Zbigniew Osiak

ANTI GRAVITY-

rS

0.5rS

0.5rS

GRAVITY

ANTI-GRAVITY

GRAVITY

2

Links to my popular-science and science publications, e-books and radio/TV auditions areavailable in ORCID database: http://orcid.org/0000-0002-5007-306X

3

Mathematics should be a servant, not a queen.

To Margaret,my daughter,

I dedicate

Zbigniew Osiak

ANTI-GRAVITYABOUT HOW GENERAL RELATIVITY

HIDES ANTI-GRAVITY – HEURISTIC APPROACH

4

Anti-gravityby Zbigniew Osiak

© Copyright 2016, 2018 by Zbigniew Osiak

e-mail: zbigniew.osiak@gmail.com

Portraits (drawings) of Newton, Gauss, Einstein and Schwarzschild – Małgorzata Osiak

Portrait of the author on back cover – Rafał Pudło

Translated by Rafał Rodziewicz

5

TABLE OF CONTENTS

TITLE PAGE

COPYRIGHT LAW PAGE

1 INTRODUTION 11

· Entry 11· Limitations for components of a metric tensor 11· Cause-and-effect link between two events 11· Physical spacetime 11· Relations between components of a metric tensor and local basis vectors 12· Dot product 12· Vector value 13· Physical (true) vector components 13· Vector value expressed by physical vector components 13· Cosine of an angle between local basis vectors 13· Stationary metric with spatial-temporal zero-components 14· Quoted works 14

2 EQUATIONS OF MOTION 15

· Quadratic Differential Form of spacetime with stationary metric with spatial-temporalzero-components 15

· 4-velocity 15· 3-velocity 15· 3-velocity value 15· Physical (true) components of 3-vector 16· Lorentz factor 16· Components of 4-vector expressed by components of 3-vector 16· Physical (true) components of 4-vector 17· Total acceleration 4-vector 17· Total acceleration 3-vector 17· Value of total acceleration 3-vector 18· Physical (true) components of total acceleration 3-vector 18· Components of total acceleration 4-vector expressed by components of total acceleration

3-vector 18· Physical (true) components of total acceleration 4-vector 19· 4-dimensional equations of motion of a test particle 20

6

3 FIELD EQUATIONS 21

· Introduction 21· Field equations 21· Quoted works 22

4 GRAVITATIONAL FIELD OUTSIDE MASS SOURCE 23

· Exterior Schwarzschild metric 23· Speed of light propagation and exterior Schwarzschild metric 23· Gravitational acceleration of free fall outside mass source 23· Gravity and anti-gravity 24· Main hypothesis 25· Quoted works 25

5 BLACK HOLE WITH MAXIMAL ANTI-GRAVITY HALO

· Black hole with maximal anti-gravity halo 26

6 GRAVITATIONAL FIELD INSIDE MASS SOURCE 27

· Spacetime metric inside mass source 27· Speed of light propagation in virtual vacuum that is inside mass source 27· Gravitational acceleration of free fall inside mass source 28

7 GRAVITATIONAL FIELD INSIDE BLACK HOLE WITH MAXIMAL ANTI-GRAVITY HALO 29

· Spacetime metric inside black hole with maximal anti-gravity halo 29· Speed of light propagation in virtual vacuum tunnel that is inside black hole with maximal

anti-gravity halo 29· Gravitational acceleration of free fall inside black hole with maximal anti-gravity halo 30

8 GRAPHIC ANALYSIS OF FULL SOLUTION 31

· Charts that show how time-time component of metric tensor and physical (true) compo-nent of gravitational acceleration of free fall depends on the distance from the center ofblack hole with maximal anti-gravity halo 31

9 FIELD EQUATIONS AND EQUATIONS OF MOTION 33

· Equations of motion are contained in field equations 33· Equations of motion and field equations in General Relativity, and 2-potential nature of

Newton’s stationary gravitational field 34

7

10 NEW TEST OF GENERAL RELATIVITY 39

· The proposition of the experiment 39

11 BLACK HOLE MODEL OF OUR UNIVERSE 40

· Our Universe as a black hole with maximal anti-gravity halo 40· Radius of Our Universe 40

12 PHOTON PARADOX 41

· Introduction 41· Gravitational redshift 41· Conformally flat spacetime 41· Schwarzschild spacetime 41· Friedman-Lemaître-Robertson-Walker spacetime 41· Photon paradox 42· Does photons have memory? 42· Energy of photon in Newton’s gravitational fiel 42· Hydrogen atom in Schwarzschild gravitational field – heuristic approach 44· How to define redshift? 44· Redshift of light which comes to Earth from the Sun 45· Redshift of light which comes to Earth from a distant galaxy 45· Hubble’s law 47· Radius of Our Universe according to Hubble’s observation 47· Average density of Our Universe according to Hubble’s observation 48· Constancy of photon energy and Pound-Rebka experiment 48· Quoted works 50

13 EARTH GRAVITATIONAL FIELD AND UNIVERSE GRAVITATIONALFIELD 51

· Influence of gravitational field on space and time distances 51· Local properties of redshift 51· Conclusions 52

14 OLBERS’ PARADOX 53

· Olbers’ paradox 53· Probability of photon hitting Earth 53· Bohr hydrogen atom in Our Universe 53· Illuminance of Earth surface at night 54· Quoted works 56

8

15 MICROVAWE BACKGROUND RADIATION 57

· Background radiation 57· Background radiation in Big-Bang theory 57· Background radiation in Black Hole Universe 57· Quoted works 58

16 AVERAGE UNIVERSE DENSITY IN FRIEDMAN’S THEORY 59

· Critical density 59· Density parameter and current average density of Friedman’s universe 59· Dark energy 59· Quoted works 59

17 ANTI-GRAVITY IN OTHER MODELS OF UNIVERSE 60

· Gravitational acceleration of free particle corresponding with F-L-R-W metric 60· Gravitational acceleration of free particle corresponding with metric of simple model of

expanding spacetime 61· Quoted works 62

18 OUR UNIVERSE AS AN EINSTEIN’S SPACE 63

· Other form of field equations – Our Universe as an Einstein’s space 63· Quoted works 63

19 ASSUMPTIONS 64

· Main postulates of General Relativity 64· 2-potential nature of stationary gravitational field 64· Poisson’s equation and 2-potential nature of stationary gravitational field 64· Signes of right-hand sides of Poisson’s equations and boundary conditions 64· Hypothesis about memory of photons 65

20 STABILITY OF THE MODEL 66

· Stability of the model 66· Gravitational Gauss law and galaxies 66

21 MAIN RESULTS 68

· Hypothesis about existence of anti-gravity 68· Black hole with maximal anti-gravity halo 68· Hypothesis about 2-potential nature of stationary gravitational field 68

9

· Field equations contain equations of motion 68· Black Hole model of Our Universe 68· Size of Our Universe 69· Density of Our Universe 69· Photon paradox 69· Hypothesis about memory of photons 69· Our Universe is an Einstein’s space 69· Anti-gravity in other cosmological models 69· New tests of General Relativity 70· Quoted works 70

22 ADDITION – MATHEMATICS 71

· Explicit form of Christoffel symbols of first and second kind 71· Explicit form of covariant Ricci curvature tensor 71· Field equations 72· Curvature scalar 73· Field equations expressed by curvature scalar 73· Laplace operator (Laplacian) in Cartesian and spherical coordinate systems 75· Used relations 75

23 ADDITION – PHYSICS 76

· Hydrogen spectral series in Earth conditions 76· Hydrogen spectral series in Our Universe 77· Electromagnetic spectrum 77· Choosen concepts, constants and units 78

24 ADDITION – HISTORY 80

· Newton’s law of gravitation 80· Gauss law of gravitation 80· Einstein’s General Theory of Relativity 81· External (vacuum) Schwarzschild solution 81

25 ENDING 82

· Author’s reflections 82

10

11

ANTI-GRAVITY

1 INTRODUCTION

· EntryCouple of reports on the topic of anti-gravity, that I presented in the circle of fellow

theoretical physicsts, met with extremely sharp critic and ignorance. Speeches for non profes-sionals, by contrast, didn’t awake any interest in listeners. I think that many „hunters” of anti-gravity wrongly assume that, this phenomenon is an exact opposite phenomenon to gravity.They justify such view by postulating an analogy to electrostatic interactions, which in allspace can be both attractive and repulsive.

Despite that, I decided to present my views about anti-gravity, because they seem to becoherent and logically correct. Main assumptions of general theory of relativity show that, ex-

ternal Schwarzschild solution is true for Sr21r ³ . Especially, for SS rrr

21

<£ it describes anti-

gravity, and for Srr > – gravity. In other words, behind event horizon of a black hole, there isarea where anti-gravity occurs. Gravity and anti-gravity have layer-like nature, so they differsignificantly from attractive and repulsive electrostatic interactions.

· Limitations for components of a metric tensorExamples served in tome „General Theory of Relativity” [1] based on an assumption, that

all components of a metric tensor are non-negative and, in relation with0g ³=× mnnm ee , ( )4,3,2,1, =nm ,

it guaranteed real values of local base vectors.

· Cause-and-effect link between two eventsTwo events ( )4321 x,x,x,x and ( )44332211 dxx,dxx,dxx,dxx ++++ stay in cause-and-

effect link, if space distance of those events is not greater than their time distance. In case ofmetric with spatial-temporal zero-components, we can write this conditions as:

( )1,2,3,ict, x,dxdxgdxdxg 44444

βα =ba=£ab .

· Physical spacetimeIn spacetime there are areas, inside which

0g ³mn , ( )4,3,2,1, =nm ,and also areas, in which

0g £mn , ( )4,3,2,1, =nm .If both events stay in cause-and-effect link, then in every of those areas we have as follows:

12

( ) ( )4,3,2,1,,0ds,0g 2 =nm£³mn

and

( ) ( )4,3,2,1,,0ds,0g 2 =nm³£mn .

Areas that fulfill above conditions, we will call physical spacetime.In areas, inside which

0g ³mn , ( ) 0ds 2 > , ( )4,3,2,1, =nmor

0g £mn , ( ) 0ds 2 < , ( )4,3,2,1, =nm ,there is no single pair of events that stays in cause-and-effect link.

· Relations between components of a metric tensor and local basis vectorsIf we want to get local basis vectors with real values in physical spacetime, then we need

to adopt new relation between those vectors and components of a metric tensor.

( ) 0gdssgn 2 ³-=× mnnm ee , ( ) 0ds 2 ¹ , ( )4,3,2,1, =nm

It causes a necessity to change a definition of dot product and associated terms. Those chan-ges, forced by physics, will cause only small complication of some formulas.

In cases when( ) 0ds 2 < and 0g ³mn ,above modifications doesn’t lead to any changes, in previously quoted by Me tome titled„General Theory of Relativity” [1].

( ) 0gdssgn 2 ³-=× mnnm ee , ( ) 0ds 2 ¹ , ( )4,3,2,1, =nm

( ) 0ds 2 <

0g ³=× mnnm ee

· Dot productDot product of vectors m

m= eA A and nn= eB A we will call phrase

nmnm ×=× eeBA BA

( ) 0gdssgn 2 ³-=× mnnm ee , ( ) 0ds 2 ¹ , ( )4,3,2,1, =nm

( ) nmmn-=× BAgdssgn 2BA

13

· Vector value

( ) nmnnm

nm -=×=× AAgdssgnAA μ2eeAA , ( ) 0ds 2 ¹ , ( )4,3,2,1, =nm

( ) ν2 AAgdssgnA mmn-=×== AAA

· Physical (true) vector components

mm= eA A

( ) ( ) mm

mmmm

--=

gdssgnAgdssgn

2

2 eA , ( ) 0ds 2 ¹ , ( )4,3,2,1, =nm

( ) μμμ

2μ AgdssgnA -= Physical vector components

( ) μμ2 gdssgne

ˆ-

== m

m

mm

eee , 1ˆ =me

m= eA ˆAμ

· Vector value expressed by physical vector components

nmnm ×=× eeAA AA

( ) ( ) νν2

μμ2

νμνμ

gdssgngdssgn

AAAA

--=

( ) 0gdssgn 2 ³-=× mnnm ee , ( ) 0ds 2 ¹ , ( )4,3,2,1, =nm

( )( ) ( ) ννμμνν

2μμ

2

2

gg

g

gdssgngdssgn

gdssgn mnmn =--

-

ννμμ

μννμ

gg

gAAA =×== AAA

· Cosine of an angle between local basis vectors

( )nmnmnm =× eeeeee ,cos

( ) 0gdssgn μν2 ³-=× nm ee , ( ) 0ds 2 ¹ , ( )4,3,2,1, =nm

( ) 0gdssgn 2 ³-= mmme

( ) 0gdssgn 2 ³-= nnne

14

( )nnmm

mn

nm

nmnm =

×=

gg

g,cos

eeee

ee

· Stationary metric with spatial-temporal zero-componentsFor the sake of simplicity, let’s limit detailed contemplations to a stationary metric with

spatial-temporal zero-components, that is:

( ) 44baab += dxdxgdxdxgds 44

2 , 0xg

4 =¶¶ ab , 0

xg

444 =

¶¶ , ( )3,2,1, =ba .

Example of such metric is external Schwarzschild metric, which depending on output coordi-nate system, can be written in equivalent forms:

( ) 44S44

1

S2

2 dxdxrrdxdx1

rr1

rxx

ds ÷÷ø

öççè

æ-d+

ïþ

ïýü

ïî

ïíì

úúû

ù

êêë

é-÷÷

ø

öççè

æ-+d= ba

-ba

ab , ( )3,2,1, =ba ,

ictx,zx,yx,xx 4321 ==== , 2S cGM2r =

or

( ) 44S2222221

S2 dxdxrr1dsinrdrdr

rr1ds ÷÷

ø

öççè

æ-+jq+q+÷÷

ø

öççè

æ-=

-

,

ictx,x,x,rx 4321 =j=q== , 2S cGM2r = .

Quoted works[1] Z. Osiak: Ogólna Teoria Względności (General Theory of Relativity).Self Publishing (2012), ISBN: 978-83-272-3515-2, http://vixra.org/abs/1804.0178

15

2 EQUATIONS OF MOTION

· Quadratic Differential Form of spacetime with stationary metric with spatial-temporal zero-components

( ) 44baab += dxdxgdxdxgds 44

2 , 0xg

4 =¶¶ ab , 0

xg

444 =

¶¶ , ( )3,2,1, =ba .

· 4-velocity

λλv~~ ev = , ( )1,2,3,4=l

dsdx

cdssgnv~ 2df l

l = , ( ) 0ds 2 ¹ , ( )1,2,3,4=l

dtγgcdtc1

dtdx

dtdx

gg

1gcds 1G442

βα

44

αβ44

--=--= , ( )1,2,3=ba,

442

44

2

g)dssgn(1

gdssgn

-=

-, 2

βα

44

αβdf

G c1

dtdx

dtdx

gg

1γ1

-=

dtdx

g)dssgn(γv~

λ

442

G

-=l , ( )1,2,3,4=l , ic

g)dssgn(γv~

442

G4

-=

· 3-velocity

aa= ev v , ( )1,2,3=a

dtdx

g)dssgn(1v

α

442

dfα

-= , ( ) 0ds 2 ¹

· 3-velocity value

vv ×=2v

aa= ev v , ( )1,2,3=a

baba

bb

aa ×=×=×= eeeevv vvvvv2 , ( )1,2,3=ba,

dtdx

g)dssgn(1v

α

442

α

-= ,

dtdx

g)dssgn(1v

442

bb

-= , ( )1,2,3=ba,

( ) 0gdssgn αβ2 ³-=× ba ee , ( ) 0ds 2 ¹

dtdx

dtdx

gg

v44

2ba

ab= ,dt

dxdt

dxgg

v44

baab=

16

· Physical (true) componets of 3-vector

aa= ev v , ( )1,2,3=a

( ) ( ) αα2

ααα

2

gdssgn vgdssgn

--= aev , ( ) 0ds 2 ¹

( )dt

dx

g

g vgdssgnv

α

44

ααααα

2α =-= Physical components of 3-vector

( ) αα2 gdssgne

ˆ-

== a

a

aa

eee , 1ˆ =ae

aaa == eev ˆ

dtdx

g

gˆvα

44

αα

· Lorentz factor

244G c

1dt

dxdt

dxgg

1ba

ab-=g1

dtdx

dtdx

gg

v44

2ba

ab= , ( )1,2,3=ba,

2

2G

cv1

1

-

=g

· Components of 4-vector expressed by components of 3-vector

dtdx

g)dssgn(γv~

λ

442

G

-=l , ( )1,2,3,4=l

dtdx

g)dssgn(1v

α

442

dfα

-= , ( )1,2,3=a

ictx4 = ,44

2

4

442

df4

g)dssgn(ic

dtdx

g)dssgn(1v

-=

-=

λG

λ vγv~ = , ( )1,2,3,4=l

17

· Physical (true) components of 4-vector

λλv~~ ev = , ( )1,2,3,4=λ

( ) ( ) λλ2

λλλλ

2

gdssgnv~gdssgn~

--=

ev , ( ) 0ds 2 ¹

( )dt

dx

g

gγv~gdssgnv~

λ

44

λλG

λλλ

2λ =-= Physical components of 4-vector

( ) λ2

λ

λ

λλ

gdssgneˆ

-==

eee , 1ˆλ =e

λ

λ

44

λλGλ

λ ˆdt

dx

g

gγˆv~~ eev ==

· Total acceleration 4-vector

λλ

λλtotaltotal a~a~~~ eeaa === , ( )1,2,3,4=l

( ) 2

λ222

dfλλ

total dsxd

cdssgna~a~ == , ( ) 0ds 2 ¹ , ( )1,2,3,4=l

dsdx

cdssgnv~ 2l

l = , ( )1,2,3,4=l

dsv~d

cdssgna~ 2l

l =

dtγgcdtc1

dtdx

dtdx

gg

1gcds 1G442

4444

-ba

ab -=--=

úúû

ù

êêë

é+-

-=

-=

dtdx

dtdγ

γ1

dtdg

dtdx

g21

dtxd

g)dssgn(γ

dtv~d

g)dssgn(γa~

λG

G

44λ

442

λ2

442

2G

λ

442

Gλ

· Total acceleration 3-vector

aa== eaa atotal

dtdv

)gdssgn(1a

α

442

dfα

-= ,

dtdx

g)dssgn(1v

α

442

α

-= , ( ) 0ds 2 ¹ , ( )1,2,3=a

aa

a

úúû

ù

êêë

é-

-=

-== eeaa

dtdg

dtdx

g21

dtxd

g)dssgn(1

dtdv

)gdssgn(1 44

λ

442

λ2

442

442total

18

· Value of total acceleration 3-vector

aa ×=2a

aa= ea a , β

βa ea = , ( )1,2,3=a

baba

bb

aa ×=×=×= eeeeaa aaaaa2 , ( )1,2,3=ba,

dtdv

)gdssgn(1a

α

442

dfα

-= ,

dtdv

)gdssgn(1a

β

442

dfβ

-= , ( )1,2,3=ba,

( ) 0gdssgn 2 ³-=× abba ee , ( ) 0ds 2 ¹

dtdv

dtdv

gg

a44

2ba

ab= ,dt

dvdt

dvgg

a44

baab=

· Physical (true) components of total acceleration 3-vector

aa= ea a , ( )1,2,3=a

( ) ( ) aa

aaaa

--=

gdssgnagdssgn

2

2 ea , ( ) 0ds 2 ¹

( )dt

dvg

gagdssgna 2

a

44

aaaaa

a =-= Physical components of total acceleration 3-vector

( ) aa

a

a

aa

-==

gdssgneˆ

2

eee , 1ˆ =ae

a

a

44

aaa

a == eea ˆdt

dvg

gˆa

· Components of total acceleration 4-vector expressed by components of total accelera-tion 3-vector

( ) dtdγ

dtdx

gdssgnγaγa~ G

λ

442

G2G -

+= aa , ( )1,2,3=a

( ) ( ) dtdγ

dtdx

gdssgnγ

dtdv

gdssgnγa~ G

4

442

G4

442

2G

-+

-=4

19

· Physical (true) components of total acceleration 4-vector

λλa~~ ea = , ( )1,2,3,4=λ

( ) ( ) λλ2

λλλλ

2

gdssgna~gdssgn~

--=

ea , ( ) 0ds 2 ¹

( ) λλλ

2λ a~gdssgna~ -=λa~ = Physical components of total acceleration 4-vector

( ) λλ2

λ

λ

λλ

gdssgneˆ

-==

eee , 1ˆλ =e

( ) 2

λ222

dfλtotal

λ

dsxd

cdssgna~a~ ==

dtv~d

g)dssgn(

γa~λ

442

Gλ

-=

λG

λ vγv~ =

úúû

ù

êêë

é+-

--

=dt

dxdt

dγγ1

dtdg

dtdx

2g1

dtxd

g)dssgn(g)dssgn(γ

a~λ

G

G

44λ

442

λ2

442

λλ22

Gλ

λλ ˆa~~ ea = ( ) λ

λλλ

2 ˆa~gdssgn~ ea -=

( ) ( ) λ2

λ222

λλ2 ˆ

dsxd

cdssgngdssgn~ ea -=

λ

λλλG ˆ

dtv~d

g

gγ~ ea44

= λGλλλG

λλλ2

G ˆdt

dγ vg

gγdt

dvg

gγ~ ea

÷÷

ø

ö

çç

è

æ+=

4444

λ

λG

G

44λ

442

λ2

442

λλ22

G ˆdt

dxdt

dγγ1

dtdg

dtdx

2g1

dtxd

g)dssgn(g)dssgn(γ~ ea

úúû

ù

êêë

é+-

--

=

20

· 4-dimensional equations of motion of a test particle

Postulated equations of motion of a test particle have interpretation as follows:

aa

a += iner&gravtotal a~mF~

a~

( ) 2

222

total dsxd

cdssgna~a~a

aa ==

( )ds

dxds

dxk~cdssgna~ 22

iner&grav

nmamn

a G-=

In case of stationary metric with spatial-temporal zero-components, which is metric of type:

( ) 4444

λκκλ

2 dxdxgdxdxgds += , 0xg

4κλ =

¶¶ , 0

xg

444 =

¶¶ , ( )3,2,1λκ, = ,

equations of motion, inter alia, can be also written in such forms:

( ) ( )1,2,3,4=nma,úúû

ù

êêë

éG-+

g= a

nmamnaa

aaa ,,ˆv~v~k~gdssgn

dtv~d

g

gm

~2

44

G eF

a

nmamn

aaa

úúû

ùG+

gg

+êêë

é-

--g

= eF ˆdt

dxdt

dxk~

dtdx

dtd1

dtdg

dtdx

2g1

dtxd

g)dssgn(g)dssgn(

m

~G

G

44λ

442

λ2

442

22G

Let’s remind:

dtdx

g)dssgn(γv~

442

Gdf a

a

-= , 2

λκ

44

κλ

G c1

dtdx

dtdx

gg1-=

g1 , ( ) aaaa -= gdssgnˆ 2ee , 1ˆ =ae

Components of acceleration 4-vector of a test particle with mass (m) in given point of1. curved Riemann spacetime or2. flat Minkowski spacetime in regard to non-inertial reference frame

are described by equations:

( ) ( ) 0gdssgn,0dxdxgds,ds

dxds

dxk~

dsxd

cdssgna~mF~ 22

2

222

df

force £¹=÷÷ø

öççè

æG+== mn

nmmn

nmamn

aa

a

aF~ = components of net force 4-vector, with ommision of „gravitational” and„inertial” forces

μνg = metric tensor of curved spacetime (components of this tensor are solutions offield equations) or metric tensor of non-inertial reference frame in flat Minkowskispacetime~

îíì-+

=sourcemassaofinside1

sourcemassaofoutside1k

Sum of components (correspondingwith index α) of gravitational andinertial acceleration of a test particle

Component (corresponding with index α)of total acceleration of a test particle

21

3 FIELD EQUATIONS

· IntroductionFrom classical physics we know, that absolute value of gravitational field strength in cen-

ter of homogeneous ball (that has constant density) is equal to zero. Together with growth ofdistance from the center – strength grows linearly, reaching its maximal value on the surfaceof a ball. With further growth of distance – it decreases inversely squared.

If we want to get the same result, within the frames of Einstein’s general theory of relati-vity, then we have to notice that stationary gravitational field is a 2-potential field.

rGEinr rp

34

-= , 2in rG rp32

-=j , 2exr r

GME -= ,r

GMex -=j .

On the surface of a ball, we have

2RGMexin =j-j , 0exin =-EE .

inE , exE = gravitational field strenght respectively inside and outside of a ballinj , exj = gravitational field potential respectively inside and outside of a ball

M = mass of a ball, R = radius of a ball, r = density~

îíì-+

=sourcemassaofinside1

sourcemassaofoutside1k

· Field equationsFirst accurate exterior and interior solutions of Einstein’s field equations [1] was served

by Schwarzschild [2, 3]. Other known interior solutions differ over form of energy momen-tum tensor, such as in Tolman’s work [4].

Equations of gravitational field we will write as:

÷øö

çèæ --= Tg

21TκR μνμνμν or μνμνμν κTRg

21R -=- ,

where

aba

bmn

abn

bmaa

amn

n

ama

mn GG-GG+¶G¶

-¶G¶

=xx

R , ÷÷ø

öççè

æ

¶¶

-¶¶

+¶¶

=G smn

mns

nmsasa

mn xg

xg

xg

g21 ,

mkgs10073.2

cG8 2

434 ×

×=p

=k - , αβαβ

dfTgT= , αβ

αβdf

RgR = , TR k= ,

rx1 = , q=2x , j=3x , ictx 4 = .

0t=

¶¶E , 0rot =E

0gradrot =j

ininin gradkgrad j-=j=~E , Rr0 <£ , 0lim in

0r=j

®

exexex gradkgrad j-=j-=~E , Rr ³ , 0lim ex

r=j

¥®

Þ

22

In case where uniformly distributed mass in a ball, is a source of a gravitational field, wepostulate existence of the solution in the following form:( ) ( ) ( ) ( ) ( )24

44222222

112 dxgdθsinrdθrdrgds +j++= ,

4411 g

1g = , 222 rg = , q= 22

33 sinrg ,11

11

g1g = ,

22

22

g1g = ,

33

33

g1g = ,

44

44

g1g = ,

abab r21

-= gcT 2 , 2df

ρ2TgT c-== abab , constρ = .

The divergence of the tensor ( αβT ) should be equal to zero, which actually happens:

( ) 0gρc21gρc

21T

;2

;

2; =-=÷

øö

çèæ-=

babb

abbab .

Adopted assumptions let us reduce number of field equations to two.

2244

44

244244

2

rc1gr

gr

rcr

grg

2r

rk21

-=-+¶¶

rk21

-=¶¶

+¶¶

Equations are fulfilled, when

Rr0 <£ , 0constρ >= , 23

S232

2244 r

2Rr1r

RcGM1r

3cG41g -=-=rp

-= ,

Rr ³ , 0ρ = ,rr1

rc2GM1g S

244 -=-= , Srr ¹ .

3pr34

= RM

R = radius of a ball, inside which a mass source is located

2S c2GMr = = Schwarzschild radius

Presented solutions of field equations fulfill below boundary values.

1glimR,r

1glimR,r0

44r

440r

=³

=<£

¥®

®

Quoted works[1] A. Einstein: Die Feldgleichungen der Gravitation. Sitzungsberichte der KöniglichPreussischen Akademie der Wissenschaften 2, 48 (1915) 844-847.[2] K. Schwarzschild: Über das Gravitationsfeld eines Massenpunktes nach der EinsteinschenTheorie. Sitzungsberichte der Königlich Preuβischen Akademie der Wissenschaften 1, 7(1916) 189-196.[3] C. Schwarzschild: Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeitnach der Einsteinschen Theorie. Sitzungsberichte der Königlich Preuβischen Akademie derWissenschaften 1, 18 (1916) 424-434.[4] Richard C. Tolman: Static Solutions of Einstein's Field Equations for Spheres of Fluid.Physical Review 55, 4 (February 15, 1939) 364-373.

23

4 GRAVITATIONAL FIELD OUTSIDE MASS SOURCE

· Exterior Schwarzschild metricSpace metric outside mass source ( Rr ³ , 0ρ = ) is being described by exteror Schwarz-

schild metric [1]:

( ) ( ) ( ) ( ) ( )24S2222221

S2 dxrr1dsinrdrdr

rr1ds ÷

øö

çèæ -+jq+q+÷

øö

çèæ -=

-

, ictx4 = , 2S cGM2rr =¹

· Speed of light propagation and exterior Schwarzschild metricExterior Schwarzschild metric, for

const=q , 0d =q , const=j , 0d =j ,

reduces itself to

( ) ( ) ( )22S21

S2 dtcrr1dr

rr1ds ÷

øö

çèæ --÷

øö

çèæ -=

-

.

We designate speed (v) of light propagation from condition:

( ) 0ds 2 =

or equivalent

2

22

2S2

22

rcGM21c

rr

1cdtdrv ÷

øö

çèæ -=÷

øö

çèæ -=÷

øö

çèæ= .

Note that

úûù

êëé ¹³Û

úúû

ù

êêë

é£÷

øö

çèæ< SS

22

rr,r21rc

dtdr0 .

It means that exterior Schwarzschild metric is correct if and only if

SS rr,r21r ¹³ .

· Gravitational acceleration of free fall outside mass sourceWe will designate radial component of gravtational acceleration of free falling test particle

by equation of motion

( ) ÷÷ø

öççè

æ×+×-==

dsdx

dsdxΓ

dsdr

dsdrΓcdssgnk~a~a~

44144

111

221r , Srr ¹ , ( ) 0ds 2 ¹ .

Taking into account, that

24

rr1g S

44 -= , 2S c2GMr = , 1k~ += ,

22

1S44

44

111 rc

GMrr1

rg

2g1Γ ×÷

øö

çèæ --=

¶¶

-=-

, 22S44

44144 rc

GMrr1

rgg

21Γ ×÷

øö

çèæ --=

¶¶

-= ,

24S

21S

dsdx

rr1

dsdr

rr11 ÷÷

ø

öççè

æ÷øö

çèæ -+÷

øö

çèæ

÷øö

çèæ -=

-

,

we get

( ) ( ) 2244

221r

rGMdssgn

rg

2cdssgnk~a~a~ =

¶¶

== .

Physical (true) component of gravitational acceleration of free fall

( ) rrr

2df

r a~gdssgna -= ,

where

1S

11rr rr1gg

-

÷øö

çèæ -== ,

finally can be written in the form:

( ) ( ) 22

rr2r

rGMdssgngdssgna -= .

· Gravity and anti-gravityAbove equation has interesting physical interpretation. For Srr > it describes gravity and

for SS rrr21

<£ – anti-gravity.

Gravity

2S cGM2rr => , 0

rr1g

1S

rr >÷øö

çèæ -=

-

, ( ) 0ds 2 < ,

rr1

1r

GMaS

2r

-×-=

Anti-gravity

2SS cGM2rr r

21

=<£ , 0rr1g

1S

rr <÷øö

çèæ -=

-

, ( ) 0ds 2 > ,1

rr1

rGMa

S2

r

-×+=

25

· Main hypothesis......Anti-gravity works in such a way that free test particle located in external gravitationalfield (of a non rotating mass source) in certain area gets acceleration directed from the centerof that mass source.

Quoted works[1] K. Schwarzschild: Über das Gravitationsfeld eines Massenpunktes nach der EinsteinschenTheorie. Sitzungsberichte der Königlich Preuβischen Akademie der Wissenschaften 1, 7(1916) 189-196.

In areas, where0g ³mn , ( ) 0ds 2 < , ( )4,3,2,1, =nm ,

graviy occurs.

In areas, where0g £mn , ( ) 0ds 2 > , ( )4,3,2,1, =nm ,

anti-graviy occurs.

26

rS

0.5rS0.5rS

GRAVITY

ANTI-GRAVITY

5 BLACK HOLE WITH MAXIMAL ANTI-GRAVITY HALO

· Black hole with maximal anti-gravity haloBlack hole with maximal anti-gravity halo we will call homogeneous ball with mass (M)

and radius (R), for which

mkg101.3466

Gc

RM 27

2

´»= .

Spatial radius of a black hole with maximal anti-gravity halo is equal to half of Schwarzschildradius. Area of outside black hole with maximal anti-gravity halo consist of two layers, infirst ( SS rr r5.0 <£ ) anti-gravity occurs and in second ( Srr > ) – gravity. Thickness of thisanti-gravity shell is equal to spatial radius of black hole with maximal anti-gravity halo.Inside of anti-gravity layer acceleration is directed from the center of mass source, and insideof gravity layer – towards the center

Inside of anti-gravity layer acceleration is directed fromthe center of a mass source and it grows to the borderwith gravity layer. Then acceleration changes direction,and its absolute value decreases together with a growthof distance from the center.This model can be called „layer model”: anti-gravity –gravity.

First time model of black hole with maximal anti-gravity halo was proposed by me in 2005 onthe V Forum of Unconventional Inventions, Constructions and Ideas in Wrocław on theoccasion of the 100th anniversary of Einstein’s formulation of special theory of relativity.This Forum was organized by Janusz Zagórski.

27

6 GRAVITATIONAL FIELD INSIDE MASS SOURCE

· Spacetime metric inside mass sourceSpacetime metric inside mass source ( Rr0 <£ , 0constρ >= ) is given by:

( ) ( ) ( ) ( ) ( )2444

22222211

2 dxgdsinrdrdrgds +jq+q+= ,

where

ictx4 = , 2S cGM2r = ,

4411 g

1g = , 23

S232

2244 r

2Rr1r

RcGM1r

3cG41g -=-=rp

-= .

· Speed of light propagation in virtual vacuum tunnel that is inside mas sourceSpacetime metric inside mass source for

const=q , 0d =q , const=j , 0d =j ,

reduces itself to a form:

( ) ( ) ( )2244

211

2 dtcgdrgds -= ,44

11 g1g = , 2

3S2

322

244 r2Rr1r

RcGM1r

3cG41g -=-=rp

-= .

We will designate speed (v) of light propagation in virtual vacuum tunnel from condition

( ) 0ds 2 =

or equivalent

22

322

22

3S2

22 r

RcGM1cr

2Rr1c

dtdrv ÷

øö

çèæ -=÷

øö

çèæ -=÷

øö

çèæ= .

Notice that

[ ]Rrr21R,c

dtdr0 S

22

<Ûúúû

ù

êêë

é³£÷

øö

çèæ< .

It means that interior metric is correct if and only if

Sr21RR,r ³< .

28

· Gravitational acceleration of free fall inside mass sourceRadial component of gravitational acceleration of freely falling test particle inside virtual

vacuum tunnel, which is located inside mass source, we will get from equation of motion

( ) ÷÷ø

öççè

æ×+×-==

dsdx

dsdxΓ

dsdr

dsdrΓcdssgnk~a~a~

44144

111

221r , Rr0 <£ , ( ) 0ds 2 ¹ .

Taking into account that

1k~ -= , 1dssgn 2 -= , Sr21R ³ , 2S c

2GMr = ,

rg

2g1Γ 44

44

111 ¶

¶-= ,

rgg

21Γ 44

44144 ¶

¶-= ,

23

S232

2244 r

2Rr1r

RcGM1r

3cG41g -=-=rp

-= ,

24

44

21

44 dsdxg

dsdrg1 ÷÷

ø

öççè

æ+÷

øö

çèæ= - ,

we get

( ) ( ) rR

GMrR

GMdssgnk~r

g2cdssgnk~a~ 33

2442

2r -=-=¶¶

= .

Physical (true) component of gravitational acceleration of free fall

( ) rrr

2df

r a~gdssgna -= ,

where

12

3S

11rr r2Rr1gg

-

÷øö

çèæ -== ,

in the end, can be written in a form:

( ) ( )r

2Rr1

1rR

GMrR

GMgdssgndssgnk~a2

3S

33rr22r

-×-=--= .

29

7 GRAVITATIONAL FIELD INSIDE BLACK HOLE WITH MAXIMAL ANTI-GRAVITY HALO

· Spacetime metric inside black hole with maximal anti-gravity haloSpacetime metric inside black hole with maximal anti-gravity halo is given by:

( ) ( ) ( ) ( ) ( )242

2222222

1

2

22 dx

Rr1dθsinrdθrdr

Rr1ds ÷÷

ø

öççè

æ-+j++÷÷

ø

öççè

æ-=

-

,

where

ictx4 = , Sr21Rr0 =<£ , 2S c

GM2r = ,44

11 g1g = , 2

2

44 Rr1g -= , 0constρ >= .

· Speed of light propagation in virtual vacuum tunnel that is inside black hole withmaximal anti-gravity halo

Spacetime metric inside black hole with maximal anti-gravity halo for

const=q , 0d =q , const=j , 0d =j ,

reduces to a form

( ) ( ) ( )222

22

1

2

22 dtc

Rr1dr

Rr1ds ÷÷

ø

öççè

æ--÷÷

ø

öççè

æ-=

-

.

We will designate speed (v) of light propagation in virtual vacuum tunnel from condition

( ) 0ds 2 =

or equivalent

2

2

22

22

Rr1c

dtdrv ÷÷

ø

öççè

æ-=÷

øö

çèæ= .

Notice that

[ ]Rrcdtdr0 2

2

<Ûúúû

ù

êêë

é£÷

øö

çèæ< .

30

· Gravitational acceleration of free fall inside black hole with maximal anti-gravityhalo

Radial component of gravitational acceleration of freely falling test particle inside virtualvacuum tunnel, which is located inside black hole with maximal anti-gravity halo, we will getfrom equation of motion

( ) ÷÷ø

öççè

æ×+×-==

dsdx

dsdxΓ

dsdr

dsdrΓcdssgnk~a~a~

44144

111

221r , Rr0 <£ , ( ) 0ds 2 ¹ .

Taking into account that

1k~ -= , 1dssgn 2 -= , Sr21R = , 2S c

2GMr = ,Gc

RM 2

= ,

rg

2g1Γ 44

44

111 ¶

¶-= ,

rgg

21Γ 44

44144 ¶

¶-= ,

2

2

44 Rr1g -= ,

24

44

21

44 dsdxg

dsdrg1 ÷÷

ø

öççè

æ+÷

øö

çèæ= - ,

we get

( ) ( ) rRcr

Rcdssgnk~

rg

2cdssgnk~a~ 2

2

2

2244

22r -=-=

¶¶

= .

Physical (true) component of gravitational acceleration of free fall

( ) rrr

2df

r a~gdssgna -= ,

where

1

2

2

11rr Rr1gg

-

÷÷ø

öççè

æ-== ,

in the end, can be written in a form:

( ) ( )Rr1

1rRcr

Rcgdssgndssgnk~a

2

22

2

2

2

rr22r

-

×-=--= .

31

8 GRAPHIC ANALYSIS OF THE FULL SOLUTION

· Charts that show how time-time component of metric tensor and physical (true)component of gravitational acceleration of free fall depends on the distance from thecenter of black hole with maximal anti-gravity halo

We will make those charts for case, where spatial radius of black hole is half of Schwarz-schild radius

2S cGMr

21R == ,

i.e. for black hole with maximal anti-gravity halo.Time-time component of metric tensor and physical (true) component of gravitational

acceleration of free fall, in three distance intervals from the center of black hole, are given bybelow relations.

GRAVITY

Sr21r0 <£ , 0

Rr1g 2

2

44 >÷÷ø

öççè

æ-= ,

Rr1

1rRca

2

22

2r

-

×-=

ANTI-GRAVITY

SS rrr21

<£ , 0rr1g S

44 <÷øö

çèæ -= ,

1rr1

rGMa

S2

r

-×+=

GRAVITY

Srr > , 0rr1g S

44 >÷øö

çèæ -= ,

rr1

1r

GMaS

2r

-×-=

1

0r

g44

R = 0.5rS

rS

1

Chart that shows how time-time component ( 44g ) of metric tensor depends on the distance (r)from the center of black hole with maximal anti-gravity halo.

32

1

0r

g44

R = 0.5rS

rS

1

Chart that shows how time-time component ( 44g ) of metric tensor depends on the distance (r)from the center of bkack hole with maximal anti-gravity halo.

ra

2rGM

0r

ra

2rGM

R = 0.5rS

rS

Chart of dependence of radial component ( ra ) of physical (true) gravitational acceleration offree fall on the distance (r) from the center of black hole with maximal anti-gravity halo.

ATTENTIONFirst graph is doublet for better clearness on the same page.

From the charts we can see that:

Sr5.0r0 ×<£ Þ gravity

Sr5.0r ×= Þ transition from gravity to anti-gravity

SS rrr5.0 <<× Þ anti-gravity

Srr = Þ transition from anti-gravity to gravity

Srr > Þ gravity

Gravity and anti-gravity has layer-like nature.

33

9 FIELD EQUATIONS AND EQUATIONS OF MOTION· Equations of motion are contained in field equations

How should one formulate equations of motion, so that unscaled radial components of4-acceleration could be served with below forms?

( ) ( ) ( ) 0dssgn1,k~R,r0r,R

GMdssgnk~rR

GMds

rdcdssgna~ 23

232

222

dfrin <-=<£-=-==

( ) ( ) ( ) 0dssgn1,k~R,r,r

GMdssgnk~r

GMds

rdcdssgna~ 22

222

222

dfrex <+=>=-==

While answering to this question, we will use second of two field equations.

22in44

in44

2in44

2

in44

2

rc1gr

gr

rcr

grg

2r

rk21

-=-+¶¶

rk21

-=¶¶

+¶¶

23

S232

22

in44 r

2Rr1r

RcGM1r

3cG41g -=-=rp

-= , 4cG8p

=k , 2S cGM2r =

rR

GMρrG34

rg

2c

3

in44

2

-=p-=¶¶

( ) ( ) rR

GMrR

GMdssgnk~ds

rdcdssgna~ 332

2

222

dfrin -=-==

( )r

g2cdssgnk~a~

in44

22r

in ¶¶

=

rg

21

dsdx

dsdxΓ

dsdr

dsdrΓ

in44

44144

111 ¶

¶-=×+×

( ) ÷÷ø

öççè

æ×+×-=

dsdx

dsdxΓ

dsdr

dsdrΓcdssgnk~a~

44144

111

22rin

Analogical considerations for component ( rexa~ ) lead to a conclusion, that equations of motion

should have the form served below:

( ) ( )ds

dxds

dxcdssgnk~ds

xdcdssgna~ 222

222

df nmamn

aa ×G-==

sourcemassaofinside1sourcemassaofoutside1

îíì-+

=k~

34

· Equations of motion and field equations in General Relativity, and 2-potential natureof Newton’s stationary gravitational field

Postulated by us, within the frames of General Relativity, equations of motion of free testparticle lead to a conclusion that Newton’s stationary gravitational field is a 2-potential field.We will show it on an example of gravitational field, which source is a mass distributedhomogeneously in a volume of a ball with radius (R).

( ) ( ) 0gdssgn,0dxdxgds,ds

dxds

dxcdssgnk~a~

mF~ 2222 £¹=G+= mn

nmmn

nmamn

aa

0F~ =a

rx1 = , q=2x , j=3x , ictx 4 =

( ) ( ) ( ) ( ) ( )2444

22222211

2 dxgdθsinrdθrdrgds +j++=

4411 g

1g = , 222 rg = , q= 22

33 sinrg

rg

21

dsdx

dsdxΓ

dsdr

dsdrΓ 44

44144

111 ¶

¶-=×+×

( )r

g2cdssgnk~a~ 44

22r

¶¶

=

rina , r

exa = radial acceleration respectively inside and outside of a ballin44g , ex

44g = metric tensor components respectively inside and outside of a ball2

3S2

322

2in44 r

2Rr1r

RcGM1r

3cG41g -=-=rp

-= , Rr0 <£ , 0constρ >=

rr1

rc2GM1g S

2ex44 -=-= , Rr ³ , Rr ³ , 0ρ =

( ) ( ) Rr0r,R

GMdssgnk~r

g2cdssgnk~a~ 3

2in44

22r

in <£-=¶¶

=

( ) ( ) Rr,r

GMdssgnk~r

g2cdssgnk~a~ 2

2ex44

22r

ex ³=¶¶

=

Srr >> , 22 cv << , 2S c2GMr =

2

2r

dtrda = , 1dssgn 2 -= ,

ballshomogeneouofnsidei1ballshomogeneouofutside1

k~

îíì-+

=o

Rr0r,R

GMr

g2ck~a 3

in44

2rin <£-=

¶¶

-=

Rr,r

GMr

g2ck~a 2

ex44

2rex ³-=

¶¶

-=

35

By substitution in last two equations

2

inin44 c

21g

j+= , 2

exex44 c

21g

j+= ,

where ( inj ) and ( exj ) are potentials of gravitational field respectively inside and outside of aball, we will get

Now lets serve definition of potentials ( inj ) and ( exj ) that corresponds with standard defini-tion of gravitational potential.

rGMdra

r2RGMdra

Rr

rex

dfex

23

Rr

0

rin

dfin

-=-=j

-==j

ò

ò¥

³

<

In the field equations inside the mass source, we will replace the time-time component( in

44g ) of the metric tensor with the potential ( inj ).

22in44

in44

2in44

2

in44

2

rc1gr

gr

rcr

grg

2r

rk21

-=-+¶¶

rk21

-=¶¶

+¶¶

2

inin44 c

21g

j+=

2inin

2in

2

in22

ρrG42r

r2

rρG4r

r2r

r

p-=j+¶j¶

p-=¶j¶

×+¶j¶

×

First of those equations is Poisson’s equation for the potential ( inj ) in spherical coordinatesystem. From classical Poisson’s equation it differs only in sign of right side. Then, from bothequations it appears that

in2

in22 2

rr j=

¶j¶

×

rR

GMrr

k~a 3

ininrin -=

¶j¶

=¶j¶

-= , Rr0 <£ , 23

in r2RGM

-=j , 0lim in

0r=j

®

2

exexrex r

GMrr

k~a -=¶j¶

-=¶j¶

-= , Rr ³ ,r

GMex -=j , 0lim ex

r=j

¥®

Poisson’s equation for the potential ( inj )

36

Now we will analize second field equation for potential ( inj ).

in2in

ρrG2r

r j-p-=¶j¶

×

23

2in r2RGMrG

32

-=rp-=j

rRMGrG

34

r 3

in

-=rp-=¶j¶

rR

GMrG34a 3

rin -=rp-=

rin

in

ar=

¶j¶

By analyzing Poisson’s equation for the potential ( inj ) we showed that the equations of mo-tion are contained in field equations inside the source mass.

In the field equations outside mass source we will substitute the time-time component( ex

44g ) of the metric tensor with the potential ( exj ).

22ex44

ex44

2ex44

2

ex44

2

rc1gr

gr

rcr

grg

2r

rk21

-=-+¶¶

rk21

-=¶¶

+¶¶

2

exex44 c

21g

j+= , 0=r

02r

r2

0r

2rr

r

exex

ex

2

ex22

=j+¶j¶

×

=¶j¶

×+¶j¶

×

First of those equations is Poisson’s equation for the potential ( exj ) in spherical coordinatesystem. Then, from both equations it appears that

ex2

ex22 2

rr j=

¶j¶

×

Equation of motion for radial component of free fall accelerationinside of mass source

Poisson’s equation for the potential ( exj )

37

Now let’s analize second field equation for potential ( exj ).

exex

rr j-=¶j¶

rGMex -=j

2

ex

rGM

r=

¶j¶

2rex r

GMa -=

rex

ex

ar

-=¶j¶

Analyzing Poisson’s equations for the potentials ( inj ) and ( exj ) we obtained analogous re-sults. The equations of motion are included in field equations.

Introducing two potentials into Newton’s theory of gravitation allowed to find solution offield and motion equations (within the frames of General Relativity), which in boundary case( Srr >> , 22 cv << ) lead to compabillity of both theories, both inside as well as outside ofmass source.

2RGM

-

RGM

-

inj exj

rR0j

2RGM

-

RGM

-

inj exj

Diagram that shows how potentials ( inj ) and ( exj ) depend on the distance (r) from the centerof mass source in case of ( Srr >> ) and ( 22 cv << ).

Equation of motion for radial component of free fall accelerationoutside of mass source

0limR,r0,r2RGM in

0r

23

in =j<£-=j®

0limR,r,r

GM ex

r

ex =j³-=j¥®

38

2RGM

-

RGM

-

inj exj

rR0j

2RGM

-

RGM

-

inj exj

Diagram that shows how potentials ( inj ) and ( exj ) depend on the distance (r) from the centerof mass source in case of ( Srr >> ) and ( 22 cv << ).

rina r

exa

rR

rina r

exa

ar

Diagram that shows how radial components of free fall acceleration ( rina ) and ( r

exa ) dependon the distance (r) from the center of mass source in case of ( Srr >> ) and ( 22 cv << ).

ATTENTIONFor clearness, we have also featured diagrams from previous page.

Rr0r,R

GMa 3rin <£-=

Rr,r

GMa 2rex ³-

0limR,r0,r2RGM in

0r

23

in =j<£-=j®

0limR,r,r

GM ex

r

ex =j³-=j¥®

39

10 NEW TEST OF GENERAL RELATIVITY

· The proposition of the experimentHow to show, in Earth conditions, 2-potential nature of gravitational field and also prove

the existence of black holes with anti-gravity halo? In this purpose we have to measure ratioof distance passed by light to the time of flight, in a vertically positioned vacuum cylinder,right under and right above the surface of Earth (of the sea level). If the difference of squaresof those measurements will be equal to the square of escape speed, then it will prove exis-tence of black holes with anti-gravity halo. This experiment would be a new test of GeneralTheory of Relativity.

Below we will justify purpose of this experiment:

?dtdr

dtdr 2

ex

2

in

=÷øö

çèæ-÷

øö

çèæ

indtdr÷øö

çèæ i

exdtdr÷øö

çèæ – ratio of path travelled by the light to the time of travel

measured respectively right under and right over Earth surface2

23

S22

in

r2Rr1c

dtdr

÷øö

çèæ -=÷

øö

çèæ

2S2

2

ex rr1c

dtdr

÷øö

çèæ -=÷

øö

çèæ

Rr »

2S cGM2r =

1RrS <<

c2dtdr

dtdr

exin

»÷øö

çèæ+÷

øö

çèæ

2

skm91.7

RGM

÷øö

çèæ=

sm103

sm102.99792458c 88 ×»×=

R2GM

dtdr

dtdr 2

ex

2

in

@÷øö

çèæ-÷

øö

çèæ

sm0.208

cRGM

dtdr

dtdr

exin

@@÷øö

çèæ-÷

øö

çèæ

40

rS

0.5rS

0.5rS

GRAVITY

ANTI-GRAVITY

GRAVITY

11 BLACK HOLE MODEL OF OUR UNIVERSE

· Our Universe as a black hole with maximal anti-gravity halo

In the center of black hole, gravitational acceleration offree particle is equal to zero which corresponds withGauss law. Later, absolute value of acceleration growstogether with growth of distance from the center. Inanti-gravity layer, gravitational acceleration is directedfrom the center of mass source and it grows to the bor-der with gravitational layer. Then, acceleration turns di-rection, and its absolute value decreases with growth ofdistance from the center.This model can be called „layer model”:gravity – anti-gravity – gravity.

Our Universe can be treated as a gigantic, homogeneous Black Hole. It is isolated from therest of universe with a space area where anti-gravity occurs. Our galaxy together with Solarsystem and Earth (which in cosmological scales can be treated merely as a point) should belocated near the center of Black Hole. Once again let’s remind that in the center of black hole,gravitational acceleration of free particle, leaving out local gravitational fields, is equal to ze-ro. In other words: Black Hole is not generating gravitational field in its center.

· Radius of Our UniverseBelow, let’s estimate radius of Our Universe, assuming that, its average density is the

same as density (np) of protons in every cubic meter.

2S cGMr

21R ==

3πρR34M =

ρ1

G43cR

2

×p

=

sm103

sm102.99792458c 88 ×»×= , 2

311

2

311

skgm106.7

skgm106.6742G

××»

××= --

327

p mkg101.67n -××»r

pn = average amount of protons in every cubic meter

kg1067262171.1m 27p

-×= = proton mass

light year m100.95 16×»

yearslight10n19.501m10

n19R 9

p

26

p

××»×»

41

12 PHOTON PARADOX

· IntroductionTheory of relativity, both special and general, is strictly connected with wave theory of

light (electomagnetic waves). Attempt to explaining gravitational redshift on a ground ofphoton light theory in spacetimes other than conformally flat leads to a paradox.

· Gravitational redshiftGravitational redshift is the phenomenon that the spectrum of light reaching the Earth

from the Sun or other stars is shifted towards the longer wavelengths relative to analogicalspectrum of light that comes from emitter which is located on the Earth.

· Conformally flat spacetimeConformally flat spacetime is a spacetime with metric type like:

( ) ( )[ ] ( ) ( ) ( ) ( )[ ]24232221243212 dxdxdxdxx,x,x,xKds +++= .

ict xz, xy, xx,x 4321 ====

( )x,x,x,xKK 4321= = conformal factor

· Schwarzschild spacetimeGravitational field outside of mass source (M) is being described by Schwarzschil metric

[3].

( ) 44S44

βα1

S2

βα

αβ2 dxdx

rrδdxdx1

rr1

rxx

δds ÷÷ø

öççè

æ-+

ïþ

ïýü

ïî

ïíì

úúû

ù

êêë

é-÷÷

ø

öççè

æ-+=

-

, ( )3,2,1, =ba

îíì

¹Û=Û

=dab βα0βα1

( ) ( ) ( )2322212 xxxr ++=

2S cGM2r =

· Friedman-Lemaître-Robertson-Walker spacetimeFLRW metric [4, 5, 6, 8, 9] describes spatially isotropic and homegeneous universe.

( ) ( ) ( )[ ] ( )24232221222 dxdxdxdxLBds +++=

ictx4 =

( )tLL = = dimensionless time scale factor

42

241

2

2

41

22

22

41 kr1

1

ar1

1

aLrL1

1B+

=+

=+

=

( ) ( ) ( )2322212 xxxr ++=

2a1k = , 1,0,1

a1sgnksgn 2 +-==

2a = square of unscaled constant radius of space curavutre

· Photon paradoxIdea of photon in the context of Schwarzschild and Friedman-Lemaître-Robertson-Walker

metrics leads to a paradox. Calculating the influence of those metrics on photon energy from

the equationThE = or equivalent

l=

hcE , we get different results, depending on used

equation. For simplification, let’s assume that .0dxdx 32 == Then we get:

This paradox was called by me „photon paradox”.

· Does photons have memory?Photon paradox doesn’t occur in conformally flat spacetimes. In other spacetimes, one of

solutions for photon paradox is an assumption that photon energy depends on point of space-time, inside which its emission happened and it remains constant during photon movement. Itmeans that photons have memory, or more scholarly – photon energy is invariant. Wherein, instronger gravitational field given source should send photons with less energy, than the samesource located in weaker field.

· Energy of photon in Newton’s gravitational fieldLet’s remind a deduction of equation for total energy ( E ) of body source with mass (M)

and the photon emitted by given atom, which in emission moment is at distance (R) from thecenter of mass souce.

constR

GMmhνE =-=

2chm n

=

constRcGM1hν

RcGMhνhνE 22 =÷

øö

çèæ -=-=

22 cRGM21

cRGM

-»-1

ThE = ,

λhcE =

KTT = , Kλλ = for conformally flat metric

TgT 44= ,44gλλ = for Schwarzschild metric

TT = , BLλλ = for FLRW metric

43

constRc

2GM1hνE 2 =-=

T1

=n ,l

=nc

22 Rc2GM1

const

Rc2GM1

Eλhc

Thhν

-=

-===

consth =constc =constE =

­­¯­ λ,T,ν,R

Interpretation of gravitational redshift (or blueshift) based on above equations, relies on anassumption that photon energy in the moment of emission doesn’t depend on gravitatio-nal field, inside which given atom was placed. It should be ephasized that those relationsaren’t generatig photon paradox.

Analogical reasoning to above, has been served for the first time by Einstein [1] and Ithink it has only heuristic meaning.

In a frame on the left, previous calculationswere repeated, based on new form of expres-sion for rest energy [13, 14].

2mc21E =

constR

GMmhνE =-=

2ch2m n

=

constRc

2GM1hνRc

2GMhνhνE 22 =÷øö

çèæ -=-=

T1

=n ,l

=nc

22 Rc2GM1

const

Rc2GM1

Eλhc

Thhν

-=

-===

consth =constc =constE =

­­¯­ λ,T,ν,R

44

· Hydrogen atom in Schwarzschild gravitational field – heuristic approachWe will estimate energy of photon emitted by hydrogen atom inside gravitational field

described by exterior Schwarzschild mertic.

r1~E -

11grr = ,1

S1

211 RR1

Rc2GM1g

--

÷øö

çèæ -=÷

øö

çèæ -= ,

1144 g

1g = , 2S cGM2R = ,

r1~E -

11gr1~E - , Eg

gEE 44

11

==

ΔEggΔEEΔ 44

11

== , 9

Earth2 101.4Rc

2GM -×»÷øö

çèæ , 6

2 104.3Rc

2GM -×»÷øö

çèæ

Sun

E = energy on allowed (stationary) orbit under the absence of gravitational fieldr = radius of allowed (stationary) orbit under the absence of gravitational fieldE = energy on allowed (stationary) orbit in exterior gravitational fieldr = radius of allowed (stationary) orbit in exterior gravitational field

11g , 44g = meric tensor components of exterior gravitational fieldR = distance of photon emission place from the center of body sourceM = mass of body sourceG = gravitational constantc = speed of light in vacuum

SR = Schwarzschild radius of body source

Einstein couldn’t serve such reasoning in 1911 [1], because Bohr atom model was formu-lated by Bohr in 1913 [2] and Schwarzschild metric showed on in 1916 [3].

· How to define redshift?

1EE

EEEz

out

lab

out

outlabdf

-=-

=*

lab11

maxlab

gEE =

out11

maxout

gEE =

1g

gz

lab11

out11 -=*

45

labE = photon energy emitted from a source that is in laboratory

outE = photon energy emitted from a source that is outside laboratory

maxE = photon energy emitted from a source in the absence of gravitational fieldlab11g = component of metric tensor in laboratory in a place of photon detectionout11g = component of metric tensor outside of laboratory in a place of photon emission

· Redshift of light which comes to Earth from the Sun

1g

gz

lab11

out11 -=*

1

S2

Sout11 Rc

2GM1g

-

úúû

ù

êêë

é÷÷ø

öççè

æ-= ,

1

E2

Elab11 Rc

2GM1g-

úû

ùêë

é÷÷ø

öççè

æ-=

6

S2

S 104.3Rc

2GM -×»÷÷ø

öççè

æ, 9

E2

E 101.4Rc

2GM -×»÷÷ø

öççè

æ

6

E2

E

S2

S

E2

E

S2

S 1015.2Rc

2GMRc

2GM41

Rc2GM

21

Rc2GM

21z -* ×»÷÷

ø

öççè

æ÷÷ø

öççè

æ-÷÷

ø

öççè

æ-÷÷

ø

öççè

æ»

Sun-light redshift is a local effect, thats why for components of metric tensor we usedequations right for exterior Schwarzschild metric, which depends on local mass sources andtheir sizes. Also we assumed that given light source is located either on Sun or Eartth surface.

· Redshift of light which comes to Earth from a distant galaxy

1g

gz

lab11

out11 -=*

1

2

2out11 R

r1g-

÷÷ø

öççè

æ-=

9

1

Earth2

Earthlab11 104.11

1Rc

2GM1g -

-

×-»ú

û

ùêë

é÷÷ø

öççè

æ-=

1

Rr1

104.111

Rr1

Rc2GM1

z

2

2

9

2

2

Earth2

Earth

-

-

×-»-

-

÷÷ø

öççè

æ-

=-

*

On the next page there is a graph of redshift (z*) dependence on the distance (r) of the sourcefrom the center of Our Universe, (R) is the radius of Our Universe.

46

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

123456

7

89

101112

1314

Z*

r/R

Diagram that shows dependence of redshift (z*) versus the distance (r) of the source from thecenter of Our Universe. [Attension: (z*) has negative values for ratio (r/R) approximate lessthan 3.74·10–5.]

Below let’s estimate redshift (z*) for case, when ( 22 Rr << ).

1

Rr1

Rr1

11

Rr1

11

Rr1

104.11z

2

2

2

2

9

-+×-

=-

-

»-

-

×-»

-*

Rr

211

Rr1

21

×+»÷øö

çèæ -

-

1

Rr1

Rr

211

z -+

×+=*

1Rr1 »+

Rr

21z ×»*

47

· Hubble’s lawFirst let’s remind a definition of redshift (z) which bases on an assumption that photon

energy doesn’t depend on place of emission and changes during the movement of photon.

1EE

EEEz

observed

emitted

observed

observedemitteddf

-=-

=

λhc

ThνhE ===

1λλ1

TT1

ννz

emitted

observed

emitted

observed

observed

emitted -=-=-=

Hubble’s law [7] is a result of connection between Hubble’s observations and non-relativisticDoppler law for light.

1λλz

emitted

observed -=

Hubble’s observations: rkz H= , 162H m100.81k --×» , Hk = Hubble’s coefficient

non-relativistic Doppler law for light:cvz =

11811H

df

H s102.43Mpcskm75ckHHr,rckv ---- ×»××»=== Hubble’s law

In literature [12] given values of Hubble’s constant (H) are contained in wide range:( ) ( )1181111811 s102.43Mpcskm57s101.944Mpcskm06H -------- ×»×׸׻××=

Let’s emphasize that proposed by us definition of redshift (z*) was basing on assumptionthat photon energy depends on place of emission and doesn’t change during movement ofphoton. Redshift values (z) and (z*) are the same.

· Radius of Our Universe according to Hubble’s observationKnowing (z*) and (r), we can designate radius of Our Universe (R).

2

1z11

rR

÷øö

çèæ

+-

=

*

22 Rr <<

*»2zrR

Hubble’s observations: rkz H= , 162H m100.81k --×» for 11 Mpcskm57H -- ××=

*= zzlight year m100.95 16×»

48

yearslightbillion6.31m106.02k

1R 26

H

»×»»

· Average density of Our Universe according to Hubble’s observationNow let’s calculate average density of Our Universe, using estimated above radius value

of Our Universe and equation from page 40.

ρ1

G43cR

2

×p

=

GH3

Gk3c

R1

G43cρ

22H

2

2

2

p=

p»×

p=

m106.0RR 26H ×»= = radius of Our Universe for 11 Mpcskm57H -- ××=

Hr=r = density of Our Universe for 11 Mpcskm57H -- ××=327

F mkg1097.4ρ -×» that is almost 3 protons per every cubic meter

Fρ = density of Friedman’s Universe according to [11] for 11 Mpcskm57H -- ××= and47.0=W

327H mkg1059.48ρ -- ××» that is almost 51 protons/m3

02.17ρρ

F

H »

· Constancy of photon energy and Pound-Rebka experimentThe purpose of Pound-Rebka experiment [10] was to prove a hypothesis, that photon

energy doesn’t depend on place of emission and changes during movement in Newton’sgravitational field. Therefore result from this hyphothesis is also: energy absorbed by atomdoesn’t depend on place of absorption.

2RcGM1

1~hν-

1ννz

absorbed

emitteddf

-=

2

2

e RcGM1

RcGM1

1~ν +»-

For Hubble’s constant 11 Mpcskm57H -- ××= and density parameter 47.0=W density ofBlack Hole Universe is over 17 times bigger than density of Friedman’s Universe.Our model doesn’t require us to assume the existence of dark energy.

This density is 8 times bigger than critical densityof Friedman’s Universe.

49

( ) 2

a

cLRGM1

1~ν

±-

R = distance from Earth’s centerL = distance between photon’s absorption and emission points± plus, when emission point was closer to the center than absorption point± minus, when emission point was further from the center than absorption point

÷øö

çèæ

±-»

LR1

R1

cGMz 2

2RGMg = = gravitational acceleration in distance (R) from Earth’s center

RL1

1cgLz 2

±±»

RL <<

2cgLz ±»

Last relation was confirmed experimentally by Pound and Rebka in 1960, usingMössbauer and Doppler phenomenons. Gamma ray emiter ( keV14.4h =n ) was Co57 , and

Fe57 was an absorber. Emitter and absorber was located alternately, in turns. First, emitter onbotton and absorber on top, then vice-versa. Gravitational redshift was being neutralized byDoppler shift. Energy differences of appropriate energetical atom levels of emitter and absor-ber (which was in the same distance from the center of Eatrth) was identical.

Now let’s make analogical deduction, granted this time, that energy of photon depends onplace of emission and is constant during its migration in Schwarzschild gravitational field.Let’s notice that energy absorbed by atom also depends on place of absorbtion.

2Rc2GM1~hν -

1νν1

EE

EEEz

emitted

absorbed

emitted

absorbed

emitted

emittedabsorbeddf

-=-=-

=*

( ) ( ) 22absorbed cLRGM1

cLR2GM1~ν

±-»

±-

2

2emitted Rc

GM1

Rc2GM1

1~ν

1+»

-

R = distance from Earth’s centerL = distance between photon’s absorption and emission points± plus, when emission point was closer to the center than absorption point± minus, when emission point was further from the center than absorption point

50

÷øö

çèæ

±-»*

LR1

R1

cGMz 2

2RGMg = = gravitational acceleration in distance (R) from Earth center

RL1

1cgLz 2

±±»*

RL <<

2cgLz ±»*

Pound-Rebka experiment does not settle which hypothesis is correct.

Quoted works[1] A. Einstein: Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes. Annalender Physik 35, 10 (1911) 898-908.[2] N. Bohr: On the Constitution of Atoms and Molecules. Part I. Philosophical Magazine 26(1913) 1-24.Niels Bohr: On the Constitution of Atoms and Molecules. Part II. Systems Containing Only aSingle Nucleus. Philosophical Magazine 26 (1913) 476-502.[3] K. Schwarzschild: Über das Gravitationsfeld eines Massenpunktes nach der EinsteinschenTheorie. Sitzungsberichte der Königlich Preuβischen Akademie der Wissenschaften 1, 7(1916) 189-196.[4] A. Friedman: Über die Krümmung des Raumes. Zeitschrift für Physik 10, 6 (1922) 377-386.[5] A. Friedmann: Über die Möglichkeit einer Welt mit konstanter negativer Krümmung desRaumes. Zeitschrift für Physik 21, 5 (1924) 326-332.[6] G. E. Lemaître: Un univers homogène de masse constante et de rayon croissant, rendantcompte de la vitesse radiale des nébuleuses extra-galactiques. Annales de la SociétéScientifique de Bruxelles A 47 (1927) 29-39.[7] E. P. Hubble: A Relation Between Distance and Radial Velocity Among Extra-galacticNebulae. Proceedings of the National Academy of Sciences of the United States of America15, 3 (March 15, 1929) 168-173.[8] H. P. Robertson: On the Foundations of Relativistic Cosmology. Proceedings of theNational Academy of Sciences of the United States of America 15, 11 (11/1929) 822-829.[9] A. G. Walker: On Milne’s theory of world-structure. Proceedings of LondonMathematical Society 42 (1937) 90-127.[10] R. V. Pound and G. A. Rebka, Jr.: Apparent weight of photons. Physical Review Letters4, 7 (April 1, 1960) 337-341.[11] J. H. Oort: The Density of the Universe. Astronomy & Astrophysics 7 (09/1970) 405-407.[12] N. Jackson: The Hubble Constant. arXiv:0709.3924v1 [astro-ph][13] Z. Osiak: Energy in Special Relativity. http://viXra.org/abs/1512.0449[14] Z. Osiak: Szczególna Teoria Względności (Special Theory of Relativity).Self Publishing (2012), ISBN: 978-83-272-3464-3, http://vixra.org/abs/1804.0179

51

13 EARTH GRAVITATIONAL FIELD AND UNIVERSE GRAVITATIONALFIELD

· Influence of gravitational field on space and time distancesGravitational fields studied by us can be clearly characterized by time-time component of

metric tensor.In big distances from the center of Our Universe, spacetime metric

( )Universe

2

2

Universe44 Rr

1g ÷÷ø

öççè

æ-=

has different form than locally near Earth

( )Earth

SEarth44 r

r1g ÷÷ø

öççè

æ-= .

From above equations results, that:1. In cosmological scale: the further from Earth, the stronger gravitational field is. Locallynear Earth we observe opposite situation.2. Space distance between two closely placed events is the greater the stronger gravitationalfield is.

( ) ( ) ­Þ¯ 2Earth11 drgr ( ) ( ) ­Þ­ 2

Universe11 drgr ( ) ( ) 14411 gg -

=

This phenomenon can be called „gravitational dilatation of space distance”.3. Time distance between two closely placed events is the lower the stronger gravitationalfield is.

( ) ( ) ¯Þ¯ 2Earth44 cdtgr ( ) ( ) ¯Þ­ 2

Universe44 cdtgr

This phenomenon can be called „gravitational dilatation of time distance”.

· Local properties of redshiftWe will determine the distance (r0) from the center of Our Universe, in which there should

be a given light source emitting photons with the same energy as an identical source locatedon the Earth surface. For this purpose, we compare the time-time components of metric ten-sors that characterize the gravitational fields of the Universe and Earth, respectively.

E

ES

2U

20

Rr1

Rr1 -=-

ESr = Schwarzschild radius for Earth

ER = radius of Earth

UR = radius of Our Universe

52

UE

ES

0 RRrr ×=

5

E

ES 1074.3

Rr -×» , m106.0R 26

U ×» , light year m1095.0 16×»

yearslight236300yearslight10363.2m10245.2r 5210 =×»×»

In distance (r0) from the center of Our Universe, time-time component of metric tensor isequal to analogical component on Earth surface.

· Conclusions1. In a distance from the center of Earth, approximately equal to (r0) redshift (z*) measured inregard to Our Planet changes sign from negative to positive.2. Light that comes to Earth from Our Galaxy, which radius is around 50000 ly, and itsthickness around 12000 ly, should be shifted towards violet in regard to the light emitted onthe surface of Earth. Wherein negative value of reshift (z*) should depend on the direction ofobservation.

53

14 OLBERS’ PARADOX

· Olbers’ paradoxOlbers’ paradox, also called photometric paradox, was formulated by Olbers in 1826 [1]:

„If universe is static, homogeneous and infinite in time and space, then why sky at night isdark?”

Olbers tried to explain this paradox, assuming that interstellar matter is absorbing lightthat is heading towards Earth.......According to commonly accepted idea, formulated within the frames of Friedman’s ex-panding universe theory [2, 3] and based on it hypothesis about Bing Bang, sky at night isdark, because age of universe is finite and light from distant stars didn’t manage to reach usyet, in addition its spectrum is redshifted.

In my view, above statement is wrong in a part reffering to the age of universe. Accordingto this statement, sky at night should get brighter over time. In turn, redshift can be explainedwithin the scope of other hypotheses.

· Probability of photon hitting EarthPortion of photons that come to Earth at night is only a negligable amount of all photons,

especially photons emitted in very far distances from Earth in comparision with Earth radius.Dividing by 4p Earth’s solid angle of view from the emission point of a photon, denoted

by (a), we obtain the probability that the photon „hit” the Earth.

2

2E

E rR2rR p

»aÞ<<

2

2E

E rR

21

4rR »

pa

Þ<<

m10371.6R 6E ×»

m106.0r 26×»

402

26

6

2

2E 10388.1

m106.0m10371.6

21

rR

21

4-×»÷÷

ø

öççè

æ××

=»pa

Probability that photon emitted on the edge of Our Universe will „hit” Earth is only4010388.1 -× .

· Bohr hydrogen atom in Our UniverseBelow we will serve a formula for hydrogen atom energy in Our Universe, when electron

is on n-th allowed (stationary) orbit.

nin44n EgE =

12n En1E =

54

1in442n Eg

n1E =

eV13.6E1 -=

2

22

3S2

322

2in44 R

r1r2Rr1r

RcGM1r

3cG41g -=-=-=rp

-=

m106.0RR 26H ×»= = Our Universe radius for 11 Mpcskm57H -- ××=

eV13.6m100.36

r1n1eV13.6

Rr1

n1eV13.6g

n1E 252

2

22

2

2in442n ×

--=--=-=

En = energy on n-th stationary orbit in the absence of gravitational field

nE = energy on n-th stationary orbit of hydrogen atom which is in distance (r) from thecenter of Our Universe

in44g = component of metric tensor of gravitational field in distance (r) from the center of

Our Universer = distance of photon emission place from the center of Our UniverseM = mass of Our UniverseR = radius of Our Universer = density of Our UniverseG = gravitational constantc = speed of light in vacuum

Sr = Schwarzschild radius of Our Universe

EXAMPLEIn what distance from Earth should hydrogen atom be, so that energy of emitted photon whichcorrespond in Earth conditions to shortwave edge of Lyman series (13.6 eV) was equal to theupper limit of energy which correspond in Earth conditions to infrared radiation (1.59 eV)?

eV13.6Rr

1eV1.59 2

2

-=

9.0Rr»

Light emitted by hydrogen atoms on the edges of Our Universe ( R0.9r ×» ) lies completly ininfrared.

REMINDERInfrared radiation (Infrared) are photons with energies from eV1024.1 -3× to eV59.1

· Illuminance of Earth surface at nightLuminous intensity of Earth’s horizontal surface by starry sky at night is lx103 4-× [4,

page 204].

55

To appoint Earth’s horizontal surface luminous intensity caused by the starry sky at nightwith light emitted by hydrogen atoms, let’s hypothetically assume that there are 51 hydrogenatoms in every cubic meter of Our Universe. Also let’s take into account energy correspon-ding to given spectral line, which would be emitted by hydrogen atom in Earth conditions,fraction of hydrogen atoms that in a time unit emitt photons which correspond to givenspectral line, probability for a photon hitting the Earth and redshift.

2

2

2

2E

ii2

H

H2E

i Rr

12rREdrr4

mR41dI -××h××p×

r×

p=

ò -hrp

=R

R

22ii

H

Hi

Z

drrRERm2

I

iiH

Hi E

mRI hr

8p

»

In the end, for the Earth’s horizontal surface luminous intensity by the starry sky at night,we will get below formula:

åå hr

8p

=»i

iiH

H

ii E

mRII

iI = Earth’s horizontal surface luminous intensity at night, with light emitted by hydro-gen atoms corresponding to given spectral lineI = Earth’s horizontal surface luminous intensity by the starry sky at night

2ER4p = Earth surface area

H

H

mr = amount of hydrogen atoms in one cubic meter

drr4 2p = spherical shell volume with thickness dr

iE = energy corresponding to given spectral line which would be emitted by hydrogenatom in Earth conditions

ih = fraction of hydrogen atoms which emit (in time unit) photons corresponding togiven spectral line[ ] 1s-=h

2

2E

rR2p = probability that photon will hit Earth

2

2

Rr

1- = time-time component of Our Universe metric tensor

Hr = Our Universe density

Hm = hydrogen atom mass

ER = Earth radiusR = Our Universe radiusr = distance from the center of Our Universe

56

Quoted works[1] H. W. M. Olbers: Über die Durchsichtigkeit des Weltraums. Berliner astronomischesJahrbuch für das Jahr 1826.[2] A. Friedman: Über die Krümmung des Raumes. Zeitschrift für Physik 10, 6 (1922) 377-386.[3] A. Friedmann: Über die Möglichkeit einer Welt mit konstanter negativer Krümmung desRaumes. Zeitschrift für Physik 21, 5 (1924) 326-332.[4] Szczepan Szczeniowski: Fizyka doświadczalna. Część IV. Opyka. (Experimental physics.Part IV. Optics.) PWN, Warszawa 1963.

57

15 MICROVAWE BACKGROUND RADIATION

· Background radiationBackground radiation is a microwave radiation that corresponds to 2.7 Kelvin degree

temperature, which comes to Earth almost equally from all directions. It’s being called relictradiation or residual radiation.

Background radiation was discovered by Penzias and Wilson in 1965 [2]. Back then, theywere working for Bell Laboratories, probing a radio connection with satelittes. In that purposethey were using 6 meter directional antenna. Noise popping out from that antenna was provento be a microwave isotropic background radiation.

Detailed studies of cosmic microwave background radiation has been done with instru-ments which was placed on a COBE satelitte. Cosmic Background Explorer has been laun-ched on 18 November 1989. Initial results of measurements were known two months later [3].It turned out that spectrum of cosmic microwave background radiation is almost identicalwith spectrum of black body radiation with 2.735 Kelvin degree temperature with 0.06 Kerror margin. According to other data from COBE [4, 5] there is a quadrupole effect in OurGalaxy, and there are only negligible fluctuactions in spatial distribution of background radia-tion temperature.

· Background radiation in Big-Bang theoryAccording to commonly accepted opinion, discovery of cosmic microwave background

radiation confirms hypothesis about existence of relict radiation as remant of Big Bang. Suchhypothesis was first formulated by Gamov in 1948 [1].

· Background radiation in Black Hole UniverseWe postulate that cosmic microwave background radiation in black hole with maximal

anti-gravity halo universe model, are photons emitted by atoms and molecules which are onthe edges of Our Universe.

EXAMPLEHere we will use expression (given in the chapter 12 of this lecture) for energy of hydrogenatom in Our Universe, when electron is on a n-th allowed (stationary) orbit.

eV13.6Rr1

n1E 2

2

2n --=

nE = energy on n-th stationary orbit of hydrogen atom which is in distance (r) from thecenter of Our Universer = photon emission distance from the center of Our UniverseR = radius of Our Universe

What distance from Earth is needed for hydrogen atom, so that energy of emitted photonwhich in Earth conditions corresponds to shortwave Lyman series limit (13.6 eV) was equalto energy eV106.6 4-× , which in Earth conditions corresponds to maximum radiationintensity of black body that is in 2,7 Kelvin degree temperature?

eV13.6Rr

1eV106.6 2

24 -=× - , R999999999.0r ×»

58

REMINDINGMicrowaves correspond to photons with energies from eV1014.4 6-× to eV1024.1 3-× .

Quoted works[1] G. Gamow: The Evolution of the Universe. Nature 162, 4122 (October 30, 1948) 680-682.[2] A. A. Penzias and R. W. Wilson: A Measurement of Excess Antenna Temperature at 4080MHz. Astrophysical Journal 142 (07/1965) 419-421.[3] COBE Group: J. C. Mather and his co-workers: A Preliminary Measurements of theCosmic Microwave Background Spectrum by the Cosmic Background Explorer (COBE)Satellite. Astrophysical Journal Letters 354 (May 10, 1990) L37-L40.[4] COBE Group: G. F. Smoot and his co-workers: First results of the COBE satellitemeasurement of the anisotropy of the cosmic microwave background radiation. Advances inSpace Research 11, 2 (1991) 193-205.[5] COBE Group: G. F. Smoot and his co-workers: Structure in the COBE differentialmicrowave radiometer first-year maps. Astrophysical Journal 396, 1 (September 1, 1992) L1-L5.

59

16 AVERAGE UNIVERSE DENSITY IN FRIEDMAN’S THEORY

· Critical densityCritical density of universe in Friedman’ theory [2] is a density, where universe becomes

spatially flat.

G83H

c3H 2

4

2

c p=

k=r

11811 s102.43Mpcskm57H ---- ×»××=

mkgs10073.2

cG8 2

434 ×

×=p

=k - ,sm103

sm102.99792458c 88 ×»×=

326

c mkg10058.1 -×»r

· Density parameter and current average density of Friedman’s universeThe density parameter is the ratio of current average density (rF) of Friedman’s Universe

to its critical density (rc).

c

Fdf

rr

=W

Knowing the value of density parameter and critical density, we can state current average den-sity of Friedman’s Universe.

cF Wr=r

47.0=W according to [1]

326

c mkg10058.1 -×»r

327

F mkg1097.4 -×»r that is almost 3 protons per cubic meter

· Dark energyInterpretation of observattional data within Friedman’s Model of Universe forced cosmo-

logists to hypothesise about existence of dark energy.

ATTENTIONFor 11 Mpcskm57H -- ××= density of Universe in Our Model is over 17 times bigger than inFriedman’s Model of Universe. Our Model doesn’t require us to assume existence of darkenergy.

Quoted works[1] J. H. Oort: The Density of the Universe. Astronomy & Astrophysics 7 (09/1970) 405-407.[2] Z. Osiak: Ogólna Teoria Względności (General Theory of Relativity).Self Publishing (2012), ISBN: 978-83-272-3515-2, http://vixra.org/abs/1804.0178

60

17 ANTI-GRAVITY IN OTHER MODELS OF UNIVERSE

· Gravitational acceleration of free particle corresponding with F-L-R-W metric [1]

( )ds

dxds

dxk~cdssgna~ 22

&inergrav

nmamn

a G-=

0dssgn 2 £

dsdx

dsdx

k~ca~

2iner&grav

nmamn

a

G=

=G=nm

1mn

1

dsdx

dsdx

k~ca~

2iner&grav

+G+G+G+G=41

114

31113

21112

11111 ds

dxds

dxds

dxds

dxds

dxds

dxds

dxds

dx

+G+G+G+G+42

124

32123

22122

12121 ds

dxds

dxds

dxds

dxds

dxds

dxds

dxds

dx

+G+G+G+G+43

134

33133

23132

13131 ds

dxds

dxds

dxds

dxds

dxds

dxds

dxds

dx

dsdx

dsdx

dsdx

dsdx

dsdx

dsdx

dsdx

dsdx 44

144

34143

24142

14141 G+G+G+G+

Expressions for amnG come from [1].

( ) ( ) ( )[ ] ( )24232221222 dxdxdxdxLBds +++=

ictx4 = , ( )tLL = = dimensionless time scale factor

241

2

2

41

22

22

41 kr1

1

ar1

1

aLrL1

1B+

=+

=+

= , ( ) ( ) ( )2322212 xxxr ++= , 2a1k =

2a = square of unscaled constant radius of space curvature

1,0,1a1sgnksgn 2 +-==

21

232221

2

22

F dtdx

dtdx

dtdx

cLB1γ

-

ïþ

ïýü

ïî

ïíì

úúû

ù

êêë

é÷÷ø

öççè

æ+÷÷

ø

öççè

æ+÷÷

ø

öççè

æ-=

dticds 1F-g=

úúû

ù

êêë

é÷÷ø

öççè

æ++-÷

÷ø

öççè

æ++g-

+¶¶

g-=

dtdx

xdt

dxx

dtdx

xdt

dxdt

dxdt

dxdt

dxdt

dxdt

dxdt

dxx

21k~kB

dtdx

tL

L1k~2a~

33

22

11

133221112

F2

12F

1&inergrav

61

Now let’s determine physical component of gravitational acceleration of free particle in spa-cetime with F-L-R-W metric.

aa = &inergrav11&inergrav a~ga~

2211 LBg =

aa = iner&graviner&grav a~BLa~

úúû

ù

êêë

é÷÷ø

öççè

æ++-÷

÷ø

öççè

æ++g-

+¶¶

g-=

dtdx

xdt

dxx

dtdx

xdt

dxdt

dxdt

dxdt

dxdt

dxdt

dxdt

dxx

21k~LkB

dtdx

tLBk~2a~

33

22

11

133221112

F3

12F

1&inergrav

In case of spatially flat spacetime (k = 0):

1k~,0tL,

dtdx

tLk~2a~ 2

F

12F

1iner&grav +=>

¶¶

g¶¶

g-=

Let’s notice, that

0a~0dt

dx 1iner&grav

1

<Þ> 0a~0dt

dx 1iner&grav

1

>Þ<

From the relation between signs of velocity components and gravitational acceleration of freeparticle, emerges that occurrence of gravity or anti-gravity depends on the velocity directionof test particle in spatially flat Friedman’s Universe.

· Gravitational acceleration of free particle corresponding with metric of simple modelof expanding spacetime [1]

( )ds

dxds

dxk~cdssgna~ 22

&inergrav

nmamn

a G-=

0dssgn 2 £

dsdx

dsdx

k~ca~

2iner&grav

nmamn

a

G=

=G=nm

1mn

1

dsdx

dsdx

k~ca~

2&inergrav

+G+G+G+G=41

114

31113

21112

11111 ds

dxds

dxds

dxds

dxds

dxds

dxds

dxds

dx

+G+G+G+G+42

124

32123

22122

12121 ds

dxds

dxds

dxds

dxds

dxds

dxds

dxds

dx

62

+G+G+G+G+43

134

33133

23132

13131 ds

dxds

dxds

dxds

dxds

dxds

dxds

dxds

dx

dsdx

dsdx

dsdx

dsdx

dsdx

dsdx

dsdx

dsdx 44

144

34143

24142

14141 G+G+G+G+

Expressions for amnG come from [1].

( ) ( ) ( ) ( )[ ]dxdxdxdxLds 2423222122 +++=

ictx4 = , ( )tLL = = dimensionless time scale factor

21

232221

2 dtdx

dtdx

dtdx

c11γ

-

ïþ

ïýü

ïî

ïíì

úúû

ù

êêë

é÷÷ø

öççè

æ+÷÷

ø

öççè

æ+÷÷

ø

öççè

æ-= , dticLds 1-g=

dtdx

tL

L1k~2a~

1

321

iner&grav ¶¶

g-=

Let’s now determine physical component of gravitational acceleration of free particle in spa-cetime with metric that represents a metric of a simple expanding spacetime model.

aa = &inergrav11&inergrav a~ga~

211 Lg =

aa = iner&graviner&grav a~La~

1k~,0tL

L1,

dtdx

tL

L1k~2a~ 2

21

221

iner&grav +=>¶¶

g¶¶

g-=

Let’s notice, that

0a~0dt

dx 1iner&grav

1

<Þ> 0a~0dt

dx 1iner&grav

1

>Þ<

From the relation between signs of velocity components and gravitational acceleration of freeparticle, emerges that occurrence of gravity or anti-gravity depends on the velocity directionof test particle in spacetime with metric that represents a metric of a simple model of expan-ding spacetime.

Quoted works[1] Z. Osiak: Ogólna Teoria Względności (General Theory of Relativity).Self Publishing (2012), ISBN: 978-83-272-3515-2, http://vixra.org/abs/1804.0178

63

18 OUR UNIVERSE AS AN EINSTEIN’S SPACE

· Other form of field equations – Our Universe as an Einstein’s space

÷øö

çèæ -k-= ababab Tg

21TR ,

mkgs102.073

cG8 2

434 ×

×=p

=k -

0gggggggggggg 433442243223411431132112 ============

4411 g

1g = , 222 rg = , θsinrg 22

33 = ,aa

aa =g1g

2ρcg21T aaaa -= , 2

dfρ2TgT c-==å

4

1=aaa

aa , 2cg21Tg

21T r=- aaaaaa

0RRRRRRRRRRRR 433442243223411431132112 ============

÷÷ø

öççè

æ

¶¶

+¶¶

= 244

244

4411 r

g21

rg

r1

g1R

rgrg1R 44

4422 ¶¶

++-=

÷÷ø

öççè

涶

++-=r

grg1θsinR 4444

233

÷÷ø

öççè

æ

¶¶

+¶¶

= 244

244

4444 rg

21

rg

r1gR

( )n¹m=nma=kr-= mnaaaa ;4,3,2,1,,,0R,gcR 221

Spacetime described by above equations, in which every component of Ricci tensor is pro-portional to adequate component of metric tensor is an Einstein’s space [1]. So spacetime ofOur Universe is an Einstein’s space.

ATTENTIONAll mixed components of Ricci tensor are identically equal to zero. Set of remaining equ-ations can be reduced to only two independent ones.

2222

1111

kr-=

kr-=

gcR

gcR2

21

221

Þ2244

44

2244

244

rc21

rgrg1

c21

rg

21

rg

r1

kr-=¶¶

++-

kr-=¶¶

+¶¶

224444

2244

244

rc21

rgrg1

rc21

rg

2r

rg

kr-=¶¶

++-

kr-=¶¶

+¶¶

Quoted works[1] А. З. Петров: Пространства Эйнштейна. Физматгиз, Москва 1961.There is an English translation of the above book:A. Z. Petrov: Einstein Spaces. Pergamon Press, Oxford 1969.

64

0t=

¶¶E

inin gradk~ j-=EGauss law:

rp-= G4div inEa+a=a graddivdiv AAA

jD=jÑ=j 2graddiv

2

2

2

2

2

2

zyx ¶¶

+¶¶

+¶¶

=D

D = Laplace operatorlaplacian

19 ASSUMPTIONS

· Main postulates of General RelativityGeneral Theory of Relativity (General Relativity) studies conclusions that emerge from

assumptions:1. Maximum speed of signals propagation is the same in every frame of reference.2. Definitions of physical quantities and physics equations can be formulated in a way so thattheir overall forms are the same in all frames of reference.3. Spacetime metric depends on density distribution of source masses.4. Inertial mass is equal to gravitational mass.

· 2-potential nature of stationary gravitational fieldStationary gravitational field is a 2-potential field:

~

îíì-+

=sourcemassaofinside1

sourcemassaofoutside1k

· Poisson’s equation and 2-potential nature of stationary gravitational field

· Signes of right-hand sides of Poisson’s equations and boundary conditionsIn Newton’s theory of gravity, boundary conditions for the gravitational potentials

0limR,r

0limR,r0ex

r

in

0r

=j³

=j<£

¥®

®

in spherical coordinates correspond to the following forms of Poisson’s equation

0t=

¶¶E , 0rot =E

0gradrot =j

ininin gradkgrad j-=j=~E , Rr0 <£ , 0lim in

0r=j

®

exexex gradkgrad j-=j-=~E , Rr ³ , 0lim ex

r=j

¥®

Þ

Poisson’s equation

rp-=rp=jD G4Gk~4in

In empty space, outside of a mass source, right-hand sideof Poisson’s equation is equal to zero.

Laplace’s equation

0ex =jD

65

0ctgsin

1r

r2r

rR,r

ρrG4ctgsin

1r

r2r

rR,r0

ex

2

ex2

2

ex2

2

ex

2

ex22

2in

2

in2

2

in2

2

in

2

in22

=q¶j¶

×q+q¶j¶

+j¶j¶

×q

+¶j¶

×+¶j¶

׳

p-=q¶j¶

×q+q¶j¶

+j¶j¶

×q

+¶j¶

×+¶j¶

×<£

Equivalents of those relations in General Relativity are:

1glimR,r

1glimR,r0

44r

440r

=³

=<£

¥®

®

0RR,r

Tg21TκRR,r0

μν

μνμνμν

=³

÷øö

çèæ --=<£

Signes of right-hand sides of Poisson’s equations depend on assumed boundary conditions,which are connected with 2-potential nature of stationary gravitational field.

· Hypothesis about memory of photonsWe assume that energy of photon depends on point in spacetime, where that photon was

emitted and it stays constant during the movement. It means that photons have „memory”, ormore scholarly – energy of photon is invariant. Wherein, in stronger gravitational field, givensource should send photons with lower energy than the same source in weaker gravitationalfield.

Photon energy, emitted in certain point of spacetime, is given by equation:

11

max

gEE =

maxE = photon energy emitted in nondeformed spacetime

11g = component of metric tensor in photon’s emission point

66

20 STABILITY OF THE MODEL

· Stability of the modelIn order for proposed by me model was real, we should additionally assume that, each

element of Universe moves in such way, so that it guarantees its stability. It’s possible whengravitational acceleration r

grava~ acts as a centripetal acceleration rcenta~ .

rcent

rgrav a~a~ =

rRca~ 2

2rgrav -= (page 30)

ρ1

G43cR

2

×p

= (page 40)

rG34a~r

grav rp-=

ra~ 2rcent w-=

rp==w G34

Rc

2

22

T2p

=w

rp

=p

=G3

cR4T 2

22

R = radius of Our Universer = distance from the center of Our UniverseG = gravitational constantc = maximal speed of signals propagationw = angular velocity of circulation of given element around the center of Our Universer = density of Our UniverseT = period of circulation of given element around the center of Our Universe

It follows from the foregoing that period of circulation of given element around the center ofOur Universe does not depend on the distance from the center.

· Gravitational Gauss law and galaxiesPerforming analogical calculations for given galaxy, within the limits of Newton’s gravi-

tation theory, we will get that: period of circulation of given star around the galactic centerdoes not depend on distance (r) of this element from the center. In below calculations we as-sumed that galaxy is a ball with constant density (r), and star with mass (m) interacts gravita-tionally with part of galaxy with radius (r) (according to Gauss law, gravitational interactionswith the remaining part of galaxy is eliminated).

67

The force of gravity gravF act as centripetal force centF .

centgrav FF =

2grav rGMmF = (Newton’s gravity force)

rp= 3r34M

Gmr34Fgrav p=

rmF 2cent w= (centripetal force)

rp=w G342

T2p

=w

rp

=G3T2

R = radius of galaxyr = distance from the galaxy centerM = part of galaxy mass with radius rm = mass of given starr = galaxy densityG = gravitational constantw = angular velocity of circulation of given star around the galactic centerT = period of circulation of given star around the galactic center

68

21 MAIN RESULTS

· Hypothesis about existence of anti-gravityQuadratic differential form of spacetime can be interpreted as square of spacetime distan-

ces of two points, called „events”, that are located very close to each other in spacetime. Spa-cetime distance between two events is not a magnitude that can be directly measured, becausewe don’t have „4 dimensional rulers”. Also, depending on the sign of the quadratic differen-tial form of spacetime, spacetime distances can be imaginary or real.

· Black hole with maximal anti-gravity haloIt has been shown that exterior (vacuum) solution of gravitational field equations, which

has been served by Schwarzschild in 1916, hides anti-gravity. Spatial radius of black holewith maximal anti-gravity halo is equal to half of Schwarzschild radius. Space outside of theblack hole with maximal anti-gravity halo consists of two layers, in first ( SS rr r5.0 <£ ) anti-gravity occurs, in second ( Srr > ) – gravity. Thickness of anti-gravity layer is equal to thespatial radius of black hole with maximal anti-gravity layer. Inside this anti-gavity layer, ac-celeration is pointed away from the center of mass source, in gravitational part – to the center.

· Hypothesis about 2-potential nature of stationary gravitational fieldAcceptance of hypothesis about 2-potential nature of stationary gravitational field made it

possible to find interior solution for field equations that correspond with gravitational Gausslaw and also made it possible to accomplish a modification of field equations.

· Field equations contain equations of motionHas been justified that equations of motion of free test particle are contained in gravitatio-

nal field equations (that is in spacetime metric equations).

· Black Hole model of Our UniverseOur Universe can be treated as a gigantic homogeneous black hole with spatial radius

equal to half of Schwarzschild radius. It is isolated from the rest of universe with an area ofspacetime where anti-gravity occurs.

Our Galaxy, together with Solar system and Earth, which in cosmological scales can beviewed only as a small point, should be located near the center of Black Hole Universe. Let’semphasize again, that in the center of black hole, gravitational acceleration of free test partic-le, ignoring local gravitational fields, is equal to zero. In other words: black hole is not crea-ting gravitational field in its center.

The gravitational field is stronger, the space is more stretched. For Earth observer, thelocal space with increasing distance from Earth is becoming less stretched. In cosmic scale wehave different situation: the further away from the Earth, the space is more stretched.

Sign change (from negative to positive) of quadratic differential form of spacetime meanstransition from gravity to anti-gravity.

GRAVITY0ds2 Û<GRAVITY-ANTI0ds2 Û>

69

· Size of Our UniverseFor 11 Mpcskm57H -- ××= radius of Our Universe has value:

yearsightl106.31m106.0R 926 ×»×»In a distance from the center of Earth approximate equal to:

yearslight236300yearslight10363.2m10245.2r 5210 =×»×»

redshift measured in regard to our planet, changes its sign from negative to positive.Light that comes to Earth from Our Galaxy (which radius is around 50000 light years and

thickness is around 12000 light years) should be shifted towards violet in regard to light thatis emitted on Earth surface. Wherein negative value of redshift should depend on direction ofobservation.

· Density of Our UniverseFor 11 Mpcskm57H -- ××= density of Universe in Our Model is over 17 times bigger than

in Friedman’s Universe Model and is: 327 mkg1059.48ρ -- ××= , that almost 51 protons per cu-bic meter.ATTENTIONOur Model doesn’t require assumption about existence of dark energy.

· Photon paradoxCommonly accepted assumption, that photon energy doesn’t depend on place of emis-

sion, leads to a paradox. In spacetimes other than conformally flat, length and oscillation pe-riod of electromagnetic wave modelled by photon depends in different way on respectivecomponents of metric tensor. However photon energy depends on length or period in analogi-cal way (that is inversely proportional).

· Hypothesis about memory of photonsWithin the model of Our Universe as a black hole with maximum anti-gravity halo, hypo-

thesis about memory of photons explains:1. photon paradox,2. Olbers’ photometric paradox,3. redshift dependence on the distance from Earth:

A. nonlinear growth of redshift for big distances from Earth,B. existence of distance from Earth, where redshift changes its sign from negative to positi-ve,

4. origin of cosmic microwave background radiation,5. why hypothesis about existence of dark energy is unnecessary.ATTENTIONIt should be highlighted that Pound-Rebka experiment doesn’t exlude hypothesis about me-mory of photons.

· Our Universe is an Einstein’s spaceIt has been shown that field equations which are modelling Our Universe and black hole

with maximum anti-gravity halo can be brought to a form describing the so-called Einstein’sspace.

· Anti-gravity in other cosmological modelsFrom the relation between signs of components of velocity and gravitational acceleration

of free test particle emerges that occurrence of gravity or anti-gravity depends on the directionof velocity of test particle in spatiallly flat Friedman’s Universe as well as in spacetime withmetric of a simple expanding spacetime model [1].

70

· New tests of General RelativityTo show in Earth conditions, existence of a black holes with anti-gravity halo and also 2-

potential nature of gravitational field, we need to measure ratio of distance passed by light tothe time of flight, in a vertically positioned vacuum cylinder, right under and right above thesurface of Earth (of the sea level). If the difference of squares of those measurements will beequal to the square of escape speed, then it will prove existence of black holes with anti-gravi-ty halo. This experiment would be a new test of General Theory of Relativity.

If Our Universe is a black hole with anti-gravity halo, then in the distance from the centerof Earth approximately equal to 236000 light years redshift measured in regard to Our Planetchanges its sign from negative to positive. Light that comes to Earth from Our Galaxy (radiusof Milky Way is around 50000 light years and its thickness is around 12000 light years)should be shifted towards violet in regard to light that is emitted on Earth. Wherein negativevalue of redshift should depend on direction of observation

Quoted works[1] Z. Osiak: Ogólna Teoria Względności (General Theory of Relativity).Self Publishing (2012), ISBN: 978-83-272-3515-2, http://vixra.org/abs/1804.0178

71

22 ADDITION – MATHEMATICS

· Explicit form of Christoffel symbols of first and second kind

rx1 = , q=2x , j=3x , ictx4 = ,44

44222222

112 dxdxgdθsinrdθrdrgds +j++= ,

4411 g

1g = , 222 rg = , θsinrg 22

33 = , 4411

g1g = , 2

22

r1g = ,

θsinr1g 22

33 = ,

( )11111 xgg =

· Explicit form of covariant Ricci curvature tensor

414

111

313

111

212

111

414

414

313

313

212

2121

414

1

313

1

212

11R GG-GG-GG-GG+GG+GG+¶G¶

+¶G¶

+¶G¶

=xxx

÷÷ø

öççè

æ

¶¶

+¶¶

= 244

244

4411 r

g21

rg

r1

g1R

414

122

313

122

111

122

323

323

122

2121

122

2

323

22 xΓ

xΓR GG-GG-GG-GG+GG+

¶¶

-¶¶

=

rgrg1R 44

4422 ¶¶

++-=

[ ]r

gg21

g1 44

44

111

11

111 ¶

¶-==G

[ ] rgg1

4422

111

122 -==G

[ ] θsinrgg1 2

4433

111

133 -==G

[ ]r

gg21

g1 44

4444

111

144 ¶

¶-==G

[ ]r1r

g1

g1

22

212

22

221

212 ===G=G

[ ] cosθsinθcosθsinθrg1

g1 2

22

332

22

233 -=-==G

[ ]r1θsinr

g1

g1 2

33

313

33

331

313 ===G=G

[ ] ctgθcosθsinθrg1

g1 2

33

323

33

332

323 ===G=G

[ ]r

g2g

1g1 44

44

414

44

441

414 ¶

¶==G=G

[ ]r

ggg2

1xg

gg21 44

4444144

4444

111 ¶

¶-=

¶¶

-=

[ ] rxg

21

12222

1 -=¶¶

-=

[ ] θsinrxg

21 2

13333

1 -=¶¶

-=

[ ] cosθsinθrxg

21 2

23333

2 -=¶¶

-=

[ ]r

g21

xg

21 44

14444

1 ¶¶

-=¶¶

-=

[ ] [ ] rxg

21

12212

2212 =

¶¶

==

[ ] [ ] θsinrxg

21 2

13313

3313 =

¶¶

==

[ ] [ ]r

g21

xg

21 44

14414

4414 ¶

¶=

¶¶

==

[ ] [ ] cosθsinθrxg

21 2

23323

332

3 =¶¶

==

72

414

133

212

133

111

133

233

323

133

3132

233

1

133

33 xΓ

xΓR GG-GG-GG-GG+GG+

¶¶

-¶¶

-=

÷÷ø

öççè

涶

++-=r

grg1θsinR 4444

233

313

144

212

144

111

144

144

4141

144

44 ΓΓΓΓΓΓΓΓR ---+¶G¶

-=x

÷÷ø

öççè

æ

¶¶

+¶¶

= 244

244

4444 rg

21

rg

r1gR

0RR

0RR

0RR

0RR

0RR

0RR

4334

4224

3223

4114

3113

2112

==

==

==

==

==

==

· Field equations

÷øö

çèæ --= aaaaaa Tg

21TκR ,

mkgs10073.2

cG8 2

434 ×

×=p

=k -

2cg21T r-= aaaa , 2

dfρ2TgT c-==å

4

1=aaa

aa , 2cg21Tg

21T r=- aaaaaa

÷÷ø

öççè

æ

¶¶

+¶¶

= 244

244

4411 r

g21

rg

r1

g1R ,

rgrg1R 44

4422 ¶¶

++-=

÷÷ø

öççè

涶

++-=r

grg1θsinR 4444

233 , ÷

÷ø

öççè

æ

¶¶

+¶¶

= 244

244

4444 rg

21

rg

r1gR

2

44244

244

44

cg1

21

rg

21

rg

r1

g1

rk-=÷÷ø

öççè

æ

¶¶

+¶¶

224444 ρcκr

21

rgrg1 -=¶¶

++-

2224444

2 θρcsinκr21

rgrg1θsin -=÷÷

ø

öççè

涶

++-

2442

442

4444 ρcκg

21

rg

21

rg

r1g -=÷

÷ø

öççè

æ

¶¶

+¶¶

Above four equations reduce to two equations.

73

224444

2244

244

rc21

rgrg1

c21

rg

21

rg

r1

kr-=¶¶

++-

kr-=¶¶

+¶¶

224444

2244

244

rc21

rgrg1

rc21

rg

2r

rg

kr-=¶¶

++-

kr-=¶¶

+¶¶

· Curvature scalar

αβαβ

dfRgR =

4444

3333

2222

1111 RgRgRgRgR +++=

4411

g1g = , 2

22

r1g = ,

θsinr1g 22

33 =

1gg 4411 =

÷÷ø

öççè

æ

¶¶

+¶¶

= 244

244

1111 rg

21

rg

r1gR ,

rgrg1R 44

4422 ¶¶

++-=

sinRR 22233 q= , 44441144 ggRR =

224444

244

2

r2

r2g

rg

r4

rgR -+

¶¶

+¶¶

=

· Field equations expressed by curvature scalar

μνμνμν κTRg21R -=-

÷÷ø

öççè

æ

¶¶

+¶¶

= 244

244

1111 rg

21

rg

r1gR ,

rgrg1R 44

4422 ¶¶

++-=

sinRR 22233 q= , 44441144 ggRR =

224444

244

2

r2

r2g

rg

r4

rgR -+

¶¶

+¶¶

=

mkgs10073.2

cG8 2

434 ×

×=p

=k -

2ρcg21T abab -=

1gg 4411 = , 222 rg = , θsinrg 22

33 =

111111 κTRg21R -=-

11211

24411 gc

21

rg

r1

rg

rg

kr=+-¶¶

- 2

After multiplying both sides of above equations by ( 244rg- ), we get

74

22kr-=-+¶¶ rc

211g

rgr 44

44

÷øö

çèæ --= 222222 Tg

21TκR

After multiplying both sides of penultimate equation by ( q2sin ), we get

qkr-=÷÷ø

öççè

æ-+

¶¶ 222

44442 sinrc

211g

rgrθsin

÷øö

çèæ --= 333333 Tg

21TκR

Now let’s examine next field equation expressed by scalar curvature.

222222 κTRg21R -=-

22244

2244 rc

21

rg

r21

rgr kr=

¶¶

-¶¶

-

After multiplying both sides of above equations by ( 211

rg

- ), we get

112

244

244

11 gc21

rg

21

rg

r1g kr-=÷

÷ø

öççè

æ

¶¶

+¶¶

÷øö

çèæ --= 111111 Tg

21TκR

442

244

244

44 gc21

rg

21

rg

r1g kr-=÷

÷ø

öççè

æ

¶¶

+¶¶

÷øö

çèæ --= 444444 Tg

21TκR

75

· Laplace operator (Laplacian) in Cartesian and spherical coordinate systems

2

2

2

2

2

2

zy x ¶¶

+¶¶

+¶¶

=D

q=jq=joq= cosrz,sinsinry,scsinrx

q¶¶

×q+q¶¶

×+j¶¶

×q

+¶¶

×+¶¶

= ctgr1

r1

sinr1

rr2

rΔ 22

2

22

2

222

2

· Used relations

444411

2 dxdxgdrdrgds +=

1ds

dxds

dxgdsdr

dsdrg

44

4411 =×+×

1gg 4411 =

rg

gg1

rg 44

4444

11

¶¶

-=¶¶

rg

21

dsdx

dsdxΓ

dsdr

dsdrΓ 44

44144

111 ¶

¶-=×+×

1gg 1111 =

1gg 4444 =

rgg

21Γ 44

11111 ¶

¶-=

rgg

21Γ 44

44144 ¶

¶-=

76

23 ADDITION – PHYSICS

· Hydrogen spectral series in Earth conditionsEnergies of photons emitted by hydrogen atom in Earth conditions are contained in the

range from 1.89 eV to 13.6 eV.

Energies of photons in Lyman series are contained in range from 10.20 eV to 13.60 eV.

¥

1

23

45

Lyman series

Energies of photons in Balmer series are contained in range from 1.89 eV to 3.40 eV.

¥Balmer series

4

2

35

6

Energies of photons in Paschen series are contained in range from 0.66 eV to 1.51 eV.

¥

3

45

67

Paschen series

Energies of photons in Brackett series are contained in the range from 0.31 eV to 0.85 eV.

¥

4

556

7

Brackett series

Energies of photons in Pfund series are contained in range from 0.17 eV to 0.54 eV.

¥

5

67

9Pfund series

77

Energies of photons in Humphreys series are contained in range from 0.10 eV to 0.38 eV.

¥

6

7910

Humphreys series

· Hydrogen spectral series in Our Universe

eV13.6Rr1

n1Eg

n1E 2

2

21in442n --==

eV13.6Rr1

m1Eg

m1E 2

2

21in442m --==

mneV,13.6n1

m1

Rr1

n1

m1EgEEE 222

2

221in44mnmn >÷

÷ø

öççè

æ--=÷

÷ø

öççè

æ--=-=®

· Electromagnetic spectrum

Radio wavescorrespond to photons with energies from eV1024.1 11-× to eV1024.1 3-× .

Microwavescorrespond to photons with energies from eV1014.4 6-× to eV1024.1 3-× .

Lightcorrespond to photons of energies in the range Ve124Ve1024.1 3 ¸× - .

Infrared radiationcorrespond to photons with energies from eV1024.1 -3× to eV59.1 .

Visible lightcorrespond to photons with energies from eV59.1 to eV26.3 .

Ultraviolet radiationcorrespond to photons with energies from eV26.3 to eV241 .

X-rayscorrespond to photons with energies from 124 eV to 12.4 keV.

Gamma radiationcorrespond to photons of energies greater than 12.4 keV.

78

· Choosen concepts, constants and units

Speed of light in vacuum

sm103

sm102.99792458c 88 ×»×=

Planck’s constantseV104.13566743sJ106.6260693h 1534 ××=××= --

Gravitational constant

2

311

2

311

skgm106.7

skgm106.6742G

××»

××= --

Einstein’s constant

mkgs10073.2

cG8 2

434 ×

×=p

=k -

Light year (ly), distance passed by light in vacuum during average sun yearm109.5m109.460536ly1 1515 ×»×=

m103.0861pc 16×=

s103.1561year 7×=

Mass of protonkg1067262171.1m 27

p-×=

Mass of the Earthkg106kg105.9736MM 2424

E ×»×== Å

Mass of the Sunkg102kg101.9891M 3030

S ×»×=

Hubble’s constant unit

s1100.324

Mpcskm 19-×=×

Critical density of Universe in Friedman’s models expresssed by Hubble’s constant (H) andgravitational constant (G)

G83H

c3H 2

4

2

c p=

k=r

Average radius of the Sunm100.696R 9

S ×=

79

Average radius of the Earthm106.371R 6

E ×=

Radius of Bohr hydrogen atomm10830.52917720R 10

B-×=

J101.60217653eV1 19-×=

1272

mkg1032.0G4

3c -××»p

, mkg10125.33c

G4 1272 ××»

p --

80

24 ADDITION – HISTORY

· Newton’s law of gravitationFirst theory of gravitation comes from Newton. It describes in a simple way, gravitational

pull effects which occur between two material points, between two homogeneous balls, bet-ween homogeneous ball and test particle.

According to Newton’s law, gravitational acceleration value of free particle (outside of amass source which homogeneous ball is) decreases inversely to the square of distance fromthe center of the ball.

· Gauss law of gravitationFrom gravitational Gauss law we know that gravitational acceleration value in center of

homogeneous ball with constant density is equal to zero. It grows linearly with growth of dis-tance from the center, reaching its maximal value on a surface of a ball. With further growthof distance – it decreases inversely squared.

Isaac Newton(1642-1727)

Carl F. Gauss(1777-1855)

81

· Einstein’s General Theory of RelativityGeneral Theory of Relativity (General Relativity) explains gravity as a result of deforma-

tion of spacetime, which depends on density distribution of source masses. The free test parti-cles move in space along the tracks, which correspond to geodetic lines in spacetime. GeneralRelativity forces us to revise, inter alia, concepts such as gravitational forces, the forces ofinertia and inertial frames.

General Relativity was finally formulated by Einstein on25 November 1915. This theory bases on four postulates:

1. Maximum speed of signals propagation is the same inevery frame of reference.

2. Definitions of physical quantities and physics equationscan be formulated in a way so that their overall forms arethe same in all frames of reference.

3. Spacetime metric depends on density distribution ofsource masses.

4. Inertial mass is equal to gravitational mass.

· External (vacuum) Schwarzschild solution

Precise external (vacuum) solution for gravitational fieldequations was served by Schwarzschild in 1916:

rr1

g1g S

1144 -== , Rr ³

M = mass of a homogeneous ball with constant densityR = radius of a ball

2S c2GMr = = Schwarzschild radius

Anti-gravity is hidden in this solution.

It is easy to see that, for Srr > sign of quadratic differen-tial form of spacetime

( ) ( ) ( )2444

211

2 dxgdrgds +=

is negative, and for Srr < this sign is positive.

Carl Schwarzschild(1873-1916)

Albert Einstein(1879-1955)

82

25 ENDING

· Author’s reflectionsThe spherical coordinate system used in this paper, for (r = 0) and (θ = 0º, 180º), generates

apparent singularities in expressions such as

222

r1g = ,

θsinr1g 22

33 =

and in the original Poisson’s equation. We applied this coordinate system despite the mentio-ned pathologies, because it is convenient in calculations.

I used general scheme of searching for solutions of field equations describing anti-gravityto known external (vacuum) Schwarzschild metric reduced by me to only two elements. Suchheuristic approach should be treated as an initial stage in exploring anti-gravity phenomenon.

As an author I’m fully aware of all imperfections and defects of a model of black holewith anti-gravity halo and Our Universe model.

Experiments proposed by me – measurement of speed of light, over and under Earth sur-face in vertical vacuum cylinder, and also observations of light spectrum that comes to Earthfrom Our Galaxy – will confirm or disprove presented here hypotheses.

83

Zb

ign

iew

Os

iak

I belong to a generation of physicists, whose idolswere Albert Einstein, Lev Davidovich Landau

and Richard P. Feynman. Einstein enslaved mewith the power of his intuition. I admire Landau

for reliability, precision, simplicity of his argumentsand instinctual feel for the essence of the problem.

Feynman magnetized me with the lightnessof narration and subtle sense of humor.