Algorithm for Describing Spherically Symmetric Metrics of a Gravitational Field ─ Using Space, Time & Angle Metric Components & Metric Coefficients

Algorithm for Describing Spherically Symmetric Metrics of a Gravitational Field ─ Using Space, Time & Angle Metric Components & Metric Coefficients

 

 

 

By

Robert Louis Kemp

 

Super Principia Mathematica

The Rage to Master Conceptual & Mathematical Physics

 

www.SuperPrincipia.com

 

www.Blog.Superprincipia.com

 

Flying Car Publishing Company

P.O Box 91861

Long Beach, CA 90809

January 25, 2013

 

 

 

Abstract

 

 

This paper described a new algorithm, for “generalized mathematical formalism” of a “Spherically Symmetric Metric” (clip_image002), that describes the Euclidean Metric, Minkowski Metric, Einstein Metric, or the Schwarzschild Metric; using Three (3) Metric Components & Three (3) Metric Coefficients; and likewise using a general algorithm which is composed of, Two (2) Metric Components & Two (2) Metric Coefficients.

 

In this paper a general introduction to basic mathematical concepts for the geometric description of Euclidean “Flat-Space” Geometry and Non-Euclidean “Curved-Space” Geometry, and Spherically Symmetric Metric equations which are used for describing the causality and motion of the “Gravitational” interaction between mass with vacuum energy space, and the mass interaction with mass.

 

This paper gives a conceptual and mathematical description of the differential geometry, of flat and curved space, space-time, or gravitational fields, using the “metric theory” mathematics of Euclidean, Minkowski, Einstein, and Schwarzschild, Spherically Symmetric metrics, and geodesic line elements.

 

PDF – Algorithm for Describing Spherically Symmetric Metrics of a Gravitational Field ─ Using Space, Time & Angle Metric Components & Metric Coefficients

 

Keywords: General Relativity, Special Relativity, Einstein Field Equation, Gravitational Field, Black Hole Event Horizon, Spherically Symmetric Metric, Euclidean Geometry, Non-Euclidean Geometry, Minkowski Metric, Einstein Metric, Schwarzschild Metric, Physical Singularity, Coordinate Singularity, Perfect Fluid Vacuum Energy, Aether, Gravity, Space-time Curvature, Flat Space-time, Curved Space-time, Geodesic, Metric Theory of Gravitation,

 

 

Contents

 

·         1.0       Introduction

 

 

·         1.1       Algorithm for Describing Spherically Symmetric Metrics of a Gravitational Field Using – Three (3) Metric Components & Three Metric Coefficients

 

 

·         1.2       Euclidean “Flat Space” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

·         1.3       Minkowski “Flat Space-Time” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

·         1.4       Schwarzschild “Curved Space-Time” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

·         1.5       Einstein “Curved Space-Time” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

·         1.6       “New” Algorithm for Describing Spherically Symmetric Metrics of a Gravitational Field Using – Two (2) Metric Components & Two (2) Metric Coefficients

 

  

 

1.   Introduction

 

 

This work is written to physicists that are interested in understanding from a conceptual view, the description of “Flat Geometry” Euclidean Space, or “Curved Geometry” Non-Euclidean Space, for describing causality for gravity motion, and the “Gravitational” interaction between mass with vacuum energy space, and the mass interaction with mass.

 

In this paper, I do weave some of my own theory, ideas, and mathematics into these well established physics concepts and mathematics; therefore, this work is written for those that have a very good basis and understanding, of the concepts of differential geometry, and General Relativity; to be able to distinguish what is newly proposed, and what is being discussed in general throughout this paper.

 

In this work I have limited the discussion only to the: Euclidean, Minkowski, Einstein, and Schwarzschild Spherically Symmetric Metrics, and geodesic line elements, of space, space-time, or the gravitational field, however there are many other geometric “metric” equations, and theories of gravitation, that are accepted by the mainstream physics. And there are many “Spherically Symmetric Metrics” that are in use in physics today.

 

In a paper written by M.S.R. Delgaty and Kayll Lake (1998) “Physical Acceptability of Isolated, Static, Spherically Symmetric, Perfect Fluid Solutions of Einstein’s Equations[1], they describe various “Spherically Symmetric Metrics” (clip_image002) equations.

 

M.S.R. Delgaty and Kayll Lake ”[1], state,

 

“It is fair to say then that most of the spherically symmetric perfect fluid “exact solutions” of Einstein’s field equations that are in the literature are of no physical interest.”

 

And likewise in a paper by, Petarpa Boonserm, Matt Visser, and Silke Weinfurtner (2005) “Generating perfect fluid spheres in general relativity[2], they describe that there are over 127 solutions to the “Spherically Symmetric Metrics”.

 

But only nine (9) of those “metric” equations satisfy the criteria for predicting actual physical measurable results.

 

The various “Spherically Symmetric Metric” (clip_image002), equations which are either Euclidean or Non-Euclidean, describes physical and observable results of gravitational interaction between mass and space, and between mass and mass, predicts that the vacuum energy, and inertial matter in motion interact, through a space, space-time, or gravitational field, that is either flat or curved, and surrounding a localized gravity source, that is either matter dependent, or matter independent, is described in the following sections of this paper.

 

The “Spherically Symmetric Metric” (clip_image002), and the “Geodesic Line Element” (clip_image005), are used for describing the “flat” or “curved” Differential Geometry of Space, Time, & Surfaces, of spherically symmetric space, space-time, or gradient gravitational field, in the presence or absence of condensed mass, matter, or energy.

 

Furthermore, the “Spherically Symmetric Metric” (clip_image002) can describe the space, space-time or a gravitational field, of or surrounding the: universe, stars, planets, galaxies, quasars, electrons, protons, neutrons, atoms, molecules, photons, etc…

 

In this work, a new algorithm, for “generalized mathematical formalism” of a “Spherically Symmetric Metric” (clip_image002), that describes the Euclidean Metric, Minkowski Metric, Einstein Metric, or the Schwarzschild Metric; using one general equation which is composed of, Three (3) Metric Components & Three (3) Metric Coefficients.

 

And likewise there is a general algorithm which is composed of, Two (2) Metric Components & Two (2) Metric Coefficients.

 

In future works, a new algorithm, that describes the Euclidean Metric, Minkowski Metric, Einstein Metric, or the Schwarzschild Metric using a general equation which is composed of:

 

·         Four (4) Metric Components & Four (4) Metric Coefficients

 

The Four (4) Metric Components & Four (4) Metric Coefficients algorithm is the current model used by the mainstream literature, and physics community; today.

 

  

1.1.       Algorithm for Describing Spherically Symmetric Metrics of a Gravitational Field Using – Three (3) Metric Components & Three (3) Metric Coefficients

 

 

A “Spherically Symmetric Metric” (clip_image002) is used for describing the “gravitational interaction” of a “flat” or “curved” Differential Geometry of Space, Time, & Surfaces, of spherically symmetric space, space-time, or gravitational field, in the presence or absence of condensed mass, matter or energy.

 

Next, a new algorithm, for “generalized mathematical formalism” of a “Spherically Symmetric Metric” (clip_image002), that describes the Euclidean Metric, Minkowski Metric, Einstein Metric, or the Schwarzschild Metric; using one general equation which is composed of, Three (3) Metric Components & Three (3) Metric Coefficients.

 

In the modern literature of General Relativity (GR), it is common mathematical formalism, to use a Einstein tensor mathematical expression, in order to describe, a generalized Spherically Symmetric (clip_image002).

 

Using the Einstein tensor mathematical expression of General Relativity, it also can satisfy the, Three (3) Metric Components & Three (3) Metric Coefficients, algorithm, for “generalized mathematical formalism” of a “Spherically Symmetric Metric” (clip_image002).

 

 

Spherically Symmetric Metric – Einstein “Tensor” Metric Expression

                                                                                                                                                                                                                                                  1.1 

clip_image007    ® clip_image009

 

 

The “Metric “Tensor” Coefficient” terms (clip_image011), can take on values of one (1) to four (4), in this algorithm:

·         Metric Coefficient (1) represents “space” ─ (clip_image013)

·         Metric Coefficient (2) represents “angular” “surface space” “latitude & longitude direction space”  ─ (clip_image015)

·         Metric Coefficient (3) represents “time” ─ (clip_image017)

For starters, let’s consider the following space, angle, and metric mathematical relations for a perfect fluid spherically symmetric gravitational field.

 

Differential Geometry “Individual” Cartesian & Spherical Coordinates, Radial Space, Latitude Space, and Longitude Space Metrics

                                                                                                                                                                                                                                                  1.1 

 

clip_image019

 

Differential Geometry – Cartesian & Spherical Coordinates, Surface Metrics

                                                                                                                                                                                                                                                  1.2 

 

 clip_image021

 

Black Hole Event Horizon – Schwarzschild Semi-Major Radius

                                                                                                                                                                                                                                                  1.3 

clip_image023      ® clip_image025

 

 Space-Time – Isotropic “square” Speed of Light

                                                                                                                                                                                                                                                  1.4 

clip_image027     ® clip_image029

 

 

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

                                                                                                                                                                                                                                                  1.5 

clip_image031     ® clip_image033

 

 

Next, we will present a new algorithm with a “classical mathematical” description of “generalized mathematical formalism” for describing the Spherically Symmetric Metric (clip_image035) that describes Differential Geometry of Space, Time, & Surfaces, of a perfect fluid, spherically symmetric space, space-time, or gravitational field.

 

Using either the Euclidean Metric, Minkowski Metric, Einstein Metric, or the Schwarzschild Metric, a Three (3) Metric Components & Three (3) Metric Coefficients, algorithm, for “generalized mathematical formalism” of a “Spherically Symmetric Metric” (clip_image002), is discussed below.

 

Three (3) Components & Three (3) Metric Coefficients Mathematical Form

 

The Metric Coefficients of the generalized Spherically Symmetric Metric (clip_image035), are given by the symbols (clip_image037).

 

The “Space” metric coefficient is given by the symbol (clip_image039).

 

The “Angular” metric coefficient is given by the symbol (clip_image041).

 

The “Time” metric coefficient is given by the symbol (clip_image043).

 

 

The Metric Components of the generalized Spherically Symmetric Metric (clip_image035), are given by the symbols (clip_image045).

 

The “Space” metric component is given by the symbol (clip_image047).

 

The “Angular” metric component is given by the symbol (clip_image049).

 

The “Time” metric component is given by the symbol (clip_image051).

 

The Spherically Symmetric Metric (clip_image035) describes Differential Geometry of Space, Time, & Surfaces, of a perfect fluid, spherically symmetric space, space-time, or gravitational field; given in its generalized three (3) components & three (3) metric coefficients mathematical forms, below:

 

Three (3) Components & Three (3) MetricCoefficients Mathematical Forms

 Spherically Symmetric Metric – function of (space, surface space, time)

– (clip_image053)

                                                                                                                                                                                                                                                  1.6 

clip_image055     ® clip_image009

 

 

Spherically Symmetric Metric – function of (space, surface angle, time)

 – (clip_image057)

                                                                                                                                                                                                                                                  1.7 

 

clip_image059     ® clip_image009

 

 

 

Spherically Symmetric Metric – function of (space, surface space, time)

– (clip_image062)

                                                                                                                                                                                                                                                  1.8 

 

clip_image064     ® clip_image009

 

 

 

Spherically Symmetric Metric – function of (space, angle, time)

– (clip_image066)

                                                                                                                                                                                                                                                  1.9 

 

 

clip_image068

 

 

 

 

Spherically Symmetric Metric – function of (Cartesian (x, y, & z) space, time)

– (clip_image070)

                                                                                                                                                                                                                                               1.10 

 

clip_image072

 

 

 Below is a table of various metric coefficients, which satisfy the Spherically Symmetric Metric (clip_image035)  equations and theories of gravitation given by: Euclidean “Flat Space”, Minkowski “Flat Space-Time”, Schwarzschild “Curved Space-time”, & Einstein “Curved Space-time”

  

Spherically Symmetric Metric Coefficients

Three (3) Components & Three (3) Metric Coefficients Math Form

“Super Principia” Metric Theory of Gravitation

 

clip_image055

 

clip_image068

 

 

Metric Coefficients

Space Coefficient

(clip_image039)

Angular Coefficient

(clip_image041)

Time Coefficient

(clip_image043)

Euclidean Metric

(Euclidean)

(Flat Space)

clip_image078

clip_image078

clip_image081

Minkowski Metric

(Pseudo-Euclidean)

(Flat Space-Time)

clip_image078

clip_image078

clip_image083

Schwarzschild Metric

(Non-Euclidean)

(Curved Space-Time)

clip_image085

 

clip_image078

clip_image087

Einstein

Metric

(Non-Euclidean)

(Static)

(Curved Space-Time)

clip_image085

 

clip_image078

clip_image083

 

The Square of the Speed of Light “Space-Time” Invariant, equation:

                                                                                                                                                                                                                                                  1.1 

clip_image090     ® clip_image029

 



Spherically Symmetric Metrics

Three (3) Metric Components & Three (3) Metric Coefficients Math Form

 “Super Principia” Metric Theory of Gravitation

clip_image055

 

Euclidean Metric

(Euclidean)

(Flat Space)

 

 

clip_image093

 

Minkowski Metric

(Pseudo-Euclidean)

(Flat Space-Time)

 

 

clip_image095

 

Einstein

Metric

(Non-Euclidean)

(Static)

(Curved Space-Time)

clip_image097

 

 

 

 

Schwarzschild Metric

(Non-Euclidean)

(Curved Space-Time)

clip_image099

 

 

 

 

 

  

1.2.       Euclidean “Flat Space” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

Next we will describe the Euclidean “Flat Space” Metric, its three (3) components and three (3) metric coefficients.

 

 

The Euclidean Metric Coefficients – Defined

                                                                                                                                                                                                                                                  1.2 

clip_image101

 

 

Euclidean Metric – Spherically Symmetric Metric (clip_image035)

                                                                                                                                                                                                                                                  1.3 

 

clip_image055

 

 

 

clip_image104

 

 

 

 

clip_image106    ® clip_image009

 

 

 

Substituting the metric coefficients

                                                                                                                                                                                                                                                  1.4 

 

clip_image108

 

 

 

 

clip_image110

 

 

 

 

clip_image112      ® clip_image009

 

 

The Euclidean Space in three-dimensional Cartesian vector space, with signature (+, +, +), (x, y, z).

                                                                                                                                                                                                                                                  1.5 

 

clip_image114     ® clip_image009

 

 

clip_image116 ® clip_image009

 

 

clip_image118

 

image

 

 

1.3.       Minkowski “Flat Space-Time” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

Next we will describe the Minkowski “Flat Space-Time” Metric, its three (3) components and three (3) metric coefficients.

 

The Minkowski Metric Coefficients – Defined

                                                                                                                                                                                                                                                  1.6 

clip_image122

 

 

 

Minkowski Metric – Spherically Symmetric Metric (clip_image035)

                                                                                                                                                                                                                                                  1.7 

clip_image055

 

 

 

clip_image125      ® clip_image009

 

 

Substituting the metric coefficients

 

                                                                                                                                                                                                                                                  1.8 

clip_image127

 

 

  

clip_image129

 

 

 

clip_image131   ® clip_image009

 

 

From the above the Euclidean Metric is derived

 

Euclidean Metric – Spherically Symmetric Metric (clip_image035)

                                                                                                                                                                                                                                                  1.9 

clip_image133

 

 

The Euclidean and the Minkowski “Metrics” (clip_image035) and geodesic “line elements” (clip_image005), are “mass independent” equations that describe the causality of “flat” space, space-time, or the gravitational field.

 

Spherically Symmetric Metric – Euclidean Metric

                                                                                                                                                                                                                                                  1.1 

 

clip_image114     ® clip_image009

 

 Spherically Symmetric Metric – Minkowski “Pseudo-Euclidean” Metric

                                                                                                                                                                                                                                                  1.2 

 clip_image138   ® clip_image009

 

 

clip_image139

 

 

The limits of integration for the Euclidean and the Minkowski “Metrics” and geodesic “line elements” (clip_image005) is described below.

 

clip_image141 

 

On an orthonormal basis the Minkowski Space is also a four-dimensional Cartesian vector space with signature (−, +, +, +), (-t, x, y, z).

                                                                                                                                                                                                                                               1.10 

 

clip_image143     ® clip_image009

 

 

 

clip_image145 ® clip_image009

 

 

clip_image118

 

image

 

 

1.4.       Schwarzschild “Curved Space-Time” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

Next we will describe the Schwarzschild Metric (Dynamic Vacuum Energy Condition), its three (3) components and three (3) metric coefficients.

 

The Schwarzschild Metric Coefficients – Defined

                                                                                                                                                                                                                                               1.11 

clip_image149

 

 

 

Schwarzschild Metric – Spherically Symmetric Metric (clip_image035)

                                                                                                                                                                                                                                               1.12 

clip_image055

 

 

 

clip_image064

 

 

 

clip_image153

 

 

Substituting the metric coefficients

 

 

clip_image155     ® clip_image157

 

clip_image159® clip_image009

 

 

 

 

On an orthonormal basis the Schwarzschild “Dynamic” Space is also a four-dimensional Cartesian vector space with signature (−, +, +, +), (-t, x, y, z).

                                                                                                                                                                                                                                               1.13 

 clip_image161 ® clip_image009

 

 

 

 

 

clip_image163

 

The “Schwarzschild” Spherically Symmetric Metric (clip_image035) “Dynamic Space-time” condition, corresponds to a gradient gravitational vortex system, where the, “Refraction/Condensing Pressure” (clip_image166) on the exterior surface, of the Black Hole Event Horizon, is zero; (clip_image168).

                                                                                                                                                                                                                                               1.14 

clip_image170

 

 

The “Schwarzschild” Spherically Symmetric Metric (clip_image035) for a “Static Vacuum Energy Space-time” predicts a “Physical Singularity” located at zero radius (clip_image173), and a “Coordinate Singularity” located at the Black Hole Event Horizon, Schwarzschild Radius (clip_image175), of the gradient gravitational field.

To avoid this problem the mainstream physics community has reject the “Schwarzschild” Spherically Symmetric Metric (clip_image035), in favor of: Kruskal–Szekeres coordinates, Eddington–Finkelstein coordinates, and Rindler coordinate; and which neither have a “Coordinate Singularity”.

 

The “Coordinate Singularity” is not a natural artifact for any Non-Euclidean metric. My goal is to find a solution to the “Coordinate Singularity” located at the Black Hole Event Horizon, Schwarzschild Radius (clip_image175), of the “Schwarzschild” Spherically Symmetric Metric (clip_image035).

 

If this “Coordinate Singularity”problem is resolved, the Schwarzschild metric is considered a valid description for the physical description of the curvature of space, space-time, or gradient gravitation field, surrounding, and in the presence of a condensed mass, matter, or energy of an isolated system mass body.

 

The Schwarzschild Metric (clip_image035) predicts the “Physical Singularity” located at zero radius, is a value that approaches zero, as the radius approaches zero. The “Physical Singularity” is a natural artifact for any Non-Euclidean metric.

                                                                                                                                                                                                                                               1.15 

clip_image178

clip_image159 ®clip_image009

                                                                                                                                                                                                                                               1.16 

clip_image181

 

 

 

 

 

The Schwarzschild Metric (clip_image002) “describes” a fluid dynamic vacuum, condition “Rarefaction/Condensing Pressure” of space, where the “Inertial Mass Gravitational Force of Attraction” and the “Isotropic Space-time Aether Gravitational Force of Attraction” interact, and where there is zero Rarefaction Pressure (clip_image183), on the surface of the Black Hole Event Horizon, for any isolated net inertial mass system body.

 

The Schwarzschild Metric (clip_image002) is a Spherically Symmetric Metric that is considered “Non-Euclidean”. The Schwarzschild Metric (clip_image002) describes the differential geometry of a “curved/warped” space-time or gravitational fields, in the presence of condensed mass and energy, for the following forces, energies, densities, and pressures:

 

clip_image185

 

 

“Dynamic” Inhomogeneous Gradient Gravitational Field ─ Volume Mass Density

 

clip_image187   ®  clip_image189

 

 

 

·         Dark Matter “Dynamic” – Rarefaction Mass Density, Force & Pressure

(Time Dependence)(clip_image191)

 

 

·         Dark Energy – Space Expansion & Gravitational Redshift

(Space Dependence)(clip_image193)

 

 

·        

Centrifugal/Centripetal Curvature/Rotation – Angular Momentum & Angular Velocity

(Angle & Space Invariance or Covariance)(clip_image195)

 

 

(clip_image197)     Then   (clip_image199)   And clip_image201

 

image

 

 

image

 

 

1.5.       Einstein “Curved Space-Time” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

Next we will describe the Einstein Metric (Static Vacuum Energy Condition), its three (3) components and three (3) metric coefficients.

 

The Einstein Metric Coefficients – Defined

                                                                                                                                                                                                                                               1.17 

clip_image207

 

 

 

Einstein Metric – Spherically Symmetric Metric (clip_image035)

                                                                                                                                                                                                                                               1.18 

 

clip_image055

 

 

 

clip_image064

 

 

 

clip_image153

 

 

 

Substituting the metric coefficients

 

 

clip_image155     ® clip_image157

 

clip_image212  ® clip_image009

 

 

 

 

On an orthonormal basis the Einstein “Static” Space is also a four-dimensional Cartesian vector space with signature (−, +, +, +), (-t, x, y, z).

                                                                                                                                                                                                                                               1.19 

 

clip_image214 ® clip_image009

 

 

 

clip_image215

 

The “Einstein” Spherically Symmetric Metric (clip_image035) corresponds to a gradient gravitational vortex system, where the, Isotropic Aether Gravitational Field Pressure (clip_image217) is equal to twice (2) the “Inertial Mass towards Mass Gravitational Attraction” (clip_image219), on the exterior surface, of the Black Hole Event Horizon, and is non zero; (clip_image168).

                                                                                                                                                                                                                                               1.20 

 

clip_image221      ®  clip_image223

 

 

The “Einstein” Spherically Symmetric Metric (clip_image035) for a “Static Vacuum Energy Space-time” predicts a “Physical Singularity” located at zero radius (clip_image173), and a “Coordinate Singularity” located at the Black Hole Event Horizon, Schwarzschild Radius (clip_image175), of the gradient gravitational field.

 

 

To avoid this problem the mainstream physics community has reject the “Einstein” Spherically Symmetric Metric(clip_image035), in favor of: Kruskal–Szekeres coordinates, Eddington–Finkelstein coordinates, and Rindler coordinate; and which neither have a “Coordinate Singularity”.

 

The “Coordinate Singularity” is not a natural artifact for any Non-Euclidean metric. My goal is to find a solution to the “Coordinate Singularity” located at the Black Hole Event Horizon, Schwarzschild Radius (clip_image175), of the “Einstein” Spherically Symmetric Metric(clip_image035).

 

If this “Coordinate Singularity”problem is resolved, the Einstein metric is considered a valid description for the physical description of the curvature of space, space-time, or gradient gravitation field, surrounding, and in the presence of a condensed mass, matter, or energy of an isolated system mass body.

 

The Einstein Metric (clip_image035) predicts the “Physical Singularity” located at zero radius, is a value that approaches zero, as the radius approaches zero. The “Physical Singularity” is a natural artifact for any Non-Euclidean metric.

                                                                                                                                                                                                                                               1.21 

 

clip_image178

 

clip_image212  ® clip_image009

 

                                                                                                                                                                                                                                               1.22 

clip_image181

 

 

 

 

 

The Einstein Metric (clip_image002) “describes” a fluid dynamic vacuum, condition “Rarefaction/Condensing Pressure” of space, where the “Inertial Mass Gravitational Force of Attraction” and the “Isotropic Space-time Aether Gravitational Force of Attraction” interact, and where there is non-zero Rarefaction Pressure (clip_image227), on the surface of the Black Hole Event Horizon, for any isolated net inertial mass system body.

 

The Einstein Metric (clip_image002) is a Spherically Symmetric Metric that is considered “Non-Euclidean”. The Einstein Metric (clip_image002) describes the differential geometry of a “curved/warped” space-time or gravitational fields, in the presence of condensed mass and energy, for the following forces, energies, and pressures:

 

 

clip_image229

 

 

“Static” Inhomogeneous Gradient Gravitational Field ─ Volume Mass Density

 

clip_image231      ®  clip_image189

 

 

·         Dark Matter “Static” Aether “Light” Isotropic Pressure – Inhomogeneous Volume Mass Density (Time Dependence)(clip_image233)

 

 

·         Dark Energy – Space Expansion & Gravitational Redshift

(Space Dependence)(clip_image193)

 

 

·       

 Centrifugal/Centripetal Angular/Rotation – Angular Momentum & Angular Velocity

(Angle & Space Invariance or Covariance)(clip_image195)

 

 

 

 

(clip_image235)     Then   (clip_image199)   And clip_image201

 

image

 

 

image

 

 

 

1.6.       “New” Algorithm for Describing Spherically Symmetric Metrics of a Gravitational Field Using – Two (2) Metric Components & Two (2) Metric Coefficients

 

 

Next, we will describe “new” two (2) components mathematical formalism for describing the Spherically Symmetric Metric (clip_image035), which describes the Differential Geometry of Space, Time, & Surfaces, of a perfect fluid spherically symmetric space, space-time, or gravitational field; and where there are gravitational interaction in consideration.

 

The “new” forms of the “Spherically Symmetric Metrics” that produce physical and observable results, of matter in motion, through a space or space-time, that is either flat or curved; or has matter present or absent, in that space or space-time.  

 

Thus, we will consider only the following Spherically Symmetric Metrics:

 

·         Euclidean Metric – “Flat Space or Flat Space-time”

·         Minkowski Metric – “Flat Space-time” (Pseudo-Euclidean)

·         Schwarzschild Metric – “Dynamic Curved Space-time” (Non-Euclidean)

·         Einstein Metric – “Static Curved Space-time” (Non-Euclidean)

 

 

Using the Einstein tensor mathematical expression of General Relativity, it also can satisfy the, Two (2) Metric Components & Two (2) Metric Coefficients, algorithm, for “generalized mathematical formalism” of a “Spherically Symmetric Metric” (clip_image002).

 

Spherically Symmetric Metric – Einstein “Tensor” Metric Expression

                                                                                                                                                                                                                                                  1.2 

 

clip_image007    ® clip_image009

 

The “Metric “Tensor” Coefficient” terms (clip_image011), can take on values of one (1) to four (4), in this algorithm:

·         Metric Coefficient (1) represents “space-time” ─ (clip_image241)

·         Metric Coefficient (2) represents “surface space” “latitude & longitude direction space”  ─ (clip_image015)

The “new” Spherically Symmetric Metric (clip_image035) algorithm, describes the Differential Geometry of Space, & Surfaces for either the Euclidean Metric, Minkowski Metric, Einstein Metric, or the Schwarzschild Metric, with two (2) components & two (2) metric coefficients.

 

Two (2) Components & Two (2) Metric Coefficients Mathematical Form

 

The Metric Coefficients of the “new” Spherically Symmetric (clip_image035) Metric, are given by the symbols (clip_image243).

 

The “Space-time” metric coefficient is given by the symbol (clip_image245).

 

The “Angular” metric coefficient is given by the symbol (clip_image041).

 

 

The Metric Components of the generalized Spherically Symmetric Metric (clip_image035), are given by the symbols (clip_image045).

 

The “Space-time” metric component is given by (clip_image247).

The “Angular” metric component is given by the symbol (clip_image049).

 

The “Space-time” metric coefficient is given by the symbol (clip_image245), is a unit-less quantity that describes the amount of curvature in a localized region of space, space-time, or the gravitational field. The curvature is given by the “Metric Coefficient” (clip_image250)

 

If there is “curvature” in a space, space-time, or the gravitational field, then the “flat-space” “Euclidean” geometry is modified and becomes “Non-Euclidean”; and this is described mathematically by varying the value of the “Metric Coefficient” (clip_image250), and multiplying by the differential Radial (clip_image252) component; to get (clip_image254).

 

By varying the “Metric Coefficient” (clip_image250) value, the Spherically Symmetric Metric (clip_image002) is generalized, and can be used to describe the various metrics: Euclidean Metric, Minkowski Metric, Einstein Metric, Schwarzschild Metric, etc…

The “new” Spherically Symmetric Metric (clip_image035) describes Differential Geometry of Space, & Surfaces, of a perfect fluid, spherically symmetric space, space-time, or gravitational field; and is given in its generalized two (2) components & two (2) metric coefficients, mathematical forms below:

 

Two (2) Components & Two (2) Metric Coefficients Mathematical Form

 

Spherically Symmetric Metric – function of (space, surface angle)

 – (clip_image257)

                                                                                                                                                                                                                                               1.23 

clip_image259     ® clip_image009

 

 Spherically Symmetric Metric – function of (space, surface angle)

 – (clip_image261)

                                                                                                                                                                                                                                               1.24 

clip_image263      ® clip_image009

 

  Spherically Symmetric Metric – function of (space, surface space, time)

– (clip_image062)

                                                                                                                                                                                                                                               1.25 

 

clip_image265     ® clip_image009

 

 Spherically Symmetric Metric – function of (space, surface angle)

 – (clip_image267)

                                                                                                                                                                                                                                               1.26 

 

 

clip_image269    ® clip_image009

 

 

 Spherically Symmetric Metric – function of (Cartesian (x, y, & z) space)

– (clip_image271)

                                                                                                                                                                                                                                               1.27 

 clip_image273

 

 

Space-time Invariant – Square of the Speed of Light 

clip_image275     ® clip_image029

 

 

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

clip_image031     ® clip_image033

 

 

Spherically Symmetric Metric “Coefficients” – “Gaussian” Algorithm

Two (2) Components & Two (2) Metric Coefficients Math Form

“Super Principia” Metric Theory of Gravitation

 

 

clip_image259

 

Metric Coefficients

Space Coefficient

(clip_image245)

Angular Coefficient

(clip_image041)

Integration Limits

(clip_image279)

Euclidean Metric

(Euclidean)

(Flat Space)

clip_image078

clip_image078

clip_image281

Minkowski Metric

(Pseudo-Euclidean)

(Flat Space-Time)

clip_image283

 

clip_image078

clip_image285

Einstein

Metric

(Non-Euclidean)

(Static)

(Curved Space-Time)

clip_image287

 

 

clip_image078

clip_image289

Schwarzschild Metric

(Non-Euclidean)

(Curved Space-Time)

clip_image291

 

clip_image078

clip_image289

 



Spherically Symmetric Metric “Gaussian” Algorithm

Two (2) Metric Components & Two (2) Metric Coefficients Math Form

 “Super Principia” Metric Theory of Gravitation

 

 

clip_image259

 

Euclidean Metric

(Euclidean)

(Flat Space)

 

clip_image093

 

Minkowski Metric

(Pseudo-Euclidean)

(Flat Space-Time)

clip_image294

 

 

 

Einstein

Metric

(Non-Euclidean)

(Static)

(Curved Space-Time)

clip_image296

 

 

 

 

Schwarzschild Metric

(Non-Euclidean)

(Curved Space-Time)

clip_image298

 

 

 

 

Next, is a set of graphs of the various Euclidean and Non-Euclidean Spherically Symmetric Metrics (clip_image035) – Exterior & Interior Black Hole Event Horizon Solutions:

(clip_image300)  ;  (clip_image302)  ;  (clip_image304)

 

image

 

 

image

 

(clip_image300)  ;  (clip_image302)  ;  (clip_image310)

 

image

 

 

image

 

 

(clip_image300)  ;  (clip_image302)  ;  (clip_image316)

 

image

 

 

image

(clip_image300)  ;  (clip_image302)  ;  (clip_image322)

 

image

 

image

(clip_image300)  ;  (clip_image302)  ;  (clip_image328)

 

image

 

image

(clip_image300)  ;  (clip_image302)  ;  (clip_image334)

 

image

 

image

(clip_image300) ;  (clip_image302)  ;  (clip_image340)

 

image

image

 

 

2.   Conclusion

 

This work was written to physicists that are interested in understanding from a conceptual view, “Flat Geometry” Euclidean Space, and “Curved Geometry” Non-Euclidean Space; as a description for causality of gravity, or general motion in a gravitational field.

 

The Euclidean and the Minkowski “Euclidean” Metrics (clip_image035) describes the causality and geometry of the “flat” space, space-time, and the gravitational field; and is independent of the condensed mass, matter, or energy absent or present, in a localized region, of a space or space-time, or gradient gravitational field under consideration.

 

The Schwarzschild and the Einstein “Non-Euclidean” Metrics (clip_image035) describes the causality and geometry of the “curvature” of space, space-time, and the gravitational field, and is used in conjunction, with a fluid mechanical model, Perfect Fluid “Static or Dynamic” Vacuum Energy Solution for the causality gravitation; and is dependent of the condensed mass, matter, or energy absent or present, in a localized region, of a space or space-time, or gradient gravitational field under consideration.

 

It was demonstrated that the “Coordinate Singularity” (clip_image346) located at the Black Hole Event Horizon, Schwarzschild Radius (clip_image175), is not a natural artifact for any Non-Euclidean metric; and is a problem to be solved.

 

If this “Coordinate Singularity”problem is resolved, the Schwarzschild and Einstein metrics are considered a valid description for the physical description of the curvature of space, space-time, or gradient gravitation field, surrounding, and in the presence of a condensed mass, matter, or energy of an isolated system mass body.

 

The “Physical Singularity” (clip_image348) located at zero radius (clip_image350), is a value that approaches zero, as the radius approaches zero. The “Physical Singularity” (clip_image346) is a natural artifact for any Non-Euclidean metric; and cannot be eliminated.

 

This paper described a new algorithm, for “generalized mathematical formalism” of a “Spherically Symmetric Metric” (clip_image002), that describes the Euclidean Metric, Minkowski Metric, Einstein Metric, or the Schwarzschild Metric; using an algorithm which is composed of, Three (3) Metric Components & Three (3) Metric Coefficients; and likewise an algorithm using Two (2) Metric Components & Two (2) Metric Coefficients.

 

In future works, a new algorithm, that describes the Euclidean Metric, Minkowski Metric, Einstein Metric, or the Schwarzschild Metric using a general equation which is composed of:

 

·         Four (4) Metric Components & Four (4) Metric Coefficients

 

The Four (4) Metric Components & Four (4) Metric Coefficients algorithm is the current model used by the mainstream literature, and physics community; today.

 

Below are the topics that were discussed in this paper:

 

 

·         1.1       Algorithm for Describing Spherically Symmetric Metrics of a Gravitational Field Using – Three (3) Metric Components & Three Metric Coefficients

 

 

·         1.2       Euclidean “Flat Space” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

·         1.3       Minkowski “Flat Space-Time” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

·         1.4       Schwarzschild “Curved Space-Time” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 

·         1.5       Einstein “Curved Space-Time” Spherically Symmetric Metric – Three (3) Metric Components & Three (3) Metric Coefficients – Algorithm

 

 1.6       “New” Algorithm for Describing Spherically Symmetric Metrics of a Gravitational Field Using – Two (2) Metric Components & Two (2) Metric Coefficients

 

 

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]        Gauss Law – Theorema Egregium:

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

 

 

[5]        List of Differential Geometry Topics (Wikipedia): http://en.wikipedia.org/wiki/List_of_differential_geometry_topics

 

 

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

 

[7]        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. http://en.wikipedia.org/wiki/Differential_geometry_of_surfaces

 

 

[8]        List Of Differential Geometry Topics:

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

[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:

http://www.gsjournal.net/old/eeuro/vankov.pdf

 

[10]      Schwarzschild, K. (1916). “Über das Gravitationsfeld eines Massenpunktes nach der Einstein’schen Theorie”. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften 1: 189–196: text of the original paper, in Wikisource: http://de.wikisource.org/wiki/%C3%9Cber_das_Gravitationsfeld_eines_Massenpunktes_nach_der_Einsteinschen_Theorie

 

[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]      Robert Louis Kemp. “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics –

            https://superprincipia.wordpress.com/2011/12/23/inertial-mass-vortex-gravitation-theory-continued-part-3/

 

 

[14]      Robert Louis Kemp. “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics –

            https://superprincipia.wordpress.com/2012/01/28/total-mechanical-energy-conservation-escape-velocity-binding-energy-einstein-field-equation/

 

 

[15]      Robert Louis Kemp. “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics –

            https://superprincipia.wordpress.com/2012/01/16/a-theory-of-gravity-for-the-21st-century-the-gravitational-force-and-potential-energy-in-consideration-with-special-relativity-general-relativity/

 

 

 

[16]      Robert Louis Kemp. “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics –

            https://superprincipia.wordpress.com/2012/05/29/newtonian-self-gravitational-force-video-lecture/

 

 

[17]      Robert Louis Kemp. “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics –

            https://superprincipia.wordpress.com/2012/11/08/conditions-for-the-absence-and-presence-of-matter-in-general-relativity-gr-einstein-field-equations-efes/

 

 

[18]      Robert Louis Kemp. “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics –

            https://superprincipia.wordpress.com/2013/01/20/what-are-the-mathematical-and-physical-concepts-of-flat-euclidean-and-curved-non-euclidean-gravitational-fields/

 

 

[19]      Robert Louis Kemp. “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics –

            https://superprincipia.wordpress.com/2013/01/20/what-are-the-mathematical-and-physical-concepts-of-flat-euclidean-and-curved-non-euclidean-gravitational-fields/

 

 

[20]      Robert Louis Kemp. “Super Principia Mathematica – The Rage To Master Conceptual & Mathematical Physics –

            https://superprincipia.wordpress.com/2013/01/20/the-perfect-fluid-vacuum-unified-gravitation-vortex-model-and-non-euclidean-spherically-symmetric-metrics/

 

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