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tech / sci.physics.relativity / I found one spherically symmetric solution of Einstein's vacuum field equations

SubjectAuthor
* I found one spherically symmetric solution of Einstein's vacuum field equationsHannu Poropudas
+* Re: I found one spherically symmetric solution of Einstein's vacuumJanPB
|`- Re: I found one spherically symmetric solution of Einstein's vacuumRoss Finlayson
+- Re: I found one spherically symmetric solution of Einstein's vacuumpatdolan
`* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
 `* Re: I found one spherically symmetric solution of Einstein's vacuum field equatiJanPB
  +- Re: I found one spherically symmetric solution of Einstein's vacuumReinel Badoff
  `* Re: I found one spherically symmetric solution of Einstein's vacuum field equatiHannu Poropudas
   +* Re: I found one spherically symmetric solution of Einstein's vacuumRoss Finlayson
   |`- Re: I found one spherically symmetric solution of Einstein's vacuummitchr...@gmail.com
   `* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
    `* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     +* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     |`* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     | `* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     |  +- Re: I found one spherically symmetric solution of Einstein's vacuummitchr...@gmail.com
     |  +* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     |  |+- Re: I found one spherically symmetric solution of Einstein's vacuumRoss Finlayson
     |  |`* Re: I found one spherically symmetric solution of Einstein's vacuumpatdolan
     |  | `* Re: I found one spherically symmetric solution of Einstein's vacuumRoss Finlayson
     |  |  `* Re: I found one spherically symmetric solution of Einstein's vacuumpatdolan
     |  |   `- Re: I found one spherically symmetric solution of Einstein's vacuummitchr...@gmail.com
     |  `* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     |   `* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     |    +* Re: I found one spherically symmetric solution of Einstein's vacuumpatdolan
     |    |`* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     |    | `- Re: I found one spherically symmetric solution of Einstein's vacuumpatdolan
     |    `* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     |     `* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     |      `* Re: I found one spherically symmetric solution of Einstein's vacuumRoss Finlayson
     |       `* Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     |        `- Re: I found one spherically symmetric solution of Einstein's vacuumHannu Poropudas
     `* Re: I found one spherically symmetric solution of Einstein's vacuumJanPB
      `- Re: I found one spherically symmetric solution of Einstein's vacuum field equatiThe Starmaker

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Re: I found one spherically symmetric solution of Einstein's vacuum field equations

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Subject: Re: I found one spherically symmetric solution of Einstein's vacuum
field equations
From: haporop...@gmail.com (Hannu Poropudas)
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 by: Hannu Poropudas - Mon, 30 Oct 2023 10:35 UTC

maanantai 30. lokakuuta 2023 klo 9.59.22 UTC+2 Hannu Poropudas kirjoitti:
> perjantai 27. lokakuuta 2023 klo 10.46.55 UTC+3 Hannu Poropudas kirjoitti:
> > torstai 26. lokakuuta 2023 klo 11.04.40 UTC+3 Hannu Poropudas kirjoitti:
> > > keskiviikko 25. lokakuuta 2023 klo 14.37.35 UTC+3 Hannu Poropudas kirjoitti:
> > > > keskiviikko 25. lokakuuta 2023 klo 12.01.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > tiistai 24. lokakuuta 2023 klo 11.56.49 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > perjantai 20. lokakuuta 2023 klo 9.54.12 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > torstai 19. lokakuuta 2023 klo 21.41.08 UTC+3 JanPB kirjoitti:
> > > > > > > > On Thursday, October 19, 2023 at 12:22:43 AM UTC-7, Hannu Poropudas wrote:
> > > > > > > > > sunnuntai 15. lokakuuta 2023 klo 11.35.22 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > Spherically symmetric metrics which satisfies
> > > > > > > > > > Einstein's vacuum field equations.
> > > > > > > > > >
> > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > >
> > > > > > > > > > matrix([[m^2/((1-m/r)^4*r^4*(1-2*m/r)), 0, 0, 0], [0, -1/(1-m/r)^2, 0, 0], [0, 0, -sin(theta)^2/(1-m/r)^2, 0], [0, 0, 0, 1-2*m/r]])])
> > > > > > > > > >
> > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > >
> > > > > > > > > > ds^2=(m^2/((1-m/r)^4*r^4*(1-2*m/r)))*dr^2-(1/(1-m/r)^2)*dtheta^2-(sin(theta)^2/(1-m/r)^2)*dphi^2+(1-2*m/r)*dt^2
> > > > > > > > > >
> > > > > > > > > > (m -> m*G/c^2 , if SI-units are used.)
> > > > > > > > > >
> > > > > > > > > > I don't know that would this solution have any astrophysical applications?
> > > > > > > > > >
> > > > > > > > > > There exist a book called something like
> > > > > > > > > > "Exact Solutions of the Einstein Field Equations",
> > > > > > > > > > which have about 740 pages and
> > > > > > > > > > I don't know if this solution is among them?
> > > > > > > > > >
> > > > > > > > > > Three singularity points of the metrics are the following:
> > > > > > > > > >
> > > > > > > > > > r = 0, r = m*G/c^2 and r = 2*m*G/c^2.
> > > > > > > > > >
> > > > > > > > > >
> > > > > > > > > > I have used generalized form of eq. (9) on page 171, when I calculated this solution with my Maple 9.
> > > > > > > > > >
> > > > > > > > > > Reference:
> > > > > > > > > > Tolman R. C., 1934.
> > > > > > > > > > Effect of inhomogeneity on cosmological models.
> > > > > > > > > > Proc. Natl. Acad. Sci. USA, 1934, Mar; 20 (30): 1679-176.
> > > > > > > > > >
> > > > > > > > > > Best Regrads,
> > > > > > > > > >
> > > > > > > > > > Hannu Poropudas
> > > > > > > > > >
> > > > > > > > > > Kolamäentie 9E
> > > > > > > > > > 90900 Kiiminki / Oulu
> > > > > > > > > > Finland
> > > > > > > > > I used random numbers for arbitrary example , which I don't know if it is sensible at all in this case
> > > > > > > > > due three integration constants from Euler-Lagrange equations does not have
> > > > > > > > > same interpretations as in Schwarzschild case (energy (constant) and angular momementum (constant) etc).
> > > > > > > > > I used these guess numbers from aphelion and perihelion of S2-star around SgrA* black hole which calculations I published
> > > > > > > > > some time ago in this sci.physics.relativity Google Group.. (I used c=1 units , c.g.s units then and I use again c=1 units and c.g.s units here)
> > > > > > > > >
> > > > > > > > > MG = 6.292090968*10^11,
> > > > > > > > > 2*MG=1.258418194*10^12.
> > > > > > > > > I found two parametric form analytic solutions (Both are primitive functions, 0<=P<=Pi/2):
> > > > > > > > >
> > > > > > > > > 2.720522631*10^11<=r<=8.306841627*10^11
> > > > > > > > > +,- sign for integral
> > > > > > > > > phi= Int(-0.8328841065*I/sqrt(1-0.5394753492*sin(P)^2),P)
> > > > > > > > > r=-2.259895064*10^23/(5.586318996*10^11*sin(P)^2-8.306841627*10^11)
> > > > > > > > >
> > > > > > > > > and
> > > > > > > > >
> > > > > > > > > -1.103327381*10^12<=rr<=0
> > > > > > > > > +,- sign for integral
> > > > > > > > > phiphi=Int(-0.8330796374*I/sqrt(1-0.4605246509*sin(P)^2),P)
> > > > > > > > > rr= (9.165165817*10^23*sin(P)^2-9.165165817*10^23)/(1.103327381*10^12*sin(P)^2+8.306841627*10^11)
> > > > > > > > >
> > > > > > > > > I calculated also these integrals but their formulae are too long to copy here.
> > > > > > > > > Both sign can be taken into account when plotting Imaginary parts 0<=P<=Pi.
> > > > > > > > > Real parts = 0 in these integrals.
> > > > > > > > > How to interpret pure imaginary phi and phiphi angles?
> > > > > > > > > How to interpret these Imaginary angle plots?
> > > > > > > > >
> > > > > > > > > Best Regards,
> > > > > > > > > Hannu Poropudas
> > > > > > > > Your solution is either:
> > > > > > > >
> > > > > > > > (a) incorrect, or:
> > > > > > > >
> > > > > > > > (b) isometric to Schwarzschild's.
> > > > > > > >
> > > > > > > > Don't waste your time.
> > > > > > > >
> > > > > > > > --
> > > > > > > > Jan
> > > > > > > Your (b) alternative seems not to be true due two separate event horizons in this metrics ?
> > > > > > >
> > > > > > > Schwarzschild metric comes also correctly, but with different sign selection in metrics than what I used.
> > > > > > >
> > > > > > > Your (a) alternative is not true due this metric satisfies Einstein's vacuum field equations,
> > > > > > > but you are correct in point of view that it may not be physically acceptable solution
> > > > > > > of these equations at our present orthodoxic physical knowledge.
> > > > > > > This is indicated by imaginary unit (I=sqrt(-1)) in these example of two analytic solutions.
> > > > > > >
> > > > > > > There exist also few other integration constants from Euler-Largrange equations,
> > > > > > > but I have selected randomly only one couple of them in this example calculation.
> > > > > > >
> > > > > > > Hannu
> > > > > > I put here those strange (NO ordinary physical interpretation) formulae of integration
> > > > > > constants from Euler-Largrange equations:
> > > > > >
> > > > > > I mark now for convenience T = coordinate time and t = proper time.
> > > > > >
> > > > > > (dphi/dt)/(1-m/r)^2 = K1 (constant of integration)
> > > > > > (1-2*m/r)*(dT/dt) = K2 (constant of integration)
> > > > > > (1-2*m/r)*(dT/dt)^2 - m^2*(dr/dt)^2 / ( (1-m/r)^4*r^4*(1-2*m/r) ) - (dphi/dt)^2 / (1-m/r)^2 = 1.
> > > > > >
> > > > > > I calculated for randomly selected numerical values of S2-star aphelion and perhelion
> > > > > > distances (c=1 units, and c.g.s units) from my earlier calculations of analytic GR solutions
> > > > > > for S2-star orbit around SgrA* black hole (sci.physics.relativity published)
> > > > > > to calculate two integration constants K1 and K2 of Euler-Largange equations
> > > > > > (NO ordinary physical interpretation), (I = sqrt(-1) = imaginary unit):
> > > > > >
> > > > > > K1 = +,- 0.7072727132*I,
> > > > > > K2 = +,- 0.5943942676 +,- 0.5943942676*I,
> > > > > >
> > > > > > And I selected here randomly as an example two constants of integration
> > > > > > in this my two analytic solutions calculation:
> > > > > >
> > > > > > K1 = - 0.7072727132*I
> > > > > > and
> > > > > > K2 = 0.5943942676 - 0.5943942676*I
> > > > > >
> > > > > > This selection gave those two pure imaginary analytic solutions which I gave here earlier.
> > > > > > (Phi(P) is pure imaginary angle and r(P) is real distance.
> > > > > > Phiphi(P) is pure imaginary angle and rr(P) is real distance).
> > > > > >
> > > > > > Plot ([Im(phi(P)),r(P),P=0..Pi]);
> > > > > > Plot ([Im(phiphi(P)),rr(P),P=0..Pi]);
> > > > > > gives both +, - solutions in both cases (P..Pi/2 gives only one branch and P..Pi gives both branches)
> > > > > >
> > > > > > Those both plots resemble somehow pendulum orbit ?
> > > > > >
> > > > > > I have NO physical interpretations of these solutions
> > > > > > and I think that these have NO real physical applications.
> > > > > >
> > > > > > Hannu Poropudas
> > > > > I investigated also question that what kind of coordinate time (T) solution would be in parametric form ?
> > > > >
> > > > > It seems to me that this integral is too complicated to calculate analytically, but it could be so
> > > > > with those above K1 and K2 (plus K3 = 0 additional integration constant in Euler-Lagrange equations)
> > > > > in this above case that the coordinate time T could be two dimensional complex number ?
> > > > >
> > > > > This also seems to support what I said above.
> > > > > I have NO physical interpretations of these solutions
> > > > > and I think at the moment that these have NO real physical applications.
> > > > >
> > > > > And we should study two dimensional complex mathematics of two dimensional
> > > > > coordinate time (T) in this complicated integral better,
> > > > > if we try to better understand this situation,
> > > > > if this would be sensible at all ?
> > > > >
> > > > > Best Regards,
> > > > > Hannu Poropudas
> > > > CORRECTION: It is proper time (t) integral in question, not coordinate time (T).
> > > > Sorry that I confused these two letters.
> > > >
> > > > Hannu
> > > I found one interesting reference, which show that there
> > > are really only few astrophysically significant exact solutions to Einstein's field equations.
> > >
> > > Ishak, M. 2015.
> > > Exact Solutions to Einstein's Equations in Astrophysics.
> > > Texas Symposium on Relativistic Astrophysics, Geneva 2015.
> > > 33 pages.
> > > https://personal.utdallas.edu/~mishak/ExactSolutionsInAstrophysics_Ishak_Final.pdf
> > >
> > > Please take a look.
> > >
> > > Best Regards,
> > > Hannu Poropudas
> > In order to me more mathematically complete I calculate also
> > approximate proper time t integral (primitive function)
> > and plotted both real part and imaginary part of it.
> > I have NO interpretations of these.
> >
> > ># Approximate proper time t integral calculated HP 27.10.2023
> > ># REMARK: My letter convenience t=proper time T=coordinate time
> > ># Real part and Imaginary part plotted
> > >#K3:=0;
> > >#K1 := -0.7072727132*I;
> > >#K2 := 0.5943942676-0.5943942676*I;
> > >#m := MG;
> > >#MG := 0.6292090968e12;
> > >#2*MG := 0.1258418194e13;
> > >#a2<=r<=a1, definition area
> > >#a2=2.720522631*10^11, a1=8.306841627*10^11
> >
> > >#Real part of primitive function t approx.
> > ># Series approx at r = MG up to 7 degree.
> +,- sign for REIF(r)
> > >REIF:=r->-0.9292411964e-8*r+0.8717127610e-20*r^2+0.4446277653e-31*(r-0.6292090968e12)^3+0.2329675135e-42*(r-0.6292090968e12)^4+0.3071201158e-47*(r-0.6292090968e12)^5+0.2827253683e-58*(r-0.6292090968e12)^6+0.2065510967e-69*(r-0.6292090968e12)^7;
> >
> > >#Imaginary part of primitive function t approx.
> > >># Series approx at r = MG up to 7 degree.
> +,- sign for IMIF(r)
> > >IMIF:=r->-0.2217185446e-7*r+0.3637453960e-19*r^2+0.1018586871e-30*(r-0.6292090968e12)^3+0.3376062792e-42*(r-0.6292090968e12)^4+0.4794788388e-47*(r-0.6292090968e12)^5+0.2475402813e-58*(r-0.6292090968e12)^6+0.1233132761e-69*(r-0.6292090968e12)^7;
> >
> > > #a2<=r<=a1, definition area
> > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > #MG := 0.6292090968e12;
> > > #2*MG := 0.1258418194e13;
> >
> +,- sign for REIF(r)
> > >plot(REIF(r),r=2.720522631*10^11..8.306841627*10^11);
> >
> > > #a2<=r<=a1, definition area
> > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > #MG := 0.6292090968e12;
> > > #2*MG := 0.1258418194e13;
> >
> +,- sign for IMIF(r)
> > >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
> >
> >
> > Best Regards,
> > Hannu Poropudas


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Re: I found one spherically symmetric solution of Einstein's vacuum field equations

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Subject: Re: I found one spherically symmetric solution of Einstein's vacuum
field equations
From: patdo...@comcast.net (patdolan)
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 by: patdolan - Mon, 30 Oct 2023 17:23 UTC

On Monday, October 30, 2023 at 3:35:05 AM UTC-7, Hannu Poropudas wrote:
> maanantai 30. lokakuuta 2023 klo 9.59.22 UTC+2 Hannu Poropudas kirjoitti:
> > perjantai 27. lokakuuta 2023 klo 10.46.55 UTC+3 Hannu Poropudas kirjoitti:
> > > torstai 26. lokakuuta 2023 klo 11.04.40 UTC+3 Hannu Poropudas kirjoitti:
> > > > keskiviikko 25. lokakuuta 2023 klo 14.37.35 UTC+3 Hannu Poropudas kirjoitti:
> > > > > keskiviikko 25. lokakuuta 2023 klo 12.01.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > tiistai 24. lokakuuta 2023 klo 11.56.49 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > perjantai 20. lokakuuta 2023 klo 9.54.12 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > torstai 19. lokakuuta 2023 klo 21.41.08 UTC+3 JanPB kirjoitti:
> > > > > > > > > On Thursday, October 19, 2023 at 12:22:43 AM UTC-7, Hannu Poropudas wrote:
> > > > > > > > > > sunnuntai 15. lokakuuta 2023 klo 11.35.22 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > Spherically symmetric metrics which satisfies
> > > > > > > > > > > Einstein's vacuum field equations.
> > > > > > > > > > >
> > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > >
> > > > > > > > > > > matrix([[m^2/((1-m/r)^4*r^4*(1-2*m/r)), 0, 0, 0], [0, -1/(1-m/r)^2, 0, 0], [0, 0, -sin(theta)^2/(1-m/r)^2, 0], [0, 0, 0, 1-2*m/r]])])
> > > > > > > > > > >
> > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > >
> > > > > > > > > > > ds^2=(m^2/((1-m/r)^4*r^4*(1-2*m/r)))*dr^2-(1/(1-m/r)^2)*dtheta^2-(sin(theta)^2/(1-m/r)^2)*dphi^2+(1-2*m/r)*dt^2
> > > > > > > > > > >
> > > > > > > > > > > (m -> m*G/c^2 , if SI-units are used.)
> > > > > > > > > > >
> > > > > > > > > > > I don't know that would this solution have any astrophysical applications?
> > > > > > > > > > >
> > > > > > > > > > > There exist a book called something like
> > > > > > > > > > > "Exact Solutions of the Einstein Field Equations",
> > > > > > > > > > > which have about 740 pages and
> > > > > > > > > > > I don't know if this solution is among them?
> > > > > > > > > > >
> > > > > > > > > > > Three singularity points of the metrics are the following:
> > > > > > > > > > >
> > > > > > > > > > > r = 0, r = m*G/c^2 and r = 2*m*G/c^2.
> > > > > > > > > > >
> > > > > > > > > > >
> > > > > > > > > > > I have used generalized form of eq. (9) on page 171, when I calculated this solution with my Maple 9.
> > > > > > > > > > >
> > > > > > > > > > > Reference:
> > > > > > > > > > > Tolman R. C., 1934.
> > > > > > > > > > > Effect of inhomogeneity on cosmological models.
> > > > > > > > > > > Proc. Natl. Acad. Sci. USA, 1934, Mar; 20 (30): 1679-176.
> > > > > > > > > > >
> > > > > > > > > > > Best Regrads,
> > > > > > > > > > >
> > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > >
> > > > > > > > > > > Kolamäentie 9E
> > > > > > > > > > > 90900 Kiiminki / Oulu
> > > > > > > > > > > Finland
> > > > > > > > > > I used random numbers for arbitrary example , which I don't know if it is sensible at all in this case
> > > > > > > > > > due three integration constants from Euler-Lagrange equations does not have
> > > > > > > > > > same interpretations as in Schwarzschild case (energy (constant) and angular momementum (constant) etc).
> > > > > > > > > > I used these guess numbers from aphelion and perihelion of S2-star around SgrA* black hole which calculations I published
> > > > > > > > > > some time ago in this sci.physics.relativity Google Group. (I used c=1 units , c.g.s units then and I use again c=1 units and c.g.s units here)
> > > > > > > > > >
> > > > > > > > > > MG = 6.292090968*10^11,
> > > > > > > > > > 2*MG=1.258418194*10^12.
> > > > > > > > > > I found two parametric form analytic solutions (Both are primitive functions, 0<=P<=Pi/2):
> > > > > > > > > >
> > > > > > > > > > 2.720522631*10^11<=r<=8.306841627*10^11
> > > > > > > > > > +,- sign for integral
> > > > > > > > > > phi= Int(-0.8328841065*I/sqrt(1-0.5394753492*sin(P)^2),P)
> > > > > > > > > > r=-2.259895064*10^23/(5.586318996*10^11*sin(P)^2-8.306841627*10^11)
> > > > > > > > > >
> > > > > > > > > > and
> > > > > > > > > >
> > > > > > > > > > -1.103327381*10^12<=rr<=0
> > > > > > > > > > +,- sign for integral
> > > > > > > > > > phiphi=Int(-0.8330796374*I/sqrt(1-0.4605246509*sin(P)^2),P)
> > > > > > > > > > rr= (9.165165817*10^23*sin(P)^2-9.165165817*10^23)/(1..103327381*10^12*sin(P)^2+8.306841627*10^11)
> > > > > > > > > >
> > > > > > > > > > I calculated also these integrals but their formulae are too long to copy here.
> > > > > > > > > > Both sign can be taken into account when plotting Imaginary parts 0<=P<=Pi.
> > > > > > > > > > Real parts = 0 in these integrals.
> > > > > > > > > > How to interpret pure imaginary phi and phiphi angles?
> > > > > > > > > > How to interpret these Imaginary angle plots?
> > > > > > > > > >
> > > > > > > > > > Best Regards,
> > > > > > > > > > Hannu Poropudas
> > > > > > > > > Your solution is either:
> > > > > > > > >
> > > > > > > > > (a) incorrect, or:
> > > > > > > > >
> > > > > > > > > (b) isometric to Schwarzschild's.
> > > > > > > > >
> > > > > > > > > Don't waste your time.
> > > > > > > > >
> > > > > > > > > --
> > > > > > > > > Jan
> > > > > > > > Your (b) alternative seems not to be true due two separate event horizons in this metrics ?
> > > > > > > >
> > > > > > > > Schwarzschild metric comes also correctly, but with different sign selection in metrics than what I used.
> > > > > > > >
> > > > > > > > Your (a) alternative is not true due this metric satisfies Einstein's vacuum field equations,
> > > > > > > > but you are correct in point of view that it may not be physically acceptable solution
> > > > > > > > of these equations at our present orthodoxic physical knowledge.
> > > > > > > > This is indicated by imaginary unit (I=sqrt(-1)) in these example of two analytic solutions.
> > > > > > > >
> > > > > > > > There exist also few other integration constants from Euler-Largrange equations,
> > > > > > > > but I have selected randomly only one couple of them in this example calculation.
> > > > > > > >
> > > > > > > > Hannu
> > > > > > > I put here those strange (NO ordinary physical interpretation) formulae of integration
> > > > > > > constants from Euler-Largrange equations:
> > > > > > >
> > > > > > > I mark now for convenience T = coordinate time and t = proper time.
> > > > > > >
> > > > > > > (dphi/dt)/(1-m/r)^2 = K1 (constant of integration)
> > > > > > > (1-2*m/r)*(dT/dt) = K2 (constant of integration)
> > > > > > > (1-2*m/r)*(dT/dt)^2 - m^2*(dr/dt)^2 / ( (1-m/r)^4*r^4*(1-2*m/r) ) - (dphi/dt)^2 / (1-m/r)^2 = 1.
> > > > > > >
> > > > > > > I calculated for randomly selected numerical values of S2-star aphelion and perhelion
> > > > > > > distances (c=1 units, and c.g.s units) from my earlier calculations of analytic GR solutions
> > > > > > > for S2-star orbit around SgrA* black hole (sci.physics.relativity published)
> > > > > > > to calculate two integration constants K1 and K2 of Euler-Largange equations
> > > > > > > (NO ordinary physical interpretation), (I = sqrt(-1) = imaginary unit):
> > > > > > >
> > > > > > > K1 = +,- 0.7072727132*I,
> > > > > > > K2 = +,- 0.5943942676 +,- 0.5943942676*I,
> > > > > > >
> > > > > > > And I selected here randomly as an example two constants of integration
> > > > > > > in this my two analytic solutions calculation:
> > > > > > >
> > > > > > > K1 = - 0.7072727132*I
> > > > > > > and
> > > > > > > K2 = 0.5943942676 - 0.5943942676*I
> > > > > > >
> > > > > > > This selection gave those two pure imaginary analytic solutions which I gave here earlier.
> > > > > > > (Phi(P) is pure imaginary angle and r(P) is real distance.
> > > > > > > Phiphi(P) is pure imaginary angle and rr(P) is real distance)..
> > > > > > >
> > > > > > > Plot ([Im(phi(P)),r(P),P=0..Pi]);
> > > > > > > Plot ([Im(phiphi(P)),rr(P),P=0..Pi]);
> > > > > > > gives both +, - solutions in both cases (P..Pi/2 gives only one branch and P..Pi gives both branches)
> > > > > > >
> > > > > > > Those both plots resemble somehow pendulum orbit ?
> > > > > > >
> > > > > > > I have NO physical interpretations of these solutions
> > > > > > > and I think that these have NO real physical applications.
> > > > > > >
> > > > > > > Hannu Poropudas
> > > > > > I investigated also question that what kind of coordinate time (T) solution would be in parametric form ?
> > > > > >
> > > > > > It seems to me that this integral is too complicated to calculate analytically, but it could be so
> > > > > > with those above K1 and K2 (plus K3 = 0 additional integration constant in Euler-Lagrange equations)
> > > > > > in this above case that the coordinate time T could be two dimensional complex number ?
> > > > > >
> > > > > > This also seems to support what I said above.
> > > > > > I have NO physical interpretations of these solutions
> > > > > > and I think at the moment that these have NO real physical applications.
> > > > > >
> > > > > > And we should study two dimensional complex mathematics of two dimensional
> > > > > > coordinate time (T) in this complicated integral better,
> > > > > > if we try to better understand this situation,
> > > > > > if this would be sensible at all ?
> > > > > >
> > > > > > Best Regards,
> > > > > > Hannu Poropudas
> > > > > CORRECTION: It is proper time (t) integral in question, not coordinate time (T).
> > > > > Sorry that I confused these two letters.
> > > > >
> > > > > Hannu
> > > > I found one interesting reference, which show that there
> > > > are really only few astrophysically significant exact solutions to Einstein's field equations.
> > > >
> > > > Ishak, M. 2015.
> > > > Exact Solutions to Einstein's Equations in Astrophysics.
> > > > Texas Symposium on Relativistic Astrophysics, Geneva 2015.
> > > > 33 pages.
> > > > https://personal.utdallas.edu/~mishak/ExactSolutionsInAstrophysics_Ishak_Final.pdf
> > > >
> > > > Please take a look.
> > > >
> > > > Best Regards,
> > > > Hannu Poropudas
> > > In order to me more mathematically complete I calculate also
> > > approximate proper time t integral (primitive function)
> > > and plotted both real part and imaginary part of it.
> > > I have NO interpretations of these.
> > >
> > > ># Approximate proper time t integral calculated HP 27.10.2023
> > > ># REMARK: My letter convenience t=proper time T=coordinate time
> > > ># Real part and Imaginary part plotted
> > > >#K3:=0;
> > > >#K1 := -0.7072727132*I;
> > > >#K2 := 0.5943942676-0.5943942676*I;
> > > >#m := MG;
> > > >#MG := 0.6292090968e12;
> > > >#2*MG := 0.1258418194e13;
> > > >#a2<=r<=a1, definition area
> > > >#a2=2.720522631*10^11, a1=8.306841627*10^11
> > >
> > > >#Real part of primitive function t approx.
> > > ># Series approx at r = MG up to 7 degree.
> > +,- sign for REIF(r)
> > > >REIF:=r->-0.9292411964e-8*r+0.8717127610e-20*r^2+0.4446277653e-31*(r-0.6292090968e12)^3+0.2329675135e-42*(r-0.6292090968e12)^4+0.3071201158e-47*(r-0.6292090968e12)^5+0.2827253683e-58*(r-0.6292090968e12)^6+0.2065510967e-69*(r-0.6292090968e12)^7;
> > >
> > > >#Imaginary part of primitive function t approx.
> > > >># Series approx at r = MG up to 7 degree.
> > +,- sign for IMIF(r)
> > > >IMIF:=r->-0.2217185446e-7*r+0.3637453960e-19*r^2+0.1018586871e-30*(r-0.6292090968e12)^3+0.3376062792e-42*(r-0.6292090968e12)^4+0.4794788388e-47*(r-0.6292090968e12)^5+0.2475402813e-58*(r-0.6292090968e12)^6+0.1233132761e-69*(r-0.6292090968e12)^7;
> > >
> > > > #a2<=r<=a1, definition area
> > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > #MG := 0.6292090968e12;
> > > > #2*MG := 0.1258418194e13;
> > >
> > +,- sign for REIF(r)
> > > >plot(REIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > >
> > > > #a2<=r<=a1, definition area
> > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > #MG := 0.6292090968e12;
> > > > #2*MG := 0.1258418194e13;
> > >
> > +,- sign for IMIF(r)
> > > >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > >
> > >
> > > Best Regards,
> > > Hannu Poropudas
> I calculated also coordinate time T series approximation up to 7 degree at r=MG,
> (REMARK: This is preliminary calculation I have not rechecked it yet):
>
> If my approximate calculations are correct, then it is possible to calculate more
> "quantities" in this strange black hole of two event horizons space-time of mine,
> if this is sensible at all?
>
> ># Two branches of coordinate time T series approx.30.10.2023 H.P.
>
> ># This coordinate time T is also complex number with two branches
> ># (real and Imaginary)
>
> >#Coordinate time T series approx. up to 7 degree
> ># function-(series approx function), not integrated here
> >#(+)branch only used error estimation (compare proper time case)
>
> ># +,- formula (Primitive function, Real part)
>
> >REIG:=r->-2.851064818*ln(abs(r-0.6292090968e12))-.8288703850*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
>
> ># error estimation max positive side about 8.3*10^(-9)
> ># error estimation max abs negative side about -4.2*10^(-9)
> ># Both max are at r=MG, other definition area error = about 0
>
> ># +,- formula (Primitive function, Imaginary part)
>
> >IMIG:=r->-1.425532409*(1-csgn(r-0.6292090968e12))*Pi+1.657740770*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
>
> ># error estimation max positive side about 3.2*10^(-9)
> ># error estimation max abs negative side about -6.4*10^(-9)
> ># Both max are at r=MG, other definition area error = about 0
>
> Best Regards,
> Hannu Poropudas


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Re: I found one spherically symmetric solution of Einstein's vacuum field equations

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Subject: Re: I found one spherically symmetric solution of Einstein's vacuum
field equations
From: haporop...@gmail.com (Hannu Poropudas)
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 by: Hannu Poropudas - Tue, 31 Oct 2023 06:31 UTC

maanantai 30. lokakuuta 2023 klo 19.23.57 UTC+2 patdolan kirjoitti:
> On Monday, October 30, 2023 at 3:35:05 AM UTC-7, Hannu Poropudas wrote:
> > maanantai 30. lokakuuta 2023 klo 9.59.22 UTC+2 Hannu Poropudas kirjoitti:
> > > perjantai 27. lokakuuta 2023 klo 10.46.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > torstai 26. lokakuuta 2023 klo 11.04.40 UTC+3 Hannu Poropudas kirjoitti:
> > > > > keskiviikko 25. lokakuuta 2023 klo 14.37.35 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > keskiviikko 25. lokakuuta 2023 klo 12.01.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > tiistai 24. lokakuuta 2023 klo 11.56.49 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > perjantai 20. lokakuuta 2023 klo 9.54.12 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > torstai 19. lokakuuta 2023 klo 21.41.08 UTC+3 JanPB kirjoitti:
> > > > > > > > > > On Thursday, October 19, 2023 at 12:22:43 AM UTC-7, Hannu Poropudas wrote:
> > > > > > > > > > > sunnuntai 15. lokakuuta 2023 klo 11.35.22 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > > Spherically symmetric metrics which satisfies
> > > > > > > > > > > > Einstein's vacuum field equations.
> > > > > > > > > > > >
> > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > >
> > > > > > > > > > > > matrix([[m^2/((1-m/r)^4*r^4*(1-2*m/r)), 0, 0, 0], [0, -1/(1-m/r)^2, 0, 0], [0, 0, -sin(theta)^2/(1-m/r)^2, 0], [0, 0, 0, 1-2*m/r]])])
> > > > > > > > > > > >
> > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > >
> > > > > > > > > > > > ds^2=(m^2/((1-m/r)^4*r^4*(1-2*m/r)))*dr^2-(1/(1-m/r)^2)*dtheta^2-(sin(theta)^2/(1-m/r)^2)*dphi^2+(1-2*m/r)*dt^2
> > > > > > > > > > > >
> > > > > > > > > > > > (m -> m*G/c^2 , if SI-units are used.)
> > > > > > > > > > > >
> > > > > > > > > > > > I don't know that would this solution have any astrophysical applications?
> > > > > > > > > > > >
> > > > > > > > > > > > There exist a book called something like
> > > > > > > > > > > > "Exact Solutions of the Einstein Field Equations",
> > > > > > > > > > > > which have about 740 pages and
> > > > > > > > > > > > I don't know if this solution is among them?
> > > > > > > > > > > >
> > > > > > > > > > > > Three singularity points of the metrics are the following:
> > > > > > > > > > > >
> > > > > > > > > > > > r = 0, r = m*G/c^2 and r = 2*m*G/c^2.
> > > > > > > > > > > >
> > > > > > > > > > > >
> > > > > > > > > > > > I have used generalized form of eq. (9) on page 171, when I calculated this solution with my Maple 9.
> > > > > > > > > > > >
> > > > > > > > > > > > Reference:
> > > > > > > > > > > > Tolman R. C., 1934.
> > > > > > > > > > > > Effect of inhomogeneity on cosmological models.
> > > > > > > > > > > > Proc. Natl. Acad. Sci. USA, 1934, Mar; 20 (30): 1679-176.
> > > > > > > > > > > >
> > > > > > > > > > > > Best Regrads,
> > > > > > > > > > > >
> > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > >
> > > > > > > > > > > > Kolamäentie 9E
> > > > > > > > > > > > 90900 Kiiminki / Oulu
> > > > > > > > > > > > Finland
> > > > > > > > > > > I used random numbers for arbitrary example , which I don't know if it is sensible at all in this case
> > > > > > > > > > > due three integration constants from Euler-Lagrange equations does not have
> > > > > > > > > > > same interpretations as in Schwarzschild case (energy (constant) and angular momementum (constant) etc).
> > > > > > > > > > > I used these guess numbers from aphelion and perihelion of S2-star around SgrA* black hole which calculations I published
> > > > > > > > > > > some time ago in this sci.physics.relativity Google Group. (I used c=1 units , c.g.s units then and I use again c=1 units and c.g.s units here)
> > > > > > > > > > >
> > > > > > > > > > > MG = 6.292090968*10^11,
> > > > > > > > > > > 2*MG=1.258418194*10^12.
> > > > > > > > > > > I found two parametric form analytic solutions (Both are primitive functions, 0<=P<=Pi/2):
> > > > > > > > > > >
> > > > > > > > > > > 2.720522631*10^11<=r<=8.306841627*10^11
> > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > phi= Int(-0.8328841065*I/sqrt(1-0.5394753492*sin(P)^2),P)
> > > > > > > > > > > r=-2.259895064*10^23/(5.586318996*10^11*sin(P)^2-8.306841627*10^11)
> > > > > > > > > > >
> > > > > > > > > > > and
> > > > > > > > > > >
> > > > > > > > > > > -1.103327381*10^12<=rr<=0
> > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > phiphi=Int(-0.8330796374*I/sqrt(1-0.4605246509*sin(P)^2),P)
> > > > > > > > > > > rr= (9.165165817*10^23*sin(P)^2-9.165165817*10^23)/(1.103327381*10^12*sin(P)^2+8.306841627*10^11)
> > > > > > > > > > >
> > > > > > > > > > > I calculated also these integrals but their formulae are too long to copy here.
> > > > > > > > > > > Both sign can be taken into account when plotting Imaginary parts 0<=P<=Pi.
> > > > > > > > > > > Real parts = 0 in these integrals.
> > > > > > > > > > > How to interpret pure imaginary phi and phiphi angles?
> > > > > > > > > > > How to interpret these Imaginary angle plots?
> > > > > > > > > > >
> > > > > > > > > > > Best Regards,
> > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > Your solution is either:
> > > > > > > > > >
> > > > > > > > > > (a) incorrect, or:
> > > > > > > > > >
> > > > > > > > > > (b) isometric to Schwarzschild's.
> > > > > > > > > >
> > > > > > > > > > Don't waste your time.
> > > > > > > > > >
> > > > > > > > > > --
> > > > > > > > > > Jan
> > > > > > > > > Your (b) alternative seems not to be true due two separate event horizons in this metrics ?
> > > > > > > > >
> > > > > > > > > Schwarzschild metric comes also correctly, but with different sign selection in metrics than what I used.
> > > > > > > > >
> > > > > > > > > Your (a) alternative is not true due this metric satisfies Einstein's vacuum field equations,
> > > > > > > > > but you are correct in point of view that it may not be physically acceptable solution
> > > > > > > > > of these equations at our present orthodoxic physical knowledge.
> > > > > > > > > This is indicated by imaginary unit (I=sqrt(-1)) in these example of two analytic solutions.
> > > > > > > > >
> > > > > > > > > There exist also few other integration constants from Euler-Largrange equations,
> > > > > > > > > but I have selected randomly only one couple of them in this example calculation.
> > > > > > > > >
> > > > > > > > > Hannu
> > > > > > > > I put here those strange (NO ordinary physical interpretation) formulae of integration
> > > > > > > > constants from Euler-Largrange equations:
> > > > > > > >
> > > > > > > > I mark now for convenience T = coordinate time and t = proper time.
> > > > > > > >
> > > > > > > > (dphi/dt)/(1-m/r)^2 = K1 (constant of integration)
> > > > > > > > (1-2*m/r)*(dT/dt) = K2 (constant of integration)
> > > > > > > > (1-2*m/r)*(dT/dt)^2 - m^2*(dr/dt)^2 / ( (1-m/r)^4*r^4*(1-2*m/r) ) - (dphi/dt)^2 / (1-m/r)^2 = 1.
> > > > > > > >
> > > > > > > > I calculated for randomly selected numerical values of S2-star aphelion and perhelion
> > > > > > > > distances (c=1 units, and c.g.s units) from my earlier calculations of analytic GR solutions
> > > > > > > > for S2-star orbit around SgrA* black hole (sci.physics.relativity published)
> > > > > > > > to calculate two integration constants K1 and K2 of Euler-Largange equations
> > > > > > > > (NO ordinary physical interpretation), (I = sqrt(-1) = imaginary unit):
> > > > > > > >
> > > > > > > > K1 = +,- 0.7072727132*I,
> > > > > > > > K2 = +,- 0.5943942676 +,- 0.5943942676*I,
> > > > > > > >
> > > > > > > > And I selected here randomly as an example two constants of integration
> > > > > > > > in this my two analytic solutions calculation:
> > > > > > > >
> > > > > > > > K1 = - 0.7072727132*I
> > > > > > > > and
> > > > > > > > K2 = 0.5943942676 - 0.5943942676*I
> > > > > > > >
> > > > > > > > This selection gave those two pure imaginary analytic solutions which I gave here earlier.
> > > > > > > > (Phi(P) is pure imaginary angle and r(P) is real distance.
> > > > > > > > Phiphi(P) is pure imaginary angle and rr(P) is real distance).
> > > > > > > >
> > > > > > > > Plot ([Im(phi(P)),r(P),P=0..Pi]);
> > > > > > > > Plot ([Im(phiphi(P)),rr(P),P=0..Pi]);
> > > > > > > > gives both +, - solutions in both cases (P..Pi/2 gives only one branch and P..Pi gives both branches)
> > > > > > > >
> > > > > > > > Those both plots resemble somehow pendulum orbit ?
> > > > > > > >
> > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > and I think that these have NO real physical applications.
> > > > > > > >
> > > > > > > > Hannu Poropudas
> > > > > > > I investigated also question that what kind of coordinate time (T) solution would be in parametric form ?
> > > > > > >
> > > > > > > It seems to me that this integral is too complicated to calculate analytically, but it could be so
> > > > > > > with those above K1 and K2 (plus K3 = 0 additional integration constant in Euler-Lagrange equations)
> > > > > > > in this above case that the coordinate time T could be two dimensional complex number ?
> > > > > > >
> > > > > > > This also seems to support what I said above.
> > > > > > > I have NO physical interpretations of these solutions
> > > > > > > and I think at the moment that these have NO real physical applications.
> > > > > > >
> > > > > > > And we should study two dimensional complex mathematics of two dimensional
> > > > > > > coordinate time (T) in this complicated integral better,
> > > > > > > if we try to better understand this situation,
> > > > > > > if this would be sensible at all ?
> > > > > > >
> > > > > > > Best Regards,
> > > > > > > Hannu Poropudas
> > > > > > CORRECTION: It is proper time (t) integral in question, not coordinate time (T).
> > > > > > Sorry that I confused these two letters.
> > > > > >
> > > > > > Hannu
> > > > > I found one interesting reference, which show that there
> > > > > are really only few astrophysically significant exact solutions to Einstein's field equations.
> > > > >
> > > > > Ishak, M. 2015.
> > > > > Exact Solutions to Einstein's Equations in Astrophysics.
> > > > > Texas Symposium on Relativistic Astrophysics, Geneva 2015.
> > > > > 33 pages.
> > > > > https://personal.utdallas.edu/~mishak/ExactSolutionsInAstrophysics_Ishak_Final.pdf
> > > > >
> > > > > Please take a look.
> > > > >
> > > > > Best Regards,
> > > > > Hannu Poropudas
> > > > In order to me more mathematically complete I calculate also
> > > > approximate proper time t integral (primitive function)
> > > > and plotted both real part and imaginary part of it.
> > > > I have NO interpretations of these.
> > > >
> > > > ># Approximate proper time t integral calculated HP 27.10.2023
> > > > ># REMARK: My letter convenience t=proper time T=coordinate time
> > > > ># Real part and Imaginary part plotted
> > > > >#K3:=0;
> > > > >#K1 := -0.7072727132*I;
> > > > >#K2 := 0.5943942676-0.5943942676*I;
> > > > >#m := MG;
> > > > >#MG := 0.6292090968e12;
> > > > >#2*MG := 0.1258418194e13;
> > > > >#a2<=r<=a1, definition area
> > > > >#a2=2.720522631*10^11, a1=8.306841627*10^11
> > > >
> > > > >#Real part of primitive function t approx.
> > > > ># Series approx at r = MG up to 7 degree.
> > > +,- sign for REIF(r)
> > > > >REIF:=r->-0.9292411964e-8*r+0.8717127610e-20*r^2+0.4446277653e-31*(r-0.6292090968e12)^3+0.2329675135e-42*(r-0.6292090968e12)^4+0.3071201158e-47*(r-0.6292090968e12)^5+0.2827253683e-58*(r-0.6292090968e12)^6+0.2065510967e-69*(r-0.6292090968e12)^7;
> > > >
> > > > >#Imaginary part of primitive function t approx.
> > > > >># Series approx at r = MG up to 7 degree.
> > > +,- sign for IMIF(r)
> > > > >IMIF:=r->-0.2217185446e-7*r+0.3637453960e-19*r^2+0.1018586871e-30*(r-0.6292090968e12)^3+0.3376062792e-42*(r-0.6292090968e12)^4+0.4794788388e-47*(r-0.6292090968e12)^5+0.2475402813e-58*(r-0.6292090968e12)^6+0.1233132761e-69*(r-0.6292090968e12)^7;
> > > >
> > > > > #a2<=r<=a1, definition area
> > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > #MG := 0.6292090968e12;
> > > > > #2*MG := 0.1258418194e13;
> > > >
> > > +,- sign for REIF(r)
> > > > >plot(REIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > >
> > > > > #a2<=r<=a1, definition area
> > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > #MG := 0.6292090968e12;
> > > > > #2*MG := 0.1258418194e13;
> > > >
> > > +,- sign for IMIF(r)
> > > > >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > >
> > > >
> > > > Best Regards,
> > > > Hannu Poropudas
> > I calculated also coordinate time T series approximation up to 7 degree at r=MG,
> > (REMARK: This is preliminary calculation I have not rechecked it yet):
> >
> > If my approximate calculations are correct, then it is possible to calculate more
> > "quantities" in this strange black hole of two event horizons space-time of mine,
> > if this is sensible at all?
> >
> > ># Two branches of coordinate time T series approx.30.10.2023 H.P.
> >
> > ># This coordinate time T is also complex number with two branches
> > ># (real and Imaginary)
> >
> > >#Coordinate time T series approx. up to 7 degree
> > ># function-(series approx function), not integrated here
> > >#(+)branch only used error estimation (compare proper time case)
> >
> > ># +,- formula (Primitive function, Real part)
> >
> > >REIG:=r->-2.851064818*ln(abs(r-0.6292090968e12))-.8288703850*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
> >
> > ># error estimation max positive side about 8.3*10^(-9)
> > ># error estimation max abs negative side about -4.2*10^(-9)
> > ># Both max are at r=MG, other definition area error = about 0
> >
> > ># +,- formula (Primitive function, Imaginary part)
> >
> > >IMIG:=r->-1.425532409*(1-csgn(r-0.6292090968e12))*Pi+1.657740770*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
> >
> > ># error estimation max positive side about 3.2*10^(-9)
> > ># error estimation max abs negative side about -6.4*10^(-9)
> > ># Both max are at r=MG, other definition area error = about 0
> >
> > Best Regards,
> > Hannu Poropudas
> Hannu, is this how you treat the discoverer of the Big Ben Paradox? Complete silence? You will not be portrayed in a very generous way when I write my autobiography.


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Re: I found one spherically symmetric solution of Einstein's vacuum field equations

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Subject: Re: I found one spherically symmetric solution of Einstein's vacuum
field equations
From: haporop...@gmail.com (Hannu Poropudas)
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 by: Hannu Poropudas - Tue, 31 Oct 2023 08:40 UTC

maanantai 30. lokakuuta 2023 klo 12.35.05 UTC+2 Hannu Poropudas kirjoitti:
> maanantai 30. lokakuuta 2023 klo 9.59.22 UTC+2 Hannu Poropudas kirjoitti:
> > perjantai 27. lokakuuta 2023 klo 10.46.55 UTC+3 Hannu Poropudas kirjoitti:
> > > torstai 26. lokakuuta 2023 klo 11.04.40 UTC+3 Hannu Poropudas kirjoitti:
> > > > keskiviikko 25. lokakuuta 2023 klo 14.37.35 UTC+3 Hannu Poropudas kirjoitti:
> > > > > keskiviikko 25. lokakuuta 2023 klo 12.01.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > tiistai 24. lokakuuta 2023 klo 11.56.49 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > perjantai 20. lokakuuta 2023 klo 9.54.12 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > torstai 19. lokakuuta 2023 klo 21.41.08 UTC+3 JanPB kirjoitti:
> > > > > > > > > On Thursday, October 19, 2023 at 12:22:43 AM UTC-7, Hannu Poropudas wrote:
> > > > > > > > > > sunnuntai 15. lokakuuta 2023 klo 11.35.22 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > Spherically symmetric metrics which satisfies
> > > > > > > > > > > Einstein's vacuum field equations.
> > > > > > > > > > >
> > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > >
> > > > > > > > > > > matrix([[m^2/((1-m/r)^4*r^4*(1-2*m/r)), 0, 0, 0], [0, -1/(1-m/r)^2, 0, 0], [0, 0, -sin(theta)^2/(1-m/r)^2, 0], [0, 0, 0, 1-2*m/r]])])
> > > > > > > > > > >
> > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > >
> > > > > > > > > > > ds^2=(m^2/((1-m/r)^4*r^4*(1-2*m/r)))*dr^2-(1/(1-m/r)^2)*dtheta^2-(sin(theta)^2/(1-m/r)^2)*dphi^2+(1-2*m/r)*dt^2
> > > > > > > > > > >
> > > > > > > > > > > (m -> m*G/c^2 , if SI-units are used.)
> > > > > > > > > > >
> > > > > > > > > > > I don't know that would this solution have any astrophysical applications?
> > > > > > > > > > >
> > > > > > > > > > > There exist a book called something like
> > > > > > > > > > > "Exact Solutions of the Einstein Field Equations",
> > > > > > > > > > > which have about 740 pages and
> > > > > > > > > > > I don't know if this solution is among them?
> > > > > > > > > > >
> > > > > > > > > > > Three singularity points of the metrics are the following:
> > > > > > > > > > >
> > > > > > > > > > > r = 0, r = m*G/c^2 and r = 2*m*G/c^2.
> > > > > > > > > > >
> > > > > > > > > > >
> > > > > > > > > > > I have used generalized form of eq. (9) on page 171, when I calculated this solution with my Maple 9.
> > > > > > > > > > >
> > > > > > > > > > > Reference:
> > > > > > > > > > > Tolman R. C., 1934.
> > > > > > > > > > > Effect of inhomogeneity on cosmological models.
> > > > > > > > > > > Proc. Natl. Acad. Sci. USA, 1934, Mar; 20 (30): 1679-176.
> > > > > > > > > > >
> > > > > > > > > > > Best Regrads,
> > > > > > > > > > >
> > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > >
> > > > > > > > > > > Kolamäentie 9E
> > > > > > > > > > > 90900 Kiiminki / Oulu
> > > > > > > > > > > Finland
> > > > > > > > > > I used random numbers for arbitrary example , which I don't know if it is sensible at all in this case
> > > > > > > > > > due three integration constants from Euler-Lagrange equations does not have
> > > > > > > > > > same interpretations as in Schwarzschild case (energy (constant) and angular momementum (constant) etc).
> > > > > > > > > > I used these guess numbers from aphelion and perihelion of S2-star around SgrA* black hole which calculations I published
> > > > > > > > > > some time ago in this sci.physics.relativity Google Group. (I used c=1 units , c.g.s units then and I use again c=1 units and c.g.s units here)
> > > > > > > > > >
> > > > > > > > > > MG = 6.292090968*10^11,
> > > > > > > > > > 2*MG=1.258418194*10^12.
> > > > > > > > > > I found two parametric form analytic solutions (Both are primitive functions, 0<=P<=Pi/2):
> > > > > > > > > >
> > > > > > > > > > 2.720522631*10^11<=r<=8.306841627*10^11
> > > > > > > > > > +,- sign for integral
> > > > > > > > > > phi= Int(-0.8328841065*I/sqrt(1-0.5394753492*sin(P)^2),P)
> > > > > > > > > > r=-2.259895064*10^23/(5.586318996*10^11*sin(P)^2-8.306841627*10^11)
> > > > > > > > > >
> > > > > > > > > > and
> > > > > > > > > >
> > > > > > > > > > -1.103327381*10^12<=rr<=0
> > > > > > > > > > +,- sign for integral
> > > > > > > > > > phiphi=Int(-0.8330796374*I/sqrt(1-0.4605246509*sin(P)^2),P)
> > > > > > > > > > rr= (9.165165817*10^23*sin(P)^2-9.165165817*10^23)/(1..103327381*10^12*sin(P)^2+8.306841627*10^11)
> > > > > > > > > >
> > > > > > > > > > I calculated also these integrals but their formulae are too long to copy here.
> > > > > > > > > > Both sign can be taken into account when plotting Imaginary parts 0<=P<=Pi.
> > > > > > > > > > Real parts = 0 in these integrals.
> > > > > > > > > > How to interpret pure imaginary phi and phiphi angles?
> > > > > > > > > > How to interpret these Imaginary angle plots?
> > > > > > > > > >
> > > > > > > > > > Best Regards,
> > > > > > > > > > Hannu Poropudas
> > > > > > > > > Your solution is either:
> > > > > > > > >
> > > > > > > > > (a) incorrect, or:
> > > > > > > > >
> > > > > > > > > (b) isometric to Schwarzschild's.
> > > > > > > > >
> > > > > > > > > Don't waste your time.
> > > > > > > > >
> > > > > > > > > --
> > > > > > > > > Jan
> > > > > > > > Your (b) alternative seems not to be true due two separate event horizons in this metrics ?
> > > > > > > >
> > > > > > > > Schwarzschild metric comes also correctly, but with different sign selection in metrics than what I used.
> > > > > > > >
> > > > > > > > Your (a) alternative is not true due this metric satisfies Einstein's vacuum field equations,
> > > > > > > > but you are correct in point of view that it may not be physically acceptable solution
> > > > > > > > of these equations at our present orthodoxic physical knowledge.
> > > > > > > > This is indicated by imaginary unit (I=sqrt(-1)) in these example of two analytic solutions.
> > > > > > > >
> > > > > > > > There exist also few other integration constants from Euler-Largrange equations,
> > > > > > > > but I have selected randomly only one couple of them in this example calculation.
> > > > > > > >
> > > > > > > > Hannu
> > > > > > > I put here those strange (NO ordinary physical interpretation) formulae of integration
> > > > > > > constants from Euler-Largrange equations:
> > > > > > >
> > > > > > > I mark now for convenience T = coordinate time and t = proper time.
> > > > > > >
> > > > > > > (dphi/dt)/(1-m/r)^2 = K1 (constant of integration)
> > > > > > > (1-2*m/r)*(dT/dt) = K2 (constant of integration)
> > > > > > > (1-2*m/r)*(dT/dt)^2 - m^2*(dr/dt)^2 / ( (1-m/r)^4*r^4*(1-2*m/r) ) - (dphi/dt)^2 / (1-m/r)^2 = 1.
> > > > > > >
> > > > > > > I calculated for randomly selected numerical values of S2-star aphelion and perhelion
> > > > > > > distances (c=1 units, and c.g.s units) from my earlier calculations of analytic GR solutions
> > > > > > > for S2-star orbit around SgrA* black hole (sci.physics.relativity published)
> > > > > > > to calculate two integration constants K1 and K2 of Euler-Largange equations
> > > > > > > (NO ordinary physical interpretation), (I = sqrt(-1) = imaginary unit):
> > > > > > >
> > > > > > > K1 = +,- 0.7072727132*I,
> > > > > > > K2 = +,- 0.5943942676 +,- 0.5943942676*I,
> > > > > > >
> > > > > > > And I selected here randomly as an example two constants of integration
> > > > > > > in this my two analytic solutions calculation:
> > > > > > >
> > > > > > > K1 = - 0.7072727132*I
> > > > > > > and
> > > > > > > K2 = 0.5943942676 - 0.5943942676*I
> > > > > > >
> > > > > > > This selection gave those two pure imaginary analytic solutions which I gave here earlier.
> > > > > > > (Phi(P) is pure imaginary angle and r(P) is real distance.
> > > > > > > Phiphi(P) is pure imaginary angle and rr(P) is real distance)..
> > > > > > >
> > > > > > > Plot ([Im(phi(P)),r(P),P=0..Pi]);
> > > > > > > Plot ([Im(phiphi(P)),rr(P),P=0..Pi]);
> > > > > > > gives both +, - solutions in both cases (P..Pi/2 gives only one branch and P..Pi gives both branches)
> > > > > > >
> > > > > > > Those both plots resemble somehow pendulum orbit ?
> > > > > > >
> > > > > > > I have NO physical interpretations of these solutions
> > > > > > > and I think that these have NO real physical applications.
> > > > > > >
> > > > > > > Hannu Poropudas
> > > > > > I investigated also question that what kind of coordinate time (T) solution would be in parametric form ?
> > > > > >
> > > > > > It seems to me that this integral is too complicated to calculate analytically, but it could be so
> > > > > > with those above K1 and K2 (plus K3 = 0 additional integration constant in Euler-Lagrange equations)
> > > > > > in this above case that the coordinate time T could be two dimensional complex number ?
> > > > > >
> > > > > > This also seems to support what I said above.
> > > > > > I have NO physical interpretations of these solutions
> > > > > > and I think at the moment that these have NO real physical applications.
> > > > > >
> > > > > > And we should study two dimensional complex mathematics of two dimensional
> > > > > > coordinate time (T) in this complicated integral better,
> > > > > > if we try to better understand this situation,
> > > > > > if this would be sensible at all ?
> > > > > >
> > > > > > Best Regards,
> > > > > > Hannu Poropudas
> > > > > CORRECTION: It is proper time (t) integral in question, not coordinate time (T).
> > > > > Sorry that I confused these two letters.
> > > > >
> > > > > Hannu
> > > > I found one interesting reference, which show that there
> > > > are really only few astrophysically significant exact solutions to Einstein's field equations.
> > > >
> > > > Ishak, M. 2015.
> > > > Exact Solutions to Einstein's Equations in Astrophysics.
> > > > Texas Symposium on Relativistic Astrophysics, Geneva 2015.
> > > > 33 pages.
> > > > https://personal.utdallas.edu/~mishak/ExactSolutionsInAstrophysics_Ishak_Final.pdf
> > > >
> > > > Please take a look.
> > > >
> > > > Best Regards,
> > > > Hannu Poropudas
> > > In order to me more mathematically complete I calculate also
> > > approximate proper time t integral (primitive function)
> > > and plotted both real part and imaginary part of it.
> > > I have NO interpretations of these.
> > >
> > > ># Approximate proper time t integral calculated HP 27.10.2023
> > > ># REMARK: My letter convenience t=proper time T=coordinate time
> > > ># Real part and Imaginary part plotted
> > > >#K3:=0;
> > > >#K1 := -0.7072727132*I;
> > > >#K2 := 0.5943942676-0.5943942676*I;
> > > >#m := MG;
> > > >#MG := 0.6292090968e12;
> > > >#2*MG := 0.1258418194e13;
> > > >#a2<=r<=a1, definition area
> > > >#a2=2.720522631*10^11, a1=8.306841627*10^11
> > >
> > > >#Real part of primitive function t approx.
> > > ># Series approx at r = MG up to 7 degree.
> > +,- sign for REIF(r)
> > > >REIF:=r->-0.9292411964e-8*r+0.8717127610e-20*r^2+0.4446277653e-31*(r-0.6292090968e12)^3+0.2329675135e-42*(r-0.6292090968e12)^4+0.3071201158e-47*(r-0.6292090968e12)^5+0.2827253683e-58*(r-0.6292090968e12)^6+0.2065510967e-69*(r-0.6292090968e12)^7;
> > >
> > > >#Imaginary part of primitive function t approx.
> > > >># Series approx at r = MG up to 7 degree.
> > +,- sign for IMIF(r)
> > > >IMIF:=r->-0.2217185446e-7*r+0.3637453960e-19*r^2+0.1018586871e-30*(r-0.6292090968e12)^3+0.3376062792e-42*(r-0.6292090968e12)^4+0.4794788388e-47*(r-0.6292090968e12)^5+0.2475402813e-58*(r-0.6292090968e12)^6+0.1233132761e-69*(r-0.6292090968e12)^7;
> > >
> > > > #a2<=r<=a1, definition area
> > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > #MG := 0.6292090968e12;
> > > > #2*MG := 0.1258418194e13;
> > >
> > +,- sign for REIF(r)
> > > >plot(REIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > >
> > > > #a2<=r<=a1, definition area
> > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > #MG := 0.6292090968e12;
> > > > #2*MG := 0.1258418194e13;
> > >
> > +,- sign for IMIF(r)
> > > >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > >
> > >
> > > Best Regards,
> > > Hannu Poropudas
> I calculated also coordinate time T series approximation up to 7 degree at r=MG,
> (REMARK: This is preliminary calculation I have not rechecked it yet):
>
> If my approximate calculations are correct, then it is possible to calculate more
> "quantities" in this strange black hole of two event horizons space-time of mine,
> if this is sensible at all?
>
> ># Two branches of coordinate time T series approx.30.10.2023 H.P.
>
> ># This coordinate time T is also complex number with two branches
> ># (real and Imaginary)
>
> >#Coordinate time T series approx. up to 7 degree
> ># function-(series approx function), not integrated here
> >#(+)branch only used error estimation (compare proper time case)
>
> ># +,- formula (Primitive function, Real part)
>
> >REIG:=r->-2.851064818*ln(abs(r-0.6292090968e12))-.8288703850*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
>
> ># error estimation max positive side about 8.3*10^(-9)
> ># error estimation max abs negative side about -4.2*10^(-9)
> ># Both max are at r=MG, other definition area error = about 0
>
> ># +,- formula (Primitive function, Imaginary part)
>
> >IMIG:=r->-1.425532409*(1-csgn(r-0.6292090968e12))*Pi+1.657740770*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
>
> ># error estimation max positive side about 3.2*10^(-9)
> ># error estimation max abs negative side about -6.4*10^(-9)
> ># Both max are at r=MG, other definition area error = about 0
>
> Best Regards,
> Hannu Poropudas


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Re: I found one spherically symmetric solution of Einstein's vacuum field equations

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Subject: Re: I found one spherically symmetric solution of Einstein's vacuum
field equations
From: haporop...@gmail.com (Hannu Poropudas)
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 by: Hannu Poropudas - Tue, 31 Oct 2023 11:41 UTC

tiistai 31. lokakuuta 2023 klo 10.40.22 UTC+2 Hannu Poropudas kirjoitti:
> maanantai 30. lokakuuta 2023 klo 12.35.05 UTC+2 Hannu Poropudas kirjoitti:
> > maanantai 30. lokakuuta 2023 klo 9.59.22 UTC+2 Hannu Poropudas kirjoitti:
> > > perjantai 27. lokakuuta 2023 klo 10.46.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > torstai 26. lokakuuta 2023 klo 11.04.40 UTC+3 Hannu Poropudas kirjoitti:
> > > > > keskiviikko 25. lokakuuta 2023 klo 14.37.35 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > keskiviikko 25. lokakuuta 2023 klo 12.01.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > tiistai 24. lokakuuta 2023 klo 11.56.49 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > perjantai 20. lokakuuta 2023 klo 9.54.12 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > torstai 19. lokakuuta 2023 klo 21.41.08 UTC+3 JanPB kirjoitti:
> > > > > > > > > > On Thursday, October 19, 2023 at 12:22:43 AM UTC-7, Hannu Poropudas wrote:
> > > > > > > > > > > sunnuntai 15. lokakuuta 2023 klo 11.35.22 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > > Spherically symmetric metrics which satisfies
> > > > > > > > > > > > Einstein's vacuum field equations.
> > > > > > > > > > > >
> > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > >
> > > > > > > > > > > > matrix([[m^2/((1-m/r)^4*r^4*(1-2*m/r)), 0, 0, 0], [0, -1/(1-m/r)^2, 0, 0], [0, 0, -sin(theta)^2/(1-m/r)^2, 0], [0, 0, 0, 1-2*m/r]])])
> > > > > > > > > > > >
> > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > >
> > > > > > > > > > > > ds^2=(m^2/((1-m/r)^4*r^4*(1-2*m/r)))*dr^2-(1/(1-m/r)^2)*dtheta^2-(sin(theta)^2/(1-m/r)^2)*dphi^2+(1-2*m/r)*dt^2
> > > > > > > > > > > >
> > > > > > > > > > > > (m -> m*G/c^2 , if SI-units are used.)
> > > > > > > > > > > >
> > > > > > > > > > > > I don't know that would this solution have any astrophysical applications?
> > > > > > > > > > > >
> > > > > > > > > > > > There exist a book called something like
> > > > > > > > > > > > "Exact Solutions of the Einstein Field Equations",
> > > > > > > > > > > > which have about 740 pages and
> > > > > > > > > > > > I don't know if this solution is among them?
> > > > > > > > > > > >
> > > > > > > > > > > > Three singularity points of the metrics are the following:
> > > > > > > > > > > >
> > > > > > > > > > > > r = 0, r = m*G/c^2 and r = 2*m*G/c^2.
> > > > > > > > > > > >
> > > > > > > > > > > >
> > > > > > > > > > > > I have used generalized form of eq. (9) on page 171, when I calculated this solution with my Maple 9.
> > > > > > > > > > > >
> > > > > > > > > > > > Reference:
> > > > > > > > > > > > Tolman R. C., 1934.
> > > > > > > > > > > > Effect of inhomogeneity on cosmological models.
> > > > > > > > > > > > Proc. Natl. Acad. Sci. USA, 1934, Mar; 20 (30): 1679-176.
> > > > > > > > > > > >
> > > > > > > > > > > > Best Regrads,
> > > > > > > > > > > >
> > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > >
> > > > > > > > > > > > Kolamäentie 9E
> > > > > > > > > > > > 90900 Kiiminki / Oulu
> > > > > > > > > > > > Finland
> > > > > > > > > > > I used random numbers for arbitrary example , which I don't know if it is sensible at all in this case
> > > > > > > > > > > due three integration constants from Euler-Lagrange equations does not have
> > > > > > > > > > > same interpretations as in Schwarzschild case (energy (constant) and angular momementum (constant) etc).
> > > > > > > > > > > I used these guess numbers from aphelion and perihelion of S2-star around SgrA* black hole which calculations I published
> > > > > > > > > > > some time ago in this sci.physics.relativity Google Group. (I used c=1 units , c.g.s units then and I use again c=1 units and c.g.s units here)
> > > > > > > > > > >
> > > > > > > > > > > MG = 6.292090968*10^11,
> > > > > > > > > > > 2*MG=1.258418194*10^12.
> > > > > > > > > > > I found two parametric form analytic solutions (Both are primitive functions, 0<=P<=Pi/2):
> > > > > > > > > > >
> > > > > > > > > > > 2.720522631*10^11<=r<=8.306841627*10^11
> > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > phi= Int(-0.8328841065*I/sqrt(1-0.5394753492*sin(P)^2),P)
> > > > > > > > > > > r=-2.259895064*10^23/(5.586318996*10^11*sin(P)^2-8.306841627*10^11)
> > > > > > > > > > >
> > > > > > > > > > > and
> > > > > > > > > > >
> > > > > > > > > > > -1.103327381*10^12<=rr<=0
> > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > phiphi=Int(-0.8330796374*I/sqrt(1-0.4605246509*sin(P)^2),P)
> > > > > > > > > > > rr= (9.165165817*10^23*sin(P)^2-9.165165817*10^23)/(1.103327381*10^12*sin(P)^2+8.306841627*10^11)
> > > > > > > > > > >
> > > > > > > > > > > I calculated also these integrals but their formulae are too long to copy here.
> > > > > > > > > > > Both sign can be taken into account when plotting Imaginary parts 0<=P<=Pi.
> > > > > > > > > > > Real parts = 0 in these integrals.
> > > > > > > > > > > How to interpret pure imaginary phi and phiphi angles?
> > > > > > > > > > > How to interpret these Imaginary angle plots?
> > > > > > > > > > >
> > > > > > > > > > > Best Regards,
> > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > Your solution is either:
> > > > > > > > > >
> > > > > > > > > > (a) incorrect, or:
> > > > > > > > > >
> > > > > > > > > > (b) isometric to Schwarzschild's.
> > > > > > > > > >
> > > > > > > > > > Don't waste your time.
> > > > > > > > > >
> > > > > > > > > > --
> > > > > > > > > > Jan
> > > > > > > > > Your (b) alternative seems not to be true due two separate event horizons in this metrics ?
> > > > > > > > >
> > > > > > > > > Schwarzschild metric comes also correctly, but with different sign selection in metrics than what I used.
> > > > > > > > >
> > > > > > > > > Your (a) alternative is not true due this metric satisfies Einstein's vacuum field equations,
> > > > > > > > > but you are correct in point of view that it may not be physically acceptable solution
> > > > > > > > > of these equations at our present orthodoxic physical knowledge.
> > > > > > > > > This is indicated by imaginary unit (I=sqrt(-1)) in these example of two analytic solutions.
> > > > > > > > >
> > > > > > > > > There exist also few other integration constants from Euler-Largrange equations,
> > > > > > > > > but I have selected randomly only one couple of them in this example calculation.
> > > > > > > > >
> > > > > > > > > Hannu
> > > > > > > > I put here those strange (NO ordinary physical interpretation) formulae of integration
> > > > > > > > constants from Euler-Largrange equations:
> > > > > > > >
> > > > > > > > I mark now for convenience T = coordinate time and t = proper time.
> > > > > > > >
> > > > > > > > (dphi/dt)/(1-m/r)^2 = K1 (constant of integration)
> > > > > > > > (1-2*m/r)*(dT/dt) = K2 (constant of integration)
> > > > > > > > (1-2*m/r)*(dT/dt)^2 - m^2*(dr/dt)^2 / ( (1-m/r)^4*r^4*(1-2*m/r) ) - (dphi/dt)^2 / (1-m/r)^2 = 1.
> > > > > > > >
> > > > > > > > I calculated for randomly selected numerical values of S2-star aphelion and perhelion
> > > > > > > > distances (c=1 units, and c.g.s units) from my earlier calculations of analytic GR solutions
> > > > > > > > for S2-star orbit around SgrA* black hole (sci.physics.relativity published)
> > > > > > > > to calculate two integration constants K1 and K2 of Euler-Largange equations
> > > > > > > > (NO ordinary physical interpretation), (I = sqrt(-1) = imaginary unit):
> > > > > > > >
> > > > > > > > K1 = +,- 0.7072727132*I,
> > > > > > > > K2 = +,- 0.5943942676 +,- 0.5943942676*I,
> > > > > > > >
> > > > > > > > And I selected here randomly as an example two constants of integration
> > > > > > > > in this my two analytic solutions calculation:
> > > > > > > >
> > > > > > > > K1 = - 0.7072727132*I
> > > > > > > > and
> > > > > > > > K2 = 0.5943942676 - 0.5943942676*I
> > > > > > > >
> > > > > > > > This selection gave those two pure imaginary analytic solutions which I gave here earlier.
> > > > > > > > (Phi(P) is pure imaginary angle and r(P) is real distance.
> > > > > > > > Phiphi(P) is pure imaginary angle and rr(P) is real distance).
> > > > > > > >
> > > > > > > > Plot ([Im(phi(P)),r(P),P=0..Pi]);
> > > > > > > > Plot ([Im(phiphi(P)),rr(P),P=0..Pi]);
> > > > > > > > gives both +, - solutions in both cases (P..Pi/2 gives only one branch and P..Pi gives both branches)
> > > > > > > >
> > > > > > > > Those both plots resemble somehow pendulum orbit ?
> > > > > > > >
> > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > and I think that these have NO real physical applications.
> > > > > > > >
> > > > > > > > Hannu Poropudas
> > > > > > > I investigated also question that what kind of coordinate time (T) solution would be in parametric form ?
> > > > > > >
> > > > > > > It seems to me that this integral is too complicated to calculate analytically, but it could be so
> > > > > > > with those above K1 and K2 (plus K3 = 0 additional integration constant in Euler-Lagrange equations)
> > > > > > > in this above case that the coordinate time T could be two dimensional complex number ?
> > > > > > >
> > > > > > > This also seems to support what I said above.
> > > > > > > I have NO physical interpretations of these solutions
> > > > > > > and I think at the moment that these have NO real physical applications.
> > > > > > >
> > > > > > > And we should study two dimensional complex mathematics of two dimensional
> > > > > > > coordinate time (T) in this complicated integral better,
> > > > > > > if we try to better understand this situation,
> > > > > > > if this would be sensible at all ?
> > > > > > >
> > > > > > > Best Regards,
> > > > > > > Hannu Poropudas
> > > > > > CORRECTION: It is proper time (t) integral in question, not coordinate time (T).
> > > > > > Sorry that I confused these two letters.
> > > > > >
> > > > > > Hannu
> > > > > I found one interesting reference, which show that there
> > > > > are really only few astrophysically significant exact solutions to Einstein's field equations.
> > > > >
> > > > > Ishak, M. 2015.
> > > > > Exact Solutions to Einstein's Equations in Astrophysics.
> > > > > Texas Symposium on Relativistic Astrophysics, Geneva 2015.
> > > > > 33 pages.
> > > > > https://personal.utdallas.edu/~mishak/ExactSolutionsInAstrophysics_Ishak_Final.pdf
> > > > >
> > > > > Please take a look.
> > > > >
> > > > > Best Regards,
> > > > > Hannu Poropudas
> > > > In order to me more mathematically complete I calculate also
> > > > approximate proper time t integral (primitive function)
> > > > and plotted both real part and imaginary part of it.
> > > > I have NO interpretations of these.
> > > >
> > > > ># Approximate proper time t integral calculated HP 27.10.2023
> > > > ># REMARK: My letter convenience t=proper time T=coordinate time
> > > > ># Real part and Imaginary part plotted
> > > > >#K3:=0;
> > > > >#K1 := -0.7072727132*I;
> > > > >#K2 := 0.5943942676-0.5943942676*I;
> > > > >#m := MG;
> > > > >#MG := 0.6292090968e12;
> > > > >#2*MG := 0.1258418194e13;
> > > > >#a2<=r<=a1, definition area
> > > > >#a2=2.720522631*10^11, a1=8.306841627*10^11
> > > >
> > > > >#Real part of primitive function t approx.
> > > > ># Series approx at r = MG up to 7 degree.
> > > +,- sign for REIF(r)
> > > > >REIF:=r->-0.9292411964e-8*r+0.8717127610e-20*r^2+0.4446277653e-31*(r-0.6292090968e12)^3+0.2329675135e-42*(r-0.6292090968e12)^4+0.3071201158e-47*(r-0.6292090968e12)^5+0.2827253683e-58*(r-0.6292090968e12)^6+0.2065510967e-69*(r-0.6292090968e12)^7;
> > > >
> > > > >#Imaginary part of primitive function t approx.
> > > > >># Series approx at r = MG up to 7 degree.
> > > +,- sign for IMIF(r)
> > > > >IMIF:=r->-0.2217185446e-7*r+0.3637453960e-19*r^2+0.1018586871e-30*(r-0.6292090968e12)^3+0.3376062792e-42*(r-0.6292090968e12)^4+0.4794788388e-47*(r-0.6292090968e12)^5+0.2475402813e-58*(r-0.6292090968e12)^6+0.1233132761e-69*(r-0.6292090968e12)^7;
> > > >
> > > > > #a2<=r<=a1, definition area
> > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > #MG := 0.6292090968e12;
> > > > > #2*MG := 0.1258418194e13;
> > > >
> > > +,- sign for REIF(r)
> > > > >plot(REIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > >
> > > > > #a2<=r<=a1, definition area
> > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > #MG := 0.6292090968e12;
> > > > > #2*MG := 0.1258418194e13;
> > > >
> > > +,- sign for IMIF(r)
> > > > >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > >
> > > >
> > > > Best Regards,
> > > > Hannu Poropudas
> > I calculated also coordinate time T series approximation up to 7 degree at r=MG,
> > (REMARK: This is preliminary calculation I have not rechecked it yet):
> >
> > If my approximate calculations are correct, then it is possible to calculate more
> > "quantities" in this strange black hole of two event horizons space-time of mine,
> > if this is sensible at all?
> >
> > ># Two branches of coordinate time T series approx.30.10.2023 H.P.
> >
> > ># This coordinate time T is also complex number with two branches
> > ># (real and Imaginary)
> >
> > >#Coordinate time T series approx. up to 7 degree
> > ># function-(series approx function), not integrated here
> > >#(+)branch only used error estimation (compare proper time case)
> >
> > ># +,- formula (Primitive function, Real part)
> >
> > >REIG:=r->-2.851064818*ln(abs(r-0.6292090968e12))-.8288703850*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
> >
> > ># error estimation max positive side about 8.3*10^(-9)
> > ># error estimation max abs negative side about -4.2*10^(-9)
> > ># Both max are at r=MG, other definition area error = about 0
> >
> > ># +,- formula (Primitive function, Imaginary part)
> >
> > >IMIG:=r->-1.425532409*(1-csgn(r-0.6292090968e12))*Pi+1.657740770*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
> >
> > ># error estimation max positive side about 3.2*10^(-9)
> > ># error estimation max abs negative side about -6.4*10^(-9)
> > ># Both max are at r=MG, other definition area error = about 0
> >
> > Best Regards,
> > Hannu Poropudas
> I'am sorry about error in 30.10.2023 posting of mine.
>
> Here is CORRECTED 30.10.2023 posting of mine
>
> ># CORRECTED. Two branches of coordinate time T series approx.31.10.2023 H.P.
> ># This coordinate time T is also complex number with two branches ># (real and Imaginary)
>
> >#Coordinate time T series approx. up to 7 degree
> ># function-(series approx function), not integrated here
> >#(+)branch only used error estimation (compare proper time case)
>
> ># +,- formula (Primitive function, Real part)
> >REIG:=r->-1.425532409*ln(abs(r-0.6292090968e12))-.4144351924*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
>
> ># error estimation max positive side about 7.5*10^(-13)
> ># error estimation max abs negative side about -1.3*10^(-12)
> ># +,- formula (Primitive function, Imaginary part)
> >IMIG:=r->-0.7127662045*(1-csgn(r-0.6292090968e12))*Pi+0.8288703849*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
>
> ># error estimation max positive side about 4.8*10^(-13)
> ># error estimation max abs negative side about -2.4*10^(-13)
>
> Best Regards,
> Hannu Poropudas


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Re: I found one spherically symmetric solution of Einstein's vacuum field equations

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Subject: Re: I found one spherically symmetric solution of Einstein's vacuum
field equations
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 by: patdolan - Tue, 31 Oct 2023 17:49 UTC

On Monday, October 30, 2023 at 11:31:23 PM UTC-7, Hannu Poropudas wrote:
> maanantai 30. lokakuuta 2023 klo 19.23.57 UTC+2 patdolan kirjoitti:
> > On Monday, October 30, 2023 at 3:35:05 AM UTC-7, Hannu Poropudas wrote:
> > > maanantai 30. lokakuuta 2023 klo 9.59.22 UTC+2 Hannu Poropudas kirjoitti:
> > > > perjantai 27. lokakuuta 2023 klo 10.46.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > torstai 26. lokakuuta 2023 klo 11.04.40 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > keskiviikko 25. lokakuuta 2023 klo 14.37.35 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > keskiviikko 25. lokakuuta 2023 klo 12.01.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > tiistai 24. lokakuuta 2023 klo 11.56.49 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > perjantai 20. lokakuuta 2023 klo 9.54.12 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > torstai 19. lokakuuta 2023 klo 21.41.08 UTC+3 JanPB kirjoitti:
> > > > > > > > > > > On Thursday, October 19, 2023 at 12:22:43 AM UTC-7, Hannu Poropudas wrote:
> > > > > > > > > > > > sunnuntai 15. lokakuuta 2023 klo 11.35.22 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > > > Spherically symmetric metrics which satisfies
> > > > > > > > > > > > > Einstein's vacuum field equations.
> > > > > > > > > > > > >
> > > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > > >
> > > > > > > > > > > > > matrix([[m^2/((1-m/r)^4*r^4*(1-2*m/r)), 0, 0, 0], [0, -1/(1-m/r)^2, 0, 0], [0, 0, -sin(theta)^2/(1-m/r)^2, 0], [0, 0, 0, 1-2*m/r]])])
> > > > > > > > > > > > >
> > > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > > >
> > > > > > > > > > > > > ds^2=(m^2/((1-m/r)^4*r^4*(1-2*m/r)))*dr^2-(1/(1-m/r)^2)*dtheta^2-(sin(theta)^2/(1-m/r)^2)*dphi^2+(1-2*m/r)*dt^2
> > > > > > > > > > > > >
> > > > > > > > > > > > > (m -> m*G/c^2 , if SI-units are used.)
> > > > > > > > > > > > >
> > > > > > > > > > > > > I don't know that would this solution have any astrophysical applications?
> > > > > > > > > > > > >
> > > > > > > > > > > > > There exist a book called something like
> > > > > > > > > > > > > "Exact Solutions of the Einstein Field Equations",
> > > > > > > > > > > > > which have about 740 pages and
> > > > > > > > > > > > > I don't know if this solution is among them?
> > > > > > > > > > > > >
> > > > > > > > > > > > > Three singularity points of the metrics are the following:
> > > > > > > > > > > > >
> > > > > > > > > > > > > r = 0, r = m*G/c^2 and r = 2*m*G/c^2.
> > > > > > > > > > > > >
> > > > > > > > > > > > >
> > > > > > > > > > > > > I have used generalized form of eq. (9) on page 171, when I calculated this solution with my Maple 9.
> > > > > > > > > > > > >
> > > > > > > > > > > > > Reference:
> > > > > > > > > > > > > Tolman R. C., 1934.
> > > > > > > > > > > > > Effect of inhomogeneity on cosmological models.
> > > > > > > > > > > > > Proc. Natl. Acad. Sci. USA, 1934, Mar; 20 (30): 1679-176.
> > > > > > > > > > > > >
> > > > > > > > > > > > > Best Regrads,
> > > > > > > > > > > > >
> > > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > > >
> > > > > > > > > > > > > Kolamäentie 9E
> > > > > > > > > > > > > 90900 Kiiminki / Oulu
> > > > > > > > > > > > > Finland
> > > > > > > > > > > > I used random numbers for arbitrary example , which I don't know if it is sensible at all in this case
> > > > > > > > > > > > due three integration constants from Euler-Lagrange equations does not have
> > > > > > > > > > > > same interpretations as in Schwarzschild case (energy (constant) and angular momementum (constant) etc).
> > > > > > > > > > > > I used these guess numbers from aphelion and perihelion of S2-star around SgrA* black hole which calculations I published
> > > > > > > > > > > > some time ago in this sci.physics.relativity Google Group. (I used c=1 units , c.g.s units then and I use again c=1 units and c.g.s units here)
> > > > > > > > > > > >
> > > > > > > > > > > > MG = 6.292090968*10^11,
> > > > > > > > > > > > 2*MG=1.258418194*10^12.
> > > > > > > > > > > > I found two parametric form analytic solutions (Both are primitive functions, 0<=P<=Pi/2):
> > > > > > > > > > > >
> > > > > > > > > > > > 2.720522631*10^11<=r<=8.306841627*10^11
> > > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > > phi= Int(-0.8328841065*I/sqrt(1-0.5394753492*sin(P)^2),P)
> > > > > > > > > > > > r=-2.259895064*10^23/(5.586318996*10^11*sin(P)^2-8.306841627*10^11)
> > > > > > > > > > > >
> > > > > > > > > > > > and
> > > > > > > > > > > >
> > > > > > > > > > > > -1.103327381*10^12<=rr<=0
> > > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > > phiphi=Int(-0.8330796374*I/sqrt(1-0.4605246509*sin(P)^2),P)
> > > > > > > > > > > > rr= (9.165165817*10^23*sin(P)^2-9.165165817*10^23)/(1.103327381*10^12*sin(P)^2+8.306841627*10^11)
> > > > > > > > > > > >
> > > > > > > > > > > > I calculated also these integrals but their formulae are too long to copy here.
> > > > > > > > > > > > Both sign can be taken into account when plotting Imaginary parts 0<=P<=Pi.
> > > > > > > > > > > > Real parts = 0 in these integrals.
> > > > > > > > > > > > How to interpret pure imaginary phi and phiphi angles?
> > > > > > > > > > > > How to interpret these Imaginary angle plots?
> > > > > > > > > > > >
> > > > > > > > > > > > Best Regards,
> > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > Your solution is either:
> > > > > > > > > > >
> > > > > > > > > > > (a) incorrect, or:
> > > > > > > > > > >
> > > > > > > > > > > (b) isometric to Schwarzschild's.
> > > > > > > > > > >
> > > > > > > > > > > Don't waste your time.
> > > > > > > > > > >
> > > > > > > > > > > --
> > > > > > > > > > > Jan
> > > > > > > > > > Your (b) alternative seems not to be true due two separate event horizons in this metrics ?
> > > > > > > > > >
> > > > > > > > > > Schwarzschild metric comes also correctly, but with different sign selection in metrics than what I used.
> > > > > > > > > >
> > > > > > > > > > Your (a) alternative is not true due this metric satisfies Einstein's vacuum field equations,
> > > > > > > > > > but you are correct in point of view that it may not be physically acceptable solution
> > > > > > > > > > of these equations at our present orthodoxic physical knowledge.
> > > > > > > > > > This is indicated by imaginary unit (I=sqrt(-1)) in these example of two analytic solutions.
> > > > > > > > > >
> > > > > > > > > > There exist also few other integration constants from Euler-Largrange equations,
> > > > > > > > > > but I have selected randomly only one couple of them in this example calculation.
> > > > > > > > > >
> > > > > > > > > > Hannu
> > > > > > > > > I put here those strange (NO ordinary physical interpretation) formulae of integration
> > > > > > > > > constants from Euler-Largrange equations:
> > > > > > > > >
> > > > > > > > > I mark now for convenience T = coordinate time and t = proper time.
> > > > > > > > >
> > > > > > > > > (dphi/dt)/(1-m/r)^2 = K1 (constant of integration)
> > > > > > > > > (1-2*m/r)*(dT/dt) = K2 (constant of integration)
> > > > > > > > > (1-2*m/r)*(dT/dt)^2 - m^2*(dr/dt)^2 / ( (1-m/r)^4*r^4*(1-2*m/r) ) - (dphi/dt)^2 / (1-m/r)^2 = 1.
> > > > > > > > >
> > > > > > > > > I calculated for randomly selected numerical values of S2-star aphelion and perhelion
> > > > > > > > > distances (c=1 units, and c.g.s units) from my earlier calculations of analytic GR solutions
> > > > > > > > > for S2-star orbit around SgrA* black hole (sci.physics.relativity published)
> > > > > > > > > to calculate two integration constants K1 and K2 of Euler-Largange equations
> > > > > > > > > (NO ordinary physical interpretation), (I = sqrt(-1) = imaginary unit):
> > > > > > > > >
> > > > > > > > > K1 = +,- 0.7072727132*I,
> > > > > > > > > K2 = +,- 0.5943942676 +,- 0.5943942676*I,
> > > > > > > > >
> > > > > > > > > And I selected here randomly as an example two constants of integration
> > > > > > > > > in this my two analytic solutions calculation:
> > > > > > > > >
> > > > > > > > > K1 = - 0.7072727132*I
> > > > > > > > > and
> > > > > > > > > K2 = 0.5943942676 - 0.5943942676*I
> > > > > > > > >
> > > > > > > > > This selection gave those two pure imaginary analytic solutions which I gave here earlier.
> > > > > > > > > (Phi(P) is pure imaginary angle and r(P) is real distance..
> > > > > > > > > Phiphi(P) is pure imaginary angle and rr(P) is real distance).
> > > > > > > > >
> > > > > > > > > Plot ([Im(phi(P)),r(P),P=0..Pi]);
> > > > > > > > > Plot ([Im(phiphi(P)),rr(P),P=0..Pi]);
> > > > > > > > > gives both +, - solutions in both cases (P..Pi/2 gives only one branch and P..Pi gives both branches)
> > > > > > > > >
> > > > > > > > > Those both plots resemble somehow pendulum orbit ?
> > > > > > > > >
> > > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > > and I think that these have NO real physical applications..
> > > > > > > > >
> > > > > > > > > Hannu Poropudas
> > > > > > > > I investigated also question that what kind of coordinate time (T) solution would be in parametric form ?
> > > > > > > >
> > > > > > > > It seems to me that this integral is too complicated to calculate analytically, but it could be so
> > > > > > > > with those above K1 and K2 (plus K3 = 0 additional integration constant in Euler-Lagrange equations)
> > > > > > > > in this above case that the coordinate time T could be two dimensional complex number ?
> > > > > > > >
> > > > > > > > This also seems to support what I said above.
> > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > and I think at the moment that these have NO real physical applications.
> > > > > > > >
> > > > > > > > And we should study two dimensional complex mathematics of two dimensional
> > > > > > > > coordinate time (T) in this complicated integral better,
> > > > > > > > if we try to better understand this situation,
> > > > > > > > if this would be sensible at all ?
> > > > > > > >
> > > > > > > > Best Regards,
> > > > > > > > Hannu Poropudas
> > > > > > > CORRECTION: It is proper time (t) integral in question, not coordinate time (T).
> > > > > > > Sorry that I confused these two letters.
> > > > > > >
> > > > > > > Hannu
> > > > > > I found one interesting reference, which show that there
> > > > > > are really only few astrophysically significant exact solutions to Einstein's field equations.
> > > > > >
> > > > > > Ishak, M. 2015.
> > > > > > Exact Solutions to Einstein's Equations in Astrophysics.
> > > > > > Texas Symposium on Relativistic Astrophysics, Geneva 2015.
> > > > > > 33 pages.
> > > > > > https://personal.utdallas.edu/~mishak/ExactSolutionsInAstrophysics_Ishak_Final.pdf
> > > > > >
> > > > > > Please take a look.
> > > > > >
> > > > > > Best Regards,
> > > > > > Hannu Poropudas
> > > > > In order to me more mathematically complete I calculate also
> > > > > approximate proper time t integral (primitive function)
> > > > > and plotted both real part and imaginary part of it.
> > > > > I have NO interpretations of these.
> > > > >
> > > > > ># Approximate proper time t integral calculated HP 27.10.2023
> > > > > ># REMARK: My letter convenience t=proper time T=coordinate time
> > > > > ># Real part and Imaginary part plotted
> > > > > >#K3:=0;
> > > > > >#K1 := -0.7072727132*I;
> > > > > >#K2 := 0.5943942676-0.5943942676*I;
> > > > > >#m := MG;
> > > > > >#MG := 0.6292090968e12;
> > > > > >#2*MG := 0.1258418194e13;
> > > > > >#a2<=r<=a1, definition area
> > > > > >#a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > >
> > > > > >#Real part of primitive function t approx.
> > > > > ># Series approx at r = MG up to 7 degree.
> > > > +,- sign for REIF(r)
> > > > > >REIF:=r->-0.9292411964e-8*r+0.8717127610e-20*r^2+0.4446277653e-31*(r-0.6292090968e12)^3+0.2329675135e-42*(r-0.6292090968e12)^4+0.3071201158e-47*(r-0.6292090968e12)^5+0.2827253683e-58*(r-0.6292090968e12)^6+0.2065510967e-69*(r-0.6292090968e12)^7;
> > > > >
> > > > > >#Imaginary part of primitive function t approx.
> > > > > >># Series approx at r = MG up to 7 degree.
> > > > +,- sign for IMIF(r)
> > > > > >IMIF:=r->-0.2217185446e-7*r+0.3637453960e-19*r^2+0.1018586871e-30*(r-0.6292090968e12)^3+0.3376062792e-42*(r-0.6292090968e12)^4+0.4794788388e-47*(r-0.6292090968e12)^5+0.2475402813e-58*(r-0.6292090968e12)^6+0.1233132761e-69*(r-0.6292090968e12)^7;
> > > > >
> > > > > > #a2<=r<=a1, definition area
> > > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > > #MG := 0.6292090968e12;
> > > > > > #2*MG := 0.1258418194e13;
> > > > >
> > > > +,- sign for REIF(r)
> > > > > >plot(REIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > > >
> > > > > > #a2<=r<=a1, definition area
> > > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > > #MG := 0.6292090968e12;
> > > > > > #2*MG := 0.1258418194e13;
> > > > >
> > > > +,- sign for IMIF(r)
> > > > > >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > > >
> > > > >
> > > > > Best Regards,
> > > > > Hannu Poropudas
> > > I calculated also coordinate time T series approximation up to 7 degree at r=MG,
> > > (REMARK: This is preliminary calculation I have not rechecked it yet):
> > >
> > > If my approximate calculations are correct, then it is possible to calculate more
> > > "quantities" in this strange black hole of two event horizons space-time of mine,
> > > if this is sensible at all?
> > >
> > > ># Two branches of coordinate time T series approx.30.10.2023 H.P.
> > >
> > > ># This coordinate time T is also complex number with two branches
> > > ># (real and Imaginary)
> > >
> > > >#Coordinate time T series approx. up to 7 degree
> > > ># function-(series approx function), not integrated here
> > > >#(+)branch only used error estimation (compare proper time case)
> > >
> > > ># +,- formula (Primitive function, Real part)
> > >
> > > >REIG:=r->-2.851064818*ln(abs(r-0.6292090968e12))-.8288703850*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
> > >
> > > ># error estimation max positive side about 8.3*10^(-9)
> > > ># error estimation max abs negative side about -4.2*10^(-9)
> > > ># Both max are at r=MG, other definition area error = about 0
> > >
> > > ># +,- formula (Primitive function, Imaginary part)
> > >
> > > >IMIG:=r->-1.425532409*(1-csgn(r-0.6292090968e12))*Pi+1.657740770*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
> > >
> > > ># error estimation max positive side about 3.2*10^(-9)
> > > ># error estimation max abs negative side about -6.4*10^(-9)
> > > ># Both max are at r=MG, other definition area error = about 0
> > >
> > > Best Regards,
> > > Hannu Poropudas
> > Hannu, is this how you treat the discoverer of the Big Ben Paradox? Complete silence? You will not be portrayed in a very generous way when I write my autobiography.
> This is question is NOT subject of my posting chain.
>
> I noticed that in the stack exchage of physics was written about "Big Ben Paradox" question:
>
> SR is not a theory of gravity.
>
> Hannu
I accept this answer, Hannu. You have been fully restored to my good graces.


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Re: I found one spherically symmetric solution of Einstein's vacuum field equations

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Subject: Re: I found one spherically symmetric solution of Einstein's vacuum
field equations
From: ross.a.f...@gmail.com (Ross Finlayson)
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 by: Ross Finlayson - Thu, 2 Nov 2023 08:55 UTC

On Tuesday, October 31, 2023 at 4:41:56 AM UTC-7, Hannu Poropudas wrote:
> tiistai 31. lokakuuta 2023 klo 10.40.22 UTC+2 Hannu Poropudas kirjoitti:
> > maanantai 30. lokakuuta 2023 klo 12.35.05 UTC+2 Hannu Poropudas kirjoitti:
> > > maanantai 30. lokakuuta 2023 klo 9.59.22 UTC+2 Hannu Poropudas kirjoitti:
> > > > perjantai 27. lokakuuta 2023 klo 10.46.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > torstai 26. lokakuuta 2023 klo 11.04.40 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > keskiviikko 25. lokakuuta 2023 klo 14.37.35 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > keskiviikko 25. lokakuuta 2023 klo 12.01.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > tiistai 24. lokakuuta 2023 klo 11.56.49 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > perjantai 20. lokakuuta 2023 klo 9.54.12 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > torstai 19. lokakuuta 2023 klo 21.41.08 UTC+3 JanPB kirjoitti:
> > > > > > > > > > > On Thursday, October 19, 2023 at 12:22:43 AM UTC-7, Hannu Poropudas wrote:
> > > > > > > > > > > > sunnuntai 15. lokakuuta 2023 klo 11.35.22 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > > > Spherically symmetric metrics which satisfies
> > > > > > > > > > > > > Einstein's vacuum field equations.
> > > > > > > > > > > > >
> > > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > > >
> > > > > > > > > > > > > matrix([[m^2/((1-m/r)^4*r^4*(1-2*m/r)), 0, 0, 0], [0, -1/(1-m/r)^2, 0, 0], [0, 0, -sin(theta)^2/(1-m/r)^2, 0], [0, 0, 0, 1-2*m/r]])])
> > > > > > > > > > > > >
> > > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > > >
> > > > > > > > > > > > > ds^2=(m^2/((1-m/r)^4*r^4*(1-2*m/r)))*dr^2-(1/(1-m/r)^2)*dtheta^2-(sin(theta)^2/(1-m/r)^2)*dphi^2+(1-2*m/r)*dt^2
> > > > > > > > > > > > >
> > > > > > > > > > > > > (m -> m*G/c^2 , if SI-units are used.)
> > > > > > > > > > > > >
> > > > > > > > > > > > > I don't know that would this solution have any astrophysical applications?
> > > > > > > > > > > > >
> > > > > > > > > > > > > There exist a book called something like
> > > > > > > > > > > > > "Exact Solutions of the Einstein Field Equations",
> > > > > > > > > > > > > which have about 740 pages and
> > > > > > > > > > > > > I don't know if this solution is among them?
> > > > > > > > > > > > >
> > > > > > > > > > > > > Three singularity points of the metrics are the following:
> > > > > > > > > > > > >
> > > > > > > > > > > > > r = 0, r = m*G/c^2 and r = 2*m*G/c^2.
> > > > > > > > > > > > >
> > > > > > > > > > > > >
> > > > > > > > > > > > > I have used generalized form of eq. (9) on page 171, when I calculated this solution with my Maple 9.
> > > > > > > > > > > > >
> > > > > > > > > > > > > Reference:
> > > > > > > > > > > > > Tolman R. C., 1934.
> > > > > > > > > > > > > Effect of inhomogeneity on cosmological models.
> > > > > > > > > > > > > Proc. Natl. Acad. Sci. USA, 1934, Mar; 20 (30): 1679-176.
> > > > > > > > > > > > >
> > > > > > > > > > > > > Best Regrads,
> > > > > > > > > > > > >
> > > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > > >
> > > > > > > > > > > > > Kolamäentie 9E
> > > > > > > > > > > > > 90900 Kiiminki / Oulu
> > > > > > > > > > > > > Finland
> > > > > > > > > > > > I used random numbers for arbitrary example , which I don't know if it is sensible at all in this case
> > > > > > > > > > > > due three integration constants from Euler-Lagrange equations does not have
> > > > > > > > > > > > same interpretations as in Schwarzschild case (energy (constant) and angular momementum (constant) etc).
> > > > > > > > > > > > I used these guess numbers from aphelion and perihelion of S2-star around SgrA* black hole which calculations I published
> > > > > > > > > > > > some time ago in this sci.physics.relativity Google Group. (I used c=1 units , c.g.s units then and I use again c=1 units and c.g.s units here)
> > > > > > > > > > > >
> > > > > > > > > > > > MG = 6.292090968*10^11,
> > > > > > > > > > > > 2*MG=1.258418194*10^12.
> > > > > > > > > > > > I found two parametric form analytic solutions (Both are primitive functions, 0<=P<=Pi/2):
> > > > > > > > > > > >
> > > > > > > > > > > > 2.720522631*10^11<=r<=8.306841627*10^11
> > > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > > phi= Int(-0.8328841065*I/sqrt(1-0.5394753492*sin(P)^2),P)
> > > > > > > > > > > > r=-2.259895064*10^23/(5.586318996*10^11*sin(P)^2-8.306841627*10^11)
> > > > > > > > > > > >
> > > > > > > > > > > > and
> > > > > > > > > > > >
> > > > > > > > > > > > -1.103327381*10^12<=rr<=0
> > > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > > phiphi=Int(-0.8330796374*I/sqrt(1-0.4605246509*sin(P)^2),P)
> > > > > > > > > > > > rr= (9.165165817*10^23*sin(P)^2-9.165165817*10^23)/(1.103327381*10^12*sin(P)^2+8.306841627*10^11)
> > > > > > > > > > > >
> > > > > > > > > > > > I calculated also these integrals but their formulae are too long to copy here.
> > > > > > > > > > > > Both sign can be taken into account when plotting Imaginary parts 0<=P<=Pi.
> > > > > > > > > > > > Real parts = 0 in these integrals.
> > > > > > > > > > > > How to interpret pure imaginary phi and phiphi angles?
> > > > > > > > > > > > How to interpret these Imaginary angle plots?
> > > > > > > > > > > >
> > > > > > > > > > > > Best Regards,
> > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > Your solution is either:
> > > > > > > > > > >
> > > > > > > > > > > (a) incorrect, or:
> > > > > > > > > > >
> > > > > > > > > > > (b) isometric to Schwarzschild's.
> > > > > > > > > > >
> > > > > > > > > > > Don't waste your time.
> > > > > > > > > > >
> > > > > > > > > > > --
> > > > > > > > > > > Jan
> > > > > > > > > > Your (b) alternative seems not to be true due two separate event horizons in this metrics ?
> > > > > > > > > >
> > > > > > > > > > Schwarzschild metric comes also correctly, but with different sign selection in metrics than what I used.
> > > > > > > > > >
> > > > > > > > > > Your (a) alternative is not true due this metric satisfies Einstein's vacuum field equations,
> > > > > > > > > > but you are correct in point of view that it may not be physically acceptable solution
> > > > > > > > > > of these equations at our present orthodoxic physical knowledge.
> > > > > > > > > > This is indicated by imaginary unit (I=sqrt(-1)) in these example of two analytic solutions.
> > > > > > > > > >
> > > > > > > > > > There exist also few other integration constants from Euler-Largrange equations,
> > > > > > > > > > but I have selected randomly only one couple of them in this example calculation.
> > > > > > > > > >
> > > > > > > > > > Hannu
> > > > > > > > > I put here those strange (NO ordinary physical interpretation) formulae of integration
> > > > > > > > > constants from Euler-Largrange equations:
> > > > > > > > >
> > > > > > > > > I mark now for convenience T = coordinate time and t = proper time.
> > > > > > > > >
> > > > > > > > > (dphi/dt)/(1-m/r)^2 = K1 (constant of integration)
> > > > > > > > > (1-2*m/r)*(dT/dt) = K2 (constant of integration)
> > > > > > > > > (1-2*m/r)*(dT/dt)^2 - m^2*(dr/dt)^2 / ( (1-m/r)^4*r^4*(1-2*m/r) ) - (dphi/dt)^2 / (1-m/r)^2 = 1.
> > > > > > > > >
> > > > > > > > > I calculated for randomly selected numerical values of S2-star aphelion and perhelion
> > > > > > > > > distances (c=1 units, and c.g.s units) from my earlier calculations of analytic GR solutions
> > > > > > > > > for S2-star orbit around SgrA* black hole (sci.physics.relativity published)
> > > > > > > > > to calculate two integration constants K1 and K2 of Euler-Largange equations
> > > > > > > > > (NO ordinary physical interpretation), (I = sqrt(-1) = imaginary unit):
> > > > > > > > >
> > > > > > > > > K1 = +,- 0.7072727132*I,
> > > > > > > > > K2 = +,- 0.5943942676 +,- 0.5943942676*I,
> > > > > > > > >
> > > > > > > > > And I selected here randomly as an example two constants of integration
> > > > > > > > > in this my two analytic solutions calculation:
> > > > > > > > >
> > > > > > > > > K1 = - 0.7072727132*I
> > > > > > > > > and
> > > > > > > > > K2 = 0.5943942676 - 0.5943942676*I
> > > > > > > > >
> > > > > > > > > This selection gave those two pure imaginary analytic solutions which I gave here earlier.
> > > > > > > > > (Phi(P) is pure imaginary angle and r(P) is real distance..
> > > > > > > > > Phiphi(P) is pure imaginary angle and rr(P) is real distance).
> > > > > > > > >
> > > > > > > > > Plot ([Im(phi(P)),r(P),P=0..Pi]);
> > > > > > > > > Plot ([Im(phiphi(P)),rr(P),P=0..Pi]);
> > > > > > > > > gives both +, - solutions in both cases (P..Pi/2 gives only one branch and P..Pi gives both branches)
> > > > > > > > >
> > > > > > > > > Those both plots resemble somehow pendulum orbit ?
> > > > > > > > >
> > > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > > and I think that these have NO real physical applications..
> > > > > > > > >
> > > > > > > > > Hannu Poropudas
> > > > > > > > I investigated also question that what kind of coordinate time (T) solution would be in parametric form ?
> > > > > > > >
> > > > > > > > It seems to me that this integral is too complicated to calculate analytically, but it could be so
> > > > > > > > with those above K1 and K2 (plus K3 = 0 additional integration constant in Euler-Lagrange equations)
> > > > > > > > in this above case that the coordinate time T could be two dimensional complex number ?
> > > > > > > >
> > > > > > > > This also seems to support what I said above.
> > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > and I think at the moment that these have NO real physical applications.
> > > > > > > >
> > > > > > > > And we should study two dimensional complex mathematics of two dimensional
> > > > > > > > coordinate time (T) in this complicated integral better,
> > > > > > > > if we try to better understand this situation,
> > > > > > > > if this would be sensible at all ?
> > > > > > > >
> > > > > > > > Best Regards,
> > > > > > > > Hannu Poropudas
> > > > > > > CORRECTION: It is proper time (t) integral in question, not coordinate time (T).
> > > > > > > Sorry that I confused these two letters.
> > > > > > >
> > > > > > > Hannu
> > > > > > I found one interesting reference, which show that there
> > > > > > are really only few astrophysically significant exact solutions to Einstein's field equations.
> > > > > >
> > > > > > Ishak, M. 2015.
> > > > > > Exact Solutions to Einstein's Equations in Astrophysics.
> > > > > > Texas Symposium on Relativistic Astrophysics, Geneva 2015.
> > > > > > 33 pages.
> > > > > > https://personal.utdallas.edu/~mishak/ExactSolutionsInAstrophysics_Ishak_Final.pdf
> > > > > >
> > > > > > Please take a look.
> > > > > >
> > > > > > Best Regards,
> > > > > > Hannu Poropudas
> > > > > In order to me more mathematically complete I calculate also
> > > > > approximate proper time t integral (primitive function)
> > > > > and plotted both real part and imaginary part of it.
> > > > > I have NO interpretations of these.
> > > > >
> > > > > ># Approximate proper time t integral calculated HP 27.10.2023
> > > > > ># REMARK: My letter convenience t=proper time T=coordinate time
> > > > > ># Real part and Imaginary part plotted
> > > > > >#K3:=0;
> > > > > >#K1 := -0.7072727132*I;
> > > > > >#K2 := 0.5943942676-0.5943942676*I;
> > > > > >#m := MG;
> > > > > >#MG := 0.6292090968e12;
> > > > > >#2*MG := 0.1258418194e13;
> > > > > >#a2<=r<=a1, definition area
> > > > > >#a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > >
> > > > > >#Real part of primitive function t approx.
> > > > > ># Series approx at r = MG up to 7 degree.
> > > > +,- sign for REIF(r)
> > > > > >REIF:=r->-0.9292411964e-8*r+0.8717127610e-20*r^2+0.4446277653e-31*(r-0.6292090968e12)^3+0.2329675135e-42*(r-0.6292090968e12)^4+0.3071201158e-47*(r-0.6292090968e12)^5+0.2827253683e-58*(r-0.6292090968e12)^6+0.2065510967e-69*(r-0.6292090968e12)^7;
> > > > >
> > > > > >#Imaginary part of primitive function t approx.
> > > > > >># Series approx at r = MG up to 7 degree.
> > > > +,- sign for IMIF(r)
> > > > > >IMIF:=r->-0.2217185446e-7*r+0.3637453960e-19*r^2+0.1018586871e-30*(r-0.6292090968e12)^3+0.3376062792e-42*(r-0.6292090968e12)^4+0.4794788388e-47*(r-0.6292090968e12)^5+0.2475402813e-58*(r-0.6292090968e12)^6+0.1233132761e-69*(r-0.6292090968e12)^7;
> > > > >
> > > > > > #a2<=r<=a1, definition area
> > > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > > #MG := 0.6292090968e12;
> > > > > > #2*MG := 0.1258418194e13;
> > > > >
> > > > +,- sign for REIF(r)
> > > > > >plot(REIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > > >
> > > > > > #a2<=r<=a1, definition area
> > > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > > #MG := 0.6292090968e12;
> > > > > > #2*MG := 0.1258418194e13;
> > > > >
> > > > +,- sign for IMIF(r)
> > > > > >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > > >
> > > > >
> > > > > Best Regards,
> > > > > Hannu Poropudas
> > > I calculated also coordinate time T series approximation up to 7 degree at r=MG,
> > > (REMARK: This is preliminary calculation I have not rechecked it yet):
> > >
> > > If my approximate calculations are correct, then it is possible to calculate more
> > > "quantities" in this strange black hole of two event horizons space-time of mine,
> > > if this is sensible at all?
> > >
> > > ># Two branches of coordinate time T series approx.30.10.2023 H.P.
> > >
> > > ># This coordinate time T is also complex number with two branches
> > > ># (real and Imaginary)
> > >
> > > >#Coordinate time T series approx. up to 7 degree
> > > ># function-(series approx function), not integrated here
> > > >#(+)branch only used error estimation (compare proper time case)
> > >
> > > ># +,- formula (Primitive function, Real part)
> > >
> > > >REIG:=r->-2.851064818*ln(abs(r-0.6292090968e12))-.8288703850*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
> > >
> > > ># error estimation max positive side about 8.3*10^(-9)
> > > ># error estimation max abs negative side about -4.2*10^(-9)
> > > ># Both max are at r=MG, other definition area error = about 0
> > >
> > > ># +,- formula (Primitive function, Imaginary part)
> > >
> > > >IMIG:=r->-1.425532409*(1-csgn(r-0.6292090968e12))*Pi+1.657740770*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
> > >
> > > ># error estimation max positive side about 3.2*10^(-9)
> > > ># error estimation max abs negative side about -6.4*10^(-9)
> > > ># Both max are at r=MG, other definition area error = about 0
> > >
> > > Best Regards,
> > > Hannu Poropudas
> > I'am sorry about error in 30.10.2023 posting of mine.
> >
> > Here is CORRECTED 30.10.2023 posting of mine
> >
> > ># CORRECTED. Two branches of coordinate time T series approx.31.10.2023 H.P.
> > ># This coordinate time T is also complex number with two branches ># (real and Imaginary)
> >
> > >#Coordinate time T series approx. up to 7 degree
> > ># function-(series approx function), not integrated here
> > >#(+)branch only used error estimation (compare proper time case)
> >
> > ># +,- formula (Primitive function, Real part)
> > >REIG:=r->-1.425532409*ln(abs(r-0.6292090968e12))-.4144351924*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
> >
> > ># error estimation max positive side about 7.5*10^(-13)
> > ># error estimation max abs negative side about -1.3*10^(-12)
> > ># +,- formula (Primitive function, Imaginary part)
> > >IMIG:=r->-0.7127662045*(1-csgn(r-0.6292090968e12))*Pi+0.8288703849*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
> >
> > ># error estimation max positive side about 4.8*10^(-13)
> > ># error estimation max abs negative side about -2.4*10^(-13)
> >
> > Best Regards,
> > Hannu Poropudas
> ONE NOTE about one "little strange" function used in my maple calculations
>
> csgn(r-0.6292090968e12) = (r-0.6292090968e12)/abs(r-0.6292090968e12)
>
> Hannu


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Re: I found one spherically symmetric solution of Einstein's vacuum field equations

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Subject: Re: I found one spherically symmetric solution of Einstein's vacuum
field equations
From: haporop...@gmail.com (Hannu Poropudas)
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 by: Hannu Poropudas - Thu, 2 Nov 2023 09:41 UTC

torstai 2. marraskuuta 2023 klo 10.55.39 UTC+2 Ross Finlayson kirjoitti:
> On Tuesday, October 31, 2023 at 4:41:56 AM UTC-7, Hannu Poropudas wrote:
> > tiistai 31. lokakuuta 2023 klo 10.40.22 UTC+2 Hannu Poropudas kirjoitti:
> > > maanantai 30. lokakuuta 2023 klo 12.35.05 UTC+2 Hannu Poropudas kirjoitti:
> > > > maanantai 30. lokakuuta 2023 klo 9.59.22 UTC+2 Hannu Poropudas kirjoitti:
> > > > > perjantai 27. lokakuuta 2023 klo 10.46.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > torstai 26. lokakuuta 2023 klo 11.04.40 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > keskiviikko 25. lokakuuta 2023 klo 14.37.35 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > keskiviikko 25. lokakuuta 2023 klo 12.01.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > tiistai 24. lokakuuta 2023 klo 11.56.49 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > perjantai 20. lokakuuta 2023 klo 9.54.12 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > torstai 19. lokakuuta 2023 klo 21.41.08 UTC+3 JanPB kirjoitti:
> > > > > > > > > > > > On Thursday, October 19, 2023 at 12:22:43 AM UTC-7, Hannu Poropudas wrote:
> > > > > > > > > > > > > sunnuntai 15. lokakuuta 2023 klo 11.35.22 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > > > > Spherically symmetric metrics which satisfies
> > > > > > > > > > > > > > Einstein's vacuum field equations.
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > matrix([[m^2/((1-m/r)^4*r^4*(1-2*m/r)), 0, 0, 0], [0, -1/(1-m/r)^2, 0, 0], [0, 0, -sin(theta)^2/(1-m/r)^2, 0], [0, 0, 0, 1-2*m/r]])])
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > ds^2=(m^2/((1-m/r)^4*r^4*(1-2*m/r)))*dr^2-(1/(1-m/r)^2)*dtheta^2-(sin(theta)^2/(1-m/r)^2)*dphi^2+(1-2*m/r)*dt^2
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > (m -> m*G/c^2 , if SI-units are used.)
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > I don't know that would this solution have any astrophysical applications?
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > There exist a book called something like
> > > > > > > > > > > > > > "Exact Solutions of the Einstein Field Equations",
> > > > > > > > > > > > > > which have about 740 pages and
> > > > > > > > > > > > > > I don't know if this solution is among them?
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > Three singularity points of the metrics are the following:
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > r = 0, r = m*G/c^2 and r = 2*m*G/c^2.
> > > > > > > > > > > > > >
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > I have used generalized form of eq. (9) on page 171, when I calculated this solution with my Maple 9.
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > Reference:
> > > > > > > > > > > > > > Tolman R. C., 1934.
> > > > > > > > > > > > > > Effect of inhomogeneity on cosmological models.
> > > > > > > > > > > > > > Proc. Natl. Acad. Sci. USA, 1934, Mar; 20 (30): 1679-176.
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > Best Regrads,
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > Kolamäentie 9E
> > > > > > > > > > > > > > 90900 Kiiminki / Oulu
> > > > > > > > > > > > > > Finland
> > > > > > > > > > > > > I used random numbers for arbitrary example , which I don't know if it is sensible at all in this case
> > > > > > > > > > > > > due three integration constants from Euler-Lagrange equations does not have
> > > > > > > > > > > > > same interpretations as in Schwarzschild case (energy (constant) and angular momementum (constant) etc).
> > > > > > > > > > > > > I used these guess numbers from aphelion and perihelion of S2-star around SgrA* black hole which calculations I published
> > > > > > > > > > > > > some time ago in this sci.physics.relativity Google Group. (I used c=1 units , c.g.s units then and I use again c=1 units and c.g.s units here)
> > > > > > > > > > > > >
> > > > > > > > > > > > > MG = 6.292090968*10^11,
> > > > > > > > > > > > > 2*MG=1.258418194*10^12.
> > > > > > > > > > > > > I found two parametric form analytic solutions (Both are primitive functions, 0<=P<=Pi/2):
> > > > > > > > > > > > >
> > > > > > > > > > > > > 2.720522631*10^11<=r<=8.306841627*10^11
> > > > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > > > phi= Int(-0.8328841065*I/sqrt(1-0.5394753492*sin(P)^2),P)
> > > > > > > > > > > > > r=-2.259895064*10^23/(5.586318996*10^11*sin(P)^2-8.306841627*10^11)
> > > > > > > > > > > > >
> > > > > > > > > > > > > and
> > > > > > > > > > > > >
> > > > > > > > > > > > > -1.103327381*10^12<=rr<=0
> > > > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > > > phiphi=Int(-0.8330796374*I/sqrt(1-0.4605246509*sin(P)^2),P)
> > > > > > > > > > > > > rr= (9.165165817*10^23*sin(P)^2-9.165165817*10^23)/(1.103327381*10^12*sin(P)^2+8.306841627*10^11)
> > > > > > > > > > > > >
> > > > > > > > > > > > > I calculated also these integrals but their formulae are too long to copy here.
> > > > > > > > > > > > > Both sign can be taken into account when plotting Imaginary parts 0<=P<=Pi.
> > > > > > > > > > > > > Real parts = 0 in these integrals.
> > > > > > > > > > > > > How to interpret pure imaginary phi and phiphi angles?
> > > > > > > > > > > > > How to interpret these Imaginary angle plots?
> > > > > > > > > > > > >
> > > > > > > > > > > > > Best Regards,
> > > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > > Your solution is either:
> > > > > > > > > > > >
> > > > > > > > > > > > (a) incorrect, or:
> > > > > > > > > > > >
> > > > > > > > > > > > (b) isometric to Schwarzschild's.
> > > > > > > > > > > >
> > > > > > > > > > > > Don't waste your time.
> > > > > > > > > > > >
> > > > > > > > > > > > --
> > > > > > > > > > > > Jan
> > > > > > > > > > > Your (b) alternative seems not to be true due two separate event horizons in this metrics ?
> > > > > > > > > > >
> > > > > > > > > > > Schwarzschild metric comes also correctly, but with different sign selection in metrics than what I used.
> > > > > > > > > > >
> > > > > > > > > > > Your (a) alternative is not true due this metric satisfies Einstein's vacuum field equations,
> > > > > > > > > > > but you are correct in point of view that it may not be physically acceptable solution
> > > > > > > > > > > of these equations at our present orthodoxic physical knowledge.
> > > > > > > > > > > This is indicated by imaginary unit (I=sqrt(-1)) in these example of two analytic solutions.
> > > > > > > > > > >
> > > > > > > > > > > There exist also few other integration constants from Euler-Largrange equations,
> > > > > > > > > > > but I have selected randomly only one couple of them in this example calculation.
> > > > > > > > > > >
> > > > > > > > > > > Hannu
> > > > > > > > > > I put here those strange (NO ordinary physical interpretation) formulae of integration
> > > > > > > > > > constants from Euler-Largrange equations:
> > > > > > > > > >
> > > > > > > > > > I mark now for convenience T = coordinate time and t = proper time.
> > > > > > > > > >
> > > > > > > > > > (dphi/dt)/(1-m/r)^2 = K1 (constant of integration)
> > > > > > > > > > (1-2*m/r)*(dT/dt) = K2 (constant of integration)
> > > > > > > > > > (1-2*m/r)*(dT/dt)^2 - m^2*(dr/dt)^2 / ( (1-m/r)^4*r^4*(1-2*m/r) ) - (dphi/dt)^2 / (1-m/r)^2 = 1.
> > > > > > > > > >
> > > > > > > > > > I calculated for randomly selected numerical values of S2-star aphelion and perhelion
> > > > > > > > > > distances (c=1 units, and c.g.s units) from my earlier calculations of analytic GR solutions
> > > > > > > > > > for S2-star orbit around SgrA* black hole (sci.physics.relativity published)
> > > > > > > > > > to calculate two integration constants K1 and K2 of Euler-Largange equations
> > > > > > > > > > (NO ordinary physical interpretation), (I = sqrt(-1) = imaginary unit):
> > > > > > > > > >
> > > > > > > > > > K1 = +,- 0.7072727132*I,
> > > > > > > > > > K2 = +,- 0.5943942676 +,- 0.5943942676*I,
> > > > > > > > > >
> > > > > > > > > > And I selected here randomly as an example two constants of integration
> > > > > > > > > > in this my two analytic solutions calculation:
> > > > > > > > > >
> > > > > > > > > > K1 = - 0.7072727132*I
> > > > > > > > > > and
> > > > > > > > > > K2 = 0.5943942676 - 0.5943942676*I
> > > > > > > > > >
> > > > > > > > > > This selection gave those two pure imaginary analytic solutions which I gave here earlier.
> > > > > > > > > > (Phi(P) is pure imaginary angle and r(P) is real distance.
> > > > > > > > > > Phiphi(P) is pure imaginary angle and rr(P) is real distance).
> > > > > > > > > >
> > > > > > > > > > Plot ([Im(phi(P)),r(P),P=0..Pi]);
> > > > > > > > > > Plot ([Im(phiphi(P)),rr(P),P=0..Pi]);
> > > > > > > > > > gives both +, - solutions in both cases (P..Pi/2 gives only one branch and P..Pi gives both branches)
> > > > > > > > > >
> > > > > > > > > > Those both plots resemble somehow pendulum orbit ?
> > > > > > > > > >
> > > > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > > > and I think that these have NO real physical applications.
> > > > > > > > > >
> > > > > > > > > > Hannu Poropudas
> > > > > > > > > I investigated also question that what kind of coordinate time (T) solution would be in parametric form ?
> > > > > > > > >
> > > > > > > > > It seems to me that this integral is too complicated to calculate analytically, but it could be so
> > > > > > > > > with those above K1 and K2 (plus K3 = 0 additional integration constant in Euler-Lagrange equations)
> > > > > > > > > in this above case that the coordinate time T could be two dimensional complex number ?
> > > > > > > > >
> > > > > > > > > This also seems to support what I said above.
> > > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > > and I think at the moment that these have NO real physical applications.
> > > > > > > > >
> > > > > > > > > And we should study two dimensional complex mathematics of two dimensional
> > > > > > > > > coordinate time (T) in this complicated integral better,
> > > > > > > > > if we try to better understand this situation,
> > > > > > > > > if this would be sensible at all ?
> > > > > > > > >
> > > > > > > > > Best Regards,
> > > > > > > > > Hannu Poropudas
> > > > > > > > CORRECTION: It is proper time (t) integral in question, not coordinate time (T).
> > > > > > > > Sorry that I confused these two letters.
> > > > > > > >
> > > > > > > > Hannu
> > > > > > > I found one interesting reference, which show that there
> > > > > > > are really only few astrophysically significant exact solutions to Einstein's field equations.
> > > > > > >
> > > > > > > Ishak, M. 2015.
> > > > > > > Exact Solutions to Einstein's Equations in Astrophysics.
> > > > > > > Texas Symposium on Relativistic Astrophysics, Geneva 2015.
> > > > > > > 33 pages.
> > > > > > > https://personal.utdallas.edu/~mishak/ExactSolutionsInAstrophysics_Ishak_Final.pdf
> > > > > > >
> > > > > > > Please take a look.
> > > > > > >
> > > > > > > Best Regards,
> > > > > > > Hannu Poropudas
> > > > > > In order to me more mathematically complete I calculate also
> > > > > > approximate proper time t integral (primitive function)
> > > > > > and plotted both real part and imaginary part of it.
> > > > > > I have NO interpretations of these.
> > > > > >
> > > > > > ># Approximate proper time t integral calculated HP 27.10.2023
> > > > > > ># REMARK: My letter convenience t=proper time T=coordinate time
> > > > > > ># Real part and Imaginary part plotted
> > > > > > >#K3:=0;
> > > > > > >#K1 := -0.7072727132*I;
> > > > > > >#K2 := 0.5943942676-0.5943942676*I;
> > > > > > >#m := MG;
> > > > > > >#MG := 0.6292090968e12;
> > > > > > >#2*MG := 0.1258418194e13;
> > > > > > >#a2<=r<=a1, definition area
> > > > > > >#a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > >
> > > > > > >#Real part of primitive function t approx.
> > > > > > ># Series approx at r = MG up to 7 degree.
> > > > > +,- sign for REIF(r)
> > > > > > >REIF:=r->-0.9292411964e-8*r+0.8717127610e-20*r^2+0.4446277653e-31*(r-0.6292090968e12)^3+0.2329675135e-42*(r-0.6292090968e12)^4+0.3071201158e-47*(r-0.6292090968e12)^5+0.2827253683e-58*(r-0.6292090968e12)^6+0.2065510967e-69*(r-0.6292090968e12)^7;
> > > > > >
> > > > > > >#Imaginary part of primitive function t approx.
> > > > > > >># Series approx at r = MG up to 7 degree.
> > > > > +,- sign for IMIF(r)
> > > > > > >IMIF:=r->-0.2217185446e-7*r+0.3637453960e-19*r^2+0.1018586871e-30*(r-0.6292090968e12)^3+0.3376062792e-42*(r-0.6292090968e12)^4+0.4794788388e-47*(r-0.6292090968e12)^5+0.2475402813e-58*(r-0.6292090968e12)^6+0.1233132761e-69*(r-0.6292090968e12)^7;
> > > > > >
> > > > > > > #a2<=r<=a1, definition area
> > > > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > > > #MG := 0.6292090968e12;
> > > > > > > #2*MG := 0.1258418194e13;
> > > > > >
> > > > > +,- sign for REIF(r)
> > > > > > >plot(REIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > > > >
> > > > > > > #a2<=r<=a1, definition area
> > > > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > > > #MG := 0.6292090968e12;
> > > > > > > #2*MG := 0.1258418194e13;
> > > > > >
> > > > > +,- sign for IMIF(r)
> > > > > > >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > > > >
> > > > > >
> > > > > > Best Regards,
> > > > > > Hannu Poropudas
> > > > I calculated also coordinate time T series approximation up to 7 degree at r=MG,
> > > > (REMARK: This is preliminary calculation I have not rechecked it yet):
> > > >
> > > > If my approximate calculations are correct, then it is possible to calculate more
> > > > "quantities" in this strange black hole of two event horizons space-time of mine,
> > > > if this is sensible at all?
> > > >
> > > > ># Two branches of coordinate time T series approx.30.10.2023 H.P.
> > > >
> > > > ># This coordinate time T is also complex number with two branches
> > > > ># (real and Imaginary)
> > > >
> > > > >#Coordinate time T series approx. up to 7 degree
> > > > ># function-(series approx function), not integrated here
> > > > >#(+)branch only used error estimation (compare proper time case)
> > > >
> > > > ># +,- formula (Primitive function, Real part)
> > > >
> > > > >REIG:=r->-2.851064818*ln(abs(r-0.6292090968e12))-.8288703850*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
> > > >
> > > > ># error estimation max positive side about 8.3*10^(-9)
> > > > ># error estimation max abs negative side about -4.2*10^(-9)
> > > > ># Both max are at r=MG, other definition area error = about 0
> > > >
> > > > ># +,- formula (Primitive function, Imaginary part)
> > > >
> > > > >IMIG:=r->-1.425532409*(1-csgn(r-0.6292090968e12))*Pi+1.657740770*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
> > > >
> > > > ># error estimation max positive side about 3.2*10^(-9)
> > > > ># error estimation max abs negative side about -6.4*10^(-9)
> > > > ># Both max are at r=MG, other definition area error = about 0
> > > >
> > > > Best Regards,
> > > > Hannu Poropudas
> > > I'am sorry about error in 30.10.2023 posting of mine.
> > >
> > > Here is CORRECTED 30.10.2023 posting of mine
> > >
> > > ># CORRECTED. Two branches of coordinate time T series approx.31.10.2023 H.P.
> > > ># This coordinate time T is also complex number with two branches ># (real and Imaginary)
> > >
> > > >#Coordinate time T series approx. up to 7 degree
> > > ># function-(series approx function), not integrated here
> > > >#(+)branch only used error estimation (compare proper time case)
> > >
> > > ># +,- formula (Primitive function, Real part)
> > > >REIG:=r->-1.425532409*ln(abs(r-0.6292090968e12))-.4144351924*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
> > >
> > > ># error estimation max positive side about 7.5*10^(-13)
> > > ># error estimation max abs negative side about -1.3*10^(-12)
> > > ># +,- formula (Primitive function, Imaginary part)
> > > >IMIG:=r->-0.7127662045*(1-csgn(r-0.6292090968e12))*Pi+0.8288703849*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
> > >
> > > ># error estimation max positive side about 4.8*10^(-13)
> > > ># error estimation max abs negative side about -2.4*10^(-13)
> > >
> > > Best Regards,
> > > Hannu Poropudas
> > ONE NOTE about one "little strange" function used in my maple calculations
> >
> > csgn(r-0.6292090968e12) = (r-0.6292090968e12)/abs(r-0.6292090968e12)
> >
> > Hannu
> You mentioned the sign term reflecting that earlier you wrote from your derivation,
> that part of it was as under-defined or de facto, a compensating term.
>
> It seems what you are integrating is power terms resolving, why the halfs either
> way have a "pseudo" product, what is a law that results why in your terms, they add
> up, if you haven't explained "why" it's legal those terms wouldn't resolve,
> radiating usually.
>
> It seems those would be waves falling so would result "why", is because, they
> are under the area terms, the absorption or radiation, how then those have to
> add up, to make the estimate, which as you note appears accurate.


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Re: I found one spherically symmetric solution of Einstein's vacuum field equations

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Subject: Re: I found one spherically symmetric solution of Einstein's vacuum
field equations
From: haporop...@gmail.com (Hannu Poropudas)
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 by: Hannu Poropudas - Fri, 3 Nov 2023 10:01 UTC

torstai 2. marraskuuta 2023 klo 11.41.57 UTC+2 Hannu Poropudas kirjoitti:
> torstai 2. marraskuuta 2023 klo 10.55.39 UTC+2 Ross Finlayson kirjoitti:
> > On Tuesday, October 31, 2023 at 4:41:56 AM UTC-7, Hannu Poropudas wrote:
> > > tiistai 31. lokakuuta 2023 klo 10.40.22 UTC+2 Hannu Poropudas kirjoitti:
> > > > maanantai 30. lokakuuta 2023 klo 12.35.05 UTC+2 Hannu Poropudas kirjoitti:
> > > > > maanantai 30. lokakuuta 2023 klo 9.59.22 UTC+2 Hannu Poropudas kirjoitti:
> > > > > > perjantai 27. lokakuuta 2023 klo 10.46.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > torstai 26. lokakuuta 2023 klo 11.04.40 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > keskiviikko 25. lokakuuta 2023 klo 14.37.35 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > keskiviikko 25. lokakuuta 2023 klo 12.01.55 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > tiistai 24. lokakuuta 2023 klo 11.56.49 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > perjantai 20. lokakuuta 2023 klo 9.54.12 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > > torstai 19. lokakuuta 2023 klo 21.41.08 UTC+3 JanPB kirjoitti:
> > > > > > > > > > > > > On Thursday, October 19, 2023 at 12:22:43 AM UTC-7, Hannu Poropudas wrote:
> > > > > > > > > > > > > > sunnuntai 15. lokakuuta 2023 klo 11.35.22 UTC+3 Hannu Poropudas kirjoitti:
> > > > > > > > > > > > > > > Spherically symmetric metrics which satisfies
> > > > > > > > > > > > > > > Einstein's vacuum field equations.
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > matrix([[m^2/((1-m/r)^4*r^4*(1-2*m/r)), 0, 0, 0], [0, -1/(1-m/r)^2, 0, 0], [0, 0, -sin(theta)^2/(1-m/r)^2, 0], [0, 0, 0, 1-2*m/r]])])
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > (c=1,G=1 units)
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > ds^2=(m^2/((1-m/r)^4*r^4*(1-2*m/r)))*dr^2-(1/(1-m/r)^2)*dtheta^2-(sin(theta)^2/(1-m/r)^2)*dphi^2+(1-2*m/r)*dt^2
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > (m -> m*G/c^2 , if SI-units are used.)
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > I don't know that would this solution have any astrophysical applications?
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > There exist a book called something like
> > > > > > > > > > > > > > > "Exact Solutions of the Einstein Field Equations",
> > > > > > > > > > > > > > > which have about 740 pages and
> > > > > > > > > > > > > > > I don't know if this solution is among them?
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > Three singularity points of the metrics are the following:
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > r = 0, r = m*G/c^2 and r = 2*m*G/c^2.
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > I have used generalized form of eq. (9) on page 171, when I calculated this solution with my Maple 9.
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > Reference:
> > > > > > > > > > > > > > > Tolman R. C., 1934.
> > > > > > > > > > > > > > > Effect of inhomogeneity on cosmological models.
> > > > > > > > > > > > > > > Proc. Natl. Acad. Sci. USA, 1934, Mar; 20 (30): 1679-176.
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > Best Regrads,
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > > > > >
> > > > > > > > > > > > > > > Kolamäentie 9E
> > > > > > > > > > > > > > > 90900 Kiiminki / Oulu
> > > > > > > > > > > > > > > Finland
> > > > > > > > > > > > > > I used random numbers for arbitrary example , which I don't know if it is sensible at all in this case
> > > > > > > > > > > > > > due three integration constants from Euler-Lagrange equations does not have
> > > > > > > > > > > > > > same interpretations as in Schwarzschild case (energy (constant) and angular momementum (constant) etc).
> > > > > > > > > > > > > > I used these guess numbers from aphelion and perihelion of S2-star around SgrA* black hole which calculations I published
> > > > > > > > > > > > > > some time ago in this sci.physics.relativity Google Group. (I used c=1 units , c.g.s units then and I use again c=1 units and c.g.s units here)
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > MG = 6.292090968*10^11,
> > > > > > > > > > > > > > 2*MG=1.258418194*10^12.
> > > > > > > > > > > > > > I found two parametric form analytic solutions (Both are primitive functions, 0<=P<=Pi/2):
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > 2.720522631*10^11<=r<=8.306841627*10^11
> > > > > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > > > > phi= Int(-0.8328841065*I/sqrt(1-0.5394753492*sin(P)^2),P)
> > > > > > > > > > > > > > r=-2.259895064*10^23/(5.586318996*10^11*sin(P)^2-8.306841627*10^11)
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > and
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > -1.103327381*10^12<=rr<=0
> > > > > > > > > > > > > > +,- sign for integral
> > > > > > > > > > > > > > phiphi=Int(-0.8330796374*I/sqrt(1-0.4605246509*sin(P)^2),P)
> > > > > > > > > > > > > > rr= (9.165165817*10^23*sin(P)^2-9.165165817*10^23)/(1.103327381*10^12*sin(P)^2+8.306841627*10^11)
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > I calculated also these integrals but their formulae are too long to copy here.
> > > > > > > > > > > > > > Both sign can be taken into account when plotting Imaginary parts 0<=P<=Pi.
> > > > > > > > > > > > > > Real parts = 0 in these integrals.
> > > > > > > > > > > > > > How to interpret pure imaginary phi and phiphi angles?
> > > > > > > > > > > > > > How to interpret these Imaginary angle plots?
> > > > > > > > > > > > > >
> > > > > > > > > > > > > > Best Regards,
> > > > > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > > > > Your solution is either:
> > > > > > > > > > > > >
> > > > > > > > > > > > > (a) incorrect, or:
> > > > > > > > > > > > >
> > > > > > > > > > > > > (b) isometric to Schwarzschild's.
> > > > > > > > > > > > >
> > > > > > > > > > > > > Don't waste your time.
> > > > > > > > > > > > >
> > > > > > > > > > > > > --
> > > > > > > > > > > > > Jan
> > > > > > > > > > > > Your (b) alternative seems not to be true due two separate event horizons in this metrics ?
> > > > > > > > > > > >
> > > > > > > > > > > > Schwarzschild metric comes also correctly, but with different sign selection in metrics than what I used.
> > > > > > > > > > > >
> > > > > > > > > > > > Your (a) alternative is not true due this metric satisfies Einstein's vacuum field equations,
> > > > > > > > > > > > but you are correct in point of view that it may not be physically acceptable solution
> > > > > > > > > > > > of these equations at our present orthodoxic physical knowledge.
> > > > > > > > > > > > This is indicated by imaginary unit (I=sqrt(-1)) in these example of two analytic solutions.
> > > > > > > > > > > >
> > > > > > > > > > > > There exist also few other integration constants from Euler-Largrange equations,
> > > > > > > > > > > > but I have selected randomly only one couple of them in this example calculation.
> > > > > > > > > > > >
> > > > > > > > > > > > Hannu
> > > > > > > > > > > I put here those strange (NO ordinary physical interpretation) formulae of integration
> > > > > > > > > > > constants from Euler-Largrange equations:
> > > > > > > > > > >
> > > > > > > > > > > I mark now for convenience T = coordinate time and t = proper time.
> > > > > > > > > > >
> > > > > > > > > > > (dphi/dt)/(1-m/r)^2 = K1 (constant of integration)
> > > > > > > > > > > (1-2*m/r)*(dT/dt) = K2 (constant of integration)
> > > > > > > > > > > (1-2*m/r)*(dT/dt)^2 - m^2*(dr/dt)^2 / ( (1-m/r)^4*r^4*(1-2*m/r) ) - (dphi/dt)^2 / (1-m/r)^2 = 1.
> > > > > > > > > > >
> > > > > > > > > > > I calculated for randomly selected numerical values of S2-star aphelion and perhelion
> > > > > > > > > > > distances (c=1 units, and c.g.s units) from my earlier calculations of analytic GR solutions
> > > > > > > > > > > for S2-star orbit around SgrA* black hole (sci.physics.relativity published)
> > > > > > > > > > > to calculate two integration constants K1 and K2 of Euler-Largange equations
> > > > > > > > > > > (NO ordinary physical interpretation), (I = sqrt(-1) = imaginary unit):
> > > > > > > > > > >
> > > > > > > > > > > K1 = +,- 0.7072727132*I,
> > > > > > > > > > > K2 = +,- 0.5943942676 +,- 0.5943942676*I,
> > > > > > > > > > >
> > > > > > > > > > > And I selected here randomly as an example two constants of integration
> > > > > > > > > > > in this my two analytic solutions calculation:
> > > > > > > > > > >
> > > > > > > > > > > K1 = - 0.7072727132*I
> > > > > > > > > > > and
> > > > > > > > > > > K2 = 0.5943942676 - 0.5943942676*I
> > > > > > > > > > >
> > > > > > > > > > > This selection gave those two pure imaginary analytic solutions which I gave here earlier.
> > > > > > > > > > > (Phi(P) is pure imaginary angle and r(P) is real distance.
> > > > > > > > > > > Phiphi(P) is pure imaginary angle and rr(P) is real distance).
> > > > > > > > > > >
> > > > > > > > > > > Plot ([Im(phi(P)),r(P),P=0..Pi]);
> > > > > > > > > > > Plot ([Im(phiphi(P)),rr(P),P=0..Pi]);
> > > > > > > > > > > gives both +, - solutions in both cases (P..Pi/2 gives only one branch and P..Pi gives both branches)
> > > > > > > > > > >
> > > > > > > > > > > Those both plots resemble somehow pendulum orbit ?
> > > > > > > > > > >
> > > > > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > > > > and I think that these have NO real physical applications.
> > > > > > > > > > >
> > > > > > > > > > > Hannu Poropudas
> > > > > > > > > > I investigated also question that what kind of coordinate time (T) solution would be in parametric form ?
> > > > > > > > > >
> > > > > > > > > > It seems to me that this integral is too complicated to calculate analytically, but it could be so
> > > > > > > > > > with those above K1 and K2 (plus K3 = 0 additional integration constant in Euler-Lagrange equations)
> > > > > > > > > > in this above case that the coordinate time T could be two dimensional complex number ?
> > > > > > > > > >
> > > > > > > > > > This also seems to support what I said above.
> > > > > > > > > > I have NO physical interpretations of these solutions
> > > > > > > > > > and I think at the moment that these have NO real physical applications.
> > > > > > > > > >
> > > > > > > > > > And we should study two dimensional complex mathematics of two dimensional
> > > > > > > > > > coordinate time (T) in this complicated integral better,
> > > > > > > > > > if we try to better understand this situation,
> > > > > > > > > > if this would be sensible at all ?
> > > > > > > > > >
> > > > > > > > > > Best Regards,
> > > > > > > > > > Hannu Poropudas
> > > > > > > > > CORRECTION: It is proper time (t) integral in question, not coordinate time (T).
> > > > > > > > > Sorry that I confused these two letters.
> > > > > > > > >
> > > > > > > > > Hannu
> > > > > > > > I found one interesting reference, which show that there
> > > > > > > > are really only few astrophysically significant exact solutions to Einstein's field equations.
> > > > > > > >
> > > > > > > > Ishak, M. 2015.
> > > > > > > > Exact Solutions to Einstein's Equations in Astrophysics.
> > > > > > > > Texas Symposium on Relativistic Astrophysics, Geneva 2015.
> > > > > > > > 33 pages.
> > > > > > > > https://personal.utdallas.edu/~mishak/ExactSolutionsInAstrophysics_Ishak_Final.pdf
> > > > > > > >
> > > > > > > > Please take a look.
> > > > > > > >
> > > > > > > > Best Regards,
> > > > > > > > Hannu Poropudas
> > > > > > > In order to me more mathematically complete I calculate also
> > > > > > > approximate proper time t integral (primitive function)
> > > > > > > and plotted both real part and imaginary part of it.
> > > > > > > I have NO interpretations of these.
> > > > > > >
> > > > > > > ># Approximate proper time t integral calculated HP 27.10.2023
> > > > > > > ># REMARK: My letter convenience t=proper time T=coordinate time
> > > > > > > ># Real part and Imaginary part plotted
> > > > > > > >#K3:=0;
> > > > > > > >#K1 := -0.7072727132*I;
> > > > > > > >#K2 := 0.5943942676-0.5943942676*I;
> > > > > > > >#m := MG;
> > > > > > > >#MG := 0.6292090968e12;
> > > > > > > >#2*MG := 0.1258418194e13;
> > > > > > > >#a2<=r<=a1, definition area
> > > > > > > >#a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > > >
> > > > > > > >#Real part of primitive function t approx.
> > > > > > > ># Series approx at r = MG up to 7 degree.
> > > > > > +,- sign for REIF(r)
> > > > > > > >REIF:=r->-0.9292411964e-8*r+0.8717127610e-20*r^2+0.4446277653e-31*(r-0.6292090968e12)^3+0.2329675135e-42*(r-0.6292090968e12)^4+0.3071201158e-47*(r-0.6292090968e12)^5+0.2827253683e-58*(r-0.6292090968e12)^6+0.2065510967e-69*(r-0.6292090968e12)^7;
> > > > > > >
> > > > > > > >#Imaginary part of primitive function t approx.
> > > > > > > >># Series approx at r = MG up to 7 degree.
> > > > > > +,- sign for IMIF(r)
> > > > > > > >IMIF:=r->-0.2217185446e-7*r+0.3637453960e-19*r^2+0.1018586871e-30*(r-0.6292090968e12)^3+0.3376062792e-42*(r-0.6292090968e12)^4+0.4794788388e-47*(r-0.6292090968e12)^5+0.2475402813e-58*(r-0.6292090968e12)^6+0.1233132761e-69*(r-0.6292090968e12)^7;
> > > > > > >
> > > > > > > > #a2<=r<=a1, definition area
> > > > > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > > > > #MG := 0.6292090968e12;
> > > > > > > > #2*MG := 0.1258418194e13;
> > > > > > >
> > > > > > +,- sign for REIF(r)
> > > > > > > >plot(REIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > > > > >
> > > > > > > > #a2<=r<=a1, definition area
> > > > > > > > #a2=2.720522631*10^11, a1=8.306841627*10^11
> > > > > > > > #MG := 0.6292090968e12;
> > > > > > > > #2*MG := 0.1258418194e13;
> > > > > > >
> > > > > > +,- sign for IMIF(r)
> > > > > > > >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
> > > > > > >
> > > > > > >
> > > > > > > Best Regards,
> > > > > > > Hannu Poropudas
> > > > > I calculated also coordinate time T series approximation up to 7 degree at r=MG,
> > > > > (REMARK: This is preliminary calculation I have not rechecked it yet):
> > > > >
> > > > > If my approximate calculations are correct, then it is possible to calculate more
> > > > > "quantities" in this strange black hole of two event horizons space-time of mine,
> > > > > if this is sensible at all?
> > > > >
> > > > > ># Two branches of coordinate time T series approx.30.10.2023 H.P..
> > > > >
> > > > > ># This coordinate time T is also complex number with two branches
> > > > > ># (real and Imaginary)
> > > > >
> > > > > >#Coordinate time T series approx. up to 7 degree
> > > > > ># function-(series approx function), not integrated here
> > > > > >#(+)branch only used error estimation (compare proper time case)
> > > > >
> > > > > ># +,- formula (Primitive function, Real part)
> > > > >
> > > > > >REIG:=r->-2.851064818*ln(abs(r-0.6292090968e12))-.8288703850*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
> > > > >
> > > > > ># error estimation max positive side about 8.3*10^(-9)
> > > > > ># error estimation max abs negative side about -4.2*10^(-9)
> > > > > ># Both max are at r=MG, other definition area error = about 0
> > > > >
> > > > > ># +,- formula (Primitive function, Imaginary part)
> > > > >
> > > > > >IMIG:=r->-1.425532409*(1-csgn(r-0.6292090968e12))*Pi+1.657740770*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
> > > > >
> > > > > ># error estimation max positive side about 3.2*10^(-9)
> > > > > ># error estimation max abs negative side about -6.4*10^(-9)
> > > > > ># Both max are at r=MG, other definition area error = about 0
> > > > >
> > > > > Best Regards,
> > > > > Hannu Poropudas
> > > > I'am sorry about error in 30.10.2023 posting of mine.
> > > >
> > > > Here is CORRECTED 30.10.2023 posting of mine
> > > >
> > > > ># CORRECTED. Two branches of coordinate time T series approx.31.10..2023 H.P.
> > > > ># This coordinate time T is also complex number with two branches ># (real and Imaginary)
> > > >
> > > > >#Coordinate time T series approx. up to 7 degree
> > > > ># function-(series approx function), not integrated here
> > > > >#(+)branch only used error estimation (compare proper time case)
> > > >
> > > > ># +,- formula (Primitive function, Real part)
> > > > >REIG:=r->-1.425532409*ln(abs(r-0.6292090968e12))-.4144351924*(1-csgn(r-0.6292090968e12))*Pi+0.3330424925e12/(r-0.6292090968e12)-0.2939012715e-11*r;
> > > >
> > > > ># error estimation max positive side about 7.5*10^(-13)
> > > > ># error estimation max abs negative side about -1.3*10^(-12)
> > > > ># +,- formula (Primitive function, Imaginary part)
> > > > >IMIG:=r->-0.7127662045*(1-csgn(r-0.6292090968e12))*Pi+0.8288703849*ln(abs(r-0.6292090968e12))-0.2312741379e12/(r-0.6292090968e12)+0.1298075147e-11*r;
> > > >
> > > > ># error estimation max positive side about 4.8*10^(-13)
> > > > ># error estimation max abs negative side about -2.4*10^(-13)
> > > >
> > > > Best Regards,
> > > > Hannu Poropudas
> > > ONE NOTE about one "little strange" function used in my maple calculations
> > >
> > > csgn(r-0.6292090968e12) = (r-0.6292090968e12)/abs(r-0.6292090968e12)
> > >
> > > Hannu
> > You mentioned the sign term reflecting that earlier you wrote from your derivation,
> > that part of it was as under-defined or de facto, a compensating term.
> >
> > It seems what you are integrating is power terms resolving, why the halfs either
> > way have a "pseudo" product, what is a law that results why in your terms, they add
> > up, if you haven't explained "why" it's legal those terms wouldn't resolve,
> > radiating usually.
> >
> > It seems those would be waves falling so would result "why", is because, they
> > are under the area terms, the absorption or radiation, how then those have to
> > add up, to make the estimate, which as you note appears accurate.
> 1. I put more clearly my incomplete error estimation procedure here:
>
> Solution was actually +,- Int(function(r),r) and only +function(r) was used
> in error estimation, -function(r) was not used in error estimation for my convenience,
> so error estimation is was not complete in this sense, but it gives correctly order of the error?
>
> Error estimation was made only of (+) branch, of function(r) and (-) branch of function(r) was not used ,
> so error estimation was not complete in that sense, but I made so for my convenience not
> to make too long posting about error estimation.
>
> function(r) - (series approx of function(r)),
>
> function(r) is NOT integrated here due it is too complicated to do that.
>
> (+) branch of the function(r) only used in error estimation (compare proper time case,
> similar way was done in this case in error estimation) .
>
> 2. It should be remembered here that all mathematics in this special example was done with
> complex numbers inside r=2*MG event horizon.
>
> 3. I only tried to point out here that how complex mathematics can be used to make calculations in
> this special example case which I have selected to calculate completely.
>
> 4. I have made here NO physical interpretations about this special example, I leave to make them to those
> who understand astrophysics better than me, if this was sensible at all ?
>
> Best Regards,
> Hannu Poropudas


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