Analytic GR solutions of S2-star orbits and precession 732” per revolution

[Hannu Poropudas]

Analytic GR solutions of S2-star orbits and precession 732” per revolution

New initial data 2020 is used. My earlier initial data was 2016 in my
postings in this sci.physics.relativity.

Please also take a look my two postings about OJ287 in sci.astro. I have got
no comments there due so heavy traffic of unbssinesslike postings from many posters.

(Only command lines of Maple 9 program is below and command line mark is >. This can be copy-pasted in Maple 9 program and run it.
Remark that for example e-2 below means 10^(-2)
or e2 below means 10^2.)

Best Regards,

Hannu Poropudas
Finland

---COPY of the Maple 9 program below------

> # RIGHT: Particle in Schwarzschild metrics H.P. 09.09.2020
> # S2 Orbit around Sagittarius A*
> # Perihelion precession of S2.
> #OK=COMPARISION to Post-Newtonian approx of Abuter R et al 2020, A&A 636,L5(2020)
> # Abuter R. et al. 2020. Detection of the Schwarzschild precession in the
> # orbit of the star S2 near the Galactic centre massive black hole.
> # Astronomy Astrophysics, 636, L5 (2020).
> # In this article: 12'/rev=720"/rev,12.1'/rev=726"/rev, 12.3'/rev=738"/rev
> # My GR calculation below, result : 731.8543470"/rev OR -676.7436912"/rev
> # (I use not rounded numbers). Approximately 732"/rev OR -677"/rev.
> # New starting values from Table E.1 Best-fit Orbit Parameters above article
> # and other constants of physics are taken from Wikipedia date 9.9.2020.
> # NEW CALCULATION x and y are roots. E and J little changed.
> # (analytic solutions defined between roots a2..a1 OR a4..a3)
>Restart;
>with(plots):with(plottools):
> # GEOMETRIZED UNIT SYSTEM IS PROBLEMATIC.
> # Ref. Formulae for E and J from Weinberg's book. (THESE FORMULAE ARE NOT USED.)
> # Reference: Weinberg, S. 1972. Gravitation and Cosmology: Principles
> # and Applications of the General Relativity. Wiley, New York. pp. 179-210.
> # (units c = 1, and c.g.s in Weinberg's book).
> # J = r^2*dPhi/dp = constant (angular momentum per unit mass), p. 186.
> # E = constant (energy per unit mass), p. 186.
> # E > 0 for material particles, E = 0 for photons. p. 186.
> # Integration limits must be determined from the problem to which apply these.
>#Int(1/(r^4*(1/J^2-E/J^2)+r^3*(2*M*G*E/J^2)-r^2+2*M*G*r)^(1/2),r);
>AU := 149597870700*10^2;
># 125.058 mas, 8246.7 pc, 1 pc =3.26 ly
>ap:= 1030.799324*AU;
>em := 0.884649;
>M := 4.261*10^6*1.98847*10^30*10^3;
># v = c
>v := 2.99792458*10^8*10^2;
>G := 6.6743015*10^(-11)*10^3;
># BH mass*G geometric units (cm) and v = c.
>M*G/v^2;
>MG := 0.6292090968e12;
> # EXAMPLE. Weinberg formulae E and J (THESE FORMULAE ARE NOT ANY MORE USED.)
> # Perihelion distance
>x := ap*(1-em);
># Aphelion distance
>y := ap*(1+em);
> # NEW CALCULATION (NEW FORMULAE OF MY OWN) x and y are roots
> #***********************************************************
>E := (2*x*y*MG+2*y^2*MG+2*x^2*MG-x^2*y-x*y^2)/((-y+2*MG)*(-2*x*MG-2*y*MG+x^2+x*y));
>J := 2^(1/2)*(-(-y+2*MG)*MG*(-2*x*MG-2*y*MG+x^2+x*y))^(1/2)*y*x/(-x^2*y+2*x^2*MG-4*x*MG^2+4*x*y*MG-x*y^2+2*y^2*MG-4*y*MG^2);
> #*****************************************************************
>
> # Weinberg's formula (NOT USED ANY MORE)
> #J := sqrt((1/(1-2*MG/y)-(1/(1-2*MG/x)))/(1/y^2-1/x^2));
> # NEW CALCULATION x and y are roots
>J := -0.4594478956e14;
> # Weinberg's formula (NOT USED ANY MORE)
> #E := (y^2/(1-2*MG/y)-x^2/(1-2*MG/x))/(y^2-x^2);
> # NEW CALCULATION x and y are roots
>E := 1.000040803;
>solve(r^4*(1/J^2-E/J^2)+r^3*(2*MG*E/J^2)-r^2+2*MG*r=0,r);
>a1 := 0.2906262559e17;
>a2 := 0.1778773310e16;
>a3 := 0.1259363679e13;
>a4 := 0;

># Primitive function, definition areas 0<=P<=Pi/2 and a2 <= r <= a1.

>e := P -> 2.000794538*EllipticF(sin(P),0.2578161452e-1);
>r := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);

> #******************************
> # P which corresponds to the perihelion distance (TWO VALUES)
>solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.1778774525e16,P);
>e(0.8532953281e-3);
>e(-0.8532953281e-3);
># Solution's definition area limit is a2-root (TWO VALUES)
>solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.1778773310e16,P);
>e(0.6516324491e-5);
>e(-0.6516324491e-5);
># P which corresponds to the aphelion distance (TWO VALUES)
>solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.2906230228e17,P);
>e(1.569945054);
>e(-1.569945054);
> # Solution's correct definition limit is a1-root (TWO VALUES)
> # COMPLEX NUMBER (SMALL COMPLEX PARTS ARE LITTLE OUTSIDE ORBIT)
>solve((0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17)=0.2906262559e17,P);
>e(1.570796327-0.1136598102e-4*I);
>e(-1.570796327+0.1136598102e-4*I);

> # Angle change, definition areas 0<=P<=Pi/2 ja a2 <= r <= a1.
> # + sign perihelion in last term

>e2 := P -> 2*abs(2.000794538*EllipticF(sin(P),0.2578161452e-1)-2.000794538*EllipticF(sin(0.8532953281e-3),0.2578161452e-1))-2*Pi;
>r2 := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);

> # Calculation of different combinations of + and - signs
> # - sign perihelion in last term

>e2B := P -> 2*abs(2.000794538*EllipticF(sin(P),0.2578161452e-1)-2.000794538*EllipticF(sin(-0.8532953281e-3),0.2578161452e-1))-2*Pi;
>r2B := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);

> # Angle change in radians per one revolution. (TWO e2 and e2B functions)
> # FOR e2
>evalf(e2(1.569945054));
># Angle change in degrees per one revolution.
>evalf(-0.3280946e-2*180/Pi);
># Angle change in arc seconds per one revolution.
>-0.1879843586*60*60;
>###(-676.7436912)
># Second root
>evalf(e2(-1.569945054));
>evalf(0.3548130e-2*180/Pi);
>0.2032928741*60*60;
>###(731.8543470)
># FOR e2B
>evalf(e2B(1.569945054));
># Angle change in degrees per one revolution.
>evalf(0.3548130e-2*180/Pi);
># Angle change in arc seconds per one revolution.
>0.2032928741*60*60;
>###(731.8543470)
># Second root
>evalf(e2B(-1.569945054));
>evalf(-0.3280946e-2*180/Pi);
>-0.1879843586*60*60;
>####(-676.7436912)
> # + and - signs for e2 or e2B does not have different results due abs-values
> # Second Primitive function, definition areas 0<=P<=Pi/2 and a4=0 <= r <= a3.
> # This has form of spiral (+ and - signs) which leads to the origin.
>
> # SUMMARY of S2 precession: 731.8543470"/revolution OR -676.7436912"/revolution.
>
> #******************************

># Second Primitive function, definition areas 0<=P<=Pi/2 and a4=0 <=
r <= a3.

>ee := P-> 2.000794538*EllipticF(sin(P),0.9996675989);
>rr := P-> (0.3660041508e29*sin(P)^2)/(0.1259363679e13*sin(P)^2+0.2906136623e17);

> #*********************************************************
>
> # Plottings only on definition area a2..a1.

> # First side of the solution (+,- solution)
>plot([r(P),e(P),P=0..Pi/2],coords=polar);

># Second side of the solution (+,- solution)
>plot([r(P),-e(P),P=0..Pi/2],coords=polar);

># Angle change picture has no other meaning than above calculated precession
>plot([r2(P),e2(P),P=0..Pi/2],coords=polar);

># Angle change picture has no other meaning than above calculated precession
>plot([r2B(P),e2B(P),P=0..Pi/2],coords=polar)

> # Plottings only on definition area a4..a3.

> # First side of the Second solution (+,- solution)
>plot([rr(P),ee(P),P=0..Pi/2],coords=polar);

># Second side of the Second solution (+,- solution)
>plot([rr(P),-ee(P),P=0..Pi/2],coords=polar);

[Hannu Poropudas]

*** 1. CORRECTION: Formula of J has + and - signs. Positive sign should be selected.
Previous - sign selection does not influence due J^2 was always used.

J := 0.4594478956e14

*** 2. -Pi/2 .. Pi/2 plots gives both sides of both solutions.
This is due
sin(-P)=-sin(P)
and
EllipticF(sin(-P),q)=EllipticF(-sin(P),q)=-EllipticF(sin(P),q).

3. I calculaled below total coordinate velocity, proper velocity and
total proper acceleration for S2-star:
(These are used only in definition areas of above analytic solutions
a2<=r<=a1 or a4<r<=a3,
a1 := 0.2906262559e17,
a2 := 0.1778773310e16,
a3 := 0.1259363679e13,
a4 := 0)

(REMARK: > is command line mark of Maple 9 program)
># Total coordinate velocity and total proper velocity S2-star
># at perihelion and at aphelion HP 12092020

> ### total coordinate velocity v^2=(dr/dt)^2+r^2*(dPhi/dt)^2
> sqrt((1-2*MG/r)^2*(1-(E+J^2/r^2)*(1-2*MG/r))+(J^2/r^2)*(1-2*MG/r)^2)
> sqrt((1-2*MG/x)^2*(1-(E+J^2/x^2)*(1-2*MG/x))+(J^2/x^2)*(1-2*MG/x)^2);
0.2581119858e-1
> # km/s
> 0.2997924580e11*0.2581119858e-1/10^5;
7738.002666
> sqrt((1-2*MG/y)^2*(1-(E+J^2/y^2)*(1-2*MG/y))+(J^2/y^2)*(1-2*MG/y)^2);
0.1581028031e-2
> # km/s
> 0.2997924580e11*0.1581028031e-2/10^5;
473.9802796
> ###
> ### total proper velocity v^2=(dr/dtau)^2+r^2*(dPhi/dtau)^2
> sqrt((1/E)*(1-(E+J^2/r^2)*(1-2*MG/r))+J^2/r^2)
> sqrt((1/E)*(1-(E+J^2/x^2)*(1-2*MG/x))+J^2/x^2);
0.2582947199e-1
> # km/s
> 0.2997924580e11*0.2582947199e-1/10^5;
7743.480897
> sqrt((1/E)*(1-(E+J^2/y^2)*(1-2*MG/y))+J^2/y^2);
0.001581096486
> # km/s
> 0.2997924580e11*0.1581096486e-2/10^5;
474.0008019
> ###

># Total proper acceleration S2 star HP 13092020

> ### Total proper acceleration (- + sign d^2Phi/dta^2)(+ - sign dr/da)

> ### A^2=(d^2r/dta^2-r*(dPhi/dta)^2)^2+(r*d^2Phi/dta^2+2*(dr/dta)*(dPhi/dta))^2

> #sqrt(((J^2/(sqrt(E)*r^3))*(1-2*MG/r)-(MG/(sqrt(E)*r^2*(1-2*MG/r)))+(MG/(sqrt(E)*r^2))*(1-(E+J^2/r^2)*(1-2*MG/r))-r*(J/r^2)^2)^2+(r*((2*J/(r^3))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))*(J/r^2))^2)

> ### total proper acceleration at perihelion (+ sign selected for both)

> sqrt(((J^2/(sqrt(E)*x^3))*(1-2*MG/x)-(MG/(sqrt(E)*x^2*(1-2*MG/x)))+(MG/(sqrt(E)*x^2))*(1-(E+J^2/x^2)*(1-2*MG/x))-x*(J/x^2)^2)^2+(x*((2*J/(x^3))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))*(J/x^2))^2);
1.992765885*10^(-19)
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.1992765885e-18;
179.1008659
> ### total proper acceleration at aphelion (+ sign selected for both)

> sqrt(((J^2/(sqrt(E)*y^3))*(1-2*MG/y)-(MG/(sqrt(E)*y^2*(1-2*MG/y)))+(MG/(sqrt(E)*y^2))*(1-(E+J^2/y^2)*(1-2*MG/y))-y*(J/y^2)^2)^2+(y*((2*J/(y^3))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))*(J/y^2))^2);
7.450050462*10^(-22)
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.7450050462e-21;
0.6695771434
> ### total proper acceleration at event horizon z = MG(+ sign selected for both)

> # ONE NOTICE: pure imaginary number acceleration for example at 3*MG , this is due 3*MG is over definition area of the second analytic solution.
> # Different selections of +, - signs (- + sign d^2Phi/dta^2)(+ - sign dr/da) are not treated here , + sign selections is made for both cases.

> z := MG;
6.292090968*10^11
> sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-z*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);
3.494379988*10^(-8)
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.3494379988e-7;
3.140592111*10^13
> ###

[Hannu Poropudas]

CORRECTION (event horizon is was not MG ):
Interesting OPEN QUESTIONS would be if S2-star somehow happens to go to my second analytic solution orbit,
which definition area is 0 < r <= 1.259363679*10^12 cm.
At r = 2*MG = = 1.258418194*10^12 cm, event horizon of the SgrA* black hole seems to
have following (OPEN QUESTION OF INTERPRETATION of the total proper velocity and the total proper acceleration):

total coordinate velocity = 1.460945827*10^(-8) , (geometric units, Weinberg 1972),
total coordinate velocity = 437.9805405 cm/s, (c.g.s units, Weinberg 1972),
total proper velocity = 36.52364512 (geometric units, Weinberg 1972),
total proper velocity = 1.094951335*10^12, cm/s (c.g.s units, Weinberg 1972)
and
total proper acceleration = 8,927256908*10^17 cm/s^2 , (c.g.s units, Weinberg 1972)

I have used definitions of Weinberg S. 1972. Gravitation and Cosmology book
and Becker 1954 Introduction to Theoretical Mechanics book
when I calculated above total coordinate velocity, total proper velocity and total proper acceleration.
(Formulae of these GR calculations are in above posting of mine).

Best Regards,,
Hannu Poropudas
Finland

[Max Keon]

On Thursday, 10 September 2020 17:10:49 UTC+10, Hannu Poropudas wrote:
> Analytic GR solutions of S2-star orbits and precession 732” per revolution
>
> New initial data 2020 is used. My earlier initial data was 2016 in my
> postings in this sci.physics.relativity.

I'm assuming that you are describing the perihelion advance of a binary star pair, where one star mass is 14.754 times that of the other?

m/M*180 = 12.2 degrees per orbit cycle is much simpler.

-----

Max Keon

[Max Keon]

It should work on black holes as well.

[Hannu Poropudas]

No, Your calculation seems to be wrong in this case
(remark that 732''/revolution = 12.2'/revolution = 0.2033 degrees/revolution and revolution = your orbit cycle)
m_S2 = 10*M_Sun-15*M_Sun and M = M_SgrA*_BH = 4.261*10^6*M_Sun, M_Sun = 1.98847*10^33 g.

Hannu

[Hannu Poropudas]

Preliminary formulae and few important numerical points calculated below:

># S2-star’s two analytic solutions of orbits around SgrA* Black Hole
># Total coordinate velocity, total proper velocity and
># Total coordinate acceleration, total proper acceleration
># formulae and numerical calculation for some points.
># OPEN QUESTIONS OF PHYSICAL INTERPRETATIONS:
># second analytic solution cases r=2*MG and
># INSIDE Black Hole 0<r<= 2*MG
># This is copy part of my Maple 9 program
># where other symbols have their numerical values H.P. 15.9.2020

> ### total coordinate velocity v^2=(dr/dt)^2+r^2*(dPhi/dt)^2

> #sqrt((1-2*MG/r)^2*(1-(E+J^2/r^2)*(1-2*MG/r))+(J^2/r^2)*(1-2*MG/r)^2)
> # total coordinate velocity at perihelion
> sqrt((1-2*MG/x)^2*(1-(E+J^2/x^2)*(1-2*MG/x))+(J^2/x^2)*(1-2*MG/x)^2);

> # 0.2581119858e-1
> # km/s
> 0.2997924580e11*0.2581119858e-1/10^5;

> # 7738.002666
> # total coordinate velocity at aphelion
> sqrt((1-2*MG/y)^2*(1-(E+J^2/y^2)*(1-2*MG/y))+(J^2/y^2)*(1-2*MG/y)^2);

> # 0.1581028031e-2
> # km/s
> 0.2997924580e11*0.1581028031e-2/10^5;

> # 473.9802796

> ###
> ### total proper velocity v^2=(dr/dtau)^2+r^2*(dPhi/dtau)^2

> #sqrt((1/E)*(1-(E+J^2/r^2)*(1-2*MG/r))+J^2/r^2)
> # total proper velocity at perihelion
> sqrt((1/E)*(1-(E+J^2/x^2)*(1-2*MG/x))+J^2/x^2);

> # 0.2582947199e-1
> # km/s
> 0.2997924580e11*0.2582947199e-1/10^5;

> # 7743.480897
> # total proper velocity at aphelion
> sqrt((1/E)*(1-(E+J^2/y^2)*(1-2*MG/y))+J^2/y^2);

> # 0.1581096486e-2
> # km/s
> 0.2997924580e11*0.1581096486e-2/10^5;

> # 474.0008019

>###

> ### Total proper acceleration (- + sign d^2Phi/dta^2)(+ - sign dr/da)
> ### A^2=(d^2r/dta^2-r*(dPhi/dta)^2)^2+(r*d^2Phi/dta^2+2*(dr/dta)*(dPhi/dta))^2
> #sqrt(((J^2/(sqrt(E)*r^3))*(1-2*MG/r)-(MG/(sqrt(E)*r^2*(1-2*MG/r)))+(MG/(sqrt(E)*r^2))*(1-(E+J^2/r^2)*(1-2*MG/r))-r*(J/r^2)^2)^2+(r*((2*J/(r^3))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/r^2)*(1-2*MG/r)))*(J/r^2))^2)
>
> ### total proper acceleration at perihelion (+ sign selected for both)
> # total proper acceleration at perihelion

> sqrt(((J^2/(sqrt(E)*x^3))*(1-2*MG/x)-(MG/(sqrt(E)*x^2*(1-2*MG/x)))+(MG/(sqrt(E)*x^2))*(1-(E+J^2/x^2)*(1-2*MG/x))-x*(J/x^2)^2)^2+(x*((2*J/(x^3))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/x^2)*(1-2*MG/x)))*(J/x^2))^2);

> # 0.1992765885e-18
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.1992765885e-18;

> # 179.1008659

> ### total proper acceleration at aphelion (+ sign selected for both)
> # total proper acceleration at aphelion

> sqrt(((J^2/(sqrt(E)*y^3))*(1-2*MG/y)-(MG/(sqrt(E)*y^2*(1-2*MG/y)))+(MG/(sqrt(E)*y^2))*(1-(E+J^2/y^2)*(1-2*MG/y))-y*(J/y^2)^2)^2+(y*((2*J/(y^3))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/y^2)*(1-2*MG/y)))*(J/y^2))^2);

> # 0.7450050462e-21
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.7450050462e-21;

> # 0.6695771434
> ####
> ### Total coordinate acceleration (- + sign d^2Phi/dt^2)(+ - sign dr/dt)
> ### A^2=(d^2r/dt^2-r*(dPhi/dt)^2)^2+(r*d^2Phi/dt^2+2*(dr/dt)*(dPhi/dt))^2
> # First component of the sqrt formula
> #((J^2/r^3)*(1-2*MG/r)^2-MG/r^2+MG*( (1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3 )^2/(r^2*(1-2*MG/r))-r*((J/r^2)*(1-2*MG/r))^2)^2:
> # Second component of the sqrt formula
> #(r*((-2/(r*(1-2*MG/r)))*((J/r^2)*(1-2*MG/r))*((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3))-2*sqrt((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3)*((J/r^2)*(1-2*MG/r)))^2:
> #sqrt(((J^2/r^3)*(1-2*MG/r)^2-MG/r^2+MG*( (1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3 )^2/(r^2*(1-2*MG/r))-r*((J/r^2)*(1-2*MG/r))^2)^2+(r*((-2/(r*(1-2*MG/r)))*((J/r^2)*(1-2*MG/r))*((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3))-2*sqrt((1-2*MG/r)^2-(E+J^2/r^2)*(1-2*MG/r)^3)*((J/r^2)*(1-2*MG/r)))^2):

> # total coordinate acceleration at perihelion
> sqrt(((J^2/x^3)*(1-2*MG/x)^2-MG/x^2+MG*( (1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3 )^2/(x^2*(1-2*MG/x))-x*((J/x^2)*(1-2*MG/x))^2)^2+(x*((-2/(x*(1-2*MG/x)))*((J/x^2)*(1-2*MG/x))*((1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3))-2*sqrt((1-2*MG/x)^2-(E+J^2/x^2)*(1-2*MG/x)^3)*((J/x^2)*(1-2*MG/x)))^2);

> # 0.1988634824e-18
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.1988634824e-18;

> # 178.7295847
> # total coordinate acceleration at aphelion
> sqrt(((J^2/y^3)*(1-2*MG/y)^2-MG/y^2+MG*( (1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3 )^2/(y^2*(1-2*MG/y))-x*((J/y^2)*(1-2*MG/y))^2)^2+(y*((-2/(y*(1-2*MG/y)))*((J/y^2)*(1-2*MG/y))*((1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3))-2*sqrt((1-2*MG/y)^2-(E+J^2/y^2)*(1-2*MG/y)^3)*((J/y^2)*(1-2*MG/y)))^2);

> # 0.6642424264e-21
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.6642424264e-21;

> # 0.5969913206
> #####################################################
> ####
> K := 2*MG;

> # K := 0.1258418194e13
> # K = 2*MG = 0.1258418194e13 cm EVENT HORIZON of SgrA* Black Hole

> # total coordinate acceleration at event horizon K=2*MG(+ sign selected for both)
> sqrt(((J^2/K^3)*(1-2*MG/K)^2-MG/K^2+MG*( (1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3 )^2/(K^2*(1-2*MG/K))-K*((J/K^2)*(1-2*MG/K))^2)^2+(K*((-2/(K*(1-2*MG/K)))*((J/K^2)*(1-2*MG/K))*((1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3))-2*sqrt((1-2*MG/K)^2-(E+J^2/K^2)*(1-2*MG/K)^3)*((J/K^2)*(1-2*MG/K)))^2);

> # 0.3973241981e-12
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.3973241981e-12;

> # 357097180.7
> ### total proper acceleration at event horizon K = 2*MG(+ sign selected for both)
> # ONE NOTICE: pure imaginary number acceleration for example at 3*MG, this is due 3*MG is over definition area of the second analytic solution.
> # Different selections of +, - signs (- + sign d^2Phi/dta^2)(+ - sign dr/da) are not treated here, + sign selections is made for both cases.
> ####
> # K = 2*MG = 0.1258418194e13 EVENT HORIZON of SgrA* Black Hole
> K := 2*MG;

> # K := 0.1258418194e13
> ### total proper acceleration
> sqrt(((J^2/(sqrt(E)*K^3))*(1-2*MG/K)-(MG/(sqrt(E)*K^2*(1-2*MG/K)))+(MG/(sqrt(E)*K^2))*(1-(E+J^2/K^2)*(1-2*MG/K))-K*(J/K^2)^2)^2+(K*((2*J/(K^3))*sqrt(1-(E+J^2/K^2)*(1-2*MG/K)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/K^2)*(1-2*MG/K)))*(J/K^2))^2);

> # 0.9932912900e-3
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.9932912900e-3;

> # 0.8927256908e18
> ### total coordinate velocity
> sqrt((1-2*MG/K)^2*(1-(E+J^2/K^2)*(1-2*MG/K))+(J^2/K^2)*(1-2*MG/K)^2);

> # 0.1460945827e-7
> # cm/s
> 2.99792458*10^8*10^2*0.1460945827e-7;

> # 437.9805405
> ### total proper velocity
> sqrt((1/E)*(1-(E+J^2/K^2)*(1-2*MG/K))+J^2/K^2);

> # 36.52364512
> # cm/s
> 2.99792458*10^8*10^2*36.52364512;

> # 0.1094951335e13
> ####
> ##########################################################
> ###### Inside SgrA* Black Hole if S2 follows somehow the second analytic solution
> z := MG;

> # z := 0.6292090968e12
> ### total coordinate velocity
> sqrt((1-2*MG/z)^2*(1-(E+J^2/z^2)*(1-2*MG/z))+(J^2/z^2)*(1-2*MG/z)^2);

> # 103.2754259
> # cm/s
> 2.99792458*10^8*10^2*103.2754259;

> # 0.3096119378e13
> ### total proper velocity
> sqrt((1/E)*(1-(E+J^2/z^2)*(1-2*MG/z))+J^2/z^2);

> # 103.2743723
> # cm/s
> 2.99792458*10^8*10^2*103.2743723;

> # 0.3096087792e13
> ### total proper acceleration
> sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-z*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);

> # 0.3494379988e-7
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.3494379988e-7;

> # 0.3140592111e14
> ### total coordinate acceleration
> sqrt(((J^2/(sqrt(E)*z^3))*(1-2*MG/z)-(MG/(sqrt(E)*z^2*(1-2*MG/z)))+(MG/(sqrt(E)*z^2))*(1-(E+J^2/z^2)*(1-2*MG/z))-K*(J/z^2)^2)^2+(z*((2*J/(z^3))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))+2*((1/sqrt(E))*sqrt(1-(E+J^2/z^2)*(1-2*MG/z)))*(J/z^2))^2);

> # 0.3790004584e-7
> # cm/s^2
> (2.99792458*10^8*10^2)^2*0.3790004584e-7;

> # 0.3406286247e14

[Wilburn Harricharan]

Hannu Poropudas <hanp...@luukku.com> wrote:

>> # 0.3406286247e14
>
> Best Regards, Hannu Poropudas Finland

any particular reason impinging you spamming 601 lines of crap?

[Hannu Poropudas]

Oh no, please no your above unbusiness like mental comments, please use only mathematics ?
Question was OPEN QUESTIONS OF PHYSICAL INTERPRETATIONS these huge numerical results INSIDE
SgrA* Black Hole's EVENT HORIZON of my second analytic General Relativity solution.

># S2-star’s two analytic solutions of orbits around SgrA* Black Hole.

># Total coordinate velocity, total proper velocity and
># Total coordinate acceleration, total proper acceleration
># formulae and numerical calculation for some points.
># OPEN QUESTIONS OF PHYSICAL INTERPRETATIONS:
># second analytic solution cases r=2*MG and
># INSIDE Black Hole 0<r<= 2*MG
># This is copy part of my Maple 9 program
># where other symbols have their numerical values H.P. 15.9.2020

P.S Maple 9 command line mark is > and Maple 9 comment line mark is >#
These comment line marks was used due copy from Maple 9 program to Word text file
does not copy past here those Maple 9 calculation results, so I gave them in comment lines also
which are visible here correctly.

Best Regards...
Hannu Poropudas
Finland

[Demetrius Vanderpool]

Hannu Poropudas wrote:

>> > Best Regards, Hannu Poropudas Finland
>> any particular reason impinging you spamming 601 lines of crap?
>
> Oh no, please no your above unbusiness like mental comments, please use
> only mathematics ? Question was OPEN QUESTIONS OF PHYSICAL
> INTERPRETATIONS these huge numerical results INSIDE SgrA* Black Hole's
> EVENT HORIZON of my second analytic General Relativity solution.

Why don't you say so, and save yourself, and us, for those 601 lines of
crap. See it now, or I need to draw a diagram?

[Ross A. Finlayson]

Please ignore such.

What gets me about black holes is that it's basically defined by a radius,
but that makes a wall. (I.e., on the one side it's expected to be cubic,
on the other side linear, but they meet.)

Hannu Poropudas is an interesting person and his posts are interesting,
and detailed. Such nym-shifting trolls as infest usenet and forget there's
a charter they can go bye-bye.

The gravitic (or, gravific?) singularity, or, regime as it were, more
or less the Chandrasekhar regime, or Schwarzschild-Chandrasekhar,
has a lot going on.

Some article: https://www.scientificamerican.com/article/the-reluctant-father-of-black-holes-2007-04/

[Ross A. Finlayson]

See also: https://royalsocietypublishing.org/doi/10.1098/rspa.1975.0112

[Ross A. Finlayson]

Re: Regge-Wheeler, Zerilli's potential, Price's potential,

http://www.phys.lsu.edu/faculty/pullin/talks/fenomec2.pdf

https://books.google.com/books?id=n0eFaWlGK_YC&lpg=PA23&ots=CaD0e9SNMx&dq=Zerilli%20potential&pg=PA25#v=onepage&q=Zerilli%20potential&f=false

[Hannu Poropudas]

Article in your above mentioned link is very well written article about history matters about black holes.

I took a quick look at S. Chandrasekhar 1983 book :The Mathematical Theory of Black Holes.
Your reference which you mentioned in the link of your posting was more recent
1991. Selected Papers, S. Chandrasekhar.
Volume 6. The Mathematical Theory of Black Holes and of Colliding Plane Waves.

In 1983 book there are chapters about Schwarzschild black hole and many analytic solutions
of General Relativity orbit equations and numerous drawings of orbits which drawings seems to contain
among them similar to these two kind of orbits which I had described in my articles 2016 and above 2020
in sci.physics.relativity (in Google User Group). I my opinion this book has not so easy mathematics to read compared to
S. Weinberg 1972 book which I have used. One reason was unit system c=1, G=1 which I don't like to use and other reason
mathematics of differential forms which I don't like to use myself
due I prefer clear old fashioned nonlinear partial differential equations system.

I just wonder why Chandrasekhar gave also post-Newtonian approximation (not General Relativity analytically)
formula of particle's perihelion precession rate ?

[Ross A. Finlayson]

They're very deep and I don't much know them, then that it
seems the point that Chandrasekhar works up solutions from
"inside the black hole" which these days is addressed in GTR
along the lines of "multipole moment", i.e., outside the theory.

One issue with having normalized the units is whether they're
the ceiling, i.e. that the value only ranges [0,1], that the projections
are still carried out about 1/n, that it's the same as writing "equals infinity".

(The NIST CODATA "running constants" that yet maintain ratio,
but with higher energy experiment become smaller and not just
more refined, reflect relevant high and resp. low energy physics
in "running constants" about NIST CODATA, "the" standard units.)

Please carry on, I hope it helps your research in physics.

[Hannu Poropudas]

Preliminary calculation PROPER TIME and COORDINATE TIME analytic GR solutions.
(This is due it seems to be useful for better understanding to compare these plots
to OJ287 binary black hole system analytic GR solutions plots (Maple 9 program used here also)
which I gave in sci.astro user group in this summer 2020)

> # S2-star around SgrA* Black Hole COORDINATE TIME t analytic solution HP 29092020
> # PROPER TIME ta analytic solution is also here
>Restart;
>with(plots):with(plottools):
># following constants are taken from my latest previous postings
># in this sci.physics.relativity user group

>MG := 0.6292090968e12;
>E := 1.000040803;
>J := 0.4594478956e14;

> # + and - signs ta=
>Int(1/sqrt((1/E-1)+2*MG/r-(J^2/E)/r^2+(2*MG*J^2/E)/r^3),r);
> # u=1/r substitution du=-dr/r^2, dr=-r^2*du=-du/u^2
> # + and - signs ta=
>-Int(1/(u^2*sqrt((1/E-1)+2*MG*u-(J^2/E)*u^2+(2*MG*J^2/E)*u^3)),u);
>solve((1-E)/(2*MG*J^2)+(E/J^2)*u-(1/(2*MG))*u^2+u^3=0,u);
>a1 := 0.7940518034e-12;
>a2 := 0.5621850829e-15;
>a3 := 0.3440855378e-16;

># PROPER TIME ta and ttaa
>#First Proper time analytic solution and its definition area
># u => a1, r = 1/u, r <= 1/a1
># 0 <= P <= Pi/2
>u := P-> (-0.5621850829e-15*sin(P)^2+0.7940518034e-12)/( -sin(P)^2+1);
># +, - signs for ta
>ta := P-> -0.1271074840e-26*((-0.1000000000e14+6646913941.*sin(P)^2)*(sin(P)^2-1.))^(1/2)*(0.3049313107e54*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticF(sin(P), 0.2578160961e-1)*sin(P)^2-0.4306966950e57*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticF(sin(P), 0.2578160961e-1)-0.5279794198e55*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticPi(sin(P), 0.7079954740e-3, 0.2578160961e-1)*sin(P)^2+0.7457384110e58*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticPi(sin(P), 0.7079954740e-3, 0.2578160961e-1)+0.4978387869e62*sin(P)+0.3309091573e59*sin(P)^5-0.4981696960e62*sin(P)^3+0.4978387869e55*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticE(sin(P), 0.2578160961e-1)*sin(P)^2-0.7031666235e58*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticE(sin(P), 0.2578160961e-1))/((0.1000000000e14+6646913941.*sin(P)^4-0.1000664691e14*sin(P)^2)^(1/2)*(5621850829.*sin(P)^2-0.7940518034e13)*cos(P)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2));

># Second proper time solution and its definition area
>### a3 <= uu <= a2, rr = 1/uu, 1/a2 <= rr <= 1/a3
># 0 <= P <= Pi/2
>uu := P-> (0.5277765291e-15*sin(P)^2+0.3440855378e-16);
># +, - signs for ttaa
>ttaa := P-> -0.2327722886e-11*((-0.1000000000e11+9993353087.*sin(P)^2)*(sin(P)^2-1.))^(1/2)*(0.1198054508e46*sin(P)+0.1197258172e46*sin(P)^5-0.2395312679e46*sin(P)^3-0.1276110124e41*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticF(sin(P), .9996675991)*sin(P)^2-0.8319639358e39*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticF(sin(P), .9996675991)+0.1198054508e41*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticE(sin(P), .9996675991)*sin(P)^2+0.7810753358e39*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticE(sin(P), .9996675991)+0.2340118355e42*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticPi(sin(P), -15.33852694, .9996675991)*sin(P)^2+0.1525647387e41*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticPi(sin(P), -15.33852694, .9996675991))/((0.2638882646e11*sin(P)^2+1720427689.)*(0.1000000000e11+9993353087.*sin(P)^4-0.1999335309e11*sin(P)^2)^(1/2)*cos(P)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2));

># COORDINATE TIME t and tt (calculated here only with ta, check definition areas and also later ttaa case)
> # ta(P), u(P), 0<=P<=Pi/2
> # dt^2=dta^2/(E*(1-2*MG/r)^2)=dta^2/(E*(1-2*MG*u)^2)
> # dt=dta/sqrt(E*(1-2*MG*u)^2)
> # dta=(ta)´*dP, ta´= dta/dP
> # dt= dP*(dta/dP)/(sqrt(E)*(1-2*MG*u))
># First coordinate time solution and its definition area
># u => a1, r = 1/u, r <= 1/a1
># 0<= P <= Pi/2
>u:=P->( -0.5621850829e-15*sin(P)^2+0.7940518034e-12)/(-sin(P)^2+1);
># +, - signs for t
> t := P-> -0.5421461469e-82*((-0.1000000000e14+6646913941.*sin(P)^2)*(sin(P)^2-1.))^(1/2)*(0.7159158346e109*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticF(sin(P), 0.2578160961e-1)*sin(P)^2-0.1011187021e113*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticF(sin(P), 0.2578160961e-1)+0.1167171344e118*sin(P)+0.7758087478e114*sin(P)^5-0.1167947153e118*sin(P)^3+0.1167171344e111*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticE(sin(P), 0.2578160961e-1)*sin(P)^2-0.1648557635e114*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticE(sin(P), 0.2578160961e-1)-0.1237936361e111*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticPi(sin(P), 0.7079954740e-3, 0.2578160961e-1)*sin(P)^2+0.1748508862e114*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticPi(sin(P), 0.7079954740e-3, 0.2578160961e-1)+0.9518083678e106*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticPi(sin(P), 1331.034836, 0.2578160961e-1)*sin(P)^2-0.1344370696e110*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2)*EllipticPi(sin(P), 1331.034836, 0.2578160961e-1))/((0.1000000000e14+6646913941.*sin(P)^4-0.1000664691e14*sin(P)^2)^(1/2)*(5621850829.*sin(P)^2-0.7940518034e13)*cos(P)*(0.1000000000e15-0.6646913941e11*sin(P)^2)^(1/2));
># Second coordinate time solution and its definition area
>### a3 <= uu <= a2, rr = 1/uu, 1/a2 <= rr <= 1/a3
># 0<= P <= Pi/2
>uu := P-> (0.5277765291e-15*sin(P)^2+0.3440855378e-16);
># +, - signs for tt
>tt := P-> -0.8557384250e-56*((-0.1000000000e11+9993353087.*sin(P)^2)*(sin(P)^2-1.))^(1/2)*(0.2126738302e78*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticPi(sin(P), 0.6641923462e-3, .9996675991)*sin(P)^2+0.1386533603e77*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticPi(sin(P), 0.6641923462e-3, .9996675991)+0.6365795964e86*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticPi(sin(P), -15.33852694, .9996675991)*sin(P)^2+0.4150200335e85*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticPi(sin(P), -15.33852694, .9996675991)+0.3258801900e90*sin(P)+0.3256635802e90*sin(P)^5-0.6515437702e90*sin(P)^3-0.3471119276e85*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticF(sin(P), .9996675991)*sin(P)^2-0.2263006930e84*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticF(sin(P), .9996675991)+0.3258801900e85*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticE(sin(P), .9996675991)*sin(P)^2+0.2124585961e84*(-1.*sin(P)^2+1.)^(1/2)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2)*EllipticE(sin(P), .9996675991))/((0.1000000000e11+9993353087.*sin(P)^4-0.1999335309e11*sin(P)^2)^(1/2)*(0.2638882646e11*sin(P)^2+1720427689.)*cos(P)*(0.1000000000e11-9993353087.*sin(P)^2)^(1/2));

>#a1 := 0.7940518034e-12;
>#a2 := 0.5621850829e-15;
>#a3 := 0.3440855378e-16;
># cm unit for following
># 1/a1 := 0.1259363678e13;
># 1/a2 := 0.1778773629e16;
># 1/a3 := 0.2906254086e17;

># PLOTS PROPER TIME (only definition areas)
>#First Proper time analytic solution and its definition area
># u => a1, r = 1/u, r <= 1/a1
># 0 <= P <= Pi/2
>plot([ta(P),1/u(P),P=0..Pi/2]);
>plot([-ta(P),1/u(P),P=0..Pi/2]);

># Second proper time solution and its definition area
>### a3 <= uu <= a2, rr = 1/uu, 1/a2 <= rr <= 1/a3
># 0 <= P <= Pi/2
>plot([ttaa(P),1/uu(P),P=0..Pi/2]);
>plot([-ttaa(P),1/uu(P),P=0..Pi/2]);

># PLOTS COORDINATE TIME (only definition areas)
># First coordinate time solution and its definition area
># u => a1, r = 1/u, r <= 1/a1
># 0<= P <= Pi/2
>plot([t(P),1/u(P),P=0..Pi/2]);
>plot([-t(P),1/u(P),P=0..Pi/2]);

># Second coordinate time solution and its definition area
>### a3 <= uu <= a2, rr = 1/uu, 1/a2 <= rr <= 1/a3
># 0<= P <= Pi/2
>plot([tt(P),1/uu(P),P=0..Pi/2]);
>plot([-tt(P),1/uu(P),P=0..Pi/2]);

Best Regards...
Hannu Poropudas..
Finland...

[Hannu Poropudas]

I notice ERROR in my total coordinate acceleration formulae, sorry for that
corrected formulae are below and ALSO PLOTTING commands
for plotting total coordinate velocity, total proper velocity, total coordinate acceleration and
total proper acceleration for both analytic orbits of S2-star around SgrA* black hole and
also for orbit which leads inside event horizon of SgrA* black hole if
S2-star somehow would go into that orbit. Maple 9 program is used here.

Best Regards,,,
Hannu Poropudas
Finland,,,

----BELOW CORRECTED----

> # CORRECTED total coordinate acceleration formulae HP 10102020
> # Plots of total coordinate velocity and total proper velocity and
> # total coordinate acceleration and total proper acceleration of
> # S2-star two analytic solutions of orbits around SgrA* black hole HP 16092020
> restart;
> with(plots):with(plottools):
> AU := 149597870700*10^2;
> ap:= 1030.799324*AU;
># ap = 0.1542053840e17
> em := 0.884649;
> MG := 0.6292090968e12;
> x := ap*(1-em);
># x = 0.1778774525e16
> y := ap*(1+em);
># y = 0.2906230228e17
> E := (2*x*y*MG+2*y^2*MG+2*x^2*MG-x^2*y-x*y^2)/((-y+2*MG)*(-2*x*MG-2*y*MG+x^2+x*y));
># E = 1.000040803
> J := evalf(-2^(1/2)*(-(-y+2*MG)*MG*(-2*x*MG-2*y*MG+x^2+x*y))^(1/2)*y*x/(-x^2*y+2*x^2*MG-4*x*MG^2+4*x*y*MG-x*y^2+2*y^2*MG-4*y*MG^2));
># J = 0.4594478956e14

> a1 := 0.2906262559e17;
> a2 := 0.1778773310e16;
> a3 := 0.1259363679e13;
> a4 := 0;
> # Primitive function, definition areas 0<=P<=Pi/2 and a2 <= r <= a1.
> e := P -> 2.000794538*EllipticF(sin(P),0.2578161452e-1);
> r := P -> (0.3436029258e29*sin(P)^2-0.5169358260e32)/(0.2728385228e17*sin(P)^2-0.2906136623e17);

> # Second Primitive function, definition areas 0<=P<=Pi/2 and a4=0 <= r <= a3.
> ee := P-> 2.000794538*EllipticF(sin(P),0.9996675989);
> rr := P-> (0.3660041508e29*sin(P)^2)/(0.1259363679e13*sin(P)^2+0.2906136623e17);

> ### total coordinate velocity v^2=(dr/dt)^2+r^2*(dPhi/dt)^2

> vC := P ->sqrt((1-2*MG/r(P))^2*(1-(E+J^2/r(P)^2)*(1-2*MG/r(P)))+(J^2/r(P)^2)*(1-2*MG/r(P))^2);
> vvC := P ->sqrt((1-2*MG/rr(P))^2*(1-(E+J^2/rr(P)^2)*(1-2*MG/rr(P)))+(J^2/rr(P)^2)*(1-2*MG/rr(P))^2);

> ### total proper velocity v^2=(dr/dtau)^2+r^2*(dPhi/dtau)^2

> vP := P ->sqrt((1/E)*(1-(E+J^2/r(P)^2)*(1-2*MG/r(P)))+J^2/r(P)^2);
> vvP := P ->sqrt((1/E)*(1-(E+J^2/rr(P)^2)*(1-2*MG/rr(P)))+J^2/rr(P)^2);

> ### Total coordinate acceleration (- + sign d^2Phi/dt^2)(+ - sign dr/dt)
> ### A^2=(d^2r/dt^2-r*(dPhi/dt)^2)^2+(r*d^2Phi/dt^2+2*(dr/dt)*(dPhi/dt))^2

> aC := P ->sqrt((J^2*(1-2*MG/r(P))^2/r(P)^3-MG/r(P)^2+MG*(1-2*MG/r(P))*(1-(E+J^2/r(P)^2)*(1-2*MG/r(P)))/r(P)^2-J^2*(1-2*MG/r(P))^2/r(P)^3)^2+(2*J*(1-2*MG/r(P))*sqrt(1-(E+J^2/r(P)^2)*(1-2*MG/r(P)))/r(P)^2+2*J*(1-2*MG/r(P))^2*sqrt(1-(E+J^2/r(P)^2)*(1-2*MG/r(P)))/r(P)^2)^2);
> aaC := P ->sqrt((J^2*(1-2*MG/rr(P))^2/rr(P)^3-MG/rr(P)^2+MG*(1-2*MG/rr(P))*(1-(E+J^2/rr(P)^2)*(1-2*MG/rr(P)))/rr(P)^2-J^2*(1-2*MG/rr(P))^2/rr(P)^3)^2+(2*J*(1-2*MG/rr(P))*sqrt(1-(E+J^2/rr(P)^2)*(1-2*MG/rr(P)))/rr(P)^2+2*J*(1-2*MG/rr(P))^2*sqrt(1-(E+J^2/rr(P)^2)*(1-2*MG/rr(P)))/rr(P)^2)^2);

> ### Total proper acceleration (- + sign d^2Phi/dta^2)(+ - sign dr/da)
> ### A^2=(d^2r/dta^2-r*(dPhi/dta)^2)^2+(r*d^2Phi/dta^2+2*(dr/dta)*(dPhi/dta))^2

> aP := P ->sqrt(((J^2/(sqrt(E)*r(P)^3))*(1-2*MG/r(P))-(MG/(sqrt(E)*r(P)^2*(1-2*MG/r(P))))+(MG/(sqrt(E)*r(P)^2))*(1-(E+J^2/r(P)^2)*(1-2*MG/r(P)))-r(P)*(J/r(P)^2)^2)^2+(r(P)*((2*J/(r(P)^3))*sqrt(1-(E+J^2/r(P)^2)*(1-2*MG/r(P))))+2*((1/sqrt(E))*sqrt(1-(E+J^2/r(P)^2)*(1-2*MG/r(P))))*(J/r(P)^2))^2);
> aaP := P ->sqrt(((J^2/(sqrt(E)*rr(P)^3))*(1-2*MG/rr(P))-(MG/(sqrt(E)*rr(P)^2*(1-2*MG/rr(P))))+(MG/(sqrt(E)*rr(P)^2))*(1-(E+J^2/rr(P)^2)*(1-2*MG/rr(P)))-rr(P)*(J/rr(P)^2)^2)^2+(rr(P)*((2*J/(rr(P)^3))*sqrt(1-(E+J^2/rr(P)^2)*(1-2*MG/rr(P))))+2*((1/sqrt(E))*sqrt(1-(E+J^2/rr(P)^2)*(1-2*MG/rr(P))))*(J/rr(P)^2))^2);
> # -Pi/2..Pi/2 plots gives both sides of both solutions.
> # This is due sin(-P)=-sin(P) and
> # EllipticF(sin(-P),k)=EllipticF(-sin(P),k)=-EllipticF(sin(P),k)

>
> # Plottings only on definition area a2..a1.

> # Plottings only on definition area a4..a3.

> plot(vC(P),P=-Pi/2..Pi/2);
> plot(vP(P),P=-Pi/2..Pi/2);
> plot(aC(P),P=-Pi/2..Pi/2);
> plot(aP(P),P=-Pi/2..Pi/2);
> plot(log10(vvC(P)),P=-Pi/2..Pi/2);
> plot(log10(vvP(P)),P=-Pi/2..Pi/2);
> plot(log10(aaC(P)),P=-Pi/2..Pi/2);
> plot(log10(aaP(P)),P=-Pi/2..Pi/2);

>

[Hannu Poropudas]

I notice that all these velocity and acceleration formulae need to be checked again
due I found one imaginary number problem if I applied some them
in case of OJ287 binary black hole system.

Hannu

[Hannu Poropudas]

It seems that velocity- and acceleration formulae gives reasonable well
behaved plottings, so they seems to be OK at my first look.

But OJ287 proper time solution and coordinate time
analytic solutions have both exponent error in MG which was copy pasted carelessly and
mass was 10 times too small for supermassive black hole in that binary black hole
system but total energy E per mas unit and total angular momentum J per mass unit
had values correct values for OJ287 system, so E and J had too large value compared
with correct MG value. This gave strange oscillations of proper time and coordinate time
solutions inside the gigantic black hole

These solutions was in my second posting chain in sci.astro posted in this summer 2020.
I'am sorry those wrong proper time and coordinate time solutions, please ignore those.

I recalculated 12.10.2020 those solutions and they does not seem to have any gravitational
wave in time dimension and their plotting does not show much more interesting things than
those valid two analytic GR solutions of orbits of smaller black hole around gigantic black hole
and also orbit which leads into the singularity at origin. These old valid solutions are in my
second posting chain in sci.astro in this summer 2020.

I don't post these recalculated analytic GR solutions any more due they have long
and complicated mathematical forms which contains also elliptic integrals (1., 2. and 3 kind).

Best Regards,.
Hannu Poropudas,.
Finland

[Hannu Poropudas]

***************************************
I calculated derivatives of these analytic proper time and coordinate time solutions.
These formulae are too long and complicated to be copied here but I try to explain
some of their properties from their plots (c.g.s and c=1 units, I use not rounded
numbers and estimated numbers are from plots).

First of all I have made some notations in the Maple 9 program which I give here.

Parametric form function names are following (+,- sign means that solutions are same
in both sides of origin of the horizontal axis):

+,- sign for solution ta(P) , r =1/u(P),
+,- sign for solution ttaa(P), rr = 1/uu(P),
+,- sign for solution t(P), r = 1/u(P),
+,- sign for solution tt(P), rr =1/uu(P).

All functions ta(P), ttaa(P), t(P), tt(P) has only one real singularity at P = Pi/2.
This singularity is at perihelion, so perihelion velocities
v_prop and v_coord are NOT computable.

Both functions 1/u(P) and 1/uu(P) have no real singularity (no real zero at denominator).

Derivatives notations :

DrDta = (dr(P)dP)/(dta(P)/dP) = dr/dta,
DrrDttaa = (drr(P)dP)/(dttaa(P)/dP) = drr/dttaa,
DrDt = (dr(P)dP)/(dt(P)/dP) = dr/dt,
DrrDtt= (drr(P)dP)/(dtt(P)/dP) = drr/dtt,

PLOTS:

plot(DrrDttaa(P), P=0..Pi/2),
max v_prop = approx. 0.00928 at approx. P = 0.64 rad,
which is v_prop = approx. 2782.07401 km/s, c=2.99792458e8*m/s.

plot([ttaa(P),DrrDttaa(P),P=0..Pi/2]),
max v_prop = approx. 0.009333333 at t = approx. 7.44e18,
which are v_prop = approx. 2798.062941 km/s, at t = approx. 7.864254148 years,
c = 2.99792458e10*cm/s.
plot([ttaa(P),log10(DrrDttaa(P)),P=0..Pi/2]), some extremal points
approx.10^(-4.6) = 0.000025118 , which is 7.530446075 km/s,
approx. 10^(-2.866666667) = 0.001359356, which is 407.5247906 km/s,
approx. 10^(-2.11) = 0.007762471, which is 2327.130311 km/s,
approx. 10^(-6.2) = 0.00000063, which is 0.189156253 km/s.


plot(DrrDtt(P), P=0..Pi/2),
max v_coord = approx. 0.00936 at approx. P = 0.62 rad,
which is v_coord = approx. 2806.057407 km/s, c=2.99792458e8*m/s.


plot([tt(P),DrrDtt(P),P=0..Pi/2]),
max v_coord = approx. 0.0094 at t = approx. 7.48e18,
which are v_coord = approx. 2818.049105 km/s, at t = approx. 7.906535084 years, c=2.99792458e8*m/s.
plot([tt(P),log10(DrrDtt(P)),P=0..Pi/2]), some extremal points
approx. 10^(-4.6) = 0.000025118, which is 7.530446075 km/s,
approx. 10^(-2.866666667) = 0.001359356, which is 407.5247906 km/s,
approx. 10^(-2.09) = 0.008128305, which is 2436.804584 km/s,
approx. 10^(-6.19) = 0.000000645, which is 0.193562268 km/s.

plot(DrDta(P), P=0..Pi/2),
max v_prop = 8.6e8, at singularity P = Pi/2,
which is v_prop = approx. 8.6e8*c, at singularity P = Pi/2.

plot([-tt(P),1/uu(P),P=0..Pi/2]), (plot comes in the positive side of the horizontal axis),
singularity at P = Pi/2,
aphelion approx. 2.9e16 cm,
perihelion approx. 1.8e15 cm.

plot([-t(P),1/u(P),P=0..Pi/2]), (plot comes in the positive side of the horizontal axis),
Schwarzschild horizon = 2*MG = approx. 1.26e12 cm,
zero point approx. t = 2e11, which is approx. t = 6.671281904 s, c = 2.99792458e10*cm/s.

plot([-ttaa(P),1/uu(P),P=0..Pi/2]), (plot comes in the positive side of the horizontal axis),
singularity at P = Pi/2,
aphelion approx. 2.9e16 cm,
perihelion approx. 2e15 cm.

plot([-ta(P),1/u(P),P=0..Pi/2]), (plot comes in the positive side of the horizontal axis),
Schwarzschild horizon = 2*MG = approx. 1.26e12 cm,
zero point approx. t = 4.3e10, which is approx. t = 1.434325609 s, c = 2.99792458e10*cm/s.



Reference:

Poropudas, Hannu, 2020.
Re: Analytic GR solutions of S2-star orbits and precession 732’’ per revolution.
sci.physics.relativity, Google Groups, 30.9.2020, local time 11:33 AM,
3 pages.


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

[Richard Hertz]

I've read your string of posts with a big sense of sorrow and pity.

A grown man wasting his time by making little programs based on equations written by one or two delusional authors 40 years ago,
to calculate unobservable phenomena, based on RIDICULOUS linearized math of 100 years old equations.

Shame on you!

Better use your time and stamina to calculate WHERE WILL BE the Moon by Sep. 26 2023, at 03:00 UTC time (visibility, phase,
radial distance to Greenwich, ecliptic coordinates, etc.). Then calculate again for Sep. 26 2073.

The above will be more useful and observable. You can take as reference the selenographic coordinates for the Apollo XI landing site,
to verify any precession that you calculate.

This would be your gift to future generations, proving Einstein right 50 years in the future.

[Hannu Poropudas]

All functions ta(P), ttaa(P), t(P), tt(P) has imaginary zero -0.I at P = Pi/2 at nominator,
when calculating with DIGITS=10 accuracy.
Maple 9 complex zero with floating-point real and imaginary components can have four sign combinations
(0.+0.I, 0.-0.I, -0.-0.I, -0.+0.I). Similarly to 0.0 =-0.0, numerically these four complex zeros are numerically
equivalent.

Hannu

[Hannu Poropudas]

OPEN QUESTION: How to interpret following?
This "-0.I/0.0 " singularity situation at P = Pi/2, (nominator/denominator at P=Pi/2), causes
"strange sawtooth like oscillation" in coordinate time dimension and in proper time dimension
in second solution, if S2-star happens somehow follow this second solution, which leads
inside event horizon of SgrA* black hole ?

[Hannu Poropudas]

CORRECTION: nominator -> numerator

> > Maple 9 complex zero with floating-point real and imaginary components can have four sign combinations
> > (0.+0.I, 0.-0.I, -0.-0.I, -0.+0.I). Similarly to 0.0 =-0.0, numerically these four complex zeros are numerically
> > equivalent.
> >
> > Hannu
> OPEN QUESTION: How to interpret following?
> This "-0.I/0.0 " singularity situation at P = Pi/2, (nominator/denominator at P=Pi/2), causes
> "strange sawtooth like oscillation" in coordinate time dimension and in proper time dimension
> in second solution, if S2-star happens somehow follow this second solution, which leads
> inside event horizon of SgrA* black hole ?
> Hannu

CORRECTION: nominator -> numerator

Digits = 20 calculation setting in Maple 9 gives no "strange sawtooth like oscillation"

in coordinate time dimension and in proper time dimension in second solution,
if S2-star happens somehow follow this second solution, which leads

inside event horizon of SgrA* black hole ? Maybe not exactly the same numerical
situation "-0.I/0.0 " singularity situation at P = Pi/2, (numerator/denominator at P=Pi/2)?
I have not checked this but I suspect that -0.I is not exactly present in this case, so Maple9
gives really singularity = infinity in this case ?

plot(DrDt(P), P=0..Pi/2) gives velocity oscillation near P = Pi/2, when set Digits=20.

[Hannu Poropudas]

I calculated S2-star GR (angular velocities) tangential velocities ONLY with Maple 9 program
and I put copy of these Maple 9 plotting commands below:

(Definitions of all these velocities are General Relativistic, from Weinberg S 1972 Gravity, Cosmology and
General Relativity book as all other GR definitions are also from this book
(sorry if I did not remember book name correctly here due I have not it at the moment
in my hands. Accurate name of the book can be found in my previous postings).)

> # Investigation of angular velocity part (tangential) only H.P. 24.10.2022
> # DIGITS=10
> # Two angular velocities (OK outside event horizon, two cases,
> # over light speed inside event horizon, two cases)
> # Two cases inside event horizon (over light speed)

> vProp := P->J/(sqrt(E)*(1/u(P)));

> vCoor := P-> J*(1-2*MG/(1/u(P)))/(1/u(P));

> # +,- sign ta(P) solution
> plot([-ta(P),vProp(P),P=0..Pi/2]);

> # +,- sign ta(P) solution
> plot([log10(-ta(P)),log10(vProp(P)),P=0..Pi/2]);

> # +,- sign t(P) solution. Selection is + sign for vCoor(P) due + sign of vProp(P)
> plot([-t(P),vCoor(P),P=0..Pi/2]);

> # +,- sign t(P) solution. Selection is + sign for vCoor(P) due + sign of vProp(P)
> plot([-t(P),-vCoor(P),P=0..Pi/2]);

> # +,- sign t(P) solution. Selection is + sign for vCoor(P) due + sign of vProp(P)
> plot([log10(-t(P)),log10(-vCoor(P)),P=0..Pi/2]);

> # Two OK cases outside event horizon

> vPropII := P->J/(sqrt(E)*(1/uu(P)));

> vCoorII := P-> J*(1-2*MG/(1/uu(P)))/(1/uu(P));

> # +,- sign for ttaa(P) solution
> plot([-ttaa(P),vPropII(P),P=0..Pi/2]);

> # +,- sign for ttaa(P) solution
> plot([log10(-ttaa(P)),log10(vPropII(P)),P=0..Pi/2]);

> # +,- sign for tt(P) solution
> plot([-tt(P),vCoorII(P),P=0..Pi/2]);

> # +,- sign for tt(P) solution
> plot([log10(-tt(P)),log10(vCoorII(P)),P=0..Pi/2]);

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

[Richard Hachel]

Le 25/10/2022 à 13:04, Hannu Poropudas a écrit :

Oh la la...

Tu parles beaucoup, toi.

R.H.

[Hannu Poropudas]

NOT OK comment

> > # +,- sign t(P) solution. Selection is + sign for vCoor(P) due + sign of vProp(P)
> > plot([-t(P),-vCoor(P),P=0..Pi/2]);
>

NOT OK comment

> > # +,- sign t(P) solution. Selection is + sign for vCoor(P) due + sign of vProp(P)
> > plot([log10(-t(P)),log10(-vCoor(P)),P=0..Pi/2]);
>

NOT OK comment and this has log10(-vCoor(P)) = -log10(0) = infinity
at event horizon, r = 2*MG. This is NOT physical singularity it is
only due log10 function singularity.

It is better to see the real situation by using abs function to get
everything visible at positive side and the use log10 function
to see details.

CORRECTION:
> # +,- sing t(P) solution. Absolute value of vCoor(P) and log10(abs(vCoor(P))).
> plot([log10(-t(P)),log10(abs(vCoor(P))), P=0..Pi/2]);

Now all negative parts are visible with abs function.

"Strange sawtooth" like oscillation partly now partly visible (DIGITS=10 case)
due "0.I/0.0" singularity of ta(P) function at P = Pi/2 , which is partly also in t(P) function?
"Strange sawtooth" oscillation is more clearly visible in the plot(DrDt(P),P=0..Pi/2),
my shorthand marking DrDt = dr/dt near P = Pi/2.

I have not investigated more closely that is this really due
"0.I/0.0" singularity of t(P), but I guessed it is something like this anyway.

Interpretation of this complicated plot is OPEN QUESTION that
what really happens inside event horizon r=2*MG if S2 star happens somehow to move
along this solution orbit.

Hannu

[Hannu Poropudas]

I investigated this coordinate time velocity tangential part and radial part and
coordinate time GR solution more closely:

S2-star orbit which leads inside envent horizon of SgrA* black hole:

># +,- sign for t(P) solution, primitive function
>plot([log10(-t(P)),log10(1/u(P)),P=0..Pi/2 ]);
Vertical axes extremal points are about 12.1 and about 6.2 .
Horizontal axes extremal points are about 11.008, 11.312, 11.82, 12.8 .

S2-star radial part ONLY (coordinate time velocity):

># +,- sign for t(P) solution, primitive function, DrDt=dr/dt, all velocities are negative
>plot([ log10(-t(P)),log10(abs(DrDt(P))),P=0..Pi/2]);
Vertical axes extremal points are about 4.8, 3.6, 1.6, 0.4, -2.2, -4.6 .
Horizontal axes extremal points are about 11, 11.312, 11.46, 12.8 .

S2-star tangential part ONLY (coordinate time velocity):

># +,- sign for t(P) solution, primitive function, all velocities are negative
>plot([ log10(-t(P)),log10(abs(vCoor(P))),P=0..Pi/2]);
Vertical axes extremal points are about 13.4, 0, -1.6, -3.6 .
Horizontal axes extremal points are about 11.32, 11.64, 12.78 .

HOW to interpret all of these are OPEN QUESTION of GR.

Hannu Poropudas

[Hannu Poropudas]

CORRECTION of above comment:
DrDt=dr/dt, all velocities are negative for upper branch which is inside event horizon.
Lower branch is outside of event horizon of SgrA* black hole and those
velocities are positive.

Hannu

> >plot([ log10(-t(P)),log10(abs(DrDt(P))),P=0..Pi/2]);
> Vertical axes extremal points are about 4.8, 3.6, 1.6, 0.4, -2.2, -4.6 .
> Horizontal axes extremal points are about 11, 11.312, 11.46, 12.8 .
>
> S2-star tangential part ONLY (coordinate time velocity):
>
> ># +,- sign for t(P) solution, primitive function, all velocities are negative

CORRECTION of above comment:
vCoor(P), all velocities are negative for upper branch which is inside event horizon.
Lower branch is outside of event horizon of SgrA* black hole and those
velocities are positive.

Hannu

[Hannu Poropudas]

IMPORTANT REMARK:

I have recalculated proper time acceleration formulae and coordinate time
acceleration formulae component by component.

My calculation work is not ready at the moment,

and

I have some difficulties with these new acceleration formulae now.

I don't recommend to use these above old
acceleration formulae now.

Best Regards, Hannu Poropudas

[Hannu Poropudas]

It seems to me that acceleration formulae in polar coordinates
are not suitable in these general relativistic calculations although
these formulae contains correctly components by component
Christoffel symbols of second kind ,
but these Christoffel symbols of second kind are correct
for polar coordinates and these are
NOT correct for Schwarzschild metrics.

The reference is below which I have used in my calculations.

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, pp. 185-188.

Best Regards, Hannu Poropudas

[Hannu Poropudas]

I found one Figure 32.1. (a) Schwarzschild coordinates on page 848 in the book of
(Misner C. W. Thorne K. S. Wheeler J. A., 1973), which is agreement of my analytic
second solution orbit of S2 star, which orbit was spiralling from outside event horizon
to inside event horizon and finally to the singularity (at origin) of SgrA* black hole
(sign changes (from + to -) of velocity of S2-star when crossing event horizon was described here
in this posting chain of mine):

The free-fall collapse of a star if initial radius R_i = 10 M as depicted [...] in
(a) Schwarzschild coordinates. [...] (Misner C. W. Thorne K. S. Wheeler J. A., 1973).

"This shows the surface of a collapsing star (the boundary of the gray region)
graphed in [...] Schwarzschild coordinates (which has the weird property
that inside the horizon, the collapsing surface is actually moving
backwards in time relative to the time coordinate) [...]."

This copy "[...]" was taken from the First Answer from Physics stack exchange
of the question:
"Formation of the event horizon seems impossible with singularity inside
seems impossible [duplicate]"
Asked 8 years, 1 month ago.
3 Answers.

the First Answer Apr 1, 2015 at 18:42
Hypnosift
https://physics.stackexchange.com/users/59406/hypnosifl
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Reference:
Misner C. W., Thorne K. S., Wheeler J. A., 1973.
Gravitation.
W.H. Freeman and Company San Francisco.

Printed in the United States of America.

1279 pages. (Figure 32.1 (a) Schwarzschild coordinates on page 848.)

Best Regards,
Hannu Poropudas