Unravelling the dark matter--dark energy paradigm.(9 Generalised Dirac Equation Relativistic Effects in 3-Space-19 Conclusions)(Report)

An analogous generalisation of the Dirac equation is also necessary giving the coupling of the spinor to the actual dynamical 3-space, and again not to the embedding space as has been the case up until now:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (34)

where [??] and [beta] are the usual Dirac matrices. Repeating the analysis in (33) for the 3-space-induced acceleration we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (35)

which generalises (33) by having a term which limits the speed of the wave packet relative to 3-space, [absolute value of [v.sub.R]], to be G c. This equation specifies the trajectory of a spinor wave packet in the dynamical 3-space. The last term causes elliptical orbits to precess--for circular orbits [absolute value of [v.sub.R]] is independent of time.

10 Deriving the Spacetime Geodesic Formalism: Local Poincare Symmetry

We find that (35) may be also obtained by extremising the time-dilated elapsed time

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (35)

with respect to the wave-packet trajectory [r.sub.0](t) [6]. This happens because of the Fermat least-time effect for waves: only along the minimal time trajectory do the quantum waves remain in phase under small variations of the path. This again emphasises that gravity is a quantum matter wave effect. We now introduce an effective Spacetime mathematical construct according to the metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (37)

which is of the Panleve-Gullstrand class of metrics [24, 251. Then we have a Local Poinacre Symmetry, namely the transformations that leave [ds.sup.2] locally invariant under a change of coordinates. As well wave effects from (10) cause 'ripples' in this induced Spacetime, giving a different account of gravitational waves. The elapsed time in (36) may then be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (38)

The minimisation of (38) leads to the geodesics of the Spacetime, which are thus equivalent to the trajectories from (36), namely (35). We may introduce the standard differential geometry curvature tensor for the induced spacetime

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (39)

where [[GAMMA].sup.[alpha].sub.mu][sigma]v] is the affine connection for the metric in (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (40)

with [g.sup.[mu]v] the matrix inverse of [g.sub.[mu]v]. In this formalism the trajectories of quantum-matter wave-packet test objects are determined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (41)

as this is equivalent to (35). In the standard treatment of GR the geodesic for classical matter in (41) is a definition, and has no explanation. Here we see that it is finally derived, but as a quantum matter effect. Hence by coupling the Dirac spinor dynamics to the dynamical 3-space we derive the geodesic formalism of General Relativity as a quantum effect, but without reference to the Hilbert-Einstein equations for the induced metric. Indeed in general the metric of this induced spacetime will not satisfy these equations as the dynamical space involves the [alpha]-dependent dynamics, and [alpha] is missing from GR. We can also define the Ricci curvature scalar

R = [g.sup.[mu]v][R.sub.mu]v] (42)

where [R.sub.mu]v] = [R.sup.[alpha].sub.[mu][alpha]v]. In general the induced spacetime in (37) has a non-zero Ricci scalar--it is a curved spacetime. We shall compute the Ricci scalar for the expanding 3-space solution below.

We can also derive the Schwarzschild metric without reference to GR. To do this we merely have to identify the induced spacetime metric corresponding to the in-flow in (13) outside of a spherical matter system, such as the earth. Then (37) becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (43)

Making the change of variables t [right arrow] t' and r [roght arrow] r' = r with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (44)

this becomes (and now dropping the prime notation)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (45)

which is one form of the the Schwarzschild metric but with the [alpha]-dynamics induced effective mass shift. Of course this is only valid outside of the spherical matter distribution, as that is the proviso also on (13). Hence in the case of the Schwarzschild metric the dynamics missing from both the Newtonian theory of gravity and General Relativity is merely hidden in a mass redefinition, and so didn't affect the various standard tests of GR, or even of Newtonian gravity. A non-spherical symmetry version of the Schwarzchild metric is used in modelling the Global Positioning System (GPS).

11 Supernova and Gamma-Ray-Burst Data

In the next section we show that the 3-space dynamics in (10) has an expanding space solution. The supernovae and gammaray bursts provide standard candles that enable observation of the expansion of the universe. To test yet further that dynamics we compare the predicted …