## Archive for November, 2009

### Generalization of Newton's Law and Derivation of the Equivalence Principle

November 29, 2009

The Newton law is generalized to eq. (7) and the vorticity law to eq. (6). The equivalence of gravitational and inertial mass is derived in eq. (11).

a140thpapernotes6.pdf

### Einsteinian Era Over

November 29, 2009

The typeset version of paper 139 will be posted shortly, and many thanks to the Alex Hill Company. This paper also contains an introductory essay for the general public and non specialist. The geometry used by Einstein unfortunately contains many sequential errors due to the basic error, the use of a symmetric connection. A more careful scholarly analysis than that given in the textbooks shows that the connection must always have the same antisymmetry as the commutator of covariant derivatives, and a commutator is antisymmetric in its indices by definition. An ECE based general relativity and cosmology has already been developed in the papers on this website, so scientists have a choice of adhering to incorrect concepts or using a plausible new general relativity based on Cartan’s differential geometry. The various demonstrations of ECE theory are quite simple to understand, even for the non specialist. Many demonstrations check themselves as described in the following postings. If errors in science are not acknowledged, the subject ceases to be science.

### Antisymmetry of the Connection

November 28, 2009

This is easy to see from the attached paper 137, eq. (4). The mu and nu indices of the connection are the same as the mu and nu indices of the commutator. When mu and nu are the same, the commutator and connection vanish. This simple proof is enough to show that any Riemannian geometry based on a symmetric connection is incorrect (paper 139 develops this conclusion). Any work that uses a symmetric connection is incorrect. This is a simple fact of geometry, eq. (4) is easy to understand even by the non-specialist. The connection must be antisymmetric because the commutator is antisymmetric. This means that when mu is replaced by nu and vice versa, the commutator changes sign. This is well known. Similarly the connection is antisymmetric in the sense that when mu is replaced by nu and vice versa, the connection changes sign. All connections of Riemann geometry are antisymmetric in their lower two indices. Further details are given in the attached notes 138(10).

a138thpapernotes10.pdf

Paper137withmath.pdf

### The Basic Simplicity of ECE Theory

November 28, 2009

The basic structure of ECE theory is the basic structure of Cartan’s differential geometry. This structure consists of the two Maurer Cartan structure equations:

T = d ^ q + omega ^ q

and

R = d ^ omega + omega ^ omega

in shorthand notation explained on this website. Here T is the torsion form, q is the tetrad form, omega is the spin connection, R is the curvature form. Given these structure equations, Cartan derived a well known identity:

d ^ T + omega ^ T := R ^ q

and this is proven in several papers of this website (notably paper 102, eq. (9.20)). Eq. (9.20) breaks out the Cartan identity into tensor notation, showing that one side of the identity is precisely identical to the other side, a self checking proof. This proof is usually an exercise for students in differential geometry. It is written out in paper 102 for the non-specialist. There also exists the Evans identity, which is

d ^ T* + omega ^ T* := R* ^ q

where * is the well known Hodge dual. The Evans identity is proven in full detail in paper 137, Eq. (30), which precisely parallels eq. (9.20) of paper 102. The Evans identity is a precise identity of differential geometry.

### Devices by the Alex Hill Company

November 28, 2009

AIAS Fellow Simon Clifford of Malvern Instruments recently visited the Alex Hill company in Mexico City (www.et3m.net) and was kindly shown a range of devices by Alex Hill. Simon Clifford was satisfied that they worked as specified, and they bring in a new era for energy hungry human kind. These devices are years ahead of any competition, and specifically, there exists an energy saving device for street lighting and similar. There are plans to build devices at Malvern and elsewhere using parts supplied by Alex Hill. These devices have also been observed by other AIAS Fellows, and also by civilians at the US Navy. The spin connection resonance theory was developed in response to these experimental results. Alex Hill kindly mentions the ECE theory on his website. The ECE theory is a plausible explanation of the devices in outline, we claim no more than that. The older Maxwell Heaviside theory has no explanation for them. Spin connection resonance is a straightforward theory based on Euler Bernoulli resonance as in the various papers on this website.

### Field Equations of Gravitation and Electrodynamics

November 26, 2009

These are written out here for ease of reference. They replace the incorrect Einstein field equation. Newtonian dynamics consists of a limit of one of the four field equations of gravitation, equations which contain the gravitational and gravitomagnetic fields. The overall structures of the equations of gravitation and electrodynamics are the same. They have been extensively tested experimentally in the past six years.

a140thpapernotes5.pdf

### Stokes and Spin Connection

November 25, 2009

The Stokes connection of the continuity equation in flow dynamics is shown to be a combination of spin connections of Cartan geometry, and the continuity equation shown to be the tetrad postulate of the density tetrad.

a140thpapernotes4.pdf

### The ECE Equations of Gravitation, the Density Tetrad

November 24, 2009

These are the four ECE equations of gravitation, introducing the density tetrad, the Newtonian limit and the static gravitomagnetic field (see papers 117 and 119). The density tetrad postulate is the continuity equation of flow dynamics, eq. (54).

a140thpapernotes3.pdf

### Derivation of the Navier Stokes Continuity Equation from Geometry

November 21, 2009

This is the derivation of the Navier Stokes continuity equation from the tetrad postulate. It is shown that the connection of the continuity equation is a spin connection of Cartan geometry.

a140thpapernotes2.pdf

### The Continuity Equation of the Navier Stokes System

November 21, 2009

It is shown that this equation is an equation of general relativity (eq. (12)) with a scalar valued connection gamma defined by eq. (11). The continuity equation is therefore a limiting form of the tetrad postulate, and the Stokes derivative is a covariant derivative of Riemann / Cartan geometry. The Navier Stokes system may therefore be derived straighforwardly as a limit of ECE theory.

a140thpapernotes1.pdf