The Imaginary Universe

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The Imaginary Universe

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Szymon Łukaszyk^*

This version is not peer-reviewed

Submitted:

19 October 2023

Posted:

20 October 2023

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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

1. Introduction

The universe began with the Big Bang, which is a current prevailing scientific opinion. But this Big Bang was not an explosion of 4-dimensional spacetime, which also is 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 countably infinitely other points 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 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}

. And so on, 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

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

(1)

where

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

. With the appearance of the first 0-dimensional point, information began to evolve [1,2,3,4,5,6].

However, the dimensional properties are not uniform. Concerning 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

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

R^{4}

[8], absent in other dimensionalities. Due 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 nondiffeomorphic (non-smooth). This allowed the variation of phenotypic traits within individuals’ populations [9] and extended the evolution of information into biological evolution. Exotic

R^{4}

solves the problem of extra dimensions of nature, and perceived space requires a natural number of dimensions [10]. Each biological cell perceives an emergent space of three real dimensions and one imaginary (time) observer-dependently [11] and at present, when

i 0 = 0

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

- c^{2}

, where c is the speed of light in vacuum. This is the emergent dimensionality (ED) [5,9,12,13,14]. Appendix F presents some arguments to support the claim that perceived dimensionality sets favourable conditions for biological evolution to emerge.

Each dimension requires certain units of measure. 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" [15]. Planck units utilize the Planck constant h that he introduced in his black-body radiation formula. However, in 1881, George Stoney derived a system of natural units [16] based on the elementary charge e (Planck’s constant was unknown then). The ratio of Stoney units to Planck units is

\sqrt{α}

, where

α

is the fine-structure constant. This study derives the 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 research. In particular, the subject of scientific research is thermodynamics in the complex plane. For example, Lee–Yang zeros [17,18] and photon-photon thermodynamic processes under negative optical temperature conditions [19] 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 the time-reversal symmetry [20]. 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 real eigenvalues of Hermitian operators. Unlike charges, negative, real masses are generally inaccessible for direct observation. However, dissipative coupling between excitons and photons in an optical microcavity leads to the formation of exciton polaritons with negative masses [21]. In Section 6 we show that negative masses also result from merging black-body objects.

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

The paper is structured as follows. Section 2 shows that 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 3 shows that by this second fine-structure constant nature endows us with the

α_{2}

-natural units. Section 4 introduces the concept of a black-body object in thermodynamic equilibrium, emitting perfect black-body radiation, and reviews its necessary properties. Section 5 introduces complex mass and charge energies expressed in terms of real and imaginary

α_{2}

-Planck units introduced in Section 3 and applies them to black-body objects. Section 6 considers 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 7 discusses fluctuations of black-body objects. Section 8 defines the complex forces. The complex force between real masses and imaginary charges is used in Section 9 to derive a black-body object surface gravity and the generalized Hawking radiation temperature, and in Section 10 discussed in the context of the Bohr model of the hydrogen atom. Section 11 summarizes the findings of this study. Certain prospects for further research are given in the Appendices.

2. The Second Fine-Structure Constant

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

π

and the fine-structure constant

α

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 vacuum permittivity (the 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 [22] (Eq. 3), whereas spectrally flat absorptance (A)

A \approx π α \approx 2.3 %

was reported [23,24] for photon energies between about

0.5

and

2.5

[eV]. T was related to reflectance (R) [25] (Eq. 53) as

R = π^{2} α^{2} T / 4

, i.e,

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

(4)

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

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

(5)

were also derived [26] (Eqs. 29-31) based on the thin film model (setting

n_{s} = 1

for substrate). The sum of transmittance (3) and the reflectance (4) at normal EMR incidence on MLG was derived [27] (Eq. 4a) as

\begin{matrix} 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, \end{matrix}

(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 [28], k is the wave vector of light in vacuum, and

χ = 0

is the electric susceptibility of vacuum. Therefore, these coefficients are well established theoretically and experimentally [22,23,24,27,29,30].

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) of MLG, modified to include its absorption.

The reflectance

R = 0.013 %

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

α

\begin{matrix} 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, \end{matrix}

(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, the 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 the 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 B).

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 D).

These dependences on

π

only between the fine-structure constants

α

and

α_{2}

suggest that they do not vary over time.

These results are also intriguing in the context of a peculiar algebraic expression for the fine-structure constant [31]

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 in passing that CODATA values are computed by averaging the measurements.

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

π

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

π

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

3. Set of $α_{2}$ -Planck Units

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

α_{2}

. We shall further call them

α_{2}

-Planck units. Natural units can be derived from numerous starting points [5,32] (cf. Appendices Appendix G and Appendix H). The central assumption in all natural unit systems is that the quotient of the unit of length

ℓ_{*}

and time

t_{*}

is a unit of speed; we call it

c = ℓ_{*} / t_{*}

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

c α

, and Rydberg units, where it is

c α / 2

2. On the other hand, 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 vacuum permeability (the 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 well known

1 = μ_{0} ϵ_{0} c^{2},

(17)

symmetric in its electric and magnetic parts [33] 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 BH potential, etc. We further note that Maxwell’s equations in vacuum are not directly dependent on the fine-structure constant(s). It is included in the magnetic constant

μ_{0}

In the following, we assume the universality of the real elementary electric charge e defining both matter and antimatter, the Planck constant h, the uncertainty principle parameter, and the 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 the Planck constant h to find the product of the Planck length

ℓ_{P}

and the speed of light in vacuum [34]. We note in passing that antimatter obeys gravity [35], 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. The reason is that 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 [5,12].

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

q_{P i}

so 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 and negative speed of light or some complementary real and 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

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

μ_{2}

, so

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

(20)

From that and from

\tilde{ϵ_{0}} c_{2} < 0

, we conclude that the product

μ_{2} c_{2} < 0

. We note that 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 which can be calculated, knowing the value of the imaginary Planck charge

q_{P i}

from the relation (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 nonprincipal 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 \times 10^{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 electrical units. Therefore, along with temperature, amount of substance, and luminous intensity, they are base units of the International System of Quantities (ISQ). We further conclude that the magnetic constant

μ_{2}

is lower than

μ_{0}

\begin{matrix} 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}] . \end{matrix}

(23)

Unlike the electric constant

ϵ_{0}

, the magnetic constants

μ

are independent of the unit of time. Furthermore, negative

α_{2}

and

c_{2}

lead to the second, also time-dependent but negative vacuum impedance

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

(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,

c_{2}

, and the fine-structure constants

α

α_{2}

c α = c_{2} α_{2},

(26)

valid for both principal and non-principal square roots of the relation (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 10. 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 [36] (cf. Appendix I). The relation (26) is not the only

α

α_{2}

relation. Along with the two

π

-like constants

π_{1}

π_{2}

(relations (A21) and (A23), cf. Appendix E)

\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, respectively, imaginary (

- c

)-Planck units and real (

- c_{2}

)-Planck units. In particular, the imaginary (

- c

)-Planck time parameterizes the HSs time relations [5,12]. We conjecture that

α_{2}

-Planck units is appropriate for espressing physical quantities of

I^{3} \times R

Euclidean space rather than

R^{3} \times I

Euclidean space that we perceive due to the minimum energy principle (cf. Appendix F). 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; the 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 since c and

c_{2}

are bivalued. In other words, both 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

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

(28)

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

(29)

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

(30)

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

(31)

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

(32)

Most Planck units derived from the

α_{2}

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

α_{2}

Planck volume

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

(33)

the

α_{2}

Planck momentum

\begin{matrix} 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 \\ \approx i 6.7504 [kg m / s] (| m_{P i} c_{n} | > | m_{P} c |), \end{matrix}

(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 \\ \approx i 2.0701 \times 10^{9} [J] (| E_{P i} | > | E_{P} |), \end{matrix}

(35)

and the

α_{2}

Planck acceleration

\begin{matrix} 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 \\ \approx \pm i 6.0198 \times 10^{51} [m / s^{2}] (| a_{P i} | > | a_{P} |) . \end{matrix}

(36)

However, the

α_{2}

-Planck density

\begin{matrix} 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 \\ \approx - 5.7735 \times 10^{96} [kg / m^{3}] (| ρ_{P 2} | > | ρ_{P} |), \end{matrix}

(37)

and the

α_{2}

-Planck area

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

(38)

are real and bivalued similarly to the Planck density

ρ_{P}

and area

ℓ_{P}^{2}

. Interestingly, both Planck forces

F_{P}

and

\begin{matrix} 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 \\ \approx 1.3251 \times 10^{44} [N] (F_{P 2} > F_{P}), \end{matrix}

(39)

are strictly positive.

We note that 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 some real or imaginary distance and

m_{P i}

is imaginary. The area of a sphere in the denominator of the Coulomb force invites further research.

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 saturate the energy-time version of Heisenberg’s uncertainty principle (HUP) taking energy from the equipartition theorem for one bit of information [5,12,37]

\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)

Neither is there a physically meaningful elementary (and imaginary) length

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

satisfying the relation (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 to (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.

Planck charge relation (18) and the 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 the inverse-square laws.

In the following, where deemed appropriate, we shall express the physical quantities by 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 respectively 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, and the subscripts i refer 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, in particular in the open research problem of the fractional quantum Hall effect (cf. Appendix I), and energy-dependent fractional charges in electron pairing [38].

4. Black Body Objects

There are only three observable objects in nature that emit perfect black-body radiation: unsupported black holes (BHs, the densest), neutron stars (NSs), supported, as accepted, by neutron degeneracy pressure, and white dwarfs (WDs), supported, as accepted, by electron degeneracy pressure (the least dense). We shall collectively call them black-body objects (BBs). The spectral density in sonoluminescence, light emission by sound-induced collapsing gas bubbles in fluids, was also shown to have the same frequency dependence as black-body radiation [39,40]. Thus, the sonoluminescence, and in particular shrimpoluminescence [41], is probably emitted by collapsing micro-BBs. Micro-BH induced in glycerin by modulating acoustic waves was reported [42].

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 [43]. Also, 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 on how galaxies formed at early times at both redshifts, even when considering observational uncertainties [44]. This is an important unresolved issue indicating that fundamental changes to the reigning

Λ

CDM model of cosmology are needed [44]. The term object as a collection of matter is a misnomer as it neglects the (quantum) nonlocality [45] that is independent of the entanglement among particles [46], as well as the Kochen-Specker contextuality [47], and increases as the number of particles grows [48,49]. Thus, we use emphasis for (perceivably indistinguishable) particle and (perceivably distinguishable) object, as well as for matter and distance. The ugly duckling theorem [50,51] 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 ED. In particular, given the observation of quasiparticles in classical systems [52]. Within the framework of ED no object is enclosed in space.

Entropic gravity [37] explains the 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 [53]), has been experimentally confirmed [54], and is decoherence-free [55]. It has been experimentally confirmed that the so-called accretion instability is a fundamental physical process [56]. We conjecture that this process, already recreated under laboratory conditions [57], is common for all BBs. As black-body radiation is radiation of global thermodynamic equilibrium, it is patternless [58] (thermal noise) radiation that depends only on one parameter. In the case of BHs, this is known as Hawking [59] radiation, and this parameter is the BH temperature

T_{BH} = T_{P} / (2 π d_{BH})

corresponding to the BH diameter [5]

D_{BH} = d_{BH} ℓ_{P}

, where

d_{BH} \in R

. Furthermore, BHs absorb patternless information [5,60]. Therefore, since 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 7).

As black-body radiation is patternless, triangulated [5] 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 [5], based on the Bekenstein-Hawking (BH) entropy [61]

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 [5,12,37,61,62], 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 [5,12].

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 [5] by comparing the BH entropy with the binary entropy variation

δ S = k_{B} N_{1} / 2

([5] Eq. (55)), valid for any holographic sphere (HS), where

N_{1} \in N

denotes the number of active Planck triangles with binary potential

δ φ_{k} = - c^{2}

. Thus, the entropy of all BBs is

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

(48)

Furthermore,

N_{1} = N_{BB} / 2

confirms the patternless thermodynamic equilibrium of BBs by maximizing Shannon entropy [5].

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

M_{BB}

as a function of

G M_{BB} / c^{2}

multiplier

k \in R, k \geq 2

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

(49)

and the generalized BB energy

E_{BB}

as a function of

M_{BB} c^{2}

multiplier

a \in R

(this definition also applies to all HSs)

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), it becomes

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

(51)

and equals the BB entropy (48) if

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

. Thus, the energy of all BBs having 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 BHs, setting the lower bound for other BBs. We shall further call the coefficient k the size-to-mass ratio (STM). It is similar to the specific volume (the reciprocal of density) of the BB. We shall derive the upper STM bound in Section 5.

According to the no-hair theorem, all BHs general relativity (GR) solutions are characterized only by three parameters: mass, electric charge, and angular momentum. However, BHs are fundamentally uncharged, since 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 [63] using Penrose processes [64,65] to extract electrostatic and/or rotational energy of BH [66]. Thus any BH is defined by only one real parameter: its diameter, mass, temperature, energy, etc., each corresponding to the other. We note that 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}

) [12,14], resulting in its information capacity

N = 4

, one unit of BH entropy [61]. This confirms the universality and applicability of the BH entropy (48) to all BBs.

Interiors of the BBs are inaccessible to an exterior observer [61], which makes them similar to interior-less mathematical points representing real numbers on a number line7. Yet, a BH can embrace this defining real number. 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, i.e., this additional parameter defines it. Thus, the only meaningful spatial notion is the Planck area triangle, which encodes one bit of classical information and its curvature.

However, it is accepted that in the case of NSs, electrons combine with protons to form neutrons, so that NSs are composed almost entirely of neutrons. But it is never the case that all electrons and all protons of an NS become neutrons. WDs are charged by definition, as they are accepted to be mostly composed of electron degenerate matter. But how can a charged BB store both the curvature and an additional parameter corresponding to its charge? Fortunately, the relation (18) ensures that the charges are the same in 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 a vertex or two vertices with this triangle in

R^{2}

. And this configuration is capable of storing both the curvature and the 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) [5], 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 since the latter introduces the notion of time, which we find redundant in the BB description of a patternless thermodynamical equilibrium.

Not only BBs are perfectly spherical. Also, their mergers, to which we shall return in Section 6, are perfectly spherical, as it has been experimentally confirmed [67] based on the registered gravitational event GW170817. One can hardly expect a collision of two perfectly spherical, patternless thermal noises to produce some aspherical pattern instead of another perfectly spherical patternless noise. Where would the information about this pattern come from at the moment of the collision? From the point of impact? No point of impact is distinct on a patternless surface.

The considerations previously discussed 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 will model a part of the energy of NS and WD that exceeds

M c^{2}

as imaginary and thus unmeasurable.

5. 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 respectively real and imaginary energy of an object having mass

M_{R}

and charge

Q_{R}

8 was proposed in ref. [68]. Equation (53) considers real masses

M_{R}

and charges

Q_{R}

. To store the surplus energy we shall modify 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)

\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 define 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

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

(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 illustrated in Figure 2.

We neglect the energy of real and imaginary charges

E_{Q Q_{i}}

, since by the relation (18), the unit of charge is the same in real and imaginary dimensions. The mass-energy equivalence relates the 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, the 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 their real and imaginary parts equal in modulus.

Proof.

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 simultaneously balanced with the 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)

□

Since by the relation (18) charges are the same in real and imaginary dimensions, squared moduli of 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

the relation (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 the relation (64). Since the mass

m_{i} \in R

, the square root argument must be nonnegative in relation (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 uncharged micro-BHs (

q = 0

) in thermodynamic equilibrium can be arbitrary. However, micro NSs and micro WDs, 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 grain of sand.

We note here that only the masses satisfying

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

have Compton wavelengths larger than Planck length [5]. We note in passing that a classical description has been ruled out on the microgram (

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

) mass scale [69]. Comparing this bound with the 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 a little 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 [5,37,61,62]. From the relation (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.

\begin{matrix} | E_{M Q_{i}} |^{2} = | E_{Q M_{i}} |^{2} = {| E_{M M_{i}} |}^{2} = \\ = (1 + \frac{α_{2}^{4}}{α^{4}}) m^{2} E_{P}^{2} = (1 + \frac{α_{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} \end{matrix}

(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 () 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)

and finally, substituting the relation (74) into the modulus (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 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 a charge surplus energy exceeding

M_{BB} c^{2}

. Similarly, we can interpret the squared generalized energy of 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 the relation (76) into the relation (77) turns 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, notably, independent of the STM. The square root argument in the relation (78) must be non-negative, since

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

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

, which are equal in modulus. In general, the relation (78) relates 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 the 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, the relations (67) and (79) expresed 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, respectively, by

d_{BB}

and

q_{BB}

) in the 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

, while 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) substituting

A = m_{BB} k / 2

from (49) into the relation (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)

at which all three energies are equal. The equilibrium

k_{eq}

(85) and the 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, the relation (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 in state of the art as ultracompact [70], where

k = 3

is a photon sphere radius9. Any object that undergoes complete gravitational collapse passes through an ultracompact stage [71], where

k < 3

. Collapse can be approached by gradual accretion, increasing the mass to the maximum stable value, or by loss of angular momentum [71]. During the loss of angular momentum, the star passes through a sequence of increasingly compact configurations until it finally collapses to become a black hole. It was also pointed out [72] that for a neutron star of constant density, the pressure at the center would become infinite if

k = 2.25

, a radius of the maximal sustainable density for gravitating spherical matter given by Buchdahl’s theorem. It was shown [73] that this limit applies to any well-behaved spherical star where density increases monotonically with 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 [74]. As the surface gravity grows, photons from further behind the NS become visible. At

k \approx 3.52

the whole NS surface becomes visible [75]. The relative increase in brightness between the maximum and minimum of a light curve are greater in the case of

k < 3

than in the case of

k > 3

[75]. Therefore the equilibrium STM ratio

k_{eq} \approx 2.7665

(85) is well within the range of radii of ultracompact objects researched in state-of-the-art within the framework of GR.

However, aside from the Schwarzschild radius, derivable from 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 [76], all the remaining significant radii of GR are only approximations10. GR neglects the value of the fine-structure constants

α

and

α_{2}

, which, similarly to

π

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

6. 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 having 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 having the Planck length diameter, the reduced Planck temperature

\frac{T_{P}}{2 π}

(the largest physically significant temperature [12]), and no tangential acceleration

a_{L L}

[5,12], produces a BH having

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

which represents the minimum BH diameter allowing for the notion of time [12]. In comparison, a collision of the latter two BHs produces a BH having

d_{BH} = \pm 2

having the triangulation defining only one precise diameter between its poles (cf. [5] Figure 3(b)), which is also recovered from HUP (cf. Appendix G).

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 more compact than neutron stars, such as quark stars, BHs, or more exotic objects [84]. 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 [85]. Furthermore, it has been hinted that outside GR, merging BHs may differ from their GR counterparts [86].

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

M_{Ch} \approx 1.4 M_{⊙}

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

M_{TOV} \approx 2.9 M_{⊙}

[88,89]. There is no accepted value of the BH mass limit. The conjectured value is

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

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

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

(slightly less than one light year) [5] and mass

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

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

M_{TOV}

. Of 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 are classified otherwise, including GW170817, GW190425, GW200105, and GW200115, listed in Table 1.

The relation (87) explains the measurements of large masses of the 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 the relation (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 [21]) 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 the inequality (89) lead to (

α_{2} < 0

), so 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) lead 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 the inequality (90) (

α_{2}^{5} < 0

), while

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

with the inequality (90) lead 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 consistent with the second inequality (88) introducing the product of positive and negative masses.

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

with the 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 [90] based on the gravitational event GW1904251 and the FRB 20190425A event13. Furthermore, IXPE14 observations show that the detected polarized X-rays from 4U 0142+61 pulsar exhibit a

90^{\circ}

linear polarization swing from low to high photon energies [91]. 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 [92]; 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 (cf. Figure 3). From the relation (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 the relation (87) that provide the measured mass

M_{{BB}_{C}}

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

k_{{BB}_{1}}

and

k_{{BB}_{2}}

of the merging components were arbitrarily selected on the basis of their masses, taking into account the limit of mass

M_{TOV}

of the NS.

7. BB Fluctuations

A relation [93] (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 [5] (Eq. (18)) to all Compton wavelengths

λ = l ℓ_{P} = \frac{2 π}{m} ℓ_{P}

(or frequencies

ν = c / λ = 1 / (l t_{P})

) and thus to all radiated Compton energies

E = m E_{P}

m \in R

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

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

(96)

The relation (96) can be further generalized, using the generalized diameter

d = 2 k \hat{m}

(49), to all BBs as

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, a BB changes its information capacity by

\begin{matrix} 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}, \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. The relation (99) shows that, depending on its mass

\hat{m}

, a BB can expand or contract by emitting or absorbing energy m [5]. However, expansion by emission (

Δ N^{E} > 0

), for example, requires energy

m > k \hat{m}

exceeding the mass-energy equivalence of BB for

k > 2

, which is consistent with the results presented in Section 5.

8. 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 an 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 complex forces (similarly to the complex energy of real masses and charges (53), [68] Eq. (7)) acting over real and imaginary distances

R, R_{i}

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

\begin{matrix} 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}), \end{matrix}

(102)

\begin{matrix} 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})), \end{matrix}

(103)

\begin{matrix} 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}), \end{matrix}

(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 a real distanceR are stronger and opposite to the corresponding forces acting on an imaginary distance

R_{i}

even though the Planck force is lower than the

α_{2}

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

α_{2}

-Planck units; STM remains the same.

9. 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

\begin{matrix} \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}), \end{matrix}

(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 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 BH temperature for all k.

In particular,

\begin{matrix} 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}), \end{matrix}

(114)

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

(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 the magnitude of the Hagedorn temperature of strings, while

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

. It seems, therefore, that a universe without

α_{2}

-imaginary dimensions (i.e., with

α_{2} = 0

) would be a black hole. Hence, the evolution of information [1,2,3,4,5,6] requires imaginary time. And we cannot zero

α_{2}

as we would have to neglect the existence of graphene.

10. 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 the 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 .

(116)

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 .

(117)

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 the 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 .

(118)

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

α

. Joining the relations (116) and (117) yields

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

(119)

which combined with (118) 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,

(120)

Thus, at the first circular orbit (

n = 1

) in this model

v_{e} = α

We shall now 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}

(121)

where

q_{e} = - 1

and

q_{p} = 1

are the electron and proton charges, and

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

m_{p} \in R

is the proton mass.

For the electron mass

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

[kg] and the proton mass

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

[kg] the equation (121) yields

v_{e} \approx 7.2993 \times 10^{- 3} - i 3.2816 \times 10^{- 21} \approx α

assuming that R is equal to the Bohr radius

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

or the radius

R \approx (5.2946 \times 10^{- 11} - i 4.7607 \times 10^{- 29}) [m] \approx a_{0}

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

v_{e} = α

We note that these values correspond to the values given by the Bohr model. We further note that neglecting the opposite signs of the charges (

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

q_{e} = q_{p} = 1

) in the relation (121) yields, respectively, an imaginary electron velocity

v_{e} \approx 3.2852 \times 10^{- 21} \pm i 7.2993 \times 10^{- 3} \approx \pm i α

and a negative radius

R \approx (- 5.2947 \times 10^{- 11} \pm i 4.7660 \times 10^{- 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 (121), 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 \times 10^{7} [J] .

(122)

Finally, we note that the relation (121) based, as the Bohr model, on the mass of the electron provides a better agreement to the Bohr radius and the fine-structure constant since

\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}

(123)

We note that none of the charged elementary particles that form atoms and antiatoms satisfies 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 the relation (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}

(124)

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

m_{e}

and

m_{p}

the system of equations (124) 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}

(125)

yielding real masses

| M_{1} | = 1.19486 \times 10^{- 09} [kg]

and

| M_{2} | = 2.8929 \times 10^{- 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 \times 10^{+ 07} [1 / m],

(126)

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),

(127)

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

11. Discussion

The reflectance of graphene under the normal incidence of electromagnetic radiation expressed as the 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 particular 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 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 to each other. Furthermore, the relation (18) shows that the elementary charge e is common for real and imaginary dimensions.

Applying

α_{2}

Planck units to a complex energy formula [68] yields complex energies (55), (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) having the generalized radius

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

exceeds the mass-energy equivalence if

k > 2

. The complex energies (55)-(57) allow storage of the excess of this energy in their imaginary parts. The results show that the 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 the relation (79). In addition, it is shown that a black-body object is in the equilibrium of complex energies if its radius

R_{eq} \approx 1.3833 R_{BH}

(85). The proposed model explains the registered (GWOSC) high masses of the neutron star mergers without resorting to any 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 [95], graphene was long considered an academic material until it was eventually pulled from graphite in 2004 [96] by means of ordinary Scotch tape19. These fifty-seven years, along with twenty-nine years (1935-1964) between the condemnation of quantum theory as incomplete [97] and Bell’s mathematical theorem [98] asserting that it is not true, and the fifty-eight years (1964-2022) between the formulation of this theorem and 2022 Nobel Prize in Physics for its experimental loophole-free confirmation, should remind us that Max Planck, the genius who discovered Planck units, has also discovered Planck’s principle.

Acknowledgments

I truly thank my wife Magdalena Bartocha for her support ever since this research [99,100] 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 the definition of the scalar product for Euclidean spaces

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

(1).

Appendix A. 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 B. 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 other quadratic equations are not independent on 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 hints that in the case of MLG, transmittance (3) and absorptance (5) should be considered together as their sum.

Appendix C. 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 just 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), whereas 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 a source respectively, while 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.

Perhaps the negative A and T exceeding 100% for

α_{2}

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

Appendix D. 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 the refractive index of liquid helium

n \approx 1.025

at 3 K. The refractive index

n_{2} \approx 0.9776

is close to the refractive index of water

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

for X-ray radiation at a photon wavelength of

0.04

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

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

at 532 nm to

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

at 633 nm at room temperature). However, since

n_{1} = 1

, the relation (A13) relates to 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

\begin{matrix} V (- \frac{α_{2}}{α}) & = - c \frac{α}{α_{2}} = - c_{2}, \\ V (- \frac{α}{α_{2}}) & = - c \frac{α_{2}}{α} = - \frac{c^{2}}{c_{2}} \approx 2.9307 \times 10^{+ 08} [m / s] \end{matrix}

(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 \\ \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 \\ \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 \\ \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 \\ \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 E. Two π-like Constants

The quadratic equation (8) that 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 having a positive Gaussian curvature, whereas the modulus of

π_{2}

(A23) - to a negative Gaussian curvature. The product

π_{1} π_{2} = π^{2}

is independent of

α_{*}

, their quotient

π_{1} / π_{2} = α_{2}^{2} / α^{2}

is not directly dependent of

π

, and

|π_{1} - π| \neq |π - π_{2}|

. It remains to be found 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 on curved surfaces, whereas

π_{a} > π_{c}

on convex surfaces, while

π_{a} < π_{c}

on saddle surfaces [101].

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

The probability of two nuclear particles a and b to 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 masses of those particles in terms of

α

- or

α_{2}

-Planck units (46) and

Z_{a}

Z_{b}

are their respective atomic numbers.

Since

{(π α)}^{2} \approx 5.2557 \times 10^{- 04}

is larger than

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

α

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

Appendix G. 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, by 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

[32], which 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 [5] 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, one obtains

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 mass

δ M_{HUP}

decreases, and

δ R_{HUP}

increases and the factor is the same for

k = 1

i.e., for orbital speed radius

δ R = G δ M / c^{2}

or the orbital speed mass

δ M = δ R c^{2} / G

Appendix H. 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 masses

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 the 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 I. Hall Effect

The fractional quantum Hall (FQHE) effect shows a stepwise dependence of the conductance on the magnetic field (as compared to a 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 similarly to 2D graphene, giving us the second negative fine-structure constant

α_{2}

1	This is, of course, a circular definition. But for clarity, it is given.
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 [77,78,79,80,81,82,83].
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 [94] 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.

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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

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

1. Introduction

2. The Second Fine-Structure Constant

3. Set of α 2 -Planck Units

4. Black Body Objects

5. BB Complex Energies

6. BB Mergers

7. BB Fluctuations

8. Complex Forces

9. BB Complex Gravity and Temperature

10. Hydrogen Atom

11. Discussion

Acknowledgments

Appendix A. Abbreviations

Appendix B. Other MLG Quadratic Equations

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

Appendix D. MLG Fresnel Equation and Euclid’s Formula

Appendix E. Two π-like Constants

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

Appendix G. Planck Units and HUP

Appendix H. The Stoney Units Derivation

Appendix I. Hall Effect

References

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3. Set of $α_{2}$ -Planck Units