APPENDIX K

APPENDIX K: CANISTER CENTRAL FIELD ORBITS

The goal of this Appendix is to find a static central-field solution of Eq. (4.22) when the source is a MDGO. Inhomogeneous solutions for all four diagonal potentials are required for orbit calculations. The canister fields will be strong enough so that orbits will have significant advance of the radial turning points.
One starts with the inhomogeneous solutions of the polar coordinate version of Eq. (4.22), using the following components: \[\begin{array}{l}V(q,r) \equiv v_0^0(q,r),\\U(q,r) \equiv – v_r^r(q,r),\\W(q,r) \equiv – v_\theta ^\theta (q,r),\\Z(q,r) \equiv – v_\phi ^\phi (q,r).\end{array}\tag{K.1}\] One must have no off-diagonal components, so \(Z = W\). A tedious calculation using Eq. (4.20) leads to three independent equations: \[{J_0}\theta = {r^{ – 2}}{\partial _r}({r^2}{\partial _r}V) – L_{gq}^{ – 2}V,\tag{K.2}\] \[\begin{array}{l}{J_0}\theta = – {\partial ^2}_rV – 2{r^{ – 1}}{\partial _r}U – L_{gq}^{ – 2}U + 2{\partial ^2}_rW + 4{r^{ – 1}}{\partial _r}W,\\{J_0}\theta = – {r^{ – 1}}{\partial _r}V – {r^{ – 1}}{\partial _r}U – 2{r^{ – 2}}U + {\partial ^2}_rW + 4{r^{ – 1}}{\partial _r}W + (2{r^{ – 2}} – L_{gq}^{ – 2})W,\end{array}\tag{K.3}\] where \({J_0}\) is a constant and \(\theta = 1\) for \(r < {R_E}\) and \(\theta = 0\) for \(r > {R_E}\). In what follows, the unit of length will be \({10^6}\) light-years, and the mass of the MDGO will be in solar masses. The potentials V, U, and W are dimensionless, and the radial length (r) in the equations above is in the \({10^6}\) light-years unit. The source term is given as follows: \[\begin{array}{l}{J_0} = \kappa M{Z_{gq}}{L_{gq}}^2{R_E}^{ – 3},\\\kappa = 9.374432x{10^{ – 19}},\end{array}\tag{K.4}\] where M is in solar masses, \({Z_{gq}}\) is dimensionless, and both \({L_{gq}}\) and \({R_E}\) are in the \({10^6}\) light-year units. The boundary conditions for these equations are as follows: \[\begin{array}{l}U(0) = W(0),\\{\partial _r}V(0) = {\partial _r}U(0) = {\partial _r}W(0) = 0,\\V(\infty ),U(\infty ),W(\infty ) \to 0.\end{array}\tag{K.5}\] Eq. (K.2) is solved in closed form using the techniques of Appendix F. Substitute Eq. (K.2) into the set of Eq. (K.3)—eliminate \({J_0}\). One discovers a simple solution, \[V(q,r) = U(q,r) = W(q,r).\tag{K.6}\] Is Eq. (K.5) the only solution? The answer is yes, but not easy to prove. Introduce two functions f and g (where \(U = fV\) and \(W = gV\)) into Eq. (K.3). A numerical solution for the resulting equations for f and g shows that f and g can be of order unity at a (growing) finite value of r only if they are (shrinkingly) infinitesimally close to \(f = g = 1\). For example, if \(r = 15{L_{gq}}\) then f and g must be less than \({10^{ – 15}}\) away from 1 otherwise f and g become very large.
Orbit calculations require the techniques of Appendix E plus Eq. (4.14). The Hamilton-Jacobi equations for the orbit of a non-zero mass object (an electron or a galaxy) moving under the influence of the gravitational fields above are modified in the following manner. Eq. (E.2) for the weak field becomes the following: \[\begin{array}{l}V\left( r \right) \equiv \sum\limits_{q = 1}^{{N_q}} {{v_{00}}\left( {q,r} \right)} {\rm{ summer,}}\\{G^{00}} = 1 – V,\\{G^{rr}} = – (1 + V),\\{G^{\phi \phi }} = – {r^{ – 2}}(1 + V).\end{array}\tag{K.7}\] The orbit lies in a plane (\(\theta = \pi /2\)), and there is no theta dependence. Eq. (E.5) becomes the following: \[\begin{array}{l}\partial I/\partial E = 0 = \tau + \partial f\,/\partial E{\rm{,}}\\\partial I/\partial A = 0 = \phi + \partial f\,/\partial A,\\{\left( {{\partial _r}f} \right)^2} = (1 – V){\left( {1 + V} \right)^{ – 1}}{c^{ – 2}}{E^2} – {A^2}{r^{ – 2}} – {m^2}{c^2}{\left( {1 + V} \right)^{ – 1}}.\end{array}\tag{K.8}\] A series of manipulations leads to the following result: \[\begin{array}{l}{(dr/d\phi )^2} = {r^4}{R_a}^{ – 2}[{\alpha ^2}(1 – V) – 1]{(1 + V)^{ – 1}} – {r^2},\\d\tau /d\phi = {r^2}\alpha {R_a}^{ – 1}(1 – V){(1 + V)^{ – 1}},\end{array}\tag{K.9}\] where \(\alpha = E/m{c^2}\) and \({R_a} = A/mc\). Alpha is dimensionless, tau has a unit of \({10^6}\) years, and both \({R_a}\) and r have units of \({10^6}\) light-years,
Any orbit for a given gravitational field (V) is completely determined by \(\alpha \) and \({R_a}\). Orbits of interest will be bounded since the escape velocity at \(r = 2\) for a typical canister is 0.05c (radial component of orbital velocity). Thus, \(\alpha < 1\). It is convenient to define \(\alpha = 1 - \varepsilon \) where \(0 < \varepsilon < < 1\). An orbit will have turning points, \({R_{OUT}} \ge r \ge {R_{IN}}\). If \({R_{OUT}} \ne {R_{IN}}\) (not circular) the orbit will not close, and the orbiting object will never revisit a specific point in space.
Fig. K1 illustrates the complexity of a non-circular orbit. The specifications for FIG. K1 are as follows: \[\begin{array}{l}M = {10^{14}}{\rm~{ see~Fig}}{\rm{. 1,}}\\{R_E} = 2,\\\varepsilon = 0.0007,\\{R_a} = 0.06,\\{R_{IN}} = 1.416,559,\\{R_{OUT}} = 2.429637,\\{\rm{period }} = 426.52{\rm~{ million~years,}}\\{\rm{advance = 53}}{\rm{.8804~degrees}}{\rm{.}}\end{array}\tag{K.10}\]

Figure K.1 Advance of galactic orbits within the MDGO of a canister.

The advance is the difference of phi for any two nearest points where \(r = {R_{0UT}}\). The period is defined as the elapsed physical time for an object to travel along the orbit between the two advance points. I believe that the advance will vanish in the limit where \({R_{IN}} \to {R_{0UT}}\) (difficult to establish using numerical techniques).
The orbit for a photon moving in the gravitational field defined above is a subject of interest—gravitational lensing. For a photon, Eq. (K.8) is modified by setting \(m = 0\). One can use the techniques of section VI(F) to obtain the orbit equation, \[d\phi /dr = \pm {[{r^4}{R_{impact}}^{ – 2}(1 – V){(1 + V)^{ – 1}} – {r^2}]^{ – 1/2}},\tag{K.11}\] where \(A = {R_{impact}}E{c^{ – 1}}\) and \(E = h\nu (r = \infty )\). There is a turning point, \({R_{turn}}\), where \(d\phi /dr = \infty \) . The tiny repulsive bump can be neglected, so all the gravitational bending of the photon path would occur for \(r < {R_0}\) where \({R_0}\) is a finite radius for which \(V({R_0}) = 0\).
V is negligible for \(r > {R_0}\), so the path of a photon is a straight line outside of the canister. A photon (leaving a source at infinity) that intersects the canister has an inbound line. If there was no gravitational field within the canister, this inbound line would have a point for which the distance to the center of the canister would be a minimum—the impact parameter, \({R_{impact}}\). The gravitational field bends the photon path so that it will exit the canister following a straight outbound line (with the same impact parameter) ending at an observer (at infinity). The locations of the source, observer, and the canister-center define a plane on which the photon moves. The angle between the inbound and outbound lines is the deflection angle (\({\theta _D}\)). \({\theta _D}\) is positive when the photon bends toward the canister center.
The deflection angle is calculated in the following manner, \[\begin{array}{l}\psi = {\int_{{R_{turn}}}^{{R_0}} {dr\left[ {{r^4}{R_{impact}}^{ – 2}(1 – V){{(1 + V)}^{ – 1}} – {r^2}} \right]} ^{ – 1/2}},\\\omega = {\sin ^{ – 1}}({R_{impact}}/{R_0}),\\{\theta _D} = 2(\psi + \omega ) – \pi .\end{array}\tag{K.12}\] For a given canister gravitation-field, \({\theta _D}\) is a function only of \({R_{impact}}\). Eq. (K.12) is accurate to first order of V. That level of accuracy should be sufficient for most applications. Higher accuracy is obtained by using Eq. (4.14). Fig.5 (section VI(F)) illustrates an example solution of Eq. (K.12)