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.
> >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.
> >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;
>
> >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;
>
> >plot(IMIF(r),r=2.720522631*10^11..8.306841627*10^11);
>
>
> Best Regards,
> Hannu Poropudas

Error estimations of these two series approximations (function - (series approx function), not integrated here):

For REIF(r) : Error is approximately = 0 when 5.2*10^11 <r<7.2*10^11,
Max negative error about -2.15 near 2.72*10^11,
Max negative error about -0.2 near 8.307*10^11.

For IMIF(r): Error is approximately = 0 when 5.4*10^1<r<7.0*10^11,
Max negative error about -1.09 near 2.72*10^11,
Max negative error about -0.1 near 8.307*10^11.

These REIF(r) and IMIF(r) seems to me to be TWO BRANCHES of proper time t ?

Best Regards,
Hannu Poropudas