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MoreonSpiralGalaxies.doc

partial f / partial t = (1/2) omega df / dtheta

The connection, torsion and Riemann curvature elements can be found for any orbit because df / dtheta can be found from the orbit and omega can either be observed directly or found by the analysis of UFT206. The mean angular velocity of a planet is of course well known with great precision. So I can now proceed to write up UFT207 with co author Horst Eckardt, who has contributed to a rigorous discussion.

a207thpapernotes8.pdf

v bold = d(r e sub r bold) / dt

where e sub r bold is the radial unit vector (unitless) and r the radial coordinate (metres). This is important because it was shown in Section 3 of UFT196 that the acceleration calculated from the ellipse in cylindrical polar coordinates does not contain a centrifugal part. This means that the Newton theory is deeply self inconsistent. All the calculations with velocity and acceleration are given there in cylindrical polar coordinates. They are by no means straightforward, which is why the problem with Newton was not discovered for three hundred years. Examination of basic concepts is a difficult task, so only a few scholars have done it down the centuries. AIAS is outstanding in that it addresses basic concepts without peer pressure to conform to dogma. The velocity squared is then

v squared = (dr/dt) squared + r squared (dtheta / dt) squared

which has the right S. I. units. This gives the kinetic energy m v squared / 2. The so called “centrifugal effective potential energy” is the second term, despite the fact that it is clearly the rotational kinetic energy. However, as shown in Section 3 of UFT196, it does not exist, the force calculated directly from the elliptical Newtonian orbit does not contain the centrifugal force. This is why I rejected Newton’s concept of force in favour of geometry (Keplerian and earlier philosophy). I would have failed my O levels at Pontardawe Grammar School if I had done that there. Now I can concentrate on the real truth.

In a message dated 07/01/2012 19:37:20 GMT Standard Time

PS: A similar problem occurs with the definition of velocity in polar cordinates. The radial component is in m/s but the angular component is in radiants/s. I guess that this is analoguous for the torsion and Riemann tensor elements.

Horst

Am 07.01.2012 15:35, schrieb EMyrone

I checked the comprehensive computer output by Horst Eckardt once more and I put the torsion elements into the right S.I. units in eqs. (1) and (2) of this note. There should be a 1 / c for T sup 1 sub 01 and a 1 / r for T sup 1 sub 12. This gives the right S. I. units of torsion – inverse metres. In the output they are in normalized or non SI units. Each torsion element is twice the relevant connection element. The output can be put in the right SI units by noting that the time derivative must contain a 1 / c wherever it occurs, and the angle derivative must contain a 1 / r wherever it occurs. I give the mathematical origin of the 1 / r in eq. (14), the definition of the divergence in cylindrical polar coordinates. The important result is that the torsion is non zero for the hyeprbolic spiral (the observed spiral of a whirlpool galaxy), but it is zero for the logarithmic spiral, which is not observed. So spacetime torsion produces a hyperbolic spiral as observed in astronomy, another major advantage of ECE theory. Eqs. (1) to (3) of this note are valid for all orbits of any kind and are in the right S. I. units.

dr / dt = (dr / dtheta)(dtheta / dt)

and in the solar system dr / dtheta is found from the observations of a precessing ellipse, the precession of the perihelion corrected by supercomputer for the gravitational effects of other objects, and other corrections. All the concepts that we have used are spectacularly correct, and congratulations on the computer algebra. These are all major advances in cosmology.

a203rdpapernotes2.pdf