Gravitating Electron Based on Overrotating Kerr-Newman Solution

Round 1

Reviewer 1 Report

The authors consider the nonperturbative electron model based on the Kerr-Newman (KH) black hole and they use the Landau-Ginzburg theory to modify Carter's model. Such ameliorations are consistent with gravity and quantum theory in particular the superconducting electron model.

The paper's subject is interesting and matches the investigation trend related to quantum gravity. But the paper presentation is not appropriate. I suggest to the authors rewrite the paper, precisely the abstract and the conclusion to illustrate their results.

In addition, the quality of the figures has to be improved.

 In the end, the presentation is really awful. The paper should be carefully rewritten. I encourage the
authors to improve heavily the quality of the presentation: check
English language, typos, and also improve the presentation of the
formulas. Motivations and the results should be more clear.

I recommend a careful rewriting of the paper. After that, I can
reconsider recommendation for publication in Universe

Author Response

Article was cardinally rewritten with improving the presentation.

In particular: 
1. the title was changed, 
2. abstract and introduction were corrected and 
3. the conclusion was fully rewritten, 
4. scales of figures were changed.

There was given also some important clarifications in the text, especially with regards to  the reason for strong gravitational interaction.
 
(1) In particular, the long important clarification is given in the end of introduction: 
"As a result, the KN solution has undergone a transformation that replaces the "negative"  sheet with a "mirror" sheet, carrying the negative charge and current coupled with mirror boundary. Under regularization, the mirror boundary acquires the mirror Wilson line.

Since Wilson lines are integral curves of the strong gravitational and electromagnetic field in a fixed world time $t$ , the evolution of these fields in time is  best described in the Kerr-Schild coordinate system.
Meanwhile, the relativistic system can also be described in the proper time $s$, where the size of the system is reduced by the Lorentz contraction. We show that the regularized boundary of KN solution represents a classical relativistic ring string taking intermediate position between the string (in word time $t$ ) and the light-like particle (in proper time $s$) that is created from this string by relativistic contraction. This  string-particle correspondence is a copy of the Heisenberg-Schr\"odinger correspondence in quantum theory and  plays very important role in quantum interpretation of the KN electron model."

(2) After the line 93 we insert the text clarifying formation of the "mirror" modification of the KN metric :

 "The "mirror" modification of the KN metric corresponds to a change in the direction of the Kerr congruence $k_\m$ on the mirror sheet $r<0$, when passing through the boundary $r=0$. The change $r \to - r ,$ coupled with a mirror change of the direction of rotation $a \to -a $   gives
 $k _\m \to k^\pm _\m = \pm dr -dt \mp a \sin^2 \theta d\phi_K,$ and we have
 \be k^\pm _dx =\pm dr - dt + a \sin^2 \theta d\phi_K \label{kpmdx} .\ee

(3) changes in section 5

line 122
"which, in accordance with the Kerr principal relation $ J = ma = \frac \hbar 2 $, is proportional to $ 2 \pi   \hbar  .$"

is replaced with

"which, in accordance with the Kerr principal relation \be J = ma = \frac \hbar 2 \label{Jhbar2}\ee, becomes  proportional to $ 2 \pi   \hbar  .$

lines 134 and 135 are  replaced by text

" equal to twice  the magnetic field quantum $\Phi_0 = h/2e $, and the Dirac monopole must be born."

line 138

"which is usually associated with mass of the Dirac electron \cite{Weissk}." is repaced by

"which is usually associated with the mass of electron \cite{Weissk} appearing 
as the mass term "m" in the Dirac equations  $(i\gamma_\m p^\m + m)\psi =0 $ and in the expression (\ref{HKN}) for KN metric.
Therefore, the strong total interaction between the KN gravitational field  and two Wilson loops $C^\pm$ does not manifest itself explicitly in the mass term "m", but act non-linearly, increasing the mass by reducing the radius $a $ in the Kerr principal relation (\ref{a})."

(4) Before subsection 7.1 , the text inserted which clarifies importance of string structure:

"As it was shown in sec.6, the vector potential of Wilson lines creates  two surface currents (\ref{Ipm}) in the core of KN electron model, which are parametrised by the Kerr angular coordinate $\phi_K$ and have a radius of $a=\hbar/2m$ equal to half of Compton wavelength. These currents turn out to be light-like and generate the model of a closed relativistic ring string, forming a minimal surface in 4d Minkowski space.
String model turns out to be very important for compatibility of the nonperturbative KN solution with quantum theory, since it removes contradiction between the extended classical particle and the point particle of quantum theory.
Description of the  KN electron in the Kerr-Schild coordinate is realized in the word time "t" at a fixed moment $t=t_0=const.$, corresponding to image of 
a closed string placed on the boundary of the KN disk.
In  quantum theory this corresponds to a state vector in Heisenberg picture. The 
relativistic ring strings are massless, and their mass-energy is created from their 
relativistic dynamics. Below we show in sec.7.2 that the length of the relativistic 
KN string is reduced by the Lorentz contraction, and the KN ring string, which is 
considered in world time "t" as an extended string of Compton size, is 
compressed to a point in the proper time "s" coordinates corresponding to
 Schr\"odinger state vector of quantum theory."

(5)  Clarification of the important role of the electrostatic component of 
the vector potential.

line 138: "which is usually associated with mass of the Dirac electron \cite{Weissk}." 
is repaced by

"which is usually associated with the mass of electron \cite{Weissk} appearing 
as the mass term "m" in the Dirac equations  $(i\gamma_\m p^\m + m)\psi =0 $ and in the expression (\ref{HKN}) for KN metric.
Therefore, the strong total interaction between the KN gravitational field  and two Wilson loops $C^\pm$ does not manifest itself explicitly in the mass term "m", but acts non-linearly, increasing the mass by reducing the radius $a $ in the Kerr principal relation (\ref{a})."

(6) Between lines 247 and 248 inserted subsection 8.2

\subsection{Consistency of modified KN structure with QED}

(7) Conclusion completely rewritten, see PDF text

Author Response File: Author Response.pdf

Reviewer 2 Report

Report is attached.

Comments for author File: Comments.pdf

Author Response

Remarks  of the  reviewer 2 are concrete, and we improve them exactly by following the text of paper besides sec. 9 Conclusion which is fully rewritten.
In all cases it is fully coincides with the requirements of referee.

The general referee remark is that the title, abstract and conclusion are to be corrected with the removal of the words "black hole".

In particular:

(1) and (2) of pages 1,2 referee remarks are  corrected;

(3) p.2,line 63 and further, line 262 and 270 in conclusion, the remarks are similar and also corrected:

(4) and (5) on page 1,

these remarks are  corrected;

(6) and (7) - explanation of the term: "light-like particle is given at lines 220 and 223,

 long explanation is given at the end of introduction, after line (86) and it  is extended in the subsection 7.2.

"As a result, the KN solution has undergone a transformation that replaces the 
"negative"  sheet with a "mirror" sheet, carrying the negative charge and current 
coupled with mirror boundary.
Under regularization, the mirror boundary acquires the mirror Wilson line.

Since Wilson lines are integral curves of the strong gravitational and 
electromagnetic field in a fixed \emph{world time $t.$ }, the evolution of these 
fields in time is  best described in the Kerr-Schild coordinate system. Meanwhile, 
the relativistic system can also be described in the \emph{"proper time $ s $"}, 
where the size of the system is reduced by the Lorentz contraction. We show that the regularized boundary of KN solution represents \emph{a classical relativistic ring string} taking intermediate position between the string (in word time $t$ ) and the light-like particle (presented in proper time $s$) that is created from this string by relativistic contraction. This  string-particle correspondence is a copy of the Heisenberg-Schr\"odinger correspondence in quantum theory and  plays very important role in quantum interpretation of the KN electron model."

-------------
minor remarks:
-------------
(8) electronic models  is corrected to electron models

(9) title of Chapter 7 is changed and replaced with

title "Emergence of the classical ring string structure"

 before subsection 7.1 inserted the text

"As it was shown in sec.6, the vector potential of Wilson lines creates  two 
surface currents (\ref{Ipm}) in the core of KN electron model, which are 
parametrized by the Kerr angular coordinate $\phi_K$ and have a radius of 
$a=\hbar/2m$ equal to half of Compton wavelength. These currents turn out 
to be light-like and generate the model of a closed relativistic ring string, 
forming a minimal surface in 4d Minkowski space.
String model turns out to be very important for compatibility of the 
nonperturbative KN solution with quantum theory, since it removes 
contradiction between the extended classical particle and the point particle 
of quantum theory.
Description of the  KN electron in the Kerr-Schild coordinate is realized 
in the word time "t" at a fixed moment $t=t_0=const.$, corresponding 
to image of a closed string placed on the boundary of the KN disk.
In  quantum theory this corresponds to a state vector in Heisenberg picture. 
The relativistic ring strings are massless, and their mass-energy is created 
from their relativistic dynamics. Below we show in sec.7.2 that the length 
of the relativistic KN string is reduced by the Lorentz contraction, and 
the KN ring string, which is considered in world time "t" as an
extended string of Compton size, is compressed to a point in the 
proper time "s" coordinates, corresponding to Schr\"odinger state 
vector of quantum theory."

(10) bottom of page 5,

 "mast" is replaced by "must"

and at 205 line, page 8

"the Israel's electron model" is replaced by "the Israel electron model"

All remarks of the second referee concerning conclusion are fulfilled, and 
conclusion is completely rewritten, see PDF text of paper.

 

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The paper has been improved now, and I recommend it for publication.