The Physics of Schwarzschild’s Original 1916 Metric Solution and Elimination of the Coordinate Singularity Anomaly

The Physics of Schwarzschild’s Original 1916 Metric Solution and Elimination of the Coordinate Singularity Anomaly

 

 

 

By

Robert Louis Kemp

 

Super Principia Mathematica

The Rage to Master Conceptual & Mathematical Physics

 

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Abstract

 

 This paper is a mathematical treatise and historical perspective of Karl Schwarzschild’s original 1916 solution and description of a Non-Euclidean Spherically Symmetric Metric equation. In the modern literature of the Schwarzschild Metric equation, it is described as predicting an output of a “Coordinate Singularity” anomaly located at the surface of the Black Hole Event Horizon. However, in Schwarzschild’s original 1916 metric solution there is not a “Coordinate Singularity” located at the Black Hole Event Horizon, but there is an actual quantitative value for space and time, that is predicted there. This paper and work compares the Schwarzschild Spherically Symmetric Metric equation original 1916 results with the results predicted by the modern literature on the Schwarzschild Metric.

 

This work also describes conceptually the physics behind the Gaussian Distance Curvature and Reduction Density equation that Schwarzschild used to eliminate and avoid a “Coordinate Singularity”. This work reveals that Schwarzschild’s original 1916 solution, predicts that the Inertial Mass Volume Density, Escape Velocity, and gravitational field acceleration is reduced at all points in the inhomogeneous gradient gravitational field. This reduced Schwarzschild Gaussian density also reduces the gravitational tangential velocity, and acceleration in the inhomogeneous gradient gravitational field.

PDF – The Physics of Schwarzschild’s Original 1916 Metric Solution and Elimination of the Coordinate Singularity Anomaly

  

Keywords: General Relativity, Einstein Field Equation, Gravitational Field, Coordinate Singularity, Physical Singularity, Gravitation, Black Hole Event Horizon, Spherically Symmetric Metric, Euclidean Geometry, Non-Euclidean Geometry, Minkowski Metric, Euclidean Metric, Schwarzschild Metric

 

 

Contents

·         1.0       Introduction

 

 

·         1.1       Spherically Symmetric Metric – Limits of Integration & Singularities – Schwarzschild Non-Euclidean Metric – “Modern Literature Solution”

 

 

·         1.2       Spherically Symmetric Metric – Limits of Integration & Singularities – Schwarzschild Non-Euclidean Metric – “Original Solution 1916”

 

 

·         1.3       Physics Explanation for the Original 1916 Schwarzschild “Non-Euclidean” Gaussian Curvature Distance (clip_image002) as it relates to the Reduced Volume Mass Density of the Gradient Gravitational Field & the Gaussian Squared Velocity Inertia

 

 

·         1.4       Physics Explanation for the Schwarzschild “Non-Euclidean” Gaussian Curvature Distance (clip_image002) as it relates to the Kepler’s Third Law, Angular Velocity, and the Acceleration of the Gradient Gravitational Field

 

 

·         2.0       Conclusion

 

 

1.   Introduction

 

This work is written to physicists with the aim of pointing out the fact that the “Modern Literature” claims conceptually and mathematically, that the Spherically Symmetric Metric Schwarzschild Metric produces two singularities, one “Physical Singularity” located at the center of the system, and a second “Coordinate Singularity” located at the Black Hole Event Horizon, located at the Schwarzschild Semi-Major Radius, of an inhomogeneous gradient gravitational field.

 

However, in Karl Schwarzschild’s original 1916 paper, he predicts conceptually and mathematically, that the Spherically Symmetric Metric Schwarzschild Metric produces one singularity, the “Physical Singularity” located at the center of the system.

 

In Karl Schwarzschild’s original 1916 paper, he does not predict a second “Coordinate Singularity” located at the Black Hole Event Horizon, like the “Modern Literature” and the physics community claims!

 

Karl Schwarzschild’s original 1916 paper [10]: “About the gravitational field of a mass point according to Einstein’s theory”.

 

The original 1916, Spherically Symmetric Metric Schwarzschild Metric equation produces, one “Physical Singularity” located at the center of the system; and Schwarzschild avoids or eliminates “Coordinate Singularity” located at the Black Hole Event Horizon, located at the Schwarzschild Semi-Major Radius (clip_image004), of an inhomogeneous gradient gravitational field, by using a Gaussian Curvature (clip_image006) Distance Radius.

 

The mainstream physics community at large currently teaches that there is no solution to the “Coordinate Singularity” anomaly located at the Black Hole Event Horizon; and they do not appear to be searching for a solution!

 

The mainstream physics community seems to be content, accepting the fact that, space and time, become unpredictable at the Black Hole Event Horizon. While they fantasize that “Electromagnetic Radiation “Light” cannot escape a Black Hole Event Horizon”. Once again, this fantasy concept that space and time become unpredictable at the Black Hole Event Horizon is not a result of the original solution or prediction, made by the inventor of the metric, Karl Schwarzschild which he published in 1916.

 

This fact was first pointed out on a website written by Stephen J Crothers [4], where he states:

 

“The so-called “Schwarzschild” solution is not due to Karl Schwarzschild at all. The experts have either not read Schwarzschild’s 1916 memoir or have otherwise ignored it.”

 

Published in Schwarzschild’s original 1916 solution, there is not a “Coordinate Singularity” located at the Black Hole Event Horizon, but there is an actual quantitative value for space and time, that is predicted there.

 

This paper and work compares the Schwarzschild Spherically Symmetric Metric equation original 1916 results with the results predicted by the modern literature on the Schwarzschild Metric.

 

This work proves that eliminating the “Coordinate Singularity” allows for smooth prediction of a “maximally extended” space-time in the form of a “geodesic” world “line element” (clip_image008), as a real curving or warping of physical space, space-time, or a gradient gravitational field, that is spherically symmetric, surrounding a condensed matter, mass (clip_image010), or energy source, even at the Black Hole Event Horizon.

 

The Schwarzschild original 1916, metric equation and coordinates are chosen because they cover the entire maximally extended Riemann manifold, without the problem of a “Coordinate Singularity” located at the Black Hole Event Horizon; and which satisfies the “Geodesic Incompleteness” and “Null Geodesic Completeness” axioms.

 

This work also describes conceptually the physics behind the Gaussian Distance Curvature and Reduction Density equation that Schwarzschild used to eliminate and avoid a “Coordinate Singularity”.

 

This work reveals that Schwarzschild’s original 1916 solution, predicts that the Inertial Mass Volume Density is reduced at all points in the inhomogeneous gradient gravitational field. This reduced density predicted by Schwarzschild’s original 1916 solution, also reduces the gravitational tangential velocity, and gravitational acceleration throughout the entirety of the inhomogeneous gradient gravitational field.

 

Furthermore, the “Spherically Symmetric Metric” (clip_image012) can describe the geometry of space, space-time or a gravitational field, of or surrounding the: universe, stars, planets, galaxies, quasars, electrons, protons, neutrons, atoms, molecules, photons, etc…

 

 

 

1.1.       Spherically Symmetric Metric – Limits of Integration & Singularities – Schwarzschild Non-Euclidean Metric “Modern Literature Solution”

 

In General Relativity, the Non-Euclidean Spherically SymmetricSchwarzschild “Mass Dependent” Metric (clip_image012) describes the geodesic, curving, or warping of a space, space-time, or inhomogeneous gradient gravitational field, in the presence of a net inertial mass/matter (clip_image015) source; with a Black Hole Event Horizon, and the “Physical Singularity” at the core source of the isolated and conserved gravitational field.

 

Spherically Symmetric – Schwarzschild Metric  

                                                                                                                                                                                                                                                  1.1 

clip_image017 ® clip_image019

 

 

With a metric signature of (-+++) being used. In all of the “modern literature” written on General Relativity, the Schwarzschild Metric (clip_image012) yields predictable results “only” in the “exterior region” of the Black Hole Event Horizon source of the gravity field. The predictable results start with any radius relative to center of the gradient gravitational field that is greater than the Schwarzschild Radius (clip_image021), and into infinite distances (clip_image023), of the gradient field.

 

clip_image025

The Exterior solution corresponds to the space outside of the Black Hole Event Horizon, and Schwarzschild Semi-Major Radius (clip_image027).

 

 

clip_image029

 

 

The Interior solution corresponds to the space within or inside the Black Hole Event Horizon, and Schwarzschild Semi-Major Radius (clip_image027).

 

 

clip_image031

 

 

The “modern literature” written on General Relativity, describes that the Schwarzschild Metric (clip_image033) predicts that a “Black Hole” has “two (2) Singularities” in the metric of a localized inhomogeneous gravitational field:

 

There is one “Physical Singularity” that is an infinitely “small” number located at zero radius (clip_image035) of the gradient gravitational field. (clip_image037).

 

And a second “Coordinate Singularity” that is an infinitely “large” number located at the Black Hole Event Horizon, Schwarzschild (clip_image039) Radius of the inhomogeneous gradient gravitational field. (clip_image037).

 

 

clip_image041        ;        clip_image043

clip_image045

 

clip_image047

 

 

The Schwarzschild Metric (clip_image033) predicts the “Physical Singularity” located at zero radius, is “extreme curvature” as the geodesic approaches an infinitely small value, (clip_image050), as the radius approaches zero (clip_image052). The “Physical Singularity” is a natural artifact for any Non-Euclidean metric.

 

The “Coordinate Singularity” is an anomaly that is predicted by the modern literature, at the surface of the Black Hole Event Horizon, and is an infinitely large value, (clip_image050), with “extreme curvature” of the geodesic as the radius approaches Schwarzschild Radius (clip_image055).  This value is an anomaly, and is not a natural artifact for any Non-Euclidean metric.

 

Furthermore,the Schwarzschild Metric (clip_image033) predicts, that as the distance relative to a spherically symmetric center, approaches infinity or an infinite distance (clip_image057) away from the “physical singularity”, center, where there is maximum or extreme curvature, the Non-Euclidean“Curved Space-time” becomes Minkowski “Flat Space-time”. This is also known as Non-Euclidean “Geodesic Incompleteness”; of a diverging gravity field.

 

Pseudo-Euclidean – Minkowski Metric 

                                                                                                                                                                                                                                                  1.2 

clip_image059      ® clip_image019

 

 

Space-time – Square of the Speed of Light

                                                                                                                                                                                                                                                  1.3 

clip_image061     ® clip_image063

 

 

Space-time – Square of the Speed of Space (Vacuum Energy Velocity)

                                                                                                                                                                                                                                                  1.4 

clip_image065     ® clip_image063

clip_image067

 

 

New Mathematical Formalism – Generalized Spherically Symmetric Metric

                                                                                                                                                                                                                                                  1.5 

  clip_image069   ® clip_image019

 

 

 

“New” Schwarzschild Metric “Curvature” Coefficient ─ (clip_image072)

                                                                                                                                                                                                                                                  1.6 

 clip_image074    ® clip_image076

New Mathematical Formalism – Generalized Spherically Symmetric Schwarzschild (Non-Euclidean) Metric

                                                                                                                                                                                                                                                  1.7 

clip_image078® clip_image019

 

clip_image069

 

clip_image080

Below is a plot of the Schwarzschild Metric (clip_image033)

 

(clip_image082)                 Then              (clip_image084)  

 

 

And clip_image086

 

image

 

 

1.2.       Spherically Symmetric Metric – Limits of Integration & Singularities – Schwarzschild Non-Euclidean Metric “Original Solution 1916”

 

Next, let’s consider Karl Schwarzschild’s original 1916 metric solution for the Non-Euclidean Spherically SymmetricMetric (clip_image012) equation.

 

In Karl Schwarzschild’s original paper in (1916), he does not have two (2) “Singularities” in his original Metric (clip_image033) equation; like that which is described in “modern literature” claims. Schwarzschild, has only the one “Physical Singularity” located at zero radius (clip_image035) of the gravitational field; described in his original solution 1916.

 

I believe that this one “Physical Singularity” result is a natural artifact of the “Non-Euclidean” geometry, which is not a feature of the “Euclidean” geometry.

 

The original form of the Schwarzschild Metric (clip_image033) equation, as presented by Schwarzschild in 1916; below with a metric signature of (-+++).

 

Schwarzschild Spherically Symmetric Metric – in “Original 1916 Form” 

                                                                                                                                                                                                                                                  1.8 

clip_image090® clip_image019

 

 

Schwarzschild Gaussian Curvature Distance (clip_image002) – in “Original 1916 Form” 

                                                                                                                                                                                                                                                  1.9 

clip_image093      ® clip_image095

 

The Schwarzschild Metric (clip_image012) radial limits which extend from the center of the gradient gravitational field (clip_image097), and into infinite (clip_image023) distances, is given below.

 

clip_image041        ;        clip_image043

 

 

The radial distance given in Schwarzschild’s original form, is given by the Gaussian Curvature Radius given by the symbol (clip_image002) and not the Semi-Major Radius given by the symbol (clip_image101); which describes a radial distance relative to the center of the gradient gravitational field.

 

Karl Schwarzschild in his paper claimed that this Gaussian curvature distance (clip_image002), is not an actual physical measureable distance, but is an artifact of mathematics of the geodesics paths in Non-Euclidean curvature and geometry.

 

It is a very notable difference in the way that modern mainstream literature presents the Schwarzschild Spherically Symmetric Metric and the way the Karl Schwarzschild presented his original solution in 1916.

 

Using the distance (clip_image002) of Gaussian curvature, given in Schwarzschild’s original work, eliminates the “Coordinate Singularity” at the Black Hole Event Horizon.

 

The main difference is that the modern mainstream physics literature claims that there is a “Coordinate Singularity” located at the surface of the Black Hole Event Horizon. And because of this “Coordinate Singularity” space and time become unpredictable at the Black Hole Event Horizon.

 

In Karl Schwarzschild original 1916 metric solution, there is not a “Coordinate Singularity” located at the surface of the Black Hole Event Horizon; but there is a quantitative value for space, distance, and time, is measured at the Black Hole Event Horizon.

 

Next it will be presented, a new form of Schwarzschild’s original 1916 metric coordinate solution, and metric equation.

 

Starting with Schwarzschild Gaussian Curvature Distance (clip_image002) – in “Original 1916 Form”, from the equation above, and taking the differential yields the following.

                                                                                                                                                                                                                                               1.10 

clip_image103        ® clip_image105

 

Schwarzschild Gaussian Curvature Differential Distance (clip_image107) – as a function of Differential Semi-Major Distance Space (clip_image109)

                                                                                                                                                                                                                                               1.11 

clip_image111      ®  clip_image019

 

 

Next, substituting the above differential equation into the original form of the Schwarzschild Spherically Symmetric Metric (clip_image033) differential equation, as presented by Schwarzschild in 1916. This will be done, so that the Schwarzschild original 1916 metric, is in a form that is similar to the modern literature form of the metric as shown below.

 

Schwarzschild Spherically Symmetric Metric – in “Original 1916 Form” function of Differential Semi-Major Distance Space (clip_image109), time (clip_image113), and Manifold/Surface Map Angle (clip_image115)

                                                                                                                                                                                                                                               1.12 

 

clip_image117      ®  clip_image019clip_image119

 

 

  

 

The above equation is a new mathematical representation of the Karl Schwarzschild’s original 1916 solution; with a metric signature of (-+++) being used.

 

This result presented in Schwarzschild’s original paper predicts that the Schwarzschild Metric (clip_image033) should only have “One (1) Physical Singularity” in the metric located at the center of the gravitational field (clip_image121).

 

Whereas, the modern literature which describes the Schwarzschild Metric in their writings, claims that there is a second “Coordinate Singularity” at the Black Hole Event Horizon (clip_image123).

 

It is obvious from the above equation, that Karl Schwarzschild’s original 1916 solution does not produce a “Coordinate Singularity” at the Black Hole Event Horizon (clip_image125).

 

Therefore using Schwarzschild’s original 1916 solution, the “Coordinate Singularity” anomaly, located at the Black Hole Event Horizon and Schwarzschild Radius (clip_image125), as predicted by modern literature is eliminated; as demonstrated in the chart and graphically below.

 

Once again, in Schwarzschild’s original solution, there is only one “Physical Singularity” located at zero radius (clip_image035) of the gravitational field.

 

The Schwarzschild Spherically Symmetric Metric (clip_image012) therefore predicts the following.

 

clip_image041        ;        clip_image043

 

clip_image127

 

The Schwarzschild Original 1916 Metric (clip_image033) predicts the “Physical Singularity” located at zero radius, is “extreme curvature” as the geodesic approaches an infinitely large value, (clip_image181), as the radius approaches zero (clip_image052). The “Physical Singularity” is a natural artifact for any Non-Euclidean metric.

 

Therefore using Schwarzschild’s original 1916 solution, the“Coordinate Singularity” is eliminated where the radius is equal to the Schwarzschild Radius (clip_image129); and a quantitative value for space, distance, and time, is measured at the Black Hole Event Horizon.

 

Furthermore,the Schwarzschild Metric (clip_image033) predicts, that as the distance relative to a spherically symmetric center, approaches infinity or an infinite distance (clip_image057) away from the “physical singularity”, center, where there is maximum or extreme curvature, the Non-Euclidean“Curved Space-time” becomes Minkowski “Flat Space-time”. This is also known as Non-Euclidean “Geodesic Incompleteness”; of a diverging gravity field.

 

Pseudo-Euclidean – Minkowski Metric 

                                                                                                                                                                                                                                               1.13 

 

clip_image059      ® clip_image019

 

 

Space-time – Square of the Speed of Light

                                                                                                                                                                                                                                               1.14 

clip_image061     ® clip_image063

 

 

Space-time – Square of the Speed of Space (Vacuum Energy Velocity)

                                                                                                                                                                                                                                               1.15 

clip_image065     ® clip_image063

clip_image067

 

 

New Generalized Mathematical Formalism Spherically Symmetric Metric – Using the Schwarzschild Original 1916 Solution

                                                                                                                                                                                                                                               1.16 

 

clip_image111

clip_image132  ® clip_image019

 

 

“New” Schwarzschild 1916 Metric “Curvature” Coefficient ─ (clip_image134)

                                                                                                                                                                                                                                               1.17 

clip_image136

® clip_image076

 

 

New Generalized Spherically Symmetric Schwarzschild 1916 (Non-Euclidean) Metric

                                                                                                                                                                                                                                               1.18 

clip_image138

clip_image132® clip_image019

 

 

New Generalized Mathematical Formalism Spherically Symmetric Metric – Using the Schwarzschild Original 1916 Solution

                                                                                                                                                                                                                                               1.19 

 

clip_image111   ® clip_image019

clip_image140  ® clip_image019

 

 

“New” Schwarzschild 1916 Metric “Curvature” Coefficient ─ (clip_image134)

                                                                                                                                                                                                                                               1.20 

 

 clip_image142   

® clip_image076

 

 

New Generalized Spherically Symmetric Schwarzschild 1916 (Non-Euclidean) Metric

                                                                                                                                                                                                                                               1.21 

clip_image144

clip_image140® clip_image019

 

 

clip_image145

 

 

Below is a plot of the Schwarzschild Metric (clip_image033)

 

(clip_image082)                 Then              (clip_image084)  

 

And clip_image086

 

The Schwarzschild Spherically Symmetric Original 1916 Metric (clip_image012), described as

 

image

 

The Schwarzschild Spherically Symmetric Original 1916 Metric (clip_image012), compared to the modern literature Schwarzschild Spherically Symmetric Metric (clip_image012), is shown in graph below.

image

image

 

Non-Euclidean Spherically Symmetric Metric

Modern Literature & Original 1916 Solution“Schwarzschild” Metric

 “Geodesic Incompleteness” of a Diverging Gravitational Field

Singularities and Infinitely Large Spaces Listed

 

clip_image041        ;        clip_image043

 

Non Euclidean “Schwarzschild” Metric

(Curved Space-time)

(clip_image154)

Spherically Symmetric Metric

(clip_image012)

Spherically Symmetric Semi-Major Radius (Distance/Space)

 

(clip_image052)

&

(clip_image157)

Spherically Symmetric Metric

(clip_image012)

Spherically Symmetric Semi-Major Radius (Distance/Space)

 

(clip_image055)

&

(clip_image129)

Spherically Symmetric Metric

(clip_image012)

Spherically Symmetric

Semi-Major Radius (Distance/Space)

 

(clip_image160)

&

(clip_image162)

Schwarzschild Metric

(Curved Space-time)

(clip_image154)

Modern Literature

clip_image164

clip_image166

clip_image168

clip_image170

(Infinitely Small)

Physical Singularity

clip_image172

(Infinitely Large)

Coordinate Singularity

clip_image172

(Infinitely Large)

Flat Space-Time

Schwarzschild Metric

(Curved Space-time)

(clip_image154)

Original 1916 Solution

clip_image164

clip_image174

clip_image175

clip_image177

(Infinitely Large)

Physical Singularity

clip_image179

(Finite Value)

clip_image172

(Infinitely Large)

Flat Space-Time

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The Schwarzschild Spherically Symmetric Original 1916 Metric (clip_image012), eliminates the “Coordinate Singularity” located at the Black Hole Event Horizon (clip_image125), as compared to the modern literature Schwarzschild Spherically Symmetric Metric (clip_image012).

 

This mathematical form, presented in Schwarzschild’s original 1916 paper where he gives a “One (1) Singularity” solution, is much different from what is currently written, accepted, and taught in literature with the “Two (2) Singularities” solution.

 

In the author’s opinion, the original solution proposed by Karl Schwarzschild’s original paper in 1916, solves the “Coordinate Singularity” anomaly at the Black Hole Event Horizon, but is not an accurate description of nature.

 

In the author’s opinion, the most accurate “Non-Euclidean” geometric metric description, of the contracting and curving of space, space-time, or a gravitational field in the presence of mater, would behave Euclidean as the radius approaches zero (clip_image052).

 

In Karl Schwarzschild’s original 1916 solution, as the spherically symmetric radius approaches zero (clip_image052), the metric is approaching an infinitely large number (clip_image181), when it should be approaching an infinitely small number (clip_image183).

 

It is assumed by many, who read literature on the original works of Schwarzschild, that the Gaussian Curvature Distance (clip_image002), as given by Schwarzschild original 1916 solution, describes as a radial distance, which originates from a source at the center of the gravitational field; it does not!

 

The Gaussian Curvature Distance (clip_image185) is not true radial distance, but is related to the Gaussian curvature and density (clip_image187) of a spherically symmetric geodesic surface, in a localized region of gradient gravitational field, related to the inertial mass volume density (clip_image189) of the gradient gravitational field, and the Black Hole Event Horizon inertial mass volume density (clip_image191); of the space, space-time, or a gravitational field.

 

Therefore Schwarzschild’s original 1916 solution predicts that the Gaussian Curvature Density (clip_image187) is a reduced Inertial Mass Volume Density (clip_image189), spread throughout the entirety of the inhomogeneous gradient gravitational field, surrounding a Net Inertial Mass (clip_image010), or energy source; given by the following.

Gaussian Density of Non-Euclidean Curvature of Gradient Gravity Field

                                                                                                                                                                                                                                               1.22 

clip_image195       ® clip_image197

clip_image199

 

 

Likewise, Schwarzschild’s original 1916 solution predicts that located at the Black Hole Event Horizon (clip_image125), the Gaussian Curvature Density(clip_image202) is equal to one half the Black Hole Event Horizon inertial mass volume density, of the space, space-time, or a gravitational field

 

Gaussian Density of Non-Euclidean Curvature of Black Hole Event Horizon

                                                                                                                                                                                                                                               1.23 

clip_image204       ® clip_image197

 

 

Inertial Volume Mass Density – Black Hole Event Horizon

                                                                                                                                                                                                                                               1.24 

clip_image206 

 

 

1.3.       Physics Explanation for the Original 1916 Schwarzschild “Non-Euclidean” Gaussian Curvature Distance (clip_image002) as it relates to the Reduced Volume Mass Density of the Gradient Gravitational Field & the Gaussian Squared Velocity Inertia

 

Now let’s look at the Gaussian Curvature Distance (clip_image002) predicted bySchwarzschild in his “Original 1916 paper. I will extend this concept to fit the concept of a Reduced Volume Mass Density and the Kepler’s Laws of Motion.

This is not a concept that Karl Schwarzschild investigated, but here you will see that this concept can be extended in the following.

                                                                                                                                                                                                                                               1.25 

clip_image208      ® clip_image095

 

The Net Inertial “Volume” Mass Density(clip_image210), of the “Black Hole Event Horizon” is a fixed local constant value, of the “inhomogeneous” gradient gravitational field, and is defined as the Net Inertial Mass (clip_image010), divided by the “Black Hole Event Horizon” Volume(clip_image213) of the gradient gravitational field.

 

Inertial Volume Mass Density – Black Hole Event Horizon

                                                                                                                                                                                                                                               1.26 

clip_image206

The Net Inertial “Volume” Mass Density(clip_image216), of the “Inhomogeneous” gradient gravitational field, is defined as the Net Inertial Mass (clip_image010), divided by an infinite series of Spherical Volume (clip_image219), of gradient energy potentials of the inhomogeneous gradient gravitational field.

 

Inertial Volume Mass Density – Inhomogeneous Gradient Gravitational Field

                                                                                                                                                                                                                                               1.27 

clip_image221

image

 

Now, let’s consider the Schwarzschild Gaussian curvature distance (clip_image002) in a Non-Euclidean space, space-time, or gradient gravitational field, and substituting the Inertial Mass Volume density terms; and deriving the Gaussian Curvature Inertial Mass Volume Density (clip_image187), yields:

                                                                                                                                                                                                                                               1.28 

clip_image227       ® clip_image095

 

clip_image229

Gaussian Density of Non-Euclidean Curvature of Gradient Gravity Field

                                                                                                                                                                                                                                               1.29 

clip_image231       ® clip_image197

 

clip_image199

 

The Gaussian “Non-Euclidean” Curvature Density (clip_image234), predicted by Schwarzschild Gaussian curvature (clip_image002) results in mass curving or warping space, predicts that at any location in the space, space-time, or gradient gravitational field, the mass density of that gradient is slightly “less dense” than would be, the normal mass volume density at that location (clip_image236).

 

Now, let’s introduce a new concept called the Gaussian curvature squared velocity “inertial potential” (clip_image238) in a Non-Euclidean Space, which is achieved by starting with the Gaussian curvature distance (clip_image002), and substituting the density equations, and solving for the squared velocity inertia:

                                                                                                                                                                                                                                               1.30 

clip_image208       ® clip_image095

 

 

The Schwarzschild Gaussian curvature distance (clip_image002) in a Non-Euclidean Space and substituting the density equations, yields:

                                                                                                                                                                                                                                               1.31 

clip_image241     ® clip_image095

clip_image243

 

Next, substituting the Gravitational Tangential Squared Velocity Inertia of the gravitational field:

                                                                                                                                                                                                                                               1.32 

 

clip_image245    ® clip_image063

 

 

Gaussian Squared Velocity Inertia (clip_image238) Potential of Non-Euclidean Curvature of Gradient Gravity Field (clip_image249)

                                                                                                                                                                                                                                               1.33 

clip_image251       ® clip_image063

 

 

 

clip_image253

 

 

 

clip_image255

 

 

The Gaussian Squared Velocity Inertial Potential (clip_image257), is predicted when the Schwarzschild Gaussian curvature distance (clip_image002) in a Non-Euclidean space, as a result of mass curving or warping space, predicts that at any location in the space, space-time, or gradient gravitational field, the squared velocity inertia of that gradient gravity field is slightly “smaller”  or “slower” than would be, the normal Gravitational Tangential Squared Velocity Inertia of the gravitational field (clip_image259) in that location.

 

 

 

1.4.       Physics Explanation for the Schwarzschild “Non-Euclidean” Gaussian Curvature Distance (clip_image002) as it relates to the Kepler’s Third Law, Angular Velocity, and the Acceleration of the Gradient Gravitational Field

 

Now, let’s consider Kepler’s Third Law, in consideration with the Schwarzschild Gaussian Curvature distance (clip_image002).

                                                                                                                                                                                                                                               1.34 

clip_image261     ® clip_image095

Evolutionary Mass Attraction Rate – Kepler’s Third Law

                                                                                                                                                                                                                                               1.35 

 

clip_image264

clip_image266

Where the Black Hole Orbital Period (clip_image268) is given by

                                                                                                                                                                                                                                               1.36 

clip_image270    ® clip_image272

 

Where the Gradient Gravitational Field Orbital Period (clip_image274) is given by

                                                                                                                                                                                                                                               1.37 

clip_image276     ® clip_image272

 

 

Now, let’s consider Kepler’s Third Law, and rewrite the Schwarzschild Gaussian Curvature distance (clip_image002) to show that it describes the volume of space, space-time, or gravitational field, due to the presence of mass with a black hole event horizon at the center of the gravitational field.

 

The “Gaussian Total Volume” equation below is related to the Schwarzschild Gaussian Curvature distance (clip_image002), and describes a volume of “Non-Euclidean” curved or warped space, due to the presence of mass with a black hole event horizon at the center of a gradient gravitational field:

                                                                                                                                                                                                                                               1.38 

clip_image278       ® clip_image280

clip_image282

clip_image284

image

 

The above “Gaussian Total Volume” (clip_image288) equation means that due to the presence of mass, with a black hole event horizon at the center of a gradient gravitational field, the total volume is the net sum of the volume (clip_image290) of any one of the infinite shells of the spherical gradient gravitational field, and sum the gravitational field Black Hole Event Horizon Volume (clip_image292).

 

Next, substituting Kepler’s Third Law equation, into The “Gaussian Volume” equation above, yields a cube of the Gaussian distance equation:

                                                                                                                                                                                                                                               1.39 

clip_image294        ® clip_image280

 

 

clip_image296

 

clip_image298

 

Next, from the equation above, we derive, a “Gaussian Angular Velocity” which describes that the gravitational angular velocity in a gravitational field potential, is “slightly” reduced or “slowed down” in rate or motion, due to the Non-Euclidean curved, space, space-time, or gravitational field.

                                                                                                                                                                                                                                               1.40 

clip_image300     ® clip_image302

 

 

 

 

clip_image304

 

 

 

clip_image306       ® clip_image302

 

 

Next, substituting the Inhomogeneous Gradient Gravitational Acceleration of the gravitational field (clip_image308):

                                                                                                                                                                                                                                                  1.1 

clip_image310    ® clip_image312

 

 

clip_image314    ® clip_image316

 

 

 

Likewise, using the Schwarzschild Gaussian Curvature distance (clip_image002) equation, we derive, a “Gaussian Gravitational Field Attraction Acceleration” (clip_image318) which describes that the gravitational field attraction acceleration in a gravitational field potential, is “slightly” reduced in the attraction rate, due to the Non-Euclidean curved, space, space-time, or gravitational field.

                                                                                                                                                                                                                                                  1.2 

 

clip_image320     ® clip_image312

 

 

 

clip_image322      ® clip_image312

 

 

clip_image324

 

clip_image326

 

 

 

The above spherically symmetric “Non-Euclidean” inhomogeneous, gradient gravitational field, space, or space-time equations are the result of using the original solution of the Schwarzschild Gaussian Curvature distance (clip_image002) predicted in 1916.

 

 

 

2.   Conclusion

 

 

The mainstream physics community at large currently teaches that there is no solution to the “Coordinate Singularity” anomaly of the Schwarzschild Spherically Symmetric Metric, located at the Black Hole Event Horizon and does not appear to be searching for a solution!

 

The mainstream physics community seems to be content, accepting the fact that, space and time, become unpredictable at the Black Hole Event Horizon; although the inventor of the metric Schwarzschild, never described unpredictability at the Black Hole Event Horizon surface.

 

Published in Schwarzschild’s original 1916 metric solution, there is not a “Coordinate Singularity” located at the Black Hole Event Horizon, but there is an actual quantitative value for space and time, that is predicted there.

 

This paper and work compared the Schwarzschild Spherically Symmetric Metric equation original 1916 results with the results predicted by the modern literature on the Schwarzschild Metric.

 

This work proved that eliminating by the “Coordinate Singularity” anomaly this allows for smooth prediction of a maximally extended “geodesic” world “line element” (clip_image008), which describes a real curving or warping of physical space, space-time, or a gradient gravitational field, that is spherically symmetric, surrounding a condensed matter, mass (clip_image010), or energy source, even at the Black Hole Event Horizon.

 

This work demonstrated that the Schwarzschild original 1916, metric equation and coordinates are a good choice for a metric system, because, they are able to cover the entire extended Schwarzschild manifold, without the problem of a “Coordinate Singularity” located at the Black Hole Event Horizon.

 

This work also described conceptually the physics behind the Gaussian Distance Curvature and Reduction Density equation that Schwarzschild used to eliminate and avoid a “Coordinate Singularity”.

 

This work revealed that Schwarzschild’s original 1916 metric solution, predicts that the Inertial Mass Volume Density is reduced at all points in the inhomogeneous gradient gravitational field.

 

This reduced density predicted by Schwarzschild’s original 1916 solution, also reduces the gravitational tangential velocity, and acceleration throughout the entirety of the inhomogeneous gradient gravitational field.

 

The author does not agree that these gravitational field relations predicted by Schwarzschild Gaussian Curvature distance (clip_image002) predicted in 1916, are actual physical results. The predictions made by these gravitational field relations would have to be measured, to be validated.

 

I only did the physics here, to show that Karl Schwarzschild’s original solution is not trivial, but yields some interesting physics. And so that the physics community will stop propagating the invalid physics that a “Coordinate Singularity”, located at the Black Hole Event Horizon, has to be true in nature. When in fact, the “Coordinate Singularity”, is a mathematical anomaly, which can be resolved with proper mathematical treatment.

 

It turns out that there can be many different solutions and mathematical forms for describing the radial distance (clip_image101) as measured from the center of the gradient gravitational field, of a Spherically Symmetric Schwarzschild “Non-Euclidean” Metric. The original form of the Schwarzschild Metric (clip_image033) equation, as presented by Schwarzschild in 1916; below.

 

Schwarzschild Metric – in “Original 1916 Form”  

                                                                                                                                                                                                                                                  1.3 

clip_image329® clip_image019

 

 

 

Schwarzschild Gaussian Curvature distance (clip_image002)

                                                                                                                                                                                                                                                  1.4 

clip_image331    ® clip_image095

 

clip_image041

 

Schwarzschild Gaussian Curvature Differential distance (clip_image107)

                                                                                                                                                                                                                                                  1.5 

clip_image335     ® clip_image019

 

 

In the author’s opinion, the original solution proposed by Karl Schwarzschild’s original paper in 1916, solves the “Coordinate Singularity” anomaly at the Black Hole Event Horizon, but is not an accurate description of nature.

 

In the author’s opinion, the most accurate “Non-Euclidean” geometric metric description, of the contracting and curving of space, space-time, or a gravitational field in the presence of condensed mass, matter or energy, would behave Euclidean, as the radius approaches zero (clip_image052), the metric should be approaching an infinitely small number ( clip_image183)!

 

In Karl Schwarzschild’s original 1916 solution, as the spherically symmetric radius approaches zero (clip_image052), the metric is approaching an infinitely large number (clip_image181), when it should be approaching an infinitely small number (clip_image183)!

 

This paper eliminates the need for all of the weird physics that is predicted by current mainstream physics, that takes place at the surface of Black Hole Event Horizon, such as light not being able to escape the black hole, time becoming infinite, and spaghettification of anyone crossing the event horizon; all of these concepts are eliminated using the Schwarzschild Gaussian Curvature distance (clip_image002) employed by Karl Schwarzschild’s original 1916 metric solution.

 

Below are the topics that were discussed in this paper:

 

 

·         1.1       Spherically Symmetric Metric – Limits of Integration & Singularities – Schwarzschild Non-Euclidean Metric – “Modern Literature Solution”

 

 

·         1.2       Spherically Symmetric Metric – Limits of Integration & Singularities – Schwarzschild Non-Euclidean Metric – “Original Solution 1916”

 

 

·         1.3       Physics Explanation for the Original 1916 Schwarzschild “Non-Euclidean” Gaussian Curvature Distance (clip_image002) as it relates to the Reduced Volume Mass Density of the Gradient Gravitational Field & the Gaussian Squared Velocity Inertia

 

 

·         1.4       Physics Explanation for the Schwarzschild “Non-Euclidean” Gaussian Curvature Distance (clip_image002) as it relates to the Kepler’s Third Law, Angular Velocity, and the Acceleration of the Gradient Gravitational Field

References

 

 

[1]        Physical Acceptability of Isolated, Static, Spherically Symmetric, Perfect Fluid Solutions of Einstein’s Equations, M.S.R. Delgaty and Kayll Lake1, Department of Physics, Queen’s University, Kingston Ontario, Canada K7L 3N6 (Dated: Septem 1998): http://arxiv.org/pdf/gr-qc/9809013v1.pdf

 

 

[2]        Generating perfect fluid spheres in general relativity, Petarpa Boonserm, Matt Visser, and Silke Weinfurtner, School of Mathematics, Statistics, and Computer Science, Victoria University of Wellington,PO Box 600, Wellington, New Zealand (Dated: March 2005 – February 6, 2008): http://arxiv.org/pdf/gr-qc/0503007v1.pdf(Pages: 13 – 16)

 

[3]        Karl Friedrich Gauss, General Investigations of Curved Surfaces of 1827 and 1825, The Project Gutenberg EBook of General Investigations of Curved Surfaces of 1827 and 1825, by Karl Friedrich Gauss: http://www.gutenberg.org/files/36856/36856-pdf.pdf

 

[4]        Stephen J Crothers; http://www.sjcrothers.plasmaresources.com/

 

[5]        Gauss Law – Theorema Egregium:

            http://en.wikipedia.org/wiki/Theorema_Egregium

[6]        List of Differential Geometry Topics (Wikipedia):

            http://en.wikipedia.org/wiki/List_of_differential_geometry_topics

 

 

[7]        D.F. Lawden. Introduction to Tensor Calculus, Relativity and Cosmology; Dover Publications; 3 edition (January 27, 2003), ISBN 978-0486425405 pp. 6-10

[8]        Minkowski, Hermann (1907/8), “Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern“, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111 *** Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies.

[9]        Einstein’s Paper: “Explanation of the Perihelion Motion of Mercury from General Relativity Theory”; Anatoli Andrei Vankov; IPPE, Obninsk, Russia; Bethany College, KS, USA:

 

 

 

[10]      Schwarzschild, K. (1916). “About the gravitational field of a mass point according to Einstein’s theory”: text of the original paper, in Wikisource:

 

 

[11]      Schwarzschild Geodesics:

            http://en.wikipedia.org/wiki/Schwarzschild_geodesics

 

 

[12]      Robert Louis Kemp. “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics – The General Theory of Relativity.” ISBN 978-0-9841518-2-0, Flying Car Publishing Company, July 2010, pp. 20-21.

 

 

[13]      Inertial-Mass-Vortex-Gravitation-Theory-Continued-Part-3, Robert Louis Kemp (2011). “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics, Flying Car Publishing Company

           

 

 

[14]      Total-Mechanical-Energy-Conservation-Escape-Velocity-Binding-Energy-Einstein-Field-Equation, Robert Louis Kemp (2012). “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics

 

 

 

[15]      A-Theory-Of-Gravity-For-The-21st-Century-The-Gravitational-Force-And-Potential-Energy-In-Consideration-With-Special-Relativity-General-Relativity, Robert Louis Kemp (2012). “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics

 

 

[16]      Newtonian-Self-Gravitational-Force-Video-Lecture, Robert Louis Kemp (2012). “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics

 

 

 

[17]      Conditions-For-The-Absence-And-Presence-Of-Matter-In-General-Relativity-Gr-Einstein-Field-Equations-Efes, Robert Louis Kemp (2012). “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics

 

  

[18]      What-Are-The-Mathematical-And-Physical-Concepts-Of-Flat-Euclidean-And-Curved-Non-Euclidean-Gravitational-Fields, Robert Louis Kemp (2013). “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics

[19]      The-Perfect-Fluid-Vacuum-Unified-Gravitation-Vortex-Model-And-Non-Euclidean-Spherically-Symmetric-Metrics, Robert Louis Kemp (2013). “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics

 

 

 

[20]      Algorithm For Describing Spherically-Symmetric Metrics Of A Gravitational Field – Using-Space-Time-Angle-Metric-Components-Metric-Coefficients, Robert Louis Kemp (2013). “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics

 

 

 

 

[21]      What is the Cosmic Censorship Hypothesis? What are Black Hole Physical and Naked Singularities?, Robert Louis Kemp (2013). “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics

 

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