General Relativity: Four Exterior Metric Solutions...

[Orion1]

In Einstein's theory of general relativity, the exterior metric or exterior 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 exterior 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.

There is a theoretical discontinuity with a Kerr metric tensor element: (ref. 5, ref. 6, pg. 238, ref. 7, pg. 3, eq. 2.7)
[math]g_{tt} \neq -\left(1 - \frac{r_{s} r}{\Sigma} \right)[/math]
This Kerr metric tensor element [math]g_{tt}[/math] is missing the [math]\Delta [/math] rotation term [math]a^{2}[/math] in the entire numerator, and a [math]\Sigma[/math] term in the entire denominator. (ref. 8` pg. 135` tbl. 1) 
[math]\color{blue}{\text{The theoretical Kerr metric tensor element presented here is:}}[/math]
[math]\boxed{g_{tt} = -\frac{\Delta}{\Sigma} = -\left(\frac{r^{2} - r_{s} r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right)}[/math]
[math]\color{blue}{\text{This theoretical Kerr metric tensor element matches with the exterior Kerr-Newman metric tensor element when } (Q = 0) \text{.}}[/math]
[math]\color{blue}{\text{And this Kerr tensor matches with all the other exterior metric tensor elements when } (J = 0) \text{ and or when } (m = 0) \text{.}}[/math]
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[math]\color{blue}{\text{Table 1. Metric tensor components and symmetry.}}[/math]
[math]\begin{array}{l*{5}c} \text{Metric tensor} & \text{theorem} & \text{identity with applied state} & \text{cited state} & \text{Symmetry} & \text{State} \\ \text{} & dt^{2} & dt^{2} & dt^{2} & \text{} \\ \text{Minkowski} & -1 & -1 & -1 & \text{Spherical} & m = 0,J = 0,Q = 0 \\ \text{Schwarzchild} & -\left(\frac{r^{2} - r_{s} r}{r^{2}} \right) & -\left(1 - \frac{r_{s}}{r} \right) & -\left(1 - \frac{r_{s}}{r} \right) & \text{Spherical} & m \neq 0,J = 0,Q = 0 \\ \text{Reissner-Nordstrom} & -\left(\frac{r^{2} - r_{s} r + r_{Q}^{2}}{r^{2}} \right) & -\left(1 - \frac{r_{s}}{r} \right) & -\left(1 - \frac{r_{s}}{r} \right) & \text{Spherical} & m \neq 0,J = 0,Q = 0 \\ \text{Ellipsoid} & -\left(\frac{r^{2} + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & -\left(\frac{r^{2} + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & -\left(\frac{r^{2} + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & \text{Ellipsoid} & m = 0,J \neq 0,Q = 0 \\ \text{Kerr} & -\left({\frac{r^{2} - r_{s}r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta}} \right)  & -\left({\frac{r^{2} - r_{s}r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta}} \right)  & \underline{-\left(1 - \frac{r_{s} r}{r^{2} + a^{2} \cos^{2} \theta} \right)} & \text{Ellipsoid} & m \neq 0,J \neq 0,Q = 0 \\ \text{Kerr } \left(\text{cited} \right) & \underline{-\left(1 - \frac{r_{s} r}{r^{2} + a^{2} \cos^{2} \theta} \right)} & \underline{-1} & \underline{-1} & \boxed{\text{Incorrect}} & m = 0,J \neq 0,Q = 0 \\ \text{Kerr-Newman} & -\left(\frac{r^{2} - r_{s} r + a^{2} + r_{Q}^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & -\left(\frac{r^{2} - r_{s} r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & -\left(\frac{r^{2} - r_{s} r + a^{2}}{r^{2} + a^{2} \cos^{2} \theta} \right) & \text{Ellipsoid} & m \neq 0, J \neq 0, Q = 0 \\ \end{array}[/math]
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[math]\color{blue}{\text{The underlined metric tensors highlight the matrix locations of the theoretical discontinuities.}}[/math]
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[math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math]
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[math]\color{blue}{\text{"You will do well to expand your horizons." - Fortune Cookie}}[/math]
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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 - Exterior Reissner-Nordstrom metric: (ref. 4)
https://en.wikipedia.org/wiki/Reissner–Nordström_metric#The_metric
Wikipedia - Exterior Kerr metric: (ref. 5)
https://en.wikipedia.org/wiki/Kerr_metric
The Racah Institute of Physics - Gravitational field of a spinning mass - Roy Kerr (ref. 6)
http://old.phys.huji.ac.il/~barak_kol/Courses/Black-holes/reading-papers/Kerr.pdf
Arxiv - The Kerr Metric - Saul A. Teukolsky: (ref. 7)
https://arxiv.org/pdf/1410.2130.pdf
Academic Journals - A derivation of the Kerr metric by ellipsoid coordinate transformation - Yu-ching Chou: (ref. 8)
https://academicjournals.org/journal/IJPS/article-full-text-pdf/AEE776964987
Wikipedia - Exterior Kerr-Newman metric: (ref. 9)
https://arxiv.org/pdf/astro-ph/9801252.pdf
Wikipedia - Exterior Kerr-Newman metric: (ref. 10)
https://en.wikipedia.org/wiki/Kerr-Newman_metric
Science Direct - Rotating black hole and Kerr metric - Pierre Binetruy: (ref. 11)
https://www.sciencedirect.com/topics/physics-and-astronomy/kerr-metric
 

Edited January 30, 2022 by Orion1
source code correction...

[Orion1]

[math]\color{blue}{\text{Page 1 updated with more detailed steps...}}[/math]
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[math]\color{blue}{\text{What distinguishes this theoretical approach from mainstream science?}}[/math]
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[math]\color{blue}{\text{Any discussions and/or peer reviews about this specific topic thread?}}[/math]
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Edited March 2, 2022 by Orion1

[swansont]

1 hour ago, Orion1 said:

What distinguishes this theoretical approach from mainstream science?

Why don't you tell us?

[Markus Hanke]

What is the question, exactly?

[Orion1]

"Why don't you tell us?" - swansont

"What is the question, exactly?" - Markus Hanke

Are there any detectable theoretical deviations from the mathematical solutions already published by mainstream science?

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

A constructive critique response under a peer review would have been preferred to potentially highlight any other detectable theoretical deviations, and I always try to provide such an opportunity for students and professors. This type of academic strategy has proven to be more effective for improved research quality than mere dictate.

The only detectable deviation from this theory is the theoretical discontinuity with the cited temporal Kerr metric tensor element, which fails to reduce into an ellipsoid when a massless parameter state is directly mathematically applied. However, the cited temporal Kerr metric tensor element published by Kerr is a mathematical identity with the temporal Kerr metric tensor element generated by this theory.

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

 

Edited July 26, 2023 by Orion1

[Orion1]

On 3/2/2022 at 6:20 AM, swansont said:

Why don't you tell us?

On 3/2/2022 at 6:47 PM, Markus Hanke said:

What is the question, exactly?

Are there any detectable theoretical deviations from the mathematical solutions already published by mainstream science?

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

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