A Quantized Metric As an Alternative to Dark Matter

The cosmological spherical symmetry background metric coefficient (g$_{44}\equiv )$ g$_{00}$= 1-2GM/c$^{2}$r should be inserted into a Dirac equation $\Sigma _{\mu }(\surd $g$_{\mu \mu }\gamma ^{\mu }\partial \psi $/$\partial $x$_{\mu })-\omega \psi $ = 0 (1,Maker) to make it generally covariant. The spin of this cosmological Dirac object is nearly unobservable due to inertial frame dragging and has rotational L(L+1) $\Delta \varepsilon $ and oscillatory $\varepsilon $ interactions with external objects at distance away r$>>$10$^{10 }$LY. The inside and outside frequencies $\omega $ match at the boundary allowing the outside metric eigenvalues to propagate inside. To include the correct 3 lepton masses in this Dirac equation we must use ansatz g$_{oo}$= e$^{i(2\varepsilon +\Delta \varepsilon )}$ with $\varepsilon $=.06, $\Delta \varepsilon $=.00058. For local metric effects our ansatz is g$_{oo}=_{ }$e$^{i\Delta \varepsilon }$. Here the metric coefficient g$_{oo}$ levels off to the quantized value e$^{i\Delta \varepsilon }$ in the galaxy halo: g$_{oo}$=1-2GM/rc$^{2}\to $ rel(e$^{i\Delta \varepsilon })$ \textbf{=}cos($\Delta \varepsilon )$= 1-($\Delta \varepsilon )^{2}$/2 $\to (\Delta \varepsilon )^{2}$/2=2GM/rc$^{2}$ for this circular motion v$^{2}$/r=GM/r$^{2}$=c$^{2}(\Delta \varepsilon )^{2}$/4r $\to $v$^{2 }$=c$^{2}(\Delta \varepsilon )^{2}$/4 =87km/sec)$^{2} \quad \approx $\textbf{100km/sec})$^{2}$. So the metric acts to quantize v. Note also there is rotational energy quantization for the $\Delta \varepsilon $ rotational states that goes as: (L(L+1)) $\propto $ $\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} $mv$^{2} \quad \to \sqrt {L(L+1)} \quad \propto $v. Thus differences in v are proportional to L, L being an integer. Therefore $\Delta $v = kL so v = 1k, v = 2k, v = 3k, v = 4k{\ldots}. v=N (the above $\sim $100km/sec) with \textit{dark matter then not required} to give these high halo velocities. Recent nearby galaxy Doppler halo velocity data \textbf{\textit{strongly support}} this velocity quantization result.