekenstein bound (47), we obtain

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

(51)

which equals BB entropy (48) if

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

. Thus, the energy of all BBs with a generalized radius (49) is

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

(52)

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 V.

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 [65] using Penrose processes [66,67] to extract the electrostatic and/or rotational energy of BH [68]. Thus, any BH is defined by only one real parameter: its diameter, mass, temperature, energy, etc., each differing to one another by a multiplicative constant. In the complex Euclidean

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

space, an n-ball (

n = a + b i \in C

) is spherical only for a vanishing imaginary dimension and for the radius

r = 1 / \sqrt{π}

(

R = ℓ_{P} / \sqrt{π}

) [5,16], resulting in its information capacity

N = 4

, one unit of BH entropy [63]. This confirms the universality and applicability of BH entropy (48) to all BBs.

The interiors of the BBs are inaccessible to an exterior observer [63], which makes them similar to interior-less mathematical points representing real numbers on a number line7. However, a BH can embrace the real number that defines it. Three points forming a Planck triangle corresponding to a bit of information on a BH surface can store this parameter, and this is intuitively comprehensible. The area of a spherical triangle is larger than that of a flat triangle defined by the same vertices, provided the curvature is nonvanishing and depends on this curvature (this additional parameter defines it). Thus, the only meaningful spatial notion is the Planck area triangle, which encodes one bit of the classical information and 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 and an additional parameter corresponding to its charge? Fortunately, the relation (18) ensures that the charges are the same in the real and imaginary dimensions. Therefore, each charged Planck triangle of a BB surface is associated with at least three

R \times I

Planck triangles, each sharing one vertex or two vertices with this triangle in

R^{2}

. This configuration is capable of storing both curvature and charge. The Planck area

ℓ_{P}^{2}

(38) and the

R \times I

imaginary Planck area

ℓ_{P} ℓ_{P i} = ℓ_{P}^{2} \sqrt{α_{2}^{3} / α^{3}} \approx \pm 0.9666 i ℓ_{P}^{2}

, which is smaller in modulus, 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 electrostatics act in a tangential direction.

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].

BBs are perfectly spherical. However, their mergers, to which we shall return in Section VI, are also perfectly spherical, as experimentally confirmed [69] based on the registered gravitational event GW170817. 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.

The previously discussed considerations may be confusing to the reader, as the energy (52) 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. In the following section, we model a part of the energy of NS and WD that exceeds

M c^{2}

as imaginary and thus unmeasurable.

V. BB 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},

(53)

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}

8 was proposed in ref. [70]. Equation (53) considers the real masses

M_{R}

and charges

Q_{R}

. To store the surplus energy, we modified it to a form involving real physical quantities expressed terms in Planck units and imaginary physical quantities expressed terms of the imaginary

α_{2}

-Planck units using relations (26), (30), (35), (46), and (25)

a s \begin{matrix} \frac{e}{2 \sqrt{π ϵ_{0}}} = \sqrt{α c ℏ} = \sqrt{α_{2} c_{2} ℏ} . \end{matrix}

(54)

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

E_{M Q_{i}} E_{M} + E_{Q_{i}} = M c^{2} + \frac{Q_{i}}{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},

(55)

of real charge and imaginary mass

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

(56)

and of real mass and imaginary mass

\begin{matrix} E_{M M_{i}} & M c^{2} + M_{i} c_{2}^{2} = (m + \sqrt{\frac{α^{5}}{α_{2}^{5}}} m_{i}) E_{P}, \end{matrix}

(57)

as shown in Figure 2.

We neglect the energy of real and imaginary charges

E_{Q Q_{i}}

, because by equation (18), the unit of charge is the same in real and imaginary dimensions. The mass-energy equivalence relates mass M or

M_{i}

to the speed of light c or

c_{2}

Energies (55) and (56) yield two different charge energies corresponding to the elementary charge, imaginary quantum

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

(58)

and the - larger in modulus - real quantum

E_{Q} (q = \pm 1) = \pm \sqrt{α_{2}} E_{P i} \approx \pm 1.7684 \times 10^{8} [J] .

(59)

Furthermore,

\forall q

α^{2} E_{Q i} = i α_{2}^{2} E_{Q}

The squared moduli of the complex energies (55)-(57), expressed in terms of the Planck energy, are

\begin{matrix} | E_{M Q_{i}} |^{2} = (M^{2} + q^{2} α m_{P}^{2}) c^{4} = (m^{2} + q^{2} α) E_{P}^{2}, \end{matrix}

(60)

\begin{matrix} | E_{Q M_{i}} |^{2} = \frac{α^{4}}{α_{2}^{4}} (q^{2} α m_{P}^{2} - M_{i}^{2}) c^{4} = \frac{α^{4}}{α_{2}^{4}} (q^{2} α - \frac{α}{α_{2}} m_{i}^{2}) E_{P}^{2}, \end{matrix}

(61)

\begin{matrix} | E_{M M_{i}} |^{2} = (M^{2} - \frac{α^{4}}{α_{2}^{4}} M_{i}^{2}) c^{4} = (m^{2} - \frac{α^{5}}{α_{2}^{5}} m_{i}^{2}) E_{P}^{2} . \end{matrix}

(62)

Theorem 1.

Complex energies (55)-(57) cannot simultaneously have real and imaginary parts equal in modulus.

Proof.

The complex energies

E_{M Q_{i}}

and

E_{Q M_{i}}

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

q^{2} α = m^{2} = - \frac{α}{α_{2}} m_{i}^{2} .

(63)

However, they cannot be balancedsimultaneously with energy

E_{M M_{i}}

, which is balanced for

m^{2} = - \frac{α^{5}}{α_{2}^{5}} m_{i}^{2} \neq - \frac{α}{α_{2}} m_{i}^{2} .

(64)

□

Because, owing to equation (18) charges are the same in real and imaginary dimensions, the squared moduli of the complex energies

E_{M Q_{i}}

and

E_{Q M_{i}}

must be equal, allowing us to obtain the value of the imaginary mass

M_{i}

as a function of mass M and charge Q in this equilibrium:

\begin{matrix} m_{i} = \pm \sqrt{\frac{α_{2}}{α} [q^{2} α (1 - \frac{α_{2}^{4}}{α^{4}}) - \frac{α_{2}^{4}}{α^{4}} m^{2}]} . \end{matrix}

(65)

In particular for

q = 0

equation (65) yields

m_{i}^{2} = - \frac{α_{2}^{5}}{α^{5}} m^{2} or M_{i} = \pm i \frac{α_{2}^{2}}{α^{2}} M \approx \pm 0.9557 i M,

(66)

which corresponds to equation (64). Because the mass

m_{i} \in R

, the square root argument must be nonnegative in equation (65)

\begin{matrix} m \geq | q | \sqrt{α (\frac{α^{4}}{α_{2}^{4}} - 1)} \approx | q | 0.0263 . \end{matrix}

(67)

This means that the masses of the uncharged micro-BHs (

q = 0

) in thermodynamic equilibrium can be arbitrary. 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 mass

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

comparable to the mass of a sand grain.

We note that only the masses satisfying

M < 2 π m_{P} \approx 1.3675 \times 10^{- 7} [kg]

have Compton wavelengths larger than the Planck length [6]. It should be noted that a classical description has been ruled out on a microgram (

1 \times 10^{- 9} [kg]

) mass scale [71]. Comparing this bound with bound (67) yields the charge multiplier q corresponding to an atomic number:

Z = ⌊\frac{2 π}{\sqrt{α (\frac{α^{4}}{α_{2}^{4}} - 1)}}⌋ = ⌊238.7580⌋ = 238,

(68)

of a hypothetical element, which, as we conjecture, sets the limit on an extended periodic table, and is higher than the accepted limit of

Z = 184

(unoctquadium). More massive elements would have Compton wavelengths smaller than the Planck length, which is physically implausible because the Planck area is the smallest area required to encode one bit of information [6,39,63,64]. From equation (67), we can also obtain the maximum wavelength

l = 2 π / m

corresponding to the charge q. For

q^{2} = 1

it is

λ < 3.8589 \times 10^{- 33} [m]

with

l < 238.7580

corresponding to the bound (68).

Theorem 2.

Complex energies (55)-(57) are equal

| 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_{2}^{4}}{α^{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}

(69)

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} .

(70)

Proof.

Direct calculation proves the relation (69) and if the squared moduli (60)-(62) are equal to some constant energy

\begin{matrix} | E_{M Q_{i}} |^{2} = | E_{Q M_{i}} |^{2} = {| E_{M M_{i}} |}^{2} A^{2} E_{P}^{2}, \end{matrix}

(71)

then subtracting

| E_{M Q_{i}} |^{2} - {| E_{Q M_{i}} |}^{2}

yields

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

(72)

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})},

(73)

which substituted into the relation (72) yields

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

(74)

Finally, substituting equation (74) into equation (60) yields

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

(75)

□

We can interpret the squared generalized energy of the BBs (52) as the squared modulus of the complex energy of the real mass

E_{M Q_{i}}

, taking the observable real energy

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

of the BB as the real part of this energy. Thus

\begin{matrix} \frac{k^{4}}{4} m_{BB}^{2} = m_{BB}^{2} + q_{BB}^{2} α, q_{BB}^{2} α = m_{BB}^{2} (\frac{k^{2}}{4} - 1), \end{matrix}

(76)

where

q_{BB}^{2} α

represents the charge surplus energy exceeding

M_{BB} c^{2}

. Similarly, we can interpret the squared generalized energy of the BBs (52) as the squared modulus of the complex energy of the imaginary mass

E_{Q M_{i}}

. Thus

\begin{matrix} \frac{k^{2}}{4} m_{BB}^{2} = \frac{α^{4}}{α_{2}^{4}} (q_{BB}^{2} α - \frac{α}{α_{2}} m_{i B B}^{2}) . \end{matrix}

(77)

Substituting

q_{BB}^{2} α

from equation (76) into equation (77) transforms the equilibrium condition (65) into a function of the STM k instead of the charge q

\begin{matrix} m_{i BB} = \pm m_{BB} \sqrt{\frac{α_{2}}{α} [\frac{k^{2}}{4} (1 - \frac{α_{2}^{4}}{α^{4}}) - 1]}, \end{matrix}

(78)

which yields the imaginary mass of a BH (for

k = 2

) and corresponds to the relation (66) between uncharged masses M and

M_{i}

, which is, remarkably, independent of the STM. The square root argument in equation (78) must be nonnegative, because

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

. This leads to the maximum STM-bound

\begin{matrix} k \leq \frac{2}{\sqrt{1 - \frac{α_{2}^{4}}{α^{4}}}} \approx 6.7933 = k_{\max} . \end{matrix}

(79)

The relations (76) and (78) are shown in Figure 3.

Furthermore, using equation (26), from equation (78) we obtain the relation between the real and imaginary BH energies

E_{BH i} = \pm i E_{BH}

, which are equal in modulus. In general, equation (78) relates the BBO energies as

E_{BBI i}^{2} = E_{BB}^{2} [\frac{α^{4}}{α_{2}^{4}} (\frac{k^{2}}{4} - 1) - \frac{k^{2}}{4}] .

(80)

The maximum STM-bound

k_{\max}

(79) sets the bounds on BB energy (52), mass, and radius (49)

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

(81)

In particular, using the relations (46),

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

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

Furthermore, equations (67) and (79) expressed in terms of the generalized radius (49)

k = d_{BB} / (2 m_{BB})

set the bound on the minimum mass of BB if

| E_{M Q_{i}} |^{2} = {| E_{Q M_{i}} |}^{2}

\begin{matrix} m_{BB} > max \{q_{BB} \sqrt{α (\frac{α^{4}}{α_{2}^{4}} - 1)}, \frac{d_{BB}}{4} \sqrt{1 - \frac{α_{2}^{4}}{α^{4}}}\}, \end{matrix}

(82)

where

q_{BB}^{2} α = \frac{d_{BB}^{2}}{16} \frac{α_{2}^{4}}{α^{4}}

(83)

defines a condition in which neither

q_{BB}

nor

d_{BB}

can be further increased to reach its counterpart (defined by

d_{BB}

and

q_{BB}

) in bound (82). Thus, for example, 1-bit BB (

d_{BB} = 1 / \sqrt{π}

) corresponds to

q_{BB} > 1.5780

π

-bit BB (

d_{BB} = 1

) corresponds to

q_{BB} > 2.7969

, whereas the conjectured heaviest element with atomic number

q_{BB}

(68) corresponds to

d_{BB} = \pm \frac{8 π}{\sqrt{1 - \frac{α_{2}^{4}}{α^{4}}}} \approx \pm 85.3666 .

(84)

In the case of a BB, we obtain the equality of all three complex energies (55)-(57) by substituting

A = m_{BB} k / 2

from (49) into equation (71) and comparing this with (69). This yields

\begin{matrix} k_{eq} = 2 \sqrt{1 + \frac{α_{2}^{4}}{α^{4}}} \approx 2.7665, \end{matrix}

(85)

where all three energies are equal. The equilibrium

k_{eq}

(85) and maximum

k_{\max}

(79) STMa satisfy

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

The BB in the energy equilibrium

k_{eq}

bearing the elementary charge (

q^{2} = 1

) would have mass

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

, imaginary mass

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

, wavelength

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

, and imaginary wavelength

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

. On the other hand, equation (76) provides the BB charge in equilibrium (71) 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[72], where

k = 3

is a photon sphere radius9. Any object that undergoes complete gravitational collapse passes through an ultracompact stage [73], 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 [73]. 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 [74] 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 [75] 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 [76]. As the surface gravity increases, photons from further behind the NS become visible. At

k \approx 3.52

the entire NS surface becomes visible [77]. The relative increase in brightness between the maximum and minimum of a light curve is greater for

k < 3

than for

k > 3

[77]. Therefore, the equilibrium STM ratio

k_{eq} \approx 2.7665

(85) is 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 approximations10. GR neglects the value of the fine-structure constants

α

and

α_{2}

, which, similar to

π

or the base of the natural logarithm, are fundamental constants of nature.

VI. BB Mergers

As the entropy (Boltzmann, Gibbs, Shannon, von Neumann) of independent systems is additive, a merger of BB₁ and BB₂ having entropies11 (48)

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

and

S_{2} = \frac{1}{4} k_{B} π d_{2}^{2}

, produces a BB_c with entropy

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

(86)

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 F).

Substituting the generalized diameter (49) into the entropy relation (86) establishes a Pythagorean relation between the generalized energies (52) 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_{k} \in {R, I} .

(87)

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 [86]. 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 [87]. Furthermore, it has been suggested that outside the GR, merging BHs may differ from their GR counterparts [88].

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

M_{Ch} \approx 1.4 M_{⊙}

[89] and the accepted value of the analogous Tolman–Oppenheimer–Volkoff NS mass limit is

M_{TOV} \approx 2.9 M_{⊙}

[90,91]. There is no accepted value for the BH mass limit. The conjectured value is

5 \times 10^{10} M_{⊙} \approx 9.95 E 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 E 15 m

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

M_{B H} \approx 3.08 E 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 are classified as BH mergers. Only a few were classified otherwise, including GW170817, GW190425, GW200105, and GW200115, as listed in Table 1.

Equation (87) 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) portal12, indicates that the total mass of a merger is the sum of the masses of the merging components. Thus

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

(88)

We can use the squared moduli

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

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

, and

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

to derive some information about the merger from equation (87). We shall initially assume

m_{k} \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 [23]) inaccessible for direct observation, unlike charges.

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

with the first inequality (88) yields:

\begin{matrix} | 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}, \end{matrix}

(89)

On the other hand,

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

with inequality (89) leads to (

α_{2} < 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} .

(90)

But

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

with the first inequality (88) 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},

(91)

contradicting inequality (90) (

α_{2}^{5} < 0

), while

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

with inequality (90) 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},

(92)

contradicting the first inequality (88) and is consistent with the second inequality (88) introducing the product of positive and negative masses.

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

with inequality (91) 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},

(93)

contradicting the inequality (90) and so on.

The additivity of the entropy (86) of statistically independent merging BBs, both in global thermodynamic equilibrium, defined by their generalized radii (49), introduces the energy relation (87). This relation, equality of charges in real and imaginary dimensions (18), and the BB complex energies (60)-(62) induce imaginary, negative, and mixed masses during the merger. Thus, the BB merger spreads in all dimensions, not only observable ones, as a gravitational event associated with a fast radio burst (FRB) event, as reported [92] based on the gravitational event GW1904251 and FRB 20190425A event13. Furthermore, IXPE14 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 [93]. 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 [94], and such extreme rotation measure reversal has never been observed before in any FRB or any astronomical object.

In the observable dimensions 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 (Figure 3). From equation (87) and the first inequality (88), 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} .

(94)

For two merging BHs

k_{1} = k_{2} = 2

and the relation (94) yields

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

Table 1 lists the mass-to-size ratios

k_{{BB}_{C}}

calculated according to equation (87), which provide the measured mass

M_{{BB}_{C}}

of the merger and satisfy inequality (94). The mass-to-size 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.

VII. BB Fluctuations

A relation [95] (p.160) describing a BH information capacity, having an initial information capacity15

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}},

(95)

was subsequently generalized [6] (Equation (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) = 16 π m^{2} \pm 8 π d m + π d^{2} .

(96)

Equation (96) can be further generalized, using the generalized diameter

d = 2 k \hat{m}

(49), 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}),

(97)

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}

(98)

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},

(99)

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 4. Equation (99) 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 V.

VIII. Complex Forces

Coulomb’s force

F_{C}

between two charges is positive or negative, depending on the sign and type (real or imaginary) of the charges, as summarized below in the case of some real distance separating the charges:

\begin{matrix} q_{1} q_{2} > 0 & q 1 q_{2} < 0 \\ Q_{k} = q_{k} e & F_{C} > 0 & F_{C} < 0 \\ Q_{k} = i q_{k} e & F_{C} < 0 & F_{C} > 0 . \end{matrix}

(100)

Newton’s law of universal gravitation is also positive or negative, depending on the sign and type of masses, as summarized below:

\begin{matrix} m_{* 1} m_{* 2} > 0 & m_{* 1} m_{* 2} < 0 \\ M_{k} = m_{k} m_{P} & F_{G} > 0 & F_{G} < 0 \\ M_{i k} = m_{i k} m_{P i} & F_{2 G} < 0 & F_{2 G} > 0 . \end{matrix}

(101)

In the case of imaginary distance, the signs of the inequalities are opposite. We do not consider mixed real or imaginary radii and mixed forces (based on real and imaginary masses/charges), as the real and imaginary dimensions are orthogonal.

Complex energies (55)-(57) define the complex forces (similarly to the complex energy of real masses and charges (53), [70] Equation (7)) acting over the real and imaginary distances

R, R_{i}

. Using the relations (46), we obtain the following products

E_{1 m q_{i}} E_{2 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}),

(102)

E_{1 q m_{i}} E_{2 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} (q_{1} m_{i 2} + q_{2} m_{i 1})),

(103)

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

(104)

defining 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_{1 a b_{i}} E_{2 a b_{i}},

(105)

and three complex forces acting over an imaginary distance

R_{i}

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

(106)

where

A, B \in {M, Q}

and

a, b \in {m, q}

, and

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

(107)

With a further simplifying assumption of

r^{2} = r_{i}^{2}

, the forces acting on the real distanceR are stronger and opposite to the corresponding forces acting on the imaginary distance

R_{i}

even though the Planck force is lower than the

α_{2}

-Planck force (39). This is a strong assumption, but it seems correct. The general radius (49) and energy (52) are the same in Planck units and in

α_{2}

-Planck units; the STM remains the same.

IX. BB Complex Gravity and Temperature

We can use the complex force

F_{M Q_{i}}

(105) with the product (102) (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 2^nd 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}),

(108)

where

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

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

. Substituting

q_{BB} \sqrt{α}

from the BB equilibrium relation (76) and the mass taken from the generalized BB radius (49)

r_{BB} = k m_{BB}

into the relation (108) yields

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

(109)

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}} .

(110)

for all k. In particular,

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

(111)

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

(112)

The BB surface gravity (109) 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}

(113)

describing the BB temperature16 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 particular,

T_{BB} (k_{\max}) = \pm \frac{T_{P}}{2 π d_{BB}} (\frac{\sqrt{α^{4} - α_{2}^{4}}}{α^{2}} \pm i \frac{α_{2}^{2}}{α^{2}}) = \pm \frac{T_{P}}{2 π^{3} d_{BB}} (\sqrt{π^{4} - π_{1}^{4}} \pm i π_{1}^{2}) = \pm \frac{T_{P}}{2 π π_{2}^{2} d_{BB}} (\sqrt{π_{2}^{4} - π^{4}} \pm i π^{2}),

(114)

T_{BB} (k_{eq}) = \pm \frac{T_{P}}{2 π d_{BB}} \frac{α^{2} \pm i α_{2}^{2}}{\sqrt{α^{4} + α_{2}^{4}}} = \pm \frac{T_{P}}{2 π d_{BB}} \frac{π^{2} \pm i π_{1}^{2}}{\sqrt{π^{4} + π_{1}^{4}}} = \pm \frac{T_{P}}{2 π d_{BB}} \frac{π_{2}^{2} \pm i π^{2}}{\sqrt{π_{2}^{4} + π^{4}}},

(115)

reduce to the BH temperature for

α_{2} = 0

. We note that for

d_{BB} = 1

Re (T_{BB} (k_{\max})) \approx 6.6387 \times 10^{30} [K]

has a magnitude of the Hagedorn temperature of strings, whereas

T_{P} / (2 π) \approx 2.2549 \times 10^{31} [K]

It was shown [5] based on the Mandelstam-Tamm [96], Margolus–Levitin [97], and Levitin-Toffoli [98] theorems on the quantum orthogonalization interval that BBs generate dissipative structures through the solid-angle correspondence. The BB entropic work

W_{BB} = T_{BB} S_{BB} = T_{BB} \frac{1}{4} k_{B} N_{BB} = T_{BB} \frac{1}{4} k_{B} π d_{BB}^{2} = \frac{E_{P} d_{BB}}{4 k} (1 \pm i \sqrt{\frac{k^{2}}{4} - 1}),

(116)

is the work done by all active Planck triangles [5] of a BB. It is the product of the BB entropy (48) and the BB temperature (113). Figure 5 shows the BB temperature (113), energy (52), and entropic work (116) for

0 \leq N_{BB} \leq 5

k_{B} | T_{BB} | / E_{BB} = 2 / N_{BB}

is a rational number for natural

N_{BB}

Therefore, it seems, that a universe without

α_{2}

-imaginary dimensions (i.e., with

α_{2} = 0

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

α_{2}

as we would have to neglect the existence of graphene.

X. Hydrogen Atom

The Bohr model of the hydrogen atom is based on three assumptions that can be conveniently expressed in terms of Planck units using relations (46). The 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 .

(117)

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 .

(118)

Finally, the 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 proton17 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 .

(119)

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

α

. Joining relations (117) and (118) yields

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

(120)

which combined with (119) and using the relation (26) yields

V_{e} = v_{e} c = \frac{1}{n} α c = \frac{1}{n} α_{2} c_{n} \Leftrightarrow v_{e} = \frac{1}{n} α,

(121)

Thus, in the first circular orbit (

n = 1

) of this model, the electron velocity factor

v_{e} = α

Now, we assume that the centripetal force acting on the electron is equal to the complex force

F_{M Q_{i}}

(105) with the product of real mass and imaginary charge energies (102) and use the reduced mass of the proton-electron system

\begin{matrix} \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}}, \\ v_{e}^{2} & = \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}}, \\ 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}}, \end{matrix}

(122)

where

q_{e} = - 1

and

q_{p} = 1

are the electron and proton charges, respectively, and

M_{p} = m_{p} m_{P}

m_{p} \in R

is the proton mass.

For the electron mass

M_{e} = 9.1094 e - 31

[kg] and the proton mass

M_{p} = 1.6726 e - 27

[kg] the equation (122) yields

v_{e} \approx 7.2993 e - 3 - i 3.2816 e - 21 \approx α

assuming that R is equal to the Bohr radius

a_{0} = 5.2918 e - 11 [m]

or the radius

R \approx (5.2946 e - 11 - i 4.7607 e - 29) [m] \approx a_{0}

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

v_{e} = α

These values correspond to those given by the Bohr model. Furthermore, neglecting the opposite signs of the charges (

q_{e} = q_{p} = - 1

q_{e} = q_{p} = 1

) in the relation (122) yields, respectively, an imaginary electron velocity

v_{e} \approx 3.2852 e - 21 \pm i 7.2993 e - 3 \approx \pm i α

and a negative radius

R \approx (- 5.2947 e - 11 \pm i 4.7660 e - 29) [m] \approx - a_{0}

. We further note that switching the signs of charges (

q_{e} = 1

q_{p} = - 1

) provides complex conjugates of the relation (122), which in this case describes the antihydrogen. Therefore, we conjecture that the energy generated during a hydrogen-antihydrogen collision is

E_{H - \bar{H}} = 2 (m_{e} m_{p} + α) E_{P} \approx 2.8549 e 7 [J] .

(123)

Finally, we note that the relation (122) based (as the Bohr model) on the mass of the electron provides a better agreement with the Bohr radius and the fine-structure constant because

\begin{matrix} m_{e} v_{e}^{2} r & = m_{e} m_{p} + α + i \sqrt{α} (m_{e} - m_{p}), \\ r & = \frac{m_{p}}{α^{2}} + \frac{1}{m_{e} α} + i α^{- 3 / 2} (1 - \frac{m_{p}}{m_{e}}) \approx \frac{1}{m_{e} α} = \frac{a_{0}}{ℓ_{P}}, \\ v_{e}^{2} & = α m_{e} m_{p} + α^{2} + i α^{3 / 2} (m_{e} - m_{p}) \approx α^{2} . \end{matrix}

(124)

Note that none of the charged elementary particles that form atoms and antiatoms satisfy the complex energy equality of Theorem 2. For example, both proton (

q_{p} = 1

) and electron (

q_{e} = - 1

) satisfy

α = α_{2}^{4} m_{e}^{2} / α^{4}

and

α = α_{2}^{4} m_{p}^{2} / α^{4}

of equation (70) but

m_{e} \neq m_{p}

. However, we can postulate the equality of the products of the complex energies (102)-(104), that is, for hydrogen and antihydrogen atoms

\begin{matrix} E_{e p m q_{i}} & = m_{e} m_{p} + α \pm i \sqrt{α} (m_{e} - m_{p}), \\ E_{e p q m_{i}} & = \frac{α^{4}}{α_{2}^{4}} (- α + \frac{α}{α_{2}} m_{i e} m_{i p} \pm \sqrt{\frac{α}{α_{2}}} \sqrt{α} (m_{i e} - m_{i p})), \\ E_{e p m m_{i}} & = m_{e} m_{p} + \frac{α}{α_{2}} m_{i e} m_{i p} \pm \sqrt{\frac{α^{5}}{α_{2}^{5}}} (m_{e} m_{i p} + m_{p} m_{i e}), \\ E_{e p m q_{i}} & = E_{e p q m_{i}} = E_{e p m m_{i}} \approx α, \end{matrix}

(125)

with "+" for hydrogen and "−" for antihydrogen. Knowing the values of

m_{e}

and

m_{p}

the system of equations (125) resolves to

\begin{matrix} m_{i e} & = {0 \pm i 0.1314, 0 \pm i 0.0543}, \\ m_{i p} & = {- m_{i e 2}, - m_{i e 1}}, \end{matrix}

(126)

yielding real masses

| M_{1} | = 1.19486 e - 09 [kg]

and

| M_{2} | = 2.8929 e - 09 [kg]

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 e + 07 [1 / m],

(127)

and the corresponding Rydberg formula can be expressed as

\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),

(128)

using the wavelength relation (46) and

α

algebraic expression (14). The coefficients

{16, 8, 9, 2, 1}

form part of the OEIS sequence A158565 for

n = 10

XI. Discussion

The reflectance of graphene under normal incidence of electromagnetic radiation, expressed as quadratic equation for the fine-structure constant

α

, includes the 2^nd negative fine-structure constant

α_{2}

. The sum of the reciprocal of this 2^nd fine-structure constant

α_{2}

with the reciprocal of the fine-structure constant

α

(2) is independent of the reflectance value R and is remarkably equal to simply

- π

. The algebraic definition of the fine-structure constant

α^{- 1} = 4 π^{3} + π^{2} + π

, containing the free term

π

, when introduced into this sum, yields

α_{2}^{- 1} = - 4 π^{3} - π^{2} - 2 π < 0

(15). The negative fine-structure constant

α_{2}

leads to the

α_{2}

-Planck units applicable to imaginary dimensions, including the imaginary

α_{2}

-Planck units (28)-(36). The real and imaginary mass and charge units (21), temperature and time units (41), and length and mass units (42) are directly related. Furthermore, equation (18) shows that the elementary charge e is common for both the real and imaginary dimensions.

Applying

α_{2}

Planck units to the complex energy formula [70] yields complex energies (55) and (56), setting the atomic number

Z = 238

as the limit on an extended periodic table. The generalized energy (52) of all perfect black-body objects (black holes, neutron stars and white dwarfs) with a generalized radius

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

exceeds the mass-energy equivalence if

k > 2

. The complex energies (55)-(57) allow storage of excess energy in their imaginary parts. The results show that perfect black-body objects other than black holes cannot have masses lower than

5.7275 \times 10^{- 10} [kg]

and that

k_{\max} \approx 6.7933

k \leq 6.7933

defined by equation (79). In addition, it has been shown that a black-body object is in equilibrium with complex energies if its radius

R_{eq} \approx 1.3833 R_{BH}

(85). The proposed model explains the registered (GWOSC) high masses of neutron star mergers without resorting to hypothetical types of exotic stellar objects.

In the context of the results of this study, monolayer graphene, a truly 2-dimensional material with no thickness18, is a keyhole to other, unperceivable dimensionalities. The history of graphene is also instructive. Discovered in 1947 [100], graphene was long considered an academic material until it was eventually pulled from graphite in 2004 [101] using ordinary Scotch tape19. These fifty-seven years, along with twenty-nine years (1935-1964) between the condemnation of quantum theory as incomplete [102] and Bell’s mathematical theorem [103] 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.

Funding

This research received no external funding.

Acknowledgments

I truly thank my wife, Magdalena Bartocha, for her support since this research [104,105] began. I thank Wawrzyniec Bieniawski for inspiring discussions and constructive ideas concerning the layout of this paper and his feedback while working on the BB mergers and BB fluctuations sections. I thank Andrzej Tomski for defining the scalar product for the Euclidean spaces

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

(1).

Conflicts of Interest

The author has declared that no competing interests exist.

Abbreviations

The following abbreviations are used in this manuscript:

ED	emergent dimensionality
EMR	electromagnetic radiation
MLG	monolayer graphene
T	transmittance
R	reflectance
A	absorptance
HUP	Heisenberg’s uncertainty principle
DOF	degree of freedom
BH	black hole
NS	neutron star
WD	white dwarf
BB	black-body object
HS	holographic sphere
STM	size-to-mass ratio
GR	general relativity

Appendix A. Other MLG Quadratic Equations

The quadratic equation for the sum of transmittance (3) and absorptance (5) of MLG under normal incidence of EMR corresponds to equation (8), substituting

R = 1 - T - A

. However, the sums of the roots of the other quadratic equations are not independent of T, A, or R. For example, the sum of

T + R

(6) expressed as the quadratic equation (substituting

C_{TR} T + R

) is

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

(A1)

and has two roots with reciprocals

α^{- 1} = \frac{π (C_{TR} - 1)}{- 2 C_{TR} + 2 \sqrt{2 C_{TR} - 1}} \approx 137.036,

(A2)

and

α_{TR}^{- 1} = \frac{π (C_{TR} - 1)}{- 2 C_{TR} - 2 \sqrt{2 C_{TR} - 1}} \approx 0.0180,

(A3)

whereas their sum

α_{{TR}_{1}}^{- 1} + α_{{TR}_{2}}^{- 1} = \frac{- π C_{TR}}{C_{TR} - 1} \approx 137.054

(A4)

is dependent on T and R, which suggests that in the case of MLG, transmittance (3) and absorptance (5) should be considered together as their sum.

Appendix B. MLG Transmittance, Absorptance, and Reflectance as Functions of π Only

With algebraic definitions of

α

(14) and

α_{2}

(15), T (3), R (4) and A (5) of MLG for normal EMR incidence can be expressed simply by

π

. For

α^{- 1} = 4 π^{3} + π^{2} + π

(14) they become

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

(A5)

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

(A6)

while for

α_{2}^{- 1} = - 4 π^{3} - π^{2} - 2 π

(15) they become

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

(A7)

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

(A8)

with

R (α) = R (α_{2}) = \frac{1}{{(8 π^{2} + 2 π + 3)}^{2}} \approx 1.2843 \times 10^{- 4} .

(A9)

(

T (α) + A (α)) + R (α) = (T (α_{2}) + A (α_{2})) + R (α_{2}) = 1

as required by the law of energy conservation (7). Each conservation law is associated with a certain symmetry, as asserted by Noether’s theorem.

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.

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

We conjecture that the negative A and T values exceeding 100% for

α_{2}

(11) and (15) could be explained in terms of spontaneous graphene emission.

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 + T = \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,

(A10)

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,

(A11)

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 Fresnel equation (A10) 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}},

(A12)

which resolves to

n_{1}

independent on

α

and two forms of

n_{2}

\begin{matrix} n_{1} & = 1, \\ n_{2} (α_{*}) & = \frac{1}{1 + π α_{*}} = \{- \frac{α_{2}}{α}, - \frac{α}{α_{2}}\} \approx \{0.9776, 1.0229\}, \end{matrix}

(A13)

where

α_{*}

indicates

α

α_{2}

, satisfying

1 + π α = 1 / (1 + π α_{2})

, which corresponds to the identity (13). 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 e - 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 (A13) relates to the absolute (

n = c / V

) refractive indices; it models MLG as a boundary between vacuum and some other bulk medium. The refractive index is related to the phase velocity of light, which does not carry information and can be faster than the speed of light in vacuum c.

Refractive indices (A13) 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 e + 08 [m / s]

(A14)

using the relation (26).

On the other hand, substituting MLG R, T+A into the Euclid formula (A11) 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}

(A15)

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}

(A16)

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}

(A17)

satisfying

π α = - π α_{2} / (π α_{2} + 1)

, which also corresponds to the identity (13), and

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

(A18)

We further note that

a (α_{*}) \approx - A (α_{*})

, (A6), (A8) and

| b (α_{*}) | \approx T (α_{*}) + 1

, (A5), (A7).

Appendix D. Two π-like Constants

Quadratic equation (8), which describes the reflectance R of MLG under the normal incidence of EMR, can also be solved for

π

, which yields two roots.

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

(A19)

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

(A20)

dependent on R and

α_{*}

, where

α_{*}

indicates

α

α_{2}

. This can be further evaluated using the MLG reflectance R (4) or (A9) (which is the same for both

α

and

α_{2}

), 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,

(A21)

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

(A22)

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 .

(A23)

The modulus of

π_{1}

(A21) corresponds to a convex surface with a positive Gaussian curvature, whereas the modulus of

π_{2}

(A23) 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 [106].

Appendix E. Why α-Space Is Better for Biological Evolution?

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}}},

(A24)

where

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

(A25)

is the Gamow energy,

m_{a}

m_{b}

are the masses of those particles in terms of

α

- or

α_{2}

-Planck units (46) and

Z_{a}

Z_{b}

are their respective atomic numbers.

{(π α)}^{2} \approx 5.2557 e - 04

is larger than

{(π α_{2})}^{2} \approx 5.0227 e - 04

, the probability (A24) is higher for the same dimensionless parameters

m_{*}

Z_{*}

. Therefore, perceivable

α

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

α_{2}

-space.

Furthermore, the

α_{2}

-Planck energy

E_{P i}

and temperature

T_{P i}

are higher than the Planck energy

E_{P}

and temperature

T_{P}

. Therefore, the perceivable

α

-space yields more favorable conditions for the evolution of information, also owing to the minimum energy principle.

Appendix F. 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},

(A26)

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

[34] yields

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

(A27)

Interpreting

δ M_{HUP} = δ R_{HUP} c^{2} / (2 G)

as the BH mass in (A27) 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 (A27) leads to

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

. In general, using the generalized radius (49) 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} .

(A28)

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 G. The Stoney Units Derivation

We assume that the elementary charge is the unit of charge

q_{S} = e

and that the speed of light is the quotient of the unit of length and time

c = l_{S} / t_{S}

. Next, we compare the Coulomb force between two elementary charges and units of mass

m_{S}

with Newton’s law of gravity, acting over the same distance:

\frac{1}{4 π ϵ_{0}} \frac{e^{2}}{R^{2}} = G \frac{m_{S}^{2}}{R^{2}} \Rightarrow m_{S} = \pm \sqrt{\frac{e^{2}}{4 π ϵ_{0} G}} .

(A29)

Finally, we compare the inertial force of the unit of mass with Newton’s law of gravity:

m_{S} \frac{ℓ_{S}}{t_{S}^{2}} = G \frac{m_{S}^{2}}{ℓ_{S}^{2}} \Rightarrow ℓ_{S} = \pm \sqrt{\frac{G e^{2}}{4 π ϵ_{0} c^{4}}},

(A30)

to derive the Stoney length

ℓ_{S}

and the remaining Stoney units.

Using the negative

c_{2}

(22) we can determine the values of

c_{2}

-Stoney units (Sn). For mass, length, time, and energy, they are

\begin{matrix} m_{S n} & = m_{S} = \sqrt{α} m_{P} \approx 0.0854 m_{P}, \\ ℓ_{S n} & = \frac{α_{2}^{2}}{α^{2}} ℓ_{S} \approx 0.9557 l_{S} \approx 0.0816 l_{P}, \\ t_{S n} & = \frac{α_{2}^{3}}{α^{3}} t_{S} \approx - 0.9343 t_{S} \approx - 0.0798 t_{P}, \\ E_{S n} & = m_{S} c_{2}^{2} = \frac{α^{2}}{α_{2}^{2}} E_{S} \approx 1.0464 E_{S} \approx 0.0894 E_{P} . \end{matrix}

(A31)

We note that the

c_{2}

-Stoney energy induced by

c_{2}

is greater than the Stoney energy and the

c_{2}

-Stoney time runs in the opposite direction. We also note that a negative value of the gravitational constant G would yield imaginary Stoney units regardless of the sign of c, as all Stoney units (except charge) contain c raised to even (4, 6) powers.

Appendix H. 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

\begin{matrix} R = \frac{h}{ν e^{2}} = \frac{2 π ℏ}{ν α 4 π ϵ_{0} ℏ c} = \frac{1}{2 ν ϵ_{0} α c} = \frac{1}{2 ν ϵ_{0} α_{2} c_{2}}, \end{matrix}

(A32)

where

ν

is an integer or fraction (for example, for

ν = 5 / 2

R = 1 / (5 ϵ_{0} α c)

). Relations (A32) and (26) suggest that 2D FQHE links real and imaginary dimensions, similar to 2D graphene, giving us the second negative fine-structure constant

α_{2}

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1	This is, of course, a circular definition. But it is given for clarity.
2	Since the square root is bivalued the unit of speed is also bivalued In Planck, Stoney, and Schrödinger units.
3	$c α$ is also the speed unit in Hartree and Schrödinger’s natural units.
4	Which inevitably enforces understanding the nature in a manner that is common to nearly all people and thus hinders its research.
5	" $⌊x⌋$ " is the floor function that yields the greatest integer less than or equal to its argument x.
6	Furthermore, the Bekenstein bound can be derived from the BH entropy: $S_{BH} = k_{B} π R R / ℓ_{P}^{2} \leq k_{B} π R \frac{2 G E}{c^{4}} \frac{c^{3}}{ℏ G}$ , where we used $M \leq \frac{R c^{2}}{2 G}$ and $E = M c^{2}$ .
7	Thus, the term object is a particularly staring misnomer if applied to BBs.
8	Charges in the cited study are defined in CGS units. Here, we adopt SI.
9	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, photons do not travel.
10	One may find constructive criticism of GR in [79,80,81,82,83,84,85].
11	We drop the HS subscripts in this section for clarity.
12	https://www.gw-openscience.org/eventapi/html/allevents
13	Data available online at the Canadian Hydrogen Intensity Mapping Experiment (CHIME) portal (https://www.chime-frb.ca/catalog).
14	X-ray Polarimetry Explorer (https://ixpe.msfc.nasa.gov).
15	We drop the HS subscripts in this section for clarity.
16	In a commonly used form it is $T_{BB} = \frac{ℏ c^{3}}{2 k^{2} π G M_{BB} k_{B}} (1 \pm i \sqrt{\frac{k^{2}}{4} - 1})$ .
17	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.
18	Thickness of MLG is reported [99] as 0.37 [nm] with other reported values up to 1.7 [nm]. However, considering that 0.335 [nm] is the established interlayer distance and consequently the thickness of bilayer graphene, these results do not seem credible: the thickness of bilayer graphene is not $2 \times 0.37 + 0.335 = 1.075$ [nm].
19	Introduced into the market in 1932.

Figure 1. MLG reflectance as a function of

β 1 / α

Figure 1. MLG reflectance as a function of

β 1 / α

Figure 2. Illustration of three complex energies linking mass m, imaginary mass

m_{i}

, and charge q.

Figure 2. Illustration of three complex energies linking mass m, imaginary mass

m_{i}

, and charge q.

Figure 3. 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

. The mass

M_{i BB}

is imaginary for

k ⪅ 6.79

. The charge

q_{BB}

is real for

k \geq 2

Figure 3. 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

. The mass

M_{i BB}

is imaginary for

k ⪅ 6.79

. The charge

q_{BB}

is real for

k \geq 2

Figure 4. BB information capacity variations

Δ N

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

k = 2

\hat{m} = 1

Figure 4. BB information capacity variations

Δ N

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

k = 2

\hat{m} = 1

Figure 5. Black body object energy

E_{BB}

(green); temperature

T_{BH}

(red),

Re [T_{BB} (k_{eq})]

(red, dash-dot),

Re [T_{BB} (k_{\max})]

(red, dash); and work

W_{BH}

(blue),

Re [W_{BB} (k_{eq})]

(blue, dash-dot),

Re [W_{BB} (k_{\max})]

(blue, dash),as a function of its information capacity

N_{BB}

in terms of Planck units, for

0 \leq N_{BB} \leq 5

Figure 5. Black body object energy

E_{BB}

(green); temperature

T_{BH}

(red),

Re [T_{BB} (k_{eq})]

(red, dash-dot),

Re [T_{BB} (k_{\max})]

(red, dash); and work

W_{BH}

(blue),

Re [W_{BB} (k_{eq})]

(blue, dash-dot),

Re [W_{BB} (k_{\max})]

(blue, dash),as a function of its information capacity

N_{BB}

in terms of Planck units, for

0 \leq N_{BB} \leq 5

Table 1. Selected BB mergers discovered with LIGO and Virgo. Masses in

M_{⊙}

Table 1. 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

Abstract

I. Introduction

II. The Second Fine-Structure Constant

III. Set of α 2 -Planck Units

IV. Black Body Objects

V. BB Complex Energies

VI. BB Mergers

VII. BB Fluctuations

VIII. Complex Forces

IX. BB Complex Gravity and Temperature

X. Hydrogen Atom

XI. Discussion

Funding

Acknowledgments

Conflicts of Interest

Abbreviations

Appendix A. Other MLG Quadratic Equations

Appendix B. MLG Transmittance, Absorptance, and Reflectance as Functions of π Only

Appendix C. MLG Fresnel Equation and Euclid’s Formula

Appendix D. Two π-like Constants

Appendix E. Why α-Space Is Better for Biological Evolution?

Appendix F. Planck Units and HUP

Appendix G. The Stoney Units Derivation

Appendix H. Hall Effect

References

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III. Set of $α_{2}$ -Planck Units