On Saturday, October 28, 2023 at 1:49:22 AM UTC-7, Hannu Poropudas wrote:
> 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

Hannu Poropudas, I have a challenge for you. I want you to use the new theory of gravitation, aka General Relativity, and your new spherical solution, to explicitly calculate the normal force of a 1.0 kg brick resting on my kitchen tabletop at an altitude of 100m above sea level in Seattle Warshington. Neither Tom Roberts nor Legion nor Gary Harnagel, nor Dono, nor Jan(s), nor Prokary, nor Pyth, nor Dirk, nor Paul B. Anderson, nor Athel will even attempt such an impossibly complex computation. Will YOU, Hannu Poropudas? If not, why not?