Interior Reissner-Nordstrom metric...

[Orion1]

In Einstein's theory of general relativity, the interior Reissner-Nordstrom metric (also interior Reissner-Nordstrom solution or interior Reissner-Nordstrom fluid solution) is an exact static solution for the gravitational field in the interior of a non-rotating charged spherically symmetric body of mass M, which consists of an incompressible fluid and constant density throughout the body and has zero pressure at the surface. This is a static solution that does not change over time.
[math]\;[/math]
[math]\eta_{\mu \nu} - \text{perturbed non-dynamical background metric}[/math]
[math]\color{blue}{\text{General Relativity Minkowski flat spacetime metric:} \; (\text{ref. 1})}[/math]
[math]\eta_{\mu \nu} = \pm \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Schwarzschild radius:} \; (\text{ref. 2})}[/math]
[math]r_{s} = \frac{2 G M}{c^{2}}[/math]
[math]\;[/math]
[math]r_{g} \text{ - radial coordinate at the body's surface}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Exterior Schwarzschild metric:} \; (\text{ref. 3, ref. 4})}[/math]
[math]ds^{2} = c^{2}{d \tau}^{2} = -\left(1 - {\frac{r_{s}}{r}} \right) c^{2} dt^{2} + \left(1 - {\frac{r_{s}}{r}} \right)^{-1} dr^{2} + r^{2} \left(d \theta^{2} + \sin^{2} \theta \; d \varphi^{2} \right)[/math]
[math]\;[/math]
[math]d\tau^{2} = -\left(1 - \frac{r_{s}}{R} \right) dt^{2} + \ldots[/math]
[math]d\tau \text{ - proper time observer clock at radial distance R from the center}[/math]
[math]dt \text{ - time measured by an observer clock at infinity}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Interior Schwarzschild metric radial parameters:} \; (\text{ref. 5})}[/math]
[math]\mathcal{R} ^{2} = \frac{r_{g}^{3}}{r_{s}}[/math]
[math]\cos \eta_{g} = \sqrt{1 - \frac{r_{g}^{2}}{\mathcal{R}^{2}}} \; \; \; \; \; \; \cos \eta = \sqrt{1 - \frac{r^{2}}{\mathcal{R}^{2}}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Interior Schwarzschild metric total redshift factor:} \; (\text{ref. 6, pg. 2, eq. 2.2})}[/math]
[math]\alpha_{T} = \frac{3 \cos \eta_{g} - \cos \eta}{2}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Interior Schwarzschild metric:} \; (\text{ref. 7})}[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\alpha_{T}^{2} c^{2} dt^{2} + \frac{dr^{2}}{\cos^{2} \eta} + r^{2}\left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\left(\frac{3 \cos \eta_{g} - \cos \eta}{2} \right)^{2} c^{2} dt^{2} + \frac{dr^{2}}{\cos^{2} \eta} + r^{2}\left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{g}^{2}}{\mathcal{R}^{2}}} - \sqrt{1 - \frac{r^{2}}{\mathcal{R}^{2}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r^{2}}{\mathcal{R}^{2}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{s}}{r_{g}}} - \sqrt{1 - \frac{r_{s} r^{2}}{r_{g}^{3}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r_{s} r^{2}}{r_{g}^{3}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]\color{blue}{\text{Reissner-Nordstrom metric charge radius:} \; (\text{ref. 8})}[/math]
[math]r_{Q}^{2} = \frac{Q^{2} G}{4 \pi \varepsilon_{0} c^{4}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Exterior Reissner-Nordstrom metric:} \; (\text{ref. 8})}[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^{2} dt^{2} + \left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right)^{-1} \; dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]d\tau^{2} = -\left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) dt^{2} + \ldots[/math]
[math]d\tau \text{ - proper time observer clock at radial distance R from the center}[/math]
[math]dt \text{ - time measured by an observer clock at infinity}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Interior Reissner-Nordstrom metric radial parameters:}}[/math]
[math]\mathcal{R} ^{2} = \frac{r_{g}^{3}}{r_{s}}[/math]
[math]\boxed{\cos \eta_{g} = \sqrt{1 - \frac{r_{g}^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}}}} \; \; \; \; \; \; \boxed{\cos \eta = \sqrt{1 - \frac{r^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}}}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Interior Reissner-Nordstrom metric total redshift factor:}}[/math]
[math]\boxed{\alpha_{T} = \frac{3 \cos \eta_{g} - \cos \eta}{2}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Interior Reissner-Nordstrom metric:}}[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\alpha_{T}^{2} c^{2} dt^{2} + \frac{dr^{2}}{\cos^{2} \eta} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\left(\frac{3 \cos \eta_{g} - \cos \eta}{2} \right)^{2} c^{2} dt^{2} + \frac{dr^{2}}{\cos^{2} \eta} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{g}^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}}} - \sqrt{1 - \frac{r^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r^{2}}{\mathcal{R}^{2}} + \frac{r_{Q}^{2}}{\mathcal{R}^{2}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]\boxed{ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{s}}{r_{g}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}}} - \sqrt{1 - \frac{r_{s} r^{2}}{r_{g}^{3}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r_{s} r^{2}}{r_{g}^{3}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Metrics summary:}}[/math]
[math]\color{blue}{\text{Exterior Schwarzschild metric:}}[/math]
[math]ds^{2} = c^{2}{d \tau}^{2} = -\left(1 - {\frac{r_{s}}{r}} \right) c^{2} dt^{2} + \left(1 - {\frac{r_{s}}{r}} \right)^{-1} dr^{2} + r^{2} \left(d \theta^{2} + \sin^{2} \theta \; d \varphi^{2} \right)[/math]
[math]\;[/math]
[math]\color{blue}{\text{Exterior Reissner-Nordstrom metric:}}[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right) c^{2} dt^{2} + \left(1 - \frac{r_{s}}{r} + \frac{r_{Q}^{2}}{r^{2}} \right)^{-1} \; dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]\color{blue}{\text{Interior Schwarzschild metric:}}[/math]
[math]ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{s}}{r_{g}}} - \sqrt{1 - \frac{r_{s} r^{2}}{r_{g}^{3}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r_{s} r^{2}}{r_{g}^{3}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)[/math]
[math]\;[/math]
[math]\color{blue}{\text{Interior Reissner-Nordstrom metric:}}[/math]
[math]\boxed{ds^{2} = c^{2} d\tau^{2} = -\frac{1}{4} \left(3 \sqrt{1 - \frac{r_{s}}{r_{g}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}}} - \sqrt{1 - \frac{r_{s} r^{2}}{r_{g}^{3}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}}} \right)^{2} c^{2} dt^{2} + \left(1 - \frac{r_{s} r^{2}}{r_{g}^{3}} + \frac{r_{s} r_{Q}^{2}}{r_{g}^{3}} \right)^{-1} dr^{2} + r^{2} \left(d\theta^{2} + \sin^{2} \theta \; d\varphi^{2} \right)}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Schwarzschild innermost stable circular orbit:} \; (\text{ref. 9})}[/math]
[math]r_{1} = 3 r_{s} = \frac{6 G M}{c^{2}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Can a stable orbit accretion disk or spherical plasma shell of charged quantum particles accumulate at the Schwarzschild innermost stable circular orbit, generating a charged black hole?}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math]
[math]\;[/math]
Reference:
Wikipedia - Four-gradient As a Jacobian matrix for the SR Minkowski metric tensor: (ref. 1)
https://en.wikipedia.org/wiki/Four-gradient#As_a_Jacobian_matrix_for_the_SR_Minkowski_metric_tensor
Wikipedia - Schwarzschild radius: (ref. 2)
https://en.wikipedia.org/wiki/Schwarzschild_radius
Wikipedia - Schwarzschild metric: (ref. 3)
https://en.wikipedia.org/wiki/Schwarzschild_metric
Wikipedia - Gravitational redshift - Spherically symmetric gravitational field: (ref. 4)
https://en.wikipedia.org/wiki/Gravitational_redshift#Spherically_symmetric_gravitational_field
Wikipedia - Interior Schwarzschild metric - Other formulations: (ref. 5)
https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Other_formulations
Austrian Reports on Gravitation - Interior Schwarzschild Solution And Free Fall - Rainer Burghardt: (ref. 6)
http://www.arg.or.at/Wpdf/WIff.pdf
Wikipedia - Interior Schwarzschild metric: (ref. 7)
https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Mathematics
Wikipedia - Exterior Reissner-Nordstrom metric: (ref. 8 )
https://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric
Wikipedia - Black hole - Innermost stable circular orbit: (ref. 9)
https://en.wikipedia.org/wiki/Black_hole#Innermost_stable_circular_orbit_(ISCO)

 

Edited December 23, 2021 by Orion1
source code correction...

[Markus Hanke]

In principle - yes, you could have charged black holes, though they wouldn’t be of the RN kind.

In practice, I doubt it would happen, since on large scales you will usually have a roughly equal amount of positive and negative charges impacting the original body over time, so the net charge will tend to be roughly zero. This is why there are no known examples of any astrophysical objects with appreciable net charge.

[Orion1]

Source code correction...

[math]d\tau^{2} = -\left(1 - \frac{r_{s}}{R} + \frac{r_{Q}^{2}}{R^{2}} \right) dt^{2} + \ldots[/math]
[math]d\tau \text{ - proper time observer clock at radial distance R from the center}[/math]
[math]dt \text{ - time measured by an observer clock at infinity}[/math]

I searched for the motivation of Hans Reissner to include charge in general relativity. This scientist wrote in the cited paper: (ref. 1)

"After Mr. Einstein, by explaining the perihelion movement of Mercury, the fertility of his new covariant field equations of gravity and thus of the having shown the postulate of the most general relativity and elsewhere given the generally covariant version of the Maxwell-Lorentz equations on the electromagnetic field, it seemed to me the next task to investigate the influence of the self-gravity of the electric field of spherical symmetry in a simple example.

I started from the hope of finding a static cohesion of elementary charges through their own gravity without having to leave the ground of Maxwell's theory. Einstein's gravitation admittedly distorts the field of the elementary electric charge in a certain, incidentally, extremely slight way, but by its very nature it cannot cancel out the mutual electrostatic repulsion of the charge elements." - Hans Reissner

On 12/23/2021 at 3:42 PM, Markus Hanke said:

In principle - yes, you could have charged black holes, though they wouldn’t be of the RN (Reissner-Nordstrom) kind.

I was able to locate a paper on electrically charged strange quark stars. The model appears to utilize the exterior Reissner-Nordstrom metric integrated with Gauss' law. (ref. 2)

I hope at least this metric solution provides a student or professor studying general relativity the opportunity to examine the effects that general relativity has on metric interiors and charge, as it did with Karl Schwarzschild and Hans Reissner, plus various other college and university students and scientists.

In the letter Schwarzschild wrote to Einstein, he concluded by saying, "As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas." - Karl Schwarzschild

Any discussions and/or peer reviews about this specific topic thread?

Reference:
About the self-gravity of the electric field according to Einstein's theory - Hans Reissner: (ref. 1)
http://old.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Reissner.pdf
https://zenodo.org/record/1447315/files/article.pdf
Electrically Charged Strange Quark Stars - Rodrigo Picanco Negreiros: (ref. 2)
https://arxiv.org/pdf/0907.5537.pdf
 

[Orion1]

[math]\color{blue}{\text{Planck force:} \; (\text{ref. 1})}[/math]
[math]F_{P} = \frac{c^4}{G}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Coulomb's law of electrostatic force:} \; (\text{ref. 2})}[/math]
[math]F_{C} = \frac{Q^{2}}{4 \pi \varepsilon_{0} r_{Q}^{2}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Planck force is equivalent to Coulomb force:}}[/math]
[math]\boxed{F_{P} = F_{C}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Planck force is equivalent to Coulomb force integration via substitution, solve for } r_{Q} \text{:}}[/math]
[math]\frac{c^4}{G} = \frac{Q^{2}}{4 \pi \varepsilon_{0} r_{Q}^{2}}[/math]
[math]r_{Q}^{2} = \frac{Q^{2} G}{4 \pi \varepsilon_{0} c^{4}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Reissner-Nordstrom black hole metric charge radius:} \; (\text{ref. 3})}[/math]
[math]\boxed{r_{Q} = \frac{Q_{bh}}{2 c^{2}} \sqrt{\frac{G}{\pi \varepsilon_{0}}}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Schwarzchild radius:} \; (\text{ref. 4})}[/math]
[math]r_{s} = \frac{2 G M}{c^{2}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Solve for a Reissner-Nordstrom one solar mass black hole event horizon maximum attainable stable charge magnitude } Q_{bh}}[/math]
[math]\color{blue}{\text{for a thin spherical shell of accumulated charged quantum particles:}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Reissner-Nordstrom black hole metric charge radius is equivalent to Schwarzchild radius:}}[/math]
[math]\boxed{r_{Q} = r_{s}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Reissner-Nordstrom metric charge radius is equivalent to Schwarzchild radius integration via substitution:}}[/math]
[math]\frac{Q_{bh}}{2 c^{2}} \sqrt{\frac{G}{\pi \varepsilon_{0}}} = \frac{2 G M_{\odot}}{c^{2}}[/math]
[math]Q_{bh} = 4 M_{\odot} \sqrt{\pi \varepsilon_{0} G} = 3.427 \cdot 10^{20} \; \text{C}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Reissner-Nordstrom one solar mass black hole event horizon maximum attainable stable charge magnitude } Q_{bh}}[/math]
[math]\color{blue}{\text{for a thin spherical shell of accumulated charged quantum particles:}}[/math]
[math]\boxed{Q_{bh} = 4 M_{\odot} \sqrt{\pi \varepsilon_{0} G}}[/math]
[math]\boxed{Q_{bh} = 3.427 \cdot 10^{20} \; \text{C}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Does a Reissner-Nordstrom black hole metric charge result from a thin spherical shell of accumulated charged quantum particles}}[/math]
[math]\color{blue}{\text{near the black hole event horizon?}}[/math]
[math]\;[/math]
[math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math]
[math]\;[/math]
"Take charge of your thoughts. You can do what you will with them." - Plato
[math]\;[/math]
Reference:
Wikipedia - Planck force: (ref. 1)
https://en.wikipedia.org/wiki/Planck_units#Derived_units
Wikipedia - Coulomb's law: (ref. 2)
https://en.wikipedia.org/wiki/Coulomb's_law
Wikipedia - Exterior Reissner-Nordstrom metric: (ref. 3)
https://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric
Wikipedia - Schwarzschild radius: (ref. 4)
https://en.wikipedia.org/wiki/Schwarzschild_radius

 

Edited January 20, 2022 by Orion1

[beecee]

All BH's should have spin.

A BH with charge, would have that charge negated in rather quick time I suggest.

Spin may also be negated but certainly over a much longer time period, (probably approaching that time period of the BH evaporating via Hawking radiation)

Wormholes are also (as far as I know) a prediction of GR, but have never been validated. The Reisner Nordstrom BH appears to be in the same catagory. The most realistic BH still seems to be the Kerr BH.

The most common form of BH then I suggest is the Kerr metric, but in a few billions/trillions of years, that could then reduce to the old garden variety Schwarzchild BH as spin is negated.

Just some thoughts so don't be too concerned. I obviously am no where near the knowledge and know how of either Orion1 or Marcus.

Edited January 20, 2022 by beecee

[beecee]

This article (today) maybe of interest......

https://phys.org/news/2022-01-black-hole-horizons-discharged.html

Study finds that black hole inner horizons can be charged or discharged:

the papers:

 DOI: 10.1103/PhysRevLett.127.231301

 

Vitor Cardoso et al, Quasinormal Modes and Strong Cosmic Censorship, Physical Review Letters (2018). DOI: 10.1103/PhysRevLett.120.031103

Oscar J C Dias et al, Strong cosmic censorship for charged de Sitter black holes with a charged scalar field, Classical and Quantum Gravity (2019). DOI: 10.1088/1361-6382/aafcf2

Vitor Cardoso et al, Strong cosmic censorship in charged black-hole spacetimes: Still subtle, Physical Review D (2018). DOI: 10.1103/PhysRevD.98.104007

Stefan Hollands et al, Quantum instability of the Cauchy horizon in Reissner–Nordström–deSitter spacetime, Classical and Quantum Gravity (2020). DOI: 10.1088/1361-6382/ab8052

Stefan Hollands et al, Quantum stress tensor at the Cauchy horizon of the Reissner–Nordström–de Sitter spacetime, Physical Review D (2020). DOI: 10.1103/PhysRevD.102.085004

Christiane Klein et al, Renormalized charged scalar current in the Reissner–Nordström–de Sitter spacetime, Physical Review D (2021). DOI: 10.1103/PhysRevD.104.025009

Journal information: Physical Review Letters  , Physical Review D 

 

hope that helps....

[Markus Hanke]

Two quick comments on this:

1. Mass, charge and spin are global properties of the entire spacetime - you cannot localise these quantities at any particular place

2. None of the metrics in the Kerr-Newman family will realistically appear in the real world in an exact way, because all four of them require asymptotic flatness - meaning these require an otherwise completely empty universe. A more realistic - yet still idealised - family of solutions would be the Vaidya spacetimes.

[Orion1]

In Einstein's theory of general relativity, the interior metric or interior fluid solution, is an exact solution to the Einstein field equations and Einstein-Maxwell equations that describes the gravitational field and the space-time geometry in the interior of a non-rotating or rotating neutral or charged spherically symmetric body of mass M, which consists of an incompressible fluid and constant density throughout the body and has zero pressure at the surface and that the electric charge and angular momentum of the mass may be zero or non-zero, and the universal cosmological constant is zero. For a non-zero charged mass, the metric takes into account the Einstein-Maxwell field energy of an electromagnetic field within the space-time geometry. The space-time geometry is in Boyer-Lindquist coordinates.

"yet still idealized family of solutions would be the Vaidya spacetimes." - Markus Hanke

Are these Vaidya metrics mathematically mapped out accurately at this point?

"Mass, charge and spin are global properties of the entire spacetime - you cannot localise these quantities at any particular place." - Markus Hanke

What if the black hole quantum charge originates from its center core? 

"We analyze the “vacuum” polarization induced by a quantum charged scalar field near the inner horizon of a charged black hole in quantum states evolving from arbitrary regular in states." 
"As the formation of charged BHs necessitates the presence of charged matter, it is actually more natural to consider a charged scalar field"
"At the event horizon HR, this current is responsible for the discharge of the BH via Hawking radiation." (ref .7)

The model in reference 7, appears to be modeling the field interaction between a black hole that is quantum charged at its core, and with the charged matter near the inner horizon, which discharges as hawking radiation.

What determines the charge magnitude and polarity of a quantum charged black hole?

The black hole core quantum charge cannot extend past the inner event horizon?

Any discussions and/or peer reviews about this specific topic thread?

"You will do well to expand your horizons." - Fortune Cookie

Reference:
Wikipedia - Four-gradient As a Jacobian matrix for the SR Minkowski metric tensor: (ref. 1)
https://en.wikipedia.org/wiki/Four-gradient#As_a_Jacobian_matrix_for_the_SR_Minkowski_metric_tensor
Wikipedia - Schwarzschild radius: (ref. 2)
https://en.wikipedia.org/wiki/Schwarzschild_radius
Wikipedia - Interior Schwarzschild metric - Other formulations: (ref. 3)
https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Other_formulations
Austrian Reports on Gravitation - Interior Schwarzschild Solution And Free Fall - Rainer Burghardt: (ref. 4)
http://www.arg.or.at/Wpdf/WIff.pdf
Wikipedia - Interior Schwarzschild metric: (ref. 5)
https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Mathematics
Wikipedia - Exterior Reissner-Nordstrom metric: (ref. 6)
https://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric
Quantum (dis)charge of black hole interiors - Christiane Klein: (ref. 7)
https://arxiv.org/pdf/2103.03714.pdf
Wikipedia - Vaidya metric: (ref. 8)
https://en.wikipedia.org/wiki/Vaidya_metric
 

Edited January 31, 2022 by Orion1

[Markus Hanke]

14 hours ago, Orion1 said:

Are these Vaidya metrics mathematically mapped out accurately at this point?

Yes, certainly. This is one of the standard solutions that can be written in closed analytic form.

14 hours ago, Orion1 said:

What if the black hole quantum charge originates from its center core? 

I don’t know what you mean by “quantum charge”. GR is a purely classical model, and the Q parameter is not quantised.

Either way, charge (along with spin and mass) is just a parameter within the metric, it’s not a function of coordinates. So it can’t be localised anywhere - it’s a global property.

14 hours ago, Orion1 said:

The model in reference 7, appears to be modeling the field interaction between a black hole that is quantum charged at its core, and with the charged matter near the inner horizon, which discharges as hawking radiation.

 

The paper you referenced introduces an additional charged scalar field, so we are no longer dealing with a vacuum solution. The original RN metric (the topic of this thread) contains no such extra fields, so I’m not sure what ref 7 has to do with this thread. 

[Orion1]

"None of the metrics in the Kerr-Newman family will realistically appear in the real world in an exact way, because all four of them require asymptotic flatness - meaning these require an otherwise completely empty universe. A more realistic - yet still idealized - family of solutions would be the Vaidya spacetimes." - Markus Hanke

It is noted that these interior solutions are pure mathematical 'artifacts' of General Relativity for a universe with asymptotic flatness.

Besides the Interior Schwarzschild metric, has any of these other interior solutions been published by scientists?

[math]\text{Interior Schwarzschild metric:} \; (J = 0,Q = 0) \; \; \; (\text{ref. 1})[/math]
[math]\text{Interior Reissner-Nordstrom metric:} \; (J = 0,Q \neq 0)[/math]
[math]\text{Interior Kerr metric:} \; (J \neq 0,Q = 0)[/math]
[math]\text{Interior Kerr-Newman metric:} \; (J \neq 0,Q \neq 0)[/math]

Would any students or professors be interested in preserving these published interior solutions in their academic library archive to survive the author's and server's longevity and for the future study of these interior solutions by students or professors?

Any discussions and/or peer reviews about this specific topic thread?

"You will do well to expand your horizons." - Fortune Cookie 

Reference:
Wikipedia - Interior Schwarzschild metric: (ref. 1)
https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Mathematics

 

 

Edited July 26, 2023 by Orion1

[Orion1]

On 1/21/2022 at 6:25 PM, Markus Hanke said:

None of the metrics in the Kerr-Newman family will realistically appear in the real world in an exact way, because all four of them require asymptotic flatness - meaning these require an otherwise completely empty universe.

Besides the Interior Schwarzschild metric, has any of these other interior solutions been published by scientists?

[math]\text{Interior Schwarzschild metric:} \; (J = 0,Q = 0) \; \; \; (\text{ref. 1})[/math]
[math]\text{Interior Reissner-Nordstrom metric:} \; (J = 0,Q \neq 0)[/math]
[math]\text{Interior Kerr metric:} \; (J \neq 0,Q = 0)[/math]
[math]\text{Interior Kerr-Newman metric:} \; (J \neq 0,Q \neq 0)[/math]

Would you interested in preserving these published interior solutions in your academic library archive to survive the author's and server's longevity and for the future study of these interior solutions by students or professors?

Any discussions and/or peer reviews about this specific topic thread?

Reference:
Wikipedia - Interior Schwarzschild metric: (ref. 1)
https://en.wikipedia.org/wiki/Interior_Schwarzschild_metric#Mathematics

 

Edited October 13, 2023 by Orion1

[Markus Hanke]

1 hour ago, Orion1 said:

Besides the Interior Schwarzschild metric, has any of these other interior solutions been published by scientists?

Yes:

Reissner-Nordström interior
Kerr interior

Kerr-Newman interior