TESTING "H-M: Electron mass is only due expansion resistance of the Universe"

[Hannu Poropudas]

Speculative formulae for electron mass dependence of
expansion resistance of the Universe.

Constant bulk viscosity formula and value is used from below reference.

Best Regards,
Hannu Poropudas
Kolamäentie 9E,
90900 Kiiminki / Oulu
Finland

--------Maple 9 program below------------------------------------------
> # Speculation of Electron mass calculation with bulk resistance F HP 15112021
> restart;
> # Ref: Avelino, A. and Nucamendi, U. 2009.
> # Can a matter dominated model with constant bulk viscosity drive the
> # accelerated expansion of the universe?
> # arXiv:0811.3253v2 [gr-qc] 11 Mar 2009.
> # 32 pages, here used pages 21-22. (c=1 units).
> # (Journal of Cosmology and Astroparticle Physics. Volume 2009, April 2009.)
> # from ref: H0 = 2.257361031*10^(-18)*s^(-1), H0 = 69.62 km*s^(-1)*Mpc^(-1)
> # H0 = 69.62+-0.59 km*s^(-1)*Mpc^(-1)
> H0 := 2.257361031*10^(-18);

> # G = 6.67388*10^(-11) N*m^2*kg^(-2)
> G := 6.67388*10^(-11);

> # c = 2.99792458*10^8 m*s^(-1)
> c := 2.99792458*10^8;

> # e = 1.602176634*10^(-19) C
> e := 1.602176634*10^(-19);

> # e0 = 8.854187813*10^(-12) N^(-1)*m^(-2)*C
> e0 := 8.854187813*10^(-12);

> # me = 9.10938356e-31 kg
> me := 9.10938356e-31;

> # 1 pc = 3.26 ly, 1 ly = 9.460528436*10^15 m, 1 Mpc =3.08413227*10^22 m
> # from ref: F = H0*c^2*1.922/(24*Pi*G) = 7.749189917*10^7*Pa*s, Pa = N/m^2
> # Error limits of constant from ref: 1.922+-0.089
> # Bulk viscosity F calculation from ref. values.
> # F := H0*c^2*1.922/(24*Pi*G);
> F := evalf(H0*c^2*1.922/(24*Pi*G));

> # F := 7.749189917*10^7;
> # Speculation Stokes Law for a sphere with bulk viscosity F and velocity c
> # and Coulomb Law is used right hand side
> # (viscosity is replaced by bulk viscosity)
> # 6*Pi*r0*F*c = (e^2/(4*Pi*e0))/r0^2
> solve(6*Pi*r0*F*c = (e^2/(4*Pi*e0))/r0^2,r0);

> # 1/6*9^(1/3)*(e^2*Pi*F^2*c^2*e0^2)^(1/3)/(Pi*F*c*e0)
> #r0 := 1/6*9^(1/3)*(e^2*Pi*F^2*c^2*e0^2)^(1/3)/(Pi*F*c*e0);

> r0 := evalf(1/6*9^(1/3)*(e^2*Pi*F^2*c^2*e0^2)^(1/3)/(Pi*F*c*e0));

> # r0 = 0.8076589702e-15 m
> # New r0 was calculated
> # Formula of classical electron radius calculation is used here with new r0
> # Speculation Einstein mass energy equation and Coulomb potential energy equation
> # m*c^2 = e^2/(4*Pi*e0*r0)
> # m = e^2/(4*Pi*e0*c^2*r0) = 0.3178284477e-29 kg
> m := evalf(e^2/(4*Pi*e0*c^2*r0));

> # m = 0.3178284477e-29 kg
> m := e^2/(4*Pi*e0*c^2*(1/6*9^(1/3)*(e^2*Pi*F^2*c^2*e0^2)^(1/3)/(Pi*F*c*e0)));

> # 1/6*e^2*9^(2/3)*F/(c*(e^2*Pi*F^2*c^2*e0^2)^(1/3))
> #simplify(1/6*e^2*9^(2/3)*F/(c*(e^2*Pi*F^2*c^2*e0^2)^(1/3)));

> #m := 1/2*e^2*3^(1/3)*F/(c*Pi^(1/3)*(e^2*F^2*c^2*e0^2)^(1/3));

> # Trying to find simplest suitable formula for calculation of electron mass
> # m/me
> m/me;

> # 3.489022562
> # approximation based above 7/2 = 3.5
> # simplest correction number (2/7) below was taken from a hat
> (2/7)*0.3178284477e-29;

> # 0.9080812791e-30
> # approximately me = (2/7)*m = (2/7)*e^2/(4*Pi*e0*c^2*r0)= 0.9080812790e-30 kg
> evalf((2/7)*e^2/(4*Pi*e0*c^2*r0));

> # 0.9080812791e-30
> # me = approximately 9.080812790e-31 kg
> # me = 9.10938356e-31 kg
> # (2/7)*1/2*e^2*3^(1/3)*F/(c*Pi^(1/3)*(e^2*F^2*c^2*e0^2)^(1/3));

> #approximately me = 1/7*e^2*3^(1/3)*F/(c*Pi^(1/3)*(e^2*F^2*c^2*e0^2)^(1/3))
> evalf(1/7*e^2*3^(1/3)*F/(c*Pi^(1/3)*(e^2*F^2*c^2*e0^2)^(1/3)));

> # 0.9080812794e-30
> # 1/7*e^2*3^(1/3)*F/(c*Pi^(1/3)*(e^2*F^2*c^2*e0^2)^(1/3))
> simplify(1/7*e^2*3^(1/3)*(H0*c^2*1.922/(24*Pi*G))/(c*Pi^(1/3)*(e^2*(H0*c^2*1.922/(24*Pi*G))^2*c^2*e0^2)^(1/3)));

> # 0.4140217692e-1*e^2*H0*c/(G*(e^2*H0^2*c^6*e0^2/G^2)^(1/3))
> 0.4140217692e-1*e^2*H0*c/(G*(e^2*H0^2*c^6*e0^2/G^2)^(1/3));

> # 0.9080812791e-30

[Arnold Sala]

Hannu Poropudas wrote:

> Speculative formulae for electron mass dependence of
> expansion resistance of the Universe.
>
> Constant bulk viscosity formula and value is used from below reference.

https://www.bitchute.com/video/xCBuQ4Q2h0mH/
ISRAEL HIT WITH COVID BLIZZARD AFTER 3RD DOSE

[hapor...@gmail.com]

The TITLE of my posting is IMPORTANT due I don't know yet how to
calculate correctly bulk viscosity frictional force for a sphere in case
which is caused by resistance of expanding Universe.

My speculation for two reasons:
1. I'am searching now correct formula for bulk viscosity case corresponding
to some kind of form of Stoke's Law ?

(Bulk viscosity frictional Force = -6*Pi*r0*F*v, v= velocity,
F=bulk viscosity, r0=radius of a sphere, Pi=3.14159...)

(v=c = speed of light in vacuum in my speculation is due
that expansion of the Universe is into the direction of time dimension)

> > # Speculation Stokes Law for a sphere with bulk viscosity F and velocity c
> > # and Coulomb Law is used right hand side
> > # (viscosity is replaced by bulk viscosity)
> > # 6*Pi*r0*F*c = (e^2/(4*Pi*e0))/r0^2

2. (2/7) insertion to the formula is from a "hat" .

Best Regards,
Hannu Poropudas
Kolamäentie 9E,
90900 Kiiminki / Oulu

Finland.

[hapor...@gmail.com]

****************************************

I calculated all solutions for mathematical completeness.
Please take a look copy of my Maple 9 program below.

Best Regards,
Hannu Poropudas

> # Maple 9 program below
> # Electron mass calculation with bulk viscosity of
> # expanding Universe H.P. 20112021

># For mathematical completeness I calculated all solutions
># (Maple 9 program gave results little more convenient forms)
># and I also calculated all solutions of the second equation
># (p. 262 from second friction equation of ref.1)

> # Ref.1. Weizel Walter, 1955.
> # Lehrbuch der Theoretischen Physik.
> # Erster Band, Zweite Verbesserte Auflage.
> # (Zweite Verbesserte Auflage, 1958, pages 805-1793)
> # Springer-Verlag, Printed in Germany, 804 pages, pp. 258-262.

> # Ref.2: Avelino, A. and Nucamendi, U. 2009.

> # Can a matter dominated model with constant bulk viscosity drive

> # the accelerated expansion of the universe?

> # arXiv:0811.3253v2 [gr-qc] 11 Mar 2009.
> # 32 pages, here used pages 21-22. (c=1 units).
> # (Journal of Cosmology and Astroparticle Physics.

> #Volume 2009, April 2009.)

> restart;
> #############################################################
> # Expansion of the Universe to direction of time dimension v = c

> # c = 2.99792458*10^8 m*s^(-1)

> # p.261 (original friction equation from ref.1)
> solve({6*Pi*c*F*r0 =(1/(4*Pi*e0))*e^2/r0^2, m*c^2=(1/(4*Pi*e0))*e^2/r0},{m,r0});

> allvalues({r0 = 1/2*RootOf(3*Pi^2*c*F*_Z^3*e0-e^2, label = _L2), m = 1/2*e^2/(c^2*Pi*e0*RootOf(3*Pi^2*c*F*_Z^3*e0-e^2, label = _L2))});



> # Little strange behavior of complex mathematics ?
> #evalc((-1)^(2/3));

> # Little strange behavior of complex mathematics ?
> #evalc((-1)^(1/3));

> {r0 = 1/6*3^(2/3)*(e^2/(Pi^2*c*F*e0))^(1/3), m = 1/2*e^2*3^(1/3)/(c^2*Pi*e0*(e^2/(Pi^2*c*F*e0))^(1/3))},
{r0 = 1/6*3^(2/3)*(e^2/(Pi^2*c*F*e0))^(1/3)*evalc((-1)^(2/3)), m = -1/2*e^2*3^(1/3)*evalc((-1)^(1/3))/(c^2*Pi*e0*(e^2/(Pi^2*c*F*e0))^(1/3))},
{m = 1/2*e^2*3^(1/3)*evalc((-1)^(2/3))/(c^2*Pi*e0*(e^2/(Pi^2*c*F*e0))^(1/3)), r0 = -1/6*3^(2/3)*(e^2/(Pi^2*c*F*e0))^(1/3)*evalc((-1)^(1/3))};



> #{m = 1/2*e^2*3^(1/3)/(c^2*Pi*e0*(e^2/(Pi^2*c*F*e0))^(1/3)), r0 = 1/6*3^(2/3)*(e^2/(Pi^2*c*F*e0))^(1/3)},
{r0 = 1/6*3^(2/3)*(e^2/(Pi^2*c*F*e0))^(1/3)*(-1/2+1/2*I*3^(1/2)), m = -1/2*e^2*3^(1/3)*(1/2+1/2*I*3^(1/2))/(c^2*Pi*e0*(e^2/(Pi^2*c*F*e0))^(1/3))},
{m = 1/2*e^2*3^(1/3)*(-1/2+1/2*I*3^(1/2))/(c^2*Pi*e0*(e^2/(Pi^2*c*F*e0))^(1/3)), r0 = -1/6*3^(2/3)*(e^2/(Pi^2*c*F*e0))^(1/3)*(1/2+1/2*I*3^(1/2))};
>
> ##################################################################
>
> # Expansion of the Universe to direction of time dimension v = c

> # c = 2.99792458*10^8 m*s^(-1)

> # p. 262 (second friction equation from ref.1)
> solve({6*Pi*c*F*r0*(1+9*m*c/(32*Pi*r0^2*F))=(1/(4*Pi*e0))*e^2/r0^2, m*c^2=(1/(4*Pi*e0))*e^2/r0},{m,r0});

> allvalues({r0 = 1/4*RootOf(6*Pi^2*c*_Z^3*e0*F+11*e^2, label = _L2), m = e^2/(Pi*c^2*RootOf(6*Pi^2*c*_Z^3*e0*F+11*e^2, label = _L2)*e0)});



> #evalc((-1)^(2/3));

> #evalc((-1)^(1/3));

> {m = e^2/(Pi*c^2*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*e0), r0 = 1/4*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)},
{r0 = 1/4*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*evalc((-1)^(2/3)), m = -e^2*evalc((-1)^(1/3))/(Pi*c^2*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*e0)},
{r0 = -1/4*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*evalc((-1)^(1/3)), m = e^2*evalc((-1)^(2/3))/(Pi*c^2*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*e0)};



> #{m = e^2/(Pi*c^2*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*e0), r0 = 1/4*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)},
{r0 = 1/4*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*(-1/2+1/2*I*3^(1/2)), m = -e^2*(1/2+1/2*I*3^(1/2))/(Pi*c^2*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*e0)},
{r0 = -1/4*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*(1/2+1/2*I*3^(1/2)), m = e^2*(-1/2+1/2*I*3^(1/2))/(Pi*c^2*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*e0)},
{m = e^2/(Pi*c^2*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)*e0), r0 = 1/4*(-11/6*e^2/(Pi^2*c*F*e0))^(1/3)};
>
> ##################################################################

> # c = 2.99792458*10^8 m*s^(-1)
> c := 2.99792458*10^8;

> # e = 1.602176634*10^(-19) C
> e := 1.602176634*10^(-19);

> # e0 = 8.854187813*10^(-12) N^(-1)*m^(-2)*C
> e0 := 8.854187813*10^(-12);

> # me = 9.10938356e-31 kg
> me := 9.10938356e-31;

> # from ref.2: F = H0*c^2*1.922/(24*Pi*G) = 7.749189917*10^7*Pa*s,
> # Pa = N/m^2.
> # Error limits of constant from ref.2: 1.922+-0.089

> # Bulk viscosity F calculation from ref. values.
> # F := H0*c^2*1.922/(24*Pi*G);

> # F := evalf(H0*c^2*1.922/(24*Pi*G));

> F := 7.749189917*10^7;

># All solutions of the original friction equation (ref.1)
> solve({6*Pi*c*F*r0 =(1/(4*Pi*e0))*e^2/r0^2, m*c^2=(1/(4*Pi*e0))*e^2/r0},{m,r0});



> #{r0 = -0.4038294851e-15-0.6994531858e-15*I, m = -0.1589142238e-29+0.2752475097e-29*I},
{r0 = -0.4038294851e-15+0.6994531858e-15*I, m = -0.1589142238e-29-0.2752475097e-29*I},
{r0 = 0.8076589702e-15, m = 0.3178284477e-29};
>
> # OK SOLUTIONS {r0 = 0.8076589702e-15, m = 0.3178284477e-29}
> # Comparison calculated mass m to the electron mass me
> #0.3178284477e-29/9.10938356e-31;

> # Imaginary numbers of solution are open question
> # that how to interpret them.
> ##################################################################
># All solutions of the second friction equation (ref.1)
> solve({6*Pi*c*F*r0*(1+9*m*c/(32*Pi*r0^2*F))=(1/(4*Pi*e0))*e^2/r0^2, m*c^2=(1/(4*Pi*e0))*e^2/r0},{m,r0});



> #{m = -0.3601100146e-29, r0 = -0.7128293752e-15},
{m = 0.1800550073e-29+0.3118644208e-29*I, r0 = 0.3564146876e-15-0.6173283475e-15*I},
{m = 0.1800550073e-29-0.3118644208e-29*I, r0 = 0.3564146876e-15+0.6173283475e-15*I};
>
> # NOT OK SOLUTIONS {m = -0.3601100146e-29, r0 = -0.7128293752e-15}
> # NEGATIVE VALUES for m and r0 are NOT ACCEPTABLE VALUES
> # for expanding case ?
> # m < 0 correspond contracting Universe case ?
> # absolute value of m < 0 case is correctly larger than m > 0 ?
> # (these m <0 can only exist for example
> # inside black hole's event horizon ?)
> # r0 < 0, I don't know that could this case correspond anything ?
> # Negative values of m and r0 of solution are open question
> # that how to interpret them ?
> # Imaginary numbers of solution are open question
># that how to interpret them.
> ###############################

[hapor...@gmail.com]

I found two articles which seems to show that my above "somekind of Stokes Law formula"
could be correct ?

OCD = the Hubble parameters, with constant bulk viscosity case
(H0 = 67.9 +- 0.004 km*s^(-1)*Mpc^(-1) and lambda0 = 9*lambda(0) = 0.053 (+0.047,-0.035))
and
SNIa = Type Ia Supernovae, with constant bulk viscosity case
(H0 = 67.6 +- 0.004 km*s^(-1)*Mpc^(-1) and lamda0 = 9*lambda(0) = 0.080 (+0.071,-0.072))
both gives correct approximate mass for electron.

Please take a look below experimental references and use my mass formula given above articles of mine.

Reference:
Herrera-Zamorano L. Hernandez-Almada A. Garcia-AAspeitia M.A. 2020.
Constraints and cosmography of LCDM in presence of viscosity.
The European Physical Journal C (2020) 80:637, 12 pages.
Table 1/ page 5 and
Table 2/ page 7 and
reference [69] for formula calculation bulk viscosity.

Reference [69] from above article:
Velten Hermano, Schwarz Dominik J. 2012.
Dissipation of dark matter.
arXiv:1206.0986v3 [astro-ph.CO] 1 Oct 2012.
10 pages. (c=1 units), Formula for calculation of
bulk viscosity is on page 3. (lambda0 = H0*c^2*ksi/(24*Pi*G)).

Best Regards,

Hannu Poropudas
Kolamäentie 9E

[hapor...@gmail.com]

Sorry this formula for bulk viscosity is correctly

H0*c^2*lambda0/(24*Pi*G)

Hannu

[hapor...@gmail.com]

I put here two best fit results from above calculations
(I use not rounded numbers):

OHD = Observational Huble Distance case (all three roots for m and r0):

F = 2.084080768*10^6 *Pa*s
m = 9.522380117*10^(-31) *kg
r0 = 2.695723061*10^(-15) *m
m/me = 1.045337487

m = -4.761190059*10^(-31) + - I*8.246623086*10^(-31)
r0 = -1.347861530*10^(-15) - + I*2.334564652*10^(-15)
abs(m) = 9.522380117*10^(-31), abs(r0) = 2.695723060*10^(-15)

SNIa = Type Ia Supernovae (all three roots for m and r0):

F = 3.131883403*10^6 *Pa*s
m = 1.090709746*10^(-30) *kg
r0 = 2.353485864*10^(-15) *m
m/me = 1.197347481

m = -5.453548728*10^(-31)+ - I*9.445823478*10^(-31)
r0 = -1.176742932*10^(-15)- + I*2.038178545*10^(-15)
abs(m) = 1.090709746*10^(-30), abs(r0) = 2.353485863*10^(-15)

(I = sqrt(-1) = imaginary unit)

In polar coordinates three roots for m and r0 seems
to represent three different phase angles which are
0 degrees, 120 degrees and 240 degrees
of the same mass m and same r0 on the complex plane.

Maybe these could indicate existence of some kind
of new dimension for mass of electron, I don't know
how to interpret them (speculation of mine)?

I don't know that could this kind of bulk viscosity
of universe expansion to the time dimension
represent also some how those Higg's particles
(speculation of mine) ?

(on H-M's one old drawing there was six colored Higg's particles.
If someone want to take a look that,
these H-M's old drawings are made public on my Facebook page
few years ago)


Best Regards, Hannu Poropudas

[hapor...@gmail.com]

Formula for bulk viscosity formula is corrected as
F = H0*c^2*lambda0/(24*Pi*G)

and this arXiv:1206.0986v3 [astro-ph.CO] 1 Oct 2012 was published:

Reference:

Velten Hermano, Schwarz Dominik J. 2012.
Dissipation of dark matter.

Physical Review D 86 , 083501 (2012),
7 pages, (c=1 units), formula (8) on page 2.

F = H0*c^2*lambda0/(24*Pi*G)

(I used letters as was before in my postings above.)

Hannu Poropudas

More speculations due I don't know enough of these new matters:

My interpretation of these above complex numbers for m and r0:
They could be interpreted as invisible DARK MATTER electron and its
invisible DARK MATTER mirror electron (positron ?) (speculation of mine)?

(we could be dealing now the invisible DARK MATTER as fifth dimension for the
(1+3) dimensions space-time ? (speculation of mine?))

(How to GR calculate these invisible dark matter orbits with real numbers
in case of Milky Way Galaxy?

Replace SgrA* Black Hole's mass M -> -M and see my earlier,
few years ago in this same sci.physics.relativity newsgroup,
postings about example of real GR calculation of orbit of the Sun's distance,
in case when SgrA* Black Hole's mass M is replaced by corresponding
negative numerical value Black Hole Mass -M (speculation of mine ?) )

-M was about -4.154*10^6*M_Sun I don't remember which value I used.

Hannu Poropudas

[hapor...@gmail.com]

I think that it could not be possible to extend flat Minkowski space-time
(3+1) dimensions by one additional dimension as fifth dimension?

Instead that I think that some kind of full mirror space-time must exist
for invisible DARK MATTER, where time dimension proceeds similar way
as our familiar time dimension (speculation of mine)?

I calculated that complex number mass, for example, when put in
spherical symmetry Schwarzchild metrics, satisfies also Einstein's field
equations for empty space, same way as positive real number mass case.

Hannu Poropudas

[hapor...@gmail.com]

I found yesterday two (4+1) dimensions (5D) spherical symmetric metrics
which satisfy Einstein Field Equations of empty space.

OPEN QUESTION is now that how useful these two metrics are ?

(r => 0, -Pi < ph <= Pi, 0 <= th <= Pi, 0 <= th <= Pi,
unique definition areas only above for the real 4D sphere case)
(t => 0 , in addition (4+1)D = 5D case)

I. (-,+,+,+,+) signature of metrics, r= +-R singularity,
SI-units.

ds^2=dt^2*(c^2*(R^2/r^2-1))+
+dr^2*(1/R^2/r^2-1)+
+dth^2*(r^2*sin(kh)^2)+
+dph^2(r^2*sin(kh)^2*sin(th)^2)+
+dkh^2*(r^2)

(c = the speed of light in vacuum,
R = constant of integration,
for example spherically symmetric metrics of Black Hole
in (4+1)D space-time, if R =( 2*M*G)/c^2), singularity point of r,
G = the constant of gravity,
M is mass of the Black Hole, M => 0 (usual case), real number or
M < 0 ?, real number or M is complex number ?).

II. (+,-,-,-,-) signature of metrtics, r=+-I*R singularity,
I=sqrt(-1) imaginary unit,
SI-units.

ds^2=dt^2*(-c^2*(R^2/r^2-1))+
-dr^2*(1/R^2/r^2-1)+
+dth^2*(-r^2*sin(kh)^2)+
+dph^2(-r^2*sin(kh)^2*sin(th)^2)+
+dkh^2*(-r^2)

(c = the speed of light in vacuum,
R = constant of integration, real number R => 0 (usual case),
or R < 0 ?, or R is complex number ?)
(4+1)D space-time).

These two metrics use and
all their definition areas of parameters are OPEN QUESTIONS?

I have thought one possibility that if these metrics could be useful
in some kind of models of galaxy ?

Best Regards,

Hannu Poropudas

[hapor...@gmail.com]

ERRORS noticed in my previous posting (20.12.2021).
CORRECTLY more carefully and detailed calculated
below new posting (20.12.2021):

Two (4+1)D = 5D ? spherically symmetric metrics,
Einstein Equations., H.P. 19.12.2021

Final form of (4+1)D = 5D ? metrics which
satisfies empty space Einstein equations.


Unique definition areas for 4D sphere
(r=>0, -Pi<ph<=Pi, 0<= th<=Pi, 0<=kh<=Pi)

Signature of metrics (+,-,-,-,-) gives singularity for r
r = 1/2*2^(1/2)*(C21*C11)^(1/2)/C21, and
r = -1/2*2^(1/2)*(C21*C11)^(1/2)/C21.
(C11 and C21 are original constants of integration from
the spherically symmetric solution of Einstein Equations
of empty space, isotropic case)

ds^2=dt^2*(1/2*(-C11+2*C21*r^2)/r^2)-dr^2*(2*r^2*C21/(-C11+2*C21*r^2))+dth^2*(-r^2*(sin(kh)^2))+dph^2*(-r^2*(sin(kh)^2*sin(th)^2))+dkh^2*(-r^2)

Simplifying
(1/2*(-C11+2*C21*r^2)/r^2) =
= -C11/(2*r^2)+C21 =
= C21*(-C11/(2*r^2*C21)+1).

ds^2=dt^2*( C21*(-C11/(2*r^2*C21)+1))-dr^2*(1/( C21*(-C11/(2*r^2*C21)+1)))+dth^2*(-r^2*(sin(kh)^2))+dph^2*(-r^2*(sin(kh)^2*sin(th)^2))+dkh^2*(-r^2)

Define new constant of integration R^2
-C11/(2*C21)=R^2.
Define second new constant of integration
C21 = 1.
Then C11=-2*R^2.

Singularity point for r
solve(-C11+2*C21*r^2=0,r)
r=1/2*2^(1/2)*(C21*C11)^(1/2)/C21, -1/2*2^(1/2)*(C21*C11)^(1/2)/C21

r = +-I*R, I=sqrt(-1)

First metrics (R^2 is new constant of integration):

ds^2=dt^2*((R^2/r^2+1))-dr^2*((1/(R^2/r^2+1))+dth^2*(-r^2*(sin(kh)^2))+dph^2*(-r^2*(sin(kh)^2*sin(th)^2))+dkh^2*(-r^2)

(t=>0,r=>0, -Pi<ph<=Pi, 0<= th<=Pi, 0<=kh<=Pi)?
(Interpretation of this metrics is OPEN QUESTION ?)

###############################################################

Second metrics (-R^2 is new constant of integration):

Define new constant of integration –R^2
-C11/(2*C21)=-R^2.
Define second new constant of integration
C21 = 1.
Then C11=2*R^2.

ds^2=dt^2*((-R^2/r^2+1))-dr^2*(1/((-R^2/r^2+1)))+dth^2*(-r^2*(sin(kh)^2))+dph^2*(-r^2*(sin(kh)^2*sin(th)^2))+dkh^2*(-r^2)

(t=>0,r=>0, -Pi<ph<=Pi, 0<= th<=Pi, 0<=kh<=Pi)

Singularity point for r:
solve(-C11+2*C21*r^2=0,r):

r=1/2*2^(1/2)*(C21*C11)^(1/2)/C21, -1/2*2^(1/2)*(C21*C11)^(1/2)/C21

r=1/2*2^(1/2)*(2*R^2)^(1/2), -1/2*2^(1/2)*(2*R^2)^(1/2)

r = +-R

(Black Hole solution in (4+1)D,
if R = 2*M*G,
M = mass of black hole and
G = Gravitational constant,
c = 1 units).

Best Regards,
Hannu Poropudas,

Kolamäentie 9E, 90900 Kiiminki / Oulu, Finland

[hapor...@gmail.com]

In this metrics NO STABLE ELLIPTICAL ORBITS found when I tested
this metrics with Milky Way's central SgrA* black hole mass M and
S2-star parameters plus modified S2-star parameters.

It seems to me that this (4+1)D space time metrics does not possible
have stable circular orbits around the origin?


Hannu

> ###############################################################
>
> Second metrics (-R^2 is new constant of integration):
>
> Define new constant of integration –R^2
> -C11/(2*C21)=-R^2.
> Define second new constant of integration
> C21 = 1.
> Then C11=2*R^2.
>
> ds^2=dt^2*((-R^2/r^2+1))-dr^2*(1/((-R^2/r^2+1)))+dth^2*(-r^2*(sin(kh)^2))+dph^2*(-r^2*(sin(kh)^2*sin(th)^2))+dkh^2*(-r^2)
>
> (t=>0,r=>0, -Pi<ph<=Pi, 0<= th<=Pi, 0<=kh<=Pi)
>
> Singularity point for r:
> solve(-C11+2*C21*r^2=0,r):
>
> r=1/2*2^(1/2)*(C21*C11)^(1/2)/C21, -1/2*2^(1/2)*(C21*C11)^(1/2)/C21
>
> r=1/2*2^(1/2)*(2*R^2)^(1/2), -1/2*2^(1/2)*(2*R^2)^(1/2)
>
> r = +-R
>
> (Black Hole solution in (4+1)D,
> if R = 2*M*G,
> M = mass of black hole and
> G = Gravitational constant,
> c = 1 units).

In this metrics NO STABLE ELLIPTICAL ORBITS found when I tested
this metrics with Milky Way's central SgrA* black hole mass M and
S2-star parameters plus modified S2-star parameters.

It seems to me that this (4+1)D space time metrics does not possible
have stable circular orbits around the black hole?


Hannu

[Ted Leo]

hapor...@gmail.com wrote:

> In this metrics NO STABLE ELLIPTICAL ORBITS found when I tested this
> metrics with Milky Way's central SgrA* black hole mass M and S2-star
> parameters plus modified S2-star parameters.
> It seems to me that this (4+1)D space time metrics does not possible
> have stable circular orbits around the black hole? Hannu

hannu, can you fucking make it shorter, next time?

[hapor...@gmail.com]

I found one strange analytical solution for S2-star around (?) SgrA* Black Hole
of (4+1)D Schwarzschild geometry:

It has imaginary value for E (constant energy) and real value for L (constant angular momentum)?

Hannu

----COPY of Maple 9 Program for plotting figures about this strange orbit (?)---

> # (4+1)D Schwarzschild geometry Maple 9 program H.P.12.01.2022
> # Analytic solution of S2-star motion around SgrA* black hole

> #(> = command line mark of Maple 9 program, # = comment mark)

># ds^2=dt^2*((-R^2/r^2+1))-dr^2*(1/((-R^2/r^2+1)))+dth^2*(-r^2*(sin(kh)^2))+dph^2*(-r^2*(sin(kh)^2*sin(th)^2))+dkh^2*(-r^2)

>#Unique def.area (t=>0, r=>0, -Pi<ph<=Pi, 0<= th<=Pi, 0<=kh<=Pi)

># (+---) signature eqs. gives same eqs. as (-+++) signature eqs.

>#(Black Hole solution in (4+1)D,

if R = 2*M*G,
M = mass of black hole and
G = Gravitational constant,
c = 1 units).

> restart;
> with(plottools):
> # MG = M*G, c=1 units
> # S2-star "elliptical" orbit around SgrA* Black Hole (NOT EXIST)
> MG := 0.5654692211e27;
> x := 0.2906230228e17;
> y := 0.1778774525e16;

> # Imaginary E value

> E := 0.3884159102e11*I;
> L := 0.1775452105e16;

> ph = Int(1/sqrt(r^4*(E^2-e)/L^2-r^2-(2*MG)^2+r^2*((2*MG)^2*e/L^2)),r);

> # e = 1 for particle

> ph = Int(1/(-0.4786037530e-9*r^4+0.4057513848e24*r^2-0.1279021760e55)^(1/2), r);

> # Roots a1 > a2 > a3 > a4


> #-0.1778774526e16, 0.1778774526e16, -0.2906230227e17, 0.2906230227e17

> #I.a2<=r<=a1

> #0<=P<=Pi/2

> ph := P->0.2964233512e-11*((-400000000+313041541*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-sin(P)^2)^(1/2)*EllipticF(sin(P), 0.8846489995)/((400000000+313041541*sin(P)^4-713041541*sin(P)^2)^(1/2)*cos(P));

>#0<=P<=Pi/2
> r := P->(-0.4853124412e32*sin(P)^2-0.5485932177e32)/(0.2728352774e17*sin(P)^2-0.3084107680e17);

> plot([ph(P),r(P),P=0..Pi/2]);
> plot([ph(P),r(P),P=-Pi/2..Pi/2]);
> plot([ph(P),r(P),P=0..Pi]);
> plot([ph(P),r(P),P=0..2*Pi]);
> plot(ph(P),P=0..Pi/2);
> plot(r(P),P=0..Pi/2);

>#II. a4<=r<=a3

>#0<=P<=Pi/2
>pphh = Int(1/(-0.4786037530e-9*r^4+0.4057513848e24*r^2-0.1279021760e55)^(1/2), r);

>#0<=P<=Pi/2
>pphh := P->0.2964233513e-11*((-0.1000000000e11+2173961469*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-sin(P)^2)^(1/2)*EllipticF(sin(P), 0.4662575971)/((0.1000000000e11+2173961469*sin(P)^4-0.1217396147e11*sin(P)^2)^(1/2)*cos(P));

>#0<=P<=Pi/2
>rr := P->(0.7929221302e33*sin(P)^2-0.8963126963e33)/(0.2728352774e17*sin(P)^2+0.2728352774e17);

>plot([pphh(P),rr(P),P=0..Pi/2]);
>plot([pphh(P),rr(P),P=-Pi/2..Pi/2]);
>plot([pphh(P),rr(P),P=0..Pi]);
>plot([pphh(P),rr(P),P=0..2*Pi]);
>plot(pphh(P),P=0..Pi/2);
>plot(rr(P),P=0..Pi/2);

[hapor...@gmail.com]

I noticed sign error in the basic integral

ph = Int(1/sqrt(r^4*(E^2-e)/L^2-r^2-(2*MG)^2+r^2*((2*MG)^2*e/L^2)),r);

CORRECT integral should be:

ph = Int(1/sqrt(r^4*(E^2-e)/L^2-r^2+(2*MG)^2+r^2*((2*MG)^2*e/L^2)),r);

I'am sorry about this, so please ignore both calculated integrals above
in this last posting of mine.

I have recalculated e= 1 case, but I'am still checking correctness
and I try to understand all plots from these new calculations.

Preliminary result is still one "strange" orbit (?) which is similar than one was above.

Hannu

[hapor...@gmail.com]

###############################################################

My final results from recalculations.

I have used now different definition of the integral.
I have taken now different definitions (I assume them to be correct) from the Weinberg 1972 book.

Open questions exist.

Plase take a look "strange" orbits mentioned below plots (copy of Maple 9 prorgam below).

Hannu

> # CORR Recalc. Equations of motion in curved (4+1)D space-time H.P. 17.01.2022
># REMARK: Different definitions used now from Weinberg S. 1972.
># OPEN QUESTIONS
> # OK
> # isotropic case
> # c = 1 units and MG = M*G
> ###
> # Weinberg 1972, p. 188 and 194-195 used to help identify integration constants E (energy/(unit mass)) and J (angular momentum/(unit mass)). (E > 0 for material particle, E = 0 for photons, p. 186)
> # Reference:
Weinberg Steven,1972.
Gravitation and Cosmology.
Principles and Applications of
The General Theory of Relativity.
John Wiley & Sons. Inc.
Printed in the United States of America.
657 pages, p. 185-188 and 194-196.
> ###

># REMARK: Different definitions used now from Weinberg S. 1972.

>#diff(r(phi), phi)^2 = E*r^4/J^2-r^2+4*MG^2-r^4/J^2+4*r^2*MG^2/J^2;

># “ellipse” NOT EXIST
>solve({E*x^4/J^2-x^2+4*MG^2-x^4/J^2+4*x^2*MG^2/J^2=0,E*y^4/J^2-y^2+4*MG^2-y^4/J^2+4*y^2*MG^2/J^2=0},{E,J});

>#{J = 2*(1/(x^2*y^2-4*x^2*MG^2-4*y^2*MG^2))^(1/2)*MG*y*x, E = (x^2-4*MG^2)*(y^2-4*MG^2)/(x^2*y^2-4*x^2*MG^2-4*y^2*MG^2)}, {J = 2*(1/(x^2*y^2-4*x^2*MG^2-4*y^2*MG^2))^(1/2)*MG*y*x, E = (x^2-4*MG^2)*(y^2-4*MG^2)/(x^2*y^2-4*x^2*MG^2-4*y^2*MG^2)}, {E = (x^2-4*MG^2)*(y^2-4*MG^2)/(x^2*y^2-4*x^2*MG^2-4*y^2*MG^2), J = -2*(1/(x^2*y^2-4*x^2*MG^2-4*y^2*MG^2))^(1/2)*MG*y*x};

># S2-star "elliptical" orbit around SgrA* Black Hole

>MG := 0.5654692211e27;
>x := 0.2906230228e17;
>y := 0.1778774525e16;

>#{J = 0.1775452105e16*I, E = -0.1508669194e22}, {J = 0.1775452105e16*I, E = -0.1508669194e22}, {J = 0.1775452105e16*I, E = -0.1508669194e22}, {E = -0.1508669194e22, J = -0.1775452105e16*I}

>#(E > 0 for material particle, E = 0 for photons, p. 186)
># E (constant energy/(unit mass))

># REMARK: Very large absolute value of E ?
># REMARK: Weinberg 1972 has not defined negative value of E ?

>E := -0.1508669194e22;

># REMARK: Imaginary J value
># J (angular momentum/(unit mass)

>J := 0.1775452105e16*I;

># REMARK: Weinberg 1972 has not defined imaginary value of J ?

># REMARK: Different definitions used now from ref. Weinberg S. 1972.

># +, - sign
>phi = Int((E*r^4/J^2-r^2+4*MG^2-r^4/J^2+4*r^2*MG^2/J^2)^(-1/2),r);

># +, - sign
>phi = Int(1/(0.4786037533e-9*r^4-0.4057513848e24*r^2+0.1279021760e55)^(1/2), r);

>solve(0.4786037533e-9*r^4-0.4057513848e24*r^2+0.1279021760e55=0,r);

>#-0.2906230227e17, 0.2906230227e17, -0.1778774526e16, 0.1778774526e16

># a1 > a2 > a3 > a4
>a1 := 0.2906230227e17;
>a2 := 0.1778774525e16;
>a3 := -0.1778774525e16;
>a4 := -0.2906230227e17;

># A = r
># I. A => a1 and A <= a4 (definition area of the first solution)
># +, - sign
># C1 = 0 (constant of integration)
># 0 <= P <= Pi/2 (definition area of solution)
>phi := P-> -0.9611316525e-28*((-2500000000+543490367*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-sin(P)^2)^(1/2)*EllipticPi(sin(P), 1.884649000, 0.4662575970)/((2500000000+543490367*sin(P)^4-3043490367*sin(P)^2)^(1/2)*cos(P));

># A = r
># 0 <= P <= Pi/2 (definition area of the first solution)
>A := P-> (0.1033905658e33*sin(P)^2-0.8963126963e33)/(0.5812460454e17*sin(P)^2-0.3084107680e17);

>plot(phi(P),P=0..Pi/2);
>plot(Im(phi(P)),P=0..Pi/2);
>plot(Re(phi(P)),P=0..Pi/2);
>plot(A(P),P=0..Pi/2);
>plot(Im(A(P)),P=0..Pi/2);
>plot(Re(A(P)),P=0..Pi/2);
>plot(log(phi(P)),P=0..Pi/2);
>plot(log(abs(phi(P))),P=0..Pi/2);
>plot([phi(P),A(P),P=0..Pi/2]);
>plot([Im(phi(P)),A(P),P=0..Pi/2]);
>plot([Re(phi(P)),A(P),P=0..Pi/2]);
>plot([abs(phi(P)),A(P),P=0..Pi/2]);
>plot([log(phi(P)),log(A(P)),P=0..Pi/2]);
>plot(phi(P),P=-Pi/2..Pi/2);
>plot(phi(P),P=0..Pi);
>plot(phi(P),P=0..2*Pi);

># “STRANGE” closed orbit ?
>plot([phi(P),A(P),P=-Pi/2..Pi/2]);
># “STRANGE” closed orbit ?
>plot([phi(P),A(P),P=0..Pi]);
># “STRANGE” closed orbit ?
>plot([phi(P),A(P),P=0..2*Pi]);

># AA = r
># II. a2 <= AA <= a3 (definition area of the second solution)
># +, - sign
># C1 = 0 (constant of integration)
># 0 <= P <= Pi/2 (definition area of the second solution)
>phi := P-> -0.9611316528e-28*((-400000000+313041541*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-sin(P)^2)^(1/2)*EllipticPi(sin(P), 0.1153510000, 0.8846489995)/((400000000+313041541*sin(P)^4-713041541*sin(P)^2)^(1/2)*cos(P));

># AA = r
># 0 <= P <= Pi/2 (definition area of the second solution)
>AA := P-> (-0.1033905658e33*sin(P)^2+0.5485932174e32)/(0.3557549050e16*sin(P)^2-0.3084107680e17);

>plot(phi(P),P=0..Pi/2);
>plot(Im(phi(P)),P=0..Pi/2);
>plot(Re(phi(P)),P=0..Pi/2);
>plot(abs(phi(P)),P=0..Pi/2);
>plot(AA(P),P=0..Pi/2);
>plot(Im(AA(P)),P=0..Pi/2);
>plot(Re(AA(P)),P=0..Pi/2);
>plot(abs(AA(P)),P=0..Pi/2);
># open orbit
>plot([phi(P),AA(P),P=0..Pi/2]);

>plot([Im(phi(P)),AA(P),P=0..Pi/2]);
>plot([Re(phi(P)),AA(P),P=0..Pi/2]);
>plot([abs(phi(P)),AA(P),P=0..Pi/2]);
>plot(log(abs(phi(P))),P=0..Pi/2);

># open orbit
>plot([phi(P),AA(P),P=-Pi/2..Pi/2]);

># “STRANGE” closed orbit ?
>plot([phi(P),AA(P),P=0..Pi]);
># “STRANGE” closed orbit ?
>plot([phi(P),AA(P),P=0..2*Pi]);

###########################################################

[hapor...@gmail.com]

#####################
#####################
I found one analytic real solution of (4+1)D Friedmann-Robertson-Walker type expanding Universe.

Analytic real solution of (4+1)D Friedmann-Robertson-Walker type Universe

MANY OPEN PROBLEMS EXIST?

I'am not sure that could this represent something of our standard FRW (3+1)D Universe?

Please real cosmologists could you take a look plots given below Maple 9 program of mine?

Hannu Poropudas

Copy below of Maple 9 program:

> # VI. 5D =(4+1)D spherically symmetric metrics with
> # a(t) Einstein eq. H.P. 27.01.2022
> # VI selection
> # (Analytic Real Solution of (4+1)D FRW type Universe)

> # QUESS2: Q_x0 = -9.175992e-5 (“dark light” negative energy?)
> # Planck Satellite 2013 results
> # Q_r0 = 9.175992e-5 (light energy)
> # Q_m0 = 0.315 (contains matter and dark matter)
> # Q_L0 = 0.685 (contains dark energy)
> # Q_x0 is accounted below sum
> # If Sum of all Q0 = Q_x0+Q_r0+Q_m0+Q_L0
> # Q_k is for curvature term
> # Q_k = 1-Q0 = 0 (flat case results with QUESS2?)
> # H0 = 2.181777e-18 (Hubble constant)
> # L = 1.08840653e-52 (Cosmological constant)
> # G = 6.67388e-11 (Gravitational constant)
> # c = 2.99792458e8 (speed of light)
> # 1/R0^2 = 2*(Q0-1)*H0^2 (R0 is radius of the space)

> # +(8*Pi*G/c^4)*rho(t)
> # k =+1 case (other cases k = 0, k = -1)

> # c = 1 units in calculations
> # Einstein tensor first component (time component)
> #-3*(1+2*R0^2*diff(a(t), t)^2)/(R0^2*a(t)^2);

> # H(t) is the Hubble parameter and H0 is the Hubble constant

> # H(t)=((da(t)/dt)/(a(t))^2

> # assumed additional term Q_x0/a^5
> # H(t)^2/H0^2=Q_x0/a^5+Q_r0/a^4+Q_m0/a^3+Q_L0-(Q0-1)/a^2

> # QUESS2 ?
> # Flat case Q_x0 = -Q_r, Q0 = 1, 1-Q0 = 0 (curvature term disappears)

> # Differential equation to be solved

>diff(a(t),t)^2/(a(t)^2)/2.181777e-18^2=-9.175992e-5/a(t)^5+9.175992e-5/a(t)^4+0.315/a(t)^3+0.685;

> # Real analytic solution
> # +,- sign

> # 0 <= P <= Pi

>t := P->-0.3739849614e17*arctanh(0.7854896124e15*(5000000000-2376843733*sin(P))/(-0.1097876573e50*(sin(P)-0.5069335353)^2-0.1113100905e50*sin(P)+0.1452910368e50)^(1/2))+0.3739849614e17*arctanh(0.7854896124e15*(5000000000+2376843733*sin(P))/(-0.1097876573e50*(sin(P)+0.5069335353)^2+0.1113100905e50*sin(P)+0.1452910368e50)^(1/2))-0.1337799330e-26*((-2500000000+2344334679*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-sin(P)^2)^(1/2)*(0.3832455655e45*EllipticF(sin(P), 0.9683665998)-0.8282633161e44*EllipticPi(sin(P), 3.891329044, 0.9683665998))/((2500000000+2344334679*sin(P)^4-4844334679*sin(P)^2)^(1/2)*cos(P))+C2;

>C2 := -0.8153703710e17;
> # C2 is constant of integration

> # 0 <= P <= Pi

> # Letter a is changed to A letter for convenience
>A := P->(-0.6472495275*cos(P)-0.5996222855)/(2.191329941*cos(P)-0.5754565064);

># REMARK: In metrics only A(t)^2 = a(t)^2 scale factor
># is present.

> # OPEN QUESTION IS HOW TO INTERPRET THESE PLOTS?

> # PLOTS (definition area)

> # 0 <= P <= Pi

>plot(A(P),P=0..Pi);

># +,- sign

> # 0 <= P <= Pi

>plot(t(P),P=0..Pi);
>plot(-t(P),P=0..Pi);

>plot([t(P),A(P),P=0..Pi]);
>plot([-t(P),A(P),P=0..Pi]);

> # PLOTS (over definition area)
>plot(A(P),P=0..2*Pi);

># +,- sign
>plot(t(P),P=0..2*Pi);
>plot(-t(P),P=0..2*Pi);

>plot([t(P),A(P),P=0..2*Pi]);
>plot([-t(P),A(P),P=0..2*Pi]);

> # Age from Big Bang to present
> # t2(A) function (not shown here) when A = 1 (present)
> # t2(1) = 0.1387844406e19
> # 0.1387844406e19/(60*60*24*365.2422);
> # 43.97907459e9 years
> # this (4+1)D Universe would be older than
> # the standard FRW (3+1)D Universe,
> # if my QUESS2 and my calculation would be correct ?

#####################
#####################

[hapor...@gmail.com]

Metrics (4+1)D of (+,-,-,-,-) signature which I have used is

ds^2=dt^2-R0^2*a(t)^2*dr^2/(1-k*r^2)-R0^2*a(t)^2*sin(kh)^2*dth^2+
-R0^2*a(t)^2*sin(kh)^2*sin(th)^2*dph^2-R0^2*a(t)^2*dkh^2

Einstein field equations have form in this metrics signature case
R_uv-(1/2)*R*g_uv+L*g_uv = -(8*Pi*G/c^4)*T_uv

Einstein tensor time component of this is (c=1 units)
-3*(1+2*R0^2*diff(a(t), t)^2)/(R0^2*a(t)^2)

Einstein tensor time component of this is (SI- units, c is now included)
-3*(c^2+2*R0^2*diff(a(t), t)^2)/(R0^2*a(t)^2)
Critical energy density would be of the form
e_c0=6*c^2*H0^2/(8*Pi*G)
,which is twice as large as in corresponding (3+1)D case.
Planck Satellite 2013 Results uses SI-units.

This is why we have no measured energy density component
of (4+1)D corresponding case as we have of (3+1)D corresponding case.

H(t)^2/H0^2=Q_x0/a^5+Q_r0/a^4+Q_m0/a^3+Q_L0-(Q0-1)/a^2

This equation which I solved analytically above is for this reason
(3+1)D case (NOT (4+1)D case which I wrongly announced above,
I'am sorry about that mistake of mine). I have used Einstein tensor
of (3+1)D case in this equation and only one guess2 term of Q_x0/a^5.

So this analytic solution only test presence of this new (4+1)D term
Q_x0/a^5 in (3+1)D.

One remark from plots above is that near Bing Bang start time
there is one strong gravitational wave in time dimension,
reason of which I don't know now, maybe parameter values
not fully suitable for (4+1)D case ?

Hannu Poropudas

Reference:
Chatterjee,S. and Bhui, B. 1990.
Homogeneous cosmological model in higher dimension.
Mon. Not. R. astr. Soc. (1990) 247, 57-61.
(c=1, G=1 units).

[hapor...@gmail.com]

CORRECTION:

ds^2=dt^2-R0^2*a(t)^2*dr^2/(1-k*r^2)-r^2*R0^2*a(t)^2*sin(kh)^2*dth^2+
-r^2*R0^2*a(t)^2*sin(kh)^2*sin(th)^2*dph^2-r^2*R0^2*a(t)^2*dkh^2

>
> Einstein field equations have form in this metrics signature case
> R_uv-(1/2)*R*g_uv+L*g_uv = -(8*Pi*G/c^4)*T_uv
>
> Einstein tensor time component of this is (c=1 units)
> -3*(1+2*R0^2*diff(a(t), t)^2)/(R0^2*a(t)^2)

CORRECTION:

-3*(k+2*R0^2*diff(a(t), t)^2)/(R0^2*a(t)^2)

>
> Einstein tensor time component of this is (SI- units, c is now included)
> -3*(c^2+2*R0^2*diff(a(t), t)^2)/(R0^2*a(t)^2)

CORRECTION:

-3*(k*c^2+2*R0^2*diff(a(t), t)^2)/(R0^2*a(t)^2)

> Critical energy density would be of the form
> e_c0=6*c^2*H0^2/(8*Pi*G)
> ,which is twice as large as in corresponding (3+1)D case.
> Planck Satellite 2013 Results uses SI-units.
>
> This is why we have no measured energy density component
> of (4+1)D corresponding case as we have of (3+1)D corresponding case.
>
> H(t)^2/H0^2=Q_x0/a^5+Q_r0/a^4+Q_m0/a^3+Q_L0-(Q0-1)/a^2
>
> This equation which I solved analytically above is for this reason
> (3+1)D case (NOT (4+1)D case which I wrongly announced above,
> I'am sorry about that mistake of mine). I have used Einstein tensor
> of (3+1)D case in this equation and only one guess2 term of Q_x0/a^5.
>
> So this analytic solution only test presence of this new (4+1)D term
> Q_x0/a^5 in (3+1)D.
>
> One remark from plots above is that near Bing Bang start time
> there is one strong gravitational wave in time dimension,
> reason of which I don't know now, maybe parameter values
> not fully suitable for (4+1)D case ?
>
> Hannu Poropudas
>
> Reference:
> Chatterjee,S. and Bhui, B. 1990.
> Homogeneous cosmological model in higher dimension.
> Mon. Not. R. astr. Soc. (1990) 247, 57-61.
> (c=1, G=1 units).

I used instead of R(t) (above reference) following:
R(t) = R0*a(t) and my letter a is changed to A letter for convenience.

Only k = +1 case is solved and Guess2 changed situation
that really k = 0 case is solved analytically and (3+1)D case.

In my article of this article chain dated 18.1.2022, local time 14:56 there
was typo error which is corrected as follows (letter a is also was
changed there to letter AA for convenience):

a3 <= AA <= a2.

I'am sorry about these errors. I hope I could correct them here.

Hannu

[hapor...@gmail.com]

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Two CORRECTIONS: 1. Meaning of R0 has also changed. It must be pure number due correct dimensions
of metric parts. It is best to use ref. Chatterjee S. and Bhui B. 1990 definition properly and
use R(t)=a(t), so we would have in my Einstein tensor components in form where R0 = 1.

2. And meaning of k has also changed this is also due reference Chatterjee S. and Bhui B. 1990.

Analytically solved (3+1)D case differential equation of mine does not change,
it used Planck Satellite 2013 results (measurements given in SI-units) and my Guess2 of additional test term for (4+1)D and it is FLAT case (1-Q0 = 0, and this differential equation
does not depend on k or R0) and it is OK solution.

Sorry about these confusions of mine. I put below explanations of different (3+1)D metrics forms
due to clarify these confusions of mine.


Different (3+1)D metric forms from refs. Tolman R.C. 1934 , Chatterjee S. and Bhui B. 1990.

(c = 1 units in my metrics forms).

ds^2=dt^2-dr^2+dth^2*(-r^2*sin(kh)^2)+dph^2*(-r^2*sin(kh)^2*sin(th)^2)+dkh^2*(-r^2)

(t=>0, unique definition area of 4D sphere, r=>0, -Pi<ph<= Pi, 0<=th<=Pi, 0<=kh<=Pi)

(4+1)D = 5D metrics (with a(t) which satisfies Einstein equations,
Einstein tensor tt-component is only used in my calculations).

ds^2=dt^2-a(t)^2*dr^2/(1-k*r^2)+dth^2*(-r^2*a(t)^2*sin(kh)^2)+dph^2*(-r^2*a(t)^2*sin(kh)^2*sin(th)^2)+dkh^2*(-r^2*a(t)^2),
k= constant of integration ,

(Ref. Chatterjee S. and Bhui B. 1990.

Homogeneous cosmological model in higher dimension.

Mon. Not. astr. Soc. (1990) 247, 57-51. (c=1, G=1 units).

Def. of constant of integration k on page 58, formulae (21) and (22) and metrics (23) and (2)
k = 2*k_bar/(n*(n-1)), k_bar is an arbitrary constant of integration
and n=3 in case of 5 dimensional space-time,

ds^2=dt^2-R(t)^2*(dr^2/(1-k*r^2)+r^2*dx_n^2), dx_n^2=dth_1^2+sin(th_1)^2*dth_2^2+…sin(th_1)^2*sin(th_2)^2…sin(th_(n-1))^2*dth_n^2,

My used letters: th_1 = kh, th_2 = th, th_3 = ph in case of 5 dimensions D = n+2, where n = 3).

(unique (?) definition area for this metrics:
t=>0 (?),1-k*r^2<>0 (?), r=>0 (?), -Pi<ph<= Pi, 0<=th<=Pi, 0<=kh<=Pi, ?)

(+,-,-,-,-) signature used here and also in my calculations.

Model from Tolman R. C. 1934: Einstein field equations form (page 376)

R_uv-(1/2)*R*g_uv+L*g_uv = -8*Pi*T_uv

Different forms of (3+1)D metrics used by Tolman R. C. 1932 (c=1, G=1 units, exp(g(t))=a(t)^2,
(-,-,-,+) signature used by Tolman , I used (+,-,-,-) signature for (3+1)D and (+,-,-,-,-) for 5D):

(Ref. Tolman, R.C. 1934. (reprint 1987). Relativity Thermodynamics and Cosmology. Dover Publications, Inc. New York. Manufactured in the United States of America.
502 pages, THIS COPY below pp. 370-372 . ThisCOPY is due that perhaps (4+1)D case
could be possible more properly to be generalized with help of this (3+1)D text of Tolman ?)

ds^2=dt^2-(a(t)^2/(1+r^2/(4*R0^2))^2)*(dr^2+r^2*dth^2+r^2*sin(th)^2*dph^2),

ds^2=dt^2-(a(t)^2/(1+r^2/(4*R0^2))^2)*(dx^2+dy^2+dz^2),
x=r*sin(th)*cos(ph),y=r*sin(th)*sin(ph),z=r*cos(ph),
r=sqrt(x^2+y^2+z^2),

ds^2=dt^2-a(t)^2*(dr^2/(1-r^2/R0^2)+ r^2*dth^2+r^2*sin(th)^2*dph^2),
r = r_ba r, r_bar = r/(1+r^2/(4*R0^2)),

ds^2=dt^2-R0^2*a(t)^2*(dkh^2+sin(kh)^2*dth^2+sin(kh)^*sin(th)^2*dph^2),
r_bar = R0*sin(kh).

ds^2=dt^2-a(t)^2*(dz1^2+dz2^2+dz3^2+dz4^2),

z1 = R0*sqrt(1-r^2/R0^2), z2 = r*sin(th)*cos(ph), z3 = r*sin(th)*sin(ph), z4 = r*cos(th),
r = r_bar.
This last metrics form at any given time t permits us to regard our original space as
embedded in a Euclidean space of a larger number of dimensions.

It will be simplest, however, as suggested by the last form of the line element, to
regard the spatial extent of this non-static universe at any given time t as the whole
three-dimensional spherical surface defined by

z1^2+z2^2+z3^2+z4^2 = R0^2,

embedded in the four-dimensional Euclidean space (z1.z2.z3,z4).
Since the proper distance at time t corresponding to the coordinate interval
dz1, would from the form of the line element

ds^2=dt^2-a(t)^2*(dz1^2+dz2^2+dz3^2+dz4^2),

would evidently be

dl0=a(t)*dz1,

with similar expressions fort he other spatial coordinates, it is evident that
the radius of this spherical surface would be

R = R0*a(t),

Hence this quantity is often spoken of as the radius of the non-static universe,
and the geometry is spoken of as being that for the surface of a sphere in
four dimensions whose radius is a function of the time.
It should be noted, however, in accordance with the equation

-c1/c2 = 1/R0^2, (c1 and c2 are constants of intgration),

by which R0 was introduced, that this radius could be real, imaginary, or
infinite.

If we assume the radius real, the total integrated proper spatial volume
of the model at any selected time t would be give in accordance with

ds^2=dt^2-R0^2*a(t)^2*(dkh^2+sin(kh)^2*dth^2+sin(kh)^*sin(th)^2*dph^2),
r_bar = R0*sin(kh).

by

v0 = int_0^2*Pi int_0^Pi int_0^Pi (R0^3*a(t)^3*sin(kh)^2*sin(th)*dkh*dth*dph) =
= 2^Pi^2*R0^3*a(t)^3

and the total integrated proper distance around the universe would be

l0 = 2*Pi*R0*a(t).

Taking the spatial geometry as elliptical rather than spherical, the corresponding
quantities would be half as great.

If we assume the radius infinite or imaginary, the model would be spatially
open rather than closed and the total proper volume could be most conveniently
calculated, in accordance with the form

ds^2=dt^2-R0^2*a(t)^2*(dkh^2+sin(kh)^2*dth^2+sin(kh)^*sin(th)^2*dph^2),
r_bar = R0*sin(kh),

for the line element, from the expression

v0 = int_0^2*Pi int_0^Pi int_0^infinity (a(t)^3/sqrt(1+r^2/A^2)*r^2*sin(th)*dr*dth*dph) =
= infinity,
r =r_bar, r_bar = r/(1+r^2/(4*R0^2)),

where A^2 is a positive quantity which can assume the value infinity, and upper limit
for r_bar can be taken as infinity without disturbing the possibility for a physical interpretation
of the line element by changing its signature. Evaluating the integral we then obtain an
infinite total proper volume for open models.

The symmetrical form

ds^2=dt^2-a(t)^2*(dz1^2+dz2^2+dz3^2+dz4^2),

which we have been able to give to the line element by the device of considering
a larger number of dimensions, is valuable in clearly showing the spatial homogeneity
of the model already mentioned in the connexion with

ds^2=dt^2-a(t)^2*b(r)^2*(dr^2+r^2*dth^2+r^2*sin(th)^2*dph^2),
(exp(f(r))=b(r)^2, exp(g(t))=a(t)^2).

It is an interesting extension of Schur’s theorem of Riemannian geometry, that
the spatial isotropy which we have assumed for observers at all points in space-time
should lead to an orthogonal separation into space and time and to homogeneity for
the sub-manifold of space. It is in accordance with this result that our present models
of the universe have been designated as non-static homogeneous cosmological models.

Hannu
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[hapor...@gmail.com]

****
ONE REMARK:

It is interesting to note that the scale factor a(t) is same for both standard Universe
FRW-model (3+1)-dimensional and also for speculative higher dimensional Universe
(4+1)-dimensional case.

This above copy from Tolmann 1934 book , if correct, would mean that my old analytic solution of Friedmann equation (published few years ago in this news group) for scale factor a(t) has this same old analytic solution for this higher dimensional case (4+1)-dimensions?

Hannu
****