OPEN QUESTION:

I have used following formulae (t=coordinate time and tau=proper time):

v^2 = (dr/dt)^2+r^2*(dphi/dt)^2,
total coordinate velocity formula,
v^2 = (dr/dtau)^2+r^2*(dphi/dtau)^2,
total proper velocity formula.

A^2=(d^2r/dt^2-r*(dphi/dt)^2)^2+
+(r*d^2phi/dt^2+2*(dr/dt)*(dphi/dt))^2,
total coordinate acceleration formula,
A^2=(d^2r/dtau^2-r*(dphi/dtau)^2)^2+
+(r*d^2phi/dtau^2+2*(dr/dtau)*(dphi/dtau))^2,
total proper acceleration formula.

Motion of S2-star around Sagittarius A* is plane motion
on equatorial plane. Formulae are plane polar coordinates
formulae. Derivatives above are calculated from Einstein's
field equations of Schwarzschild metrics.

OPEN QUESTION: I got some huge numbers inside event horizon
of Sagittarius A* black hole in second of two analytic solutions
if S2-star happens somehow to follow this second analytic solution
into the black hole. Inside event horizon that is space-like situation,
which means velocities greater than speed of light c. In my calculations
I have used Weinberg S. 1972. Gravitation and Cosmology book geometric
units where c = 1.

QUESTION: Are above formulae correct if calculations are totally general relativistic on equatorial plane of Schwarzschild black hole?
If not what are then correct formulae instead above formulae ?

References:

1. Poropudas Hannu, 2020.
Analytic GR solutions of S2-star orbits and
precession 732'' per revolution.
sci.physics.relativity, Google Groups,
Postings chain of 10 postings
10.9.2020 10:10:49 - 12.10.2020 13:32:44.

(Other references related above main reference are:

2. Poropudas Hannu, 2018, 2019.
S2 Analytic Orbit Formula around Sagittarius A*
and S2 Perihelion Shift per one revolution.
sci.physics.relativity, Google Groups,
Postings chain of 9 postings
5.12.2018 10:06:23 - 29.11.2019 9:56:13.

3. Poropudas Hannu, 2019.
How to INTERPRET the following minus sign result
of perihelion precession of S2-star ?
sci.physics.relativity, Google Groups,
Postings chain of 4 postings
12.9.2019 11:43:33 - 17.9.2019 8:55:40.

4. Poropudas Hannu, 2016, 2017.
TESTING Weinberg formulae for E and J of General Relativity.
sci.physics.relativity, Google Groups,
Postings chain of 6 postings
19.12.2016 17:12:43 - 11.1.2017 10:57:49. )

Best Regards,

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

keskiviikko 31. elokuuta 2022 klo 11.26.20 UTC+3 Hannu Poropudas kirjoitti:
> OPEN QUESTION:
>
> I have used following formulae (t=coordinate time and tau=proper time):
>
> v^2 = (dr/dt)^2+r^2*(dphi/dt)^2,
> total coordinate velocity formula,

I have thought about possibility of following formula for total coordinate velocity: ?
(m=M*G, in c=1 units)

v^2=(g(r(t)))^2*(dr(t)/dt)^2+(f(r(t)))^2*(dphi(t)/dt)^2

f(r)=r^(1/2)*(r-2*m)^(1/2)+2*m*ln(r^(1/2)+(r-2*m)^(1/2))

f(r) = proper distance in Schwarzschild metrics and
r = coordinate distance in Schwarzschild metrics.

f(r) = int(1/(1-2*m)^(1/2),r), int= Maple 9 program integration

g(r)=(r-2*m)^(1/2)*(dr/dt)/(2*r^(1/2))+r^(1/2)*(dr/dt)/(2*(r-2*m)^(1/2))+
+2*m*((dr/dt)/(2*r^(1/2))+(dr/dt)/(2*(r-2*m)^(1/2)))/(r^(1/2)+(r-2*m)^(1/2))

( Remark: g(r)=(df(r)/dr)*(dr/dt) )

I have thought about possibility of following formula for total proper velocity: ?
(m=M*G, in c=1 units)
> v^2 = (dr/dtau)^2+r^2*(dphi/dtau)^2,
> total proper velocity formula.

v^2=(g(r(tau)))^2*(dr(tau)/dtau)^2+(f(r(tau)))^2*(dphi(tau)/dt)^2

f(r)=r^(1/2)*(r-2*m)^(1/2)+2*m*ln(r^(1/2)+(r-2*m)^(1/2))

g(r)=(r-2*m)^(1/2)*(dr/dt)/(2*r^(1/2))+r^(1/2)*(dr/dt)/(2*(r-2*m)^(1/2))+
+2*m*((dr/dt)/(2*r^(1/2))+(dr/dt)/(2*(r-2*m)^(1/2)))/(r^(1/2)+(r-2*m)^(1/2))

( Remark: g(r)=(df(r)/dr)*(dr/dtau) )

I have not tested reasonability of these yet.

Hannu

>
> A^2=(d^2r/dt^2-r*(dphi/dt)^2)^2+
> +(r*d^2phi/dt^2+2*(dr/dt)*(dphi/dt))^2,
> total coordinate acceleration formula,
> A^2=(d^2r/dtau^2-r*(dphi/dtau)^2)^2+
> +(r*d^2phi/dtau^2+2*(dr/dtau)*(dphi/dtau))^2,
> total proper acceleration formula.
>
> Motion of S2-star around Sagittarius A* is plane motion
> on equatorial plane. Formulae are plane polar coordinates
> formulae. Derivatives above are calculated from Einstein's
> field equations of Schwarzschild metrics.
>
> OPEN QUESTION: I got some huge numbers inside event horizon
> of Sagittarius A* black hole in second of two analytic solutions
> if S2-star happens somehow to follow this second analytic solution
> into the black hole. Inside event horizon that is space-like situation,
> which means velocities greater than speed of light c. In my calculations
> I have used Weinberg S. 1972. Gravitation and Cosmology book geometric
> units where c = 1.
>
> QUESTION: Are above formulae correct if calculations are totally general relativistic on equatorial plane of Schwarzschild black hole?
> If not what are then correct formulae instead above formulae ?
>
> References:
>
> 1. Poropudas Hannu, 2020.
> Analytic GR solutions of S2-star orbits and
> precession 732'' per revolution.
> sci.physics.relativity, Google Groups,
> Postings chain of 10 postings
> 10.9.2020 10:10:49 - 12.10.2020 13:32:44.
>
> (Other references related above main reference are:
>
> 2. Poropudas Hannu, 2018, 2019.
> S2 Analytic Orbit Formula around Sagittarius A*
> and S2 Perihelion Shift per one revolution.
> sci.physics.relativity, Google Groups,
> Postings chain of 9 postings
> 5.12.2018 10:06:23 - 29.11.2019 9:56:13.
>
> 3. Poropudas Hannu, 2019.
> How to INTERPRET the following minus sign result
> of perihelion precession of S2-star ?
> sci.physics.relativity, Google Groups,
> Postings chain of 4 postings
> 12.9.2019 11:43:33 - 17.9.2019 8:55:40.
>
> 4. Poropudas Hannu, 2016, 2017.
> TESTING Weinberg formulae for E and J of General Relativity.
> sci.physics.relativity, Google Groups,
> Postings chain of 6 postings
> 19.12.2016 17:12:43 - 11.1.2017 10:57:49. )
>
> Best Regards,
>
> Hannu Poropudas.
> Kolamäentie 9E, 90900 Kiiminki / Oulu, Finland.

keskiviikko 31. elokuuta 2022 klo 11.26.20 UTC+3 Hannu Poropudas kirjoitti:
> OPEN QUESTION:
>
> I have used following formulae (t=coordinate time and tau=proper time):
>
> v^2 = (dr/dt)^2+r^2*(dphi/dt)^2,
> total coordinate velocity formula,
> v^2 = (dr/dtau)^2+r^2*(dphi/dtau)^2,
> total proper velocity formula.
>
> A^2=(d^2r/dt^2-r*(dphi/dt)^2)^2+
> +(r*d^2phi/dt^2+2*(dr/dt)*(dphi/dt))^2,
> total coordinate acceleration formula,
> A^2=(d^2r/dtau^2-r*(dphi/dtau)^2)^2+
> +(r*d^2phi/dtau^2+2*(dr/dtau)*(dphi/dtau))^2,
> total proper acceleration formula.
>
> Motion of S2-star around Sagittarius A* is plane motion
> on equatorial plane. Formulae are plane polar coordinates
> formulae. Derivatives above are calculated from Einstein's
> field equations of Schwarzschild metrics.
>
> OPEN QUESTION: I got some huge numbers inside event horizon
> of Sagittarius A* black hole in second of two analytic solutions
> if S2-star happens somehow to follow this second analytic solution
> into the black hole. Inside event horizon that is space-like situation,
> which means velocities greater than speed of light c. In my calculations
> I have used Weinberg S. 1972. Gravitation and Cosmology book geometric
> units where c = 1.
>
> QUESTION: Are above formulae correct if calculations are totally general relativistic on equatorial plane of Schwarzschild black hole?
> If not what are then correct formulae instead above formulae ?
>
> References:
>
> 1. Poropudas Hannu, 2020.
> Analytic GR solutions of S2-star orbits and
> precession 732'' per revolution.
> sci.physics.relativity, Google Groups,
> Postings chain of 10 postings
> 10.9.2020 10:10:49 - 12.10.2020 13:32:44.
>

One addition to above reference 1.
These are notes of concerning mainly analytic proper time and coordinate time GR solutions
which all four analytic GR solutions have one real singularity at P = Pi/2.
Postings chain of 4 postings
26.9.2022 9:39:22 - 30.9.2022 10:21:52.

Two .pdf files of all PLOTS have also sent to sci.physics.research,
which are results of GR analytic solutions which way be used to
as help to give answers to above QUESTION ?

Hannu

> (Other references related above main reference are:
>
> 2. Poropudas Hannu, 2018, 2019.
> S2 Analytic Orbit Formula around Sagittarius A*
> and S2 Perihelion Shift per one revolution.
> sci.physics.relativity, Google Groups,
> Postings chain of 9 postings
> 5.12.2018 10:06:23 - 29.11.2019 9:56:13.
>
> 3. Poropudas Hannu, 2019.
> How to INTERPRET the following minus sign result
> of perihelion precession of S2-star ?
> sci.physics.relativity, Google Groups,
> Postings chain of 4 postings
> 12.9.2019 11:43:33 - 17.9.2019 8:55:40.
>
> 4. Poropudas Hannu, 2016, 2017.
> TESTING Weinberg formulae for E and J of General Relativity.
> sci.physics.relativity, Google Groups,
> Postings chain of 6 postings
> 19.12.2016 17:12:43 - 11.1.2017 10:57:49. )
>
> Best Regards,
>
> Hannu Poropudas.
> Kolamäentie 9E, 90900 Kiiminki / Oulu, Finland.