The Imaginary Universe

I. 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 implying 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 multiples of real units (ones) or imaginary units ($a\in \mathbb{R}\iff a=1b$1, $a\in \mathbb{I}\iff a=ib,b\in \mathbb{R}$). 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 ${\mathbb{R}}^2$, ${\mathbb{I}}^2$, and $\mathbb{R}\times \mathbb{I}$ Euclidean planes, with ${\mathbb{I}}^2$ being a mirror reflection of ${\mathbb{R}}^2$. And so on, forming n-dimensional Euclidean spaces ${\mathbb{R}}^a\times {\mathbb{I}}^b$ with $a\in \mathbb{N}$ real and $b\in \mathbb{N}$ imaginary lines, $n:=a+b$, and the scalar product defined by

$$\begin{array}{cc}\hfill \mathbf{x}·\mathbf{y}& =\left(x_1,\dots ,x_a,i{x}_1^{\prime },\dots ,i{x}_b^{\prime }\right)\left(y_1,\dots ,y_a,i{y}_1^{\prime },\dots ,i{y}_b^{\prime }\right):=\hfill \\ \hfill & :=\sum _{k=1}^a{x}_k{y}_k+\sum _{l=1}^b{x}_l^{\prime }\overline{y_l^{\prime }},\hfill \end{array}$$

(1)

where $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^a\times {\mathbb{I}}^b$.

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

However, 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 ${\mathbb{R}}^4$ [8]. This property allowed for variation of phenotypic traits within populations of individuals [9] perceiving emergent Euclidean ${\mathbb{R}}^3\times \mathbb{I}$ space of three real and one imaginary (time) dimension observer-dependently [10] and at present [11] when $i0=0$ is real. The evolution of information extended into biological evolution.

Each dimension requires certain units of measure. In real dimensions, the natural units of measure were introduced by Max Planck in 1899 as "independent of special bodies or substances, thereby necessarily retaining their meaning for all times and for all civilizations, including extraterrestrial and non-human ones" [12].

This study introduces the complementary set of Planck units applicable for imaginary dimensions, including the imaginary base units, and outlines certain prospects for their research. As the speed of electromagnetic radiation is the product of its wavelength and frequency (which are real and positive), and both these quantities are imaginary in imaginary dimensions, some real but negative parameter $c_n={\nu }_i{\lambda }_i$ corresponding to the speed of light in vacuum c (i.e., the Planck speed) is also necessary as $i^2=-1$. It turns out that the imaginary Planck energy $E_{\mathrm{P}i}$ and temperature $T_{\mathrm{P}i}$ are larger in moduli than the Planck energy $E_{\mathrm{P}}$ and temperature $T_{\mathrm{P}}$ Thus, the minimum energy principle sets more favorable conditions for biological evolution to emerge in ${\mathbb{R}}^3\times \mathbb{I}$ Euclidean space than in ${\mathbb{I}}^3\times \mathbb{R}$ Euclidean one.

The study shows that the energies of neutron stars and white dwarfs exceed their mass–energy equivalences. Therefore, the excess of these energies must be stored in imaginary dimensions and is inaccessible to direct observations. This results in the upper bound on the slope of the radius of their cores as a function of their masses.

The paper is structured as follows. Section II shows that Fresnel coefficients for the normal incidence of electromagnetic radiation on monolayer graphene introduce the second, negative fine-structure constant ${\alpha }_2$ as a fundamental constant of nature. Section III shows that nature endows us with the imaginary base Planck units by this second fine-structure constant. Section IV introduces the concept of a black-body object in thermodynamic equilibrium emitting black-body radiation and discusses its necessary properties. Section V introduces two complex energies of masses and charges and applies them to black-body objects. Section VI introduces four additional complex energies of masses, charges, and wavelengths to derive the black-body object equilibrium, correcting the photon sphere radius of general relativity. Section VII summarizes the findings of this study.

VI. MASS, CHARGE, EMR - PHOTON SPHERE RADIUS

Besides complex energies of masses and charges (48), (49)

$$E_{M{Q}_i}:=M{c}^2+\frac{Q_i{c}^2}{2\sqrt{\pi {ϵ}_{0}G}}=\left(m+iq\sqrt{\alpha }\right)E_{\mathrm{P}},$$

(65)

$$E_{M{Q}_i}:=\frac{Q_i{c}^2}{2\sqrt{\pi {ϵ}_{0}G}}+M{c}^2=\frac{\sqrt{a}{a}_2}{a_2^2}\left(q+\frac{m_1}{\sqrt{a_2}}\right)E_{\mathrm{P}},$$

(66)

we can also define the complex energies of real wavelength and imaginary mass/charge E_RMi, E_RQi

$$\begin{array}{cc}\hfill E_{R{M}_i}:=& \frac{hc}{\lambda }+M_i{c}_n^2=\frac{2\pi E_{\mathrm{P}}}{l}+m_i{E}_{\mathrm{P}i}=\hfill \\ \hfill & =\left(\frac{2\pi }{l}+\sqrt{\frac{{\alpha }^5}{{\alpha }_2^5}}{m}_i\right)E_{\mathrm{P}},\hfill \end{array}$$

(67)

$$E_{R{Q}_i}:=\frac{hc}{\lambda }+\frac{Q_i{c}^2}{2\sqrt{\pi {ϵ}_{0}G}}=\left(\frac{2\pi }{l}+iq\sqrt{\alpha }\right)E_{\mathrm{P}},$$

(68)

and of imaginary wavelength and real mass/charge E_MRi, E_QRi,

$$\begin{array}{cc}\hfill E_{M{R}_i}:=& M{c}^2+\frac{h{c}_n}{{\lambda }_i}=m{E}_{\mathrm{P}}+\frac{2\pi E_{\mathrm{P}i}}{l_i}=\hfill \\ \hfill & =\left(m+\sqrt{\frac{{\alpha }^5}{{\alpha }_2^5}}\frac{2\pi }{l_i}\right)E_{\mathrm{P}},\hfill \end{array}$$

(68)

$$\begin{array}{cc}\hfill E_{Q{R}_i}:=& \frac{Q{c}_n^2}{2\sqrt{\pi {ϵ}_{0}G}}+\frac{h{c}_n}{{\lambda }_i}=\frac{{\alpha }^2}{{\alpha }_2^2}q\sqrt{\alpha }{E}_{\mathrm{P}}+\frac{2\pi }{l_i}{E}_{\mathrm{P}i}=\hfill \\ \hfill & =\frac{{\alpha }^2}{{\alpha }_2^2}\left(q\sqrt{\alpha }+\sqrt{\frac{\alpha }{{\alpha }_2}}\frac{2\pi }{l_i}\right)E_{\mathrm{P}},\hfill \end{array}$$

(70)

where we applied discretizations (47). The energies (67)-(70) link the electromagnetic radiation with gravity/inertia and electrodynamics within the framework of emergent dimensionality.

Complex energies (65)-(70) are real-to-imaginary balanced if moduli of their real and imaginary parts are equal. This holds for

$$\begin{array}{cc}\hfill & \left|M\right|=i\left|M_i\right|=\left|m_{\mathrm{P}}q\sqrt{\alpha }\right|,\hfill \\ \hfill & \left|M_i\right|=i\left|\frac{{\alpha }_2^2}{{\alpha }^2}\frac{h}{c\lambda }\right|,\left|M\right|=i\left|\frac{\alpha }{{\alpha }_2}\frac{h}{c{\lambda }_i}\right|,\hfill \\ \hfill & \left|\lambda \right|=\left|\frac{2\pi l_{\mathrm{P}}}{q\sqrt{\alpha }}\right|,\left|{\lambda }_i\right|=i\left|\frac{2\pi l_{\mathrm{P}}{\alpha }_2}{q\sqrt{{\alpha }^3}}\right|.\hfill \end{array}$$

(71)

Furthermore, complex energies (65)-(70) define complex forces between two objects acting over real and imaginary distances $R,R_i$ (cf. Appendix E).

This time, comparing the squared moduli of the energies(65)-(70) with the squared BBO energy (41) yield a solvable system of six equations with six unknowns k, q, m, m_i, l, l_i

$$\begin{array}{cc}\hfill & |E_{M{Q}_i}{|}^2\Rightarrow q^2\alpha =m^2\left(\frac{k^2}{4}-1\right),\hfill \\ \hfill & |E_{Q{M}_i}{|}^2\Rightarrow \frac{{\alpha }^4}{{\alpha }_2^4}{q}^2\alpha -\frac{{\alpha }^5}{{\alpha }_2^5}{m}_i^2=\frac{k^2}{4}{m}^2,\hfill \\ \hfill & |E_{R{M}_i}{|}^2\Rightarrow \frac{4{\pi }^2}{l^2}-\frac{{\alpha }^5}{{\alpha }_2^5}{m}_i^2=\frac{k^2}{4}{m}^2,\hfill \\ \hfill & |E_{R{Q}_i}{|}^2\Rightarrow \frac{4{\pi }^2}{l^2}+q^2\alpha =\frac{k^2}{4}{m}^2,\hfill \\ \hfill & |E_{M{R}_i}{|}^2\Rightarrow m^2\left(1-\frac{k^2}{4}\right)=\frac{{\alpha }^5}{{\alpha }_2^5}\frac{4{\pi }^2}{l_i^2},\hfill \\ \hfill & |E_{Q{R}_i}{|}^2\Rightarrow \frac{{\alpha }^4}{{\alpha }_2^4}{q}^2\alpha -\frac{4{\pi }^2}{l_i^2}\frac{{\alpha }^5}{{\alpha }_2^5}=\frac{k^2}{4}{m}^2.\hfill \end{array}$$

(72)

Substituting $q^2\alpha =m^2\left(\frac{k^2}{4}-1\right)$ from $|E_{M{Q}_i}{|}^2$ to $|E_{R{Q}_i}{|}^2$ recovers the Compton wavelength of the BBO, ${\lambda }_{\mathit{BBO}}=\frac{h}{M_{\mathit{BBO}}c}$, in its discrete form $l^2=\frac{4{\pi }^2}{m^2}$.

Furthermore, by substituting $q^2\alpha$ and the Compton mass $m^2=\frac{4{\pi }^2}{l^2}$ into $|E_{Q{M}_i}{|}^2$, and comparing the LHSs of $|E_{Q{M}_i}{|}^2$ and $|E_{R{M}_i}{|}^2$ we obtain the BBO equilibrium multiplier

$$\begin{array}{cc}\hfill & \frac{k_{\mathrm{eq}}^2}{4}=\frac{{\alpha }_2^4}{{\alpha }^4}+1\Rightarrow \left|k_{\mathrm{eq}}\right|=2\sqrt{1+\frac{{\alpha }_2^4}{{\alpha }^4}}\approx 2.7665,\hfill \end{array}$$

(73)

at which all the moduli (72) are equal. The equilibrium multiplier $k_{\mathrm{eq}}$ (73) is related to the bound $k_{\max }$ (62) by $k_{\mathrm{eq}}^2+16/k_{\max }^2=8$. Also, the following relations can be derived from the relations (72) for the BBO in the equilibrium $k_{\mathrm{eq}}$ (73)

$$m_i^2=-\frac{{\alpha }_2^9}{{\alpha }^9}{m}^2\iff M_{i{\mathit{BBO}}_{\mathrm{eq}}}=\pm i\frac{{\alpha }_2^4}{{\alpha }^4}{M}_{{\mathit{BBO}}_{\mathrm{eq}}},$$

(74)

$$l_i^2=-\frac{{\alpha }^9}{{\alpha }_2^9}{l}^2\iff {\lambda }_{i{\mathit{BBO}}_{\mathrm{eq}}}=\pm i\frac{{\alpha }^3}{{\alpha }_2^3}{\lambda }_{{\mathit{BBO}}_{\mathrm{eq}}},$$

(75)

$$l^2=\frac{4{\pi }^2}{m^2}\iff {\lambda }_{{\mathit{BBO}}_{\mathrm{eq}}}=\frac{h}{M_{{\mathit{BBO}}_{\mathrm{eq}}}c}$$

(76)

$$q^2\alpha =\frac{{\alpha }_2^4}{{\alpha }^4}{m}^2\iff \frac{{\alpha }_2^4}{{\alpha }^4}=-{\beta }_{Qi{\mathit{BBO}}_{\mathrm{eq}}}^2=\pm i\frac{M_{i{\mathit{BBO}}_{\mathrm{eq}}}}{M_{{\mathit{BBO}}_{\mathrm{eq}}}},$$

(77)

where in the last relation, we used the definition (50) and applied the relation (74).

Notably, $2.25<k_{\mathrm{eq}}<3$, where $9/4$ is the multiplier of a radius of the maximal sustainable density for gravitating spherical matter given by Buchdahl’s theorem, and 3 is the multiplier of a BH photon sphere radius.

This shows that $k_{\mathrm{eq}}\approx 2.766$ is a true photon sphere radius, where BBO gravity, charge, and photon energies remain at equilibrium. Aside from the Schwarzschild radius (derivable from escape velocity $v_{esc}^2=2GM/R$ of mass M by setting $v_{esc}^2=c^2$), all the remaining thresholds of general relativity, such as Buchdahl’s threshold ($k=9/4$) or a photon sphere radius ($k=3$), are only crude approximations. It must be so, since general relativity neglects the value of the fine-structure constants $\alpha$ and ${\alpha }_2$, which, similarly as $\pi$ or the base of the natural logarithm, are the fundamental constants of nature.