Section VI – The New Cosmology and the New Gravity

VI. THE STRONG FIELD LIMIT OF THE NEW GRAVITY

The primary goal of this section is to determine the function \({H_{\mu \nu }}\) (defined in Eq. (4.15)) which governs the strong field limit for NG. I have chosen a specific form for H that is simple, especially when \({V_{\mu \nu }}\) is diagonal. There are more complex possibilities that could be considered as possible perturbations—beyond the scope of this document.
Strong gravitational fields can only occur close to sources that are massive and compact. Only three types of sufficiently compact sources are known—white dwarfs, neutron stars (pulsars), and black holes. New equations are needed to develop the NCNG version of these objects.

A. Static central field orbit equations

One requires an inhomogeneous version of Eq. (5.13) as the starting point for this study. Assume that the source of the local central field is a spherically symmetric mass distribution contained within radius \(R_{sorce}^{}\). Since the source is assumed to be static, the only change to Eq. (5.13) is that \({\Omega _{00}}(r) \ne 0\) for \(r < R_{sorce}^{}\). The Hamilton-Jacobi solution follows: \[\begin{array}{l}\partial I/\partial E = 0 = \tau + \partial f\,/\partial E{\rm{,}}\\\partial I/\partial A = 0 = \phi + \partial f\,/\partial A,\\{[1 + H( - U)]^{ - 1}}{\left( {{\partial _r}f} \right)^2} = {[1 + H(V)]^{ - 1}}{c^{ - 2}}{E^2} - {A^2}{r^{ - 2}} - {m^2}{c^2},\end{array}\tag{6.1}\] where E is the conserved (total) energy of the orbiting object, A is the conserved angular momentum, m is the orbiting object mass, and \(\tau \) is physical time in the rest frame of a far-distant observer and the source. If \(E < m{c^2}\), the object is bound to the source. The weak field limit requires that \(H(0) = 0\).
For orbits that lie outside of the source, \(U = V = – b/r\). As the mass of the source (and b) goes to infinity, there will be a range (\(R_{sorce}^{} + \varepsilon > r > R_{sorce}^{}\)) for which V can go to minus infinity as well. Such a source cannot grow unless there are orbits which intersect with the source—f is real for \(V \to – \infty \). Thus, one requires \(H(V) > – 1\) for \(V \le 0\) (the physical region) otherwise a source cannot accrete mass. If orbits are to have the observed advance of perihelion and bending of starlight, it is necessary that \(\partial H/\partial V = 1\) at \(V = 0\). Thus, \(H > 0\) at least in a portion of the unphysical region (\(V > 0\)). In GR, \(V = – 1\) is the point where a singularity arises (dividing the physical region into two subregions). \(H(V)\) is a continuous function for all values of V in NG, but \(V = – 1\) will still represent a (nonpathological) boundary dividing the physical region into two sectors.
Eq. (6.1) gives rise to the orbit equations, \[\begin{array}{l}dr/d\tau = – {[\partial ({\partial _r}f)/\partial E]^{ – 1}},\\d\phi /d\tau = – (dr/d\tau )[\partial ({\partial _r}f)/\partial A].\end{array}\tag{6.2}\] Eq. (6.2) can be rewritten: \[\begin{array}{l}dr/d\tau = \pm {[1 + H( – U)]^{ – 1/2}}[1 + H(V)]{E^{ – 1}}{c^2}K,\\K = {\{ {[1 + H(V)]^{ – 1}}{c^{ – 2}}{E^2} – {A^2}{r^{ – 2}} – {m^2}{c^2}\} ^{1/2}},\\d\phi /d\tau = – A{E^{ – 1}}{c^2}[1 + H(V)]{r^{ – 2}},\end{array}\tag{6.3}\] where plus means the orbiting object is moving away from perihelion and minus is toward perihelion. Note that the advance of perihelion depends on \({A^2}\)—independent of counterclockwise or clockwise observed motion.

B. The strong central field source

The strong (red summer) source for Eq. (5.14) is given by one change, \[\begin{array}{l}{\Omega _{00}} = 8\pi G{c^{ – 2}}\rho \,(\partial {G^{00}}/\partial V),\\{G^{00}} = {[1 + H(V)]^{ – 1}}.\end{array}\tag{6.4}\] Eq. (5.14) with alteration Eq. (6.4) represents the inhomogeneous central field equations that will be used in what follows. These equations are approximations. The \({L_{gq}}\) terms have been neglected since \(r < < {L_{gq}}\) for examples of interest. The effects of pressure (P) within the source are neglected since \((P/\rho {c^2}) < < 1\) for most examples of interest. The resulting equation for V is nonlinear and represents what the source does to the field.

C. Hydrostatic equilibrium

What does the field do to the source? The field defined by modified Eq. (5.14) exists inside the source. In turn, the orbit equation for a particle within the source is given by Eq. (6.3). The source particle is stationary (\(dr/d\tau = 0\)), but it is subject to a force ( \({d^2}r/d{\tau ^2} \ne 0\)). This force is countered by a pressure gradient, \[\begin{array}{l}\partial P/\partial r = \rho {a_g},\\{a_g} = – 0.5{c^2}(\partial V/\partial r)[\partial H(V)/\partial V]{[1 + H( – U)]^{ – 1}}.\end{array}\tag{6.5}\] The radial acceleration (\({a_g}\)) is calculated using Eq. (6.3) for zero angular momentum, \[{a_g} = {\left[ {\frac{\partial }{{\partial \tau }}\left( {\frac{{dr}}{{d\tau }}} \right)} \right]_{dr/d\tau = 0}} = {\left[ {\left( {\frac{{dr}}{{d\tau }}} \right)\frac{\partial }{{\partial r}}\left( {\frac{{dr}}{{d\tau }}} \right)} \right]_{dr/d\tau = 0}}{\rm{ }}{\rm{.}}\tag{6.6}\] In the weak field limit, \({a_g} \to – \partial \mathcal{A}_N / \partial r\). Eq. (6.5) is also an approximation—valid when \((P/\rho {c^2}) < < 1\).

D. Equation of state and boundary conditions

The (summer) field/source equations can be summarized, \[\begin{array}{l}U = – r(\partial V/\partial r),\\\partial V/\partial r = 8\pi G{c^{ – 2}}{r^{ – 2}}\int_0^r {dr'{{r’}^2}\{ \rho \,(\partial H/\partial V){{[1 + H(V)]}^{ – 2}}} {\} _{r’}}{\rm{ ,}}\\\partial \ln (\rho )/\partial r = – 0.5{c^2}{(\partial P/\partial \rho )^{ – 1}}(\partial V/\partial r)(\partial H/\partial V){[1 + H( – U)]^{ – 1}},\end{array}\tag{6.7}\] and they will need boundary conditions. The equation of state (EOS) for matter within the source allows the elimination of one unknown (pressure).
The thermodynamics of the source material comes into play if the EOS depends on temperature. An additional equation is required in that case: \[\partial T/\partial \tau = {r^{ – 2}}{\partial _r}({r^2}{K_T}{\rho ^{ – 1}}{S_T}^{ – 1}{\partial _r}T) + {J_T}{\rho ^{ – 1}}{S_T}^{ – 1},\tag{6.8}\] where \(T(r,\tau )\) is the temperature of the source material (\(^\circ\) kelvin). Eq. (6.8) must be well-behaved as \(r \to 0\), and \(\partial T/\partial \tau \to 0\) for all applications of interest except core collapse. \({K_T}\) is the heat conductivity of the source material, and \({S_T}\) is the specific heat capacity—both can depend on temperature and composition of the source material. \({J_T}\) is the sum of the (positive) power/volume of any nuclear fusion within the source and the (negative) heat loss caused by any neutrinos fleeing the source. It is possible for a small amount of proton-proton fusion to occur where hydrogen material abuts helium material in the source, but this heat source can be neglected. There are conditions where a zone in the source can have protons (in the underlying nuclides) undergo electron capture (producing a neutron and an electron flavor neutrino). This heat loss can be significant. The asymptotic static version of Eq. (6.8) follows, \[{\partial _r}T \propto – {r^{ – 2}}\rho {S_T}{K_T}^{ – 1}{\rm~~{ as }}~r \to {r_{edge}}\tag{6.9}\] One should expect that \({K_T}\) could be large near the center of the source and very small near the outer edge.
Static means that the source composition, temperature profile, mass, gravitational field, etc., change very slowly after the brief formation era of the source—evolution. The solution of Eq. (6.7) is a snapshot of an instant in the lengthy object evolution.
The method of solution starts with forming an EOS that spans a large range of densities and corresponding pressures (providing \(\partial P/\partial \rho \)). The EOS can have phase changes, so \(\partial P/\partial \rho \) can have discontinuities. The source composition profile will depend upon the age of the object, so some plausible evolution is needed.
The three strong-gravitational source types are all formed during the death of a star. In NCNG there are only three kinds of source material, electron degenerate matter (EDM), neutron degenerate matter (NDM), and proton-neutron matter (PNM) —known as neutron pasta. EDM also includes partial degeneracy as does NDM. EDM has been well understood for more than eighty years. A white dwarf is made of EDM, and I believe that NGNC will not provide any useful (observable) white dwarf information about the strong limit. The pulsar and black-hole objects are a different story. These two objects are created in a very similar manner—core collapse. They are cinders remaining from the death of a summer-time star.
The NCNG versions of the two types need new names: CCNS (core collapse neutron star) replaces pulsar and CCBO (core collapse boundary object) replaces black hole. A boundary takes the place of an event horizon—described below. The composition of CCNS and CCBO is a mixture of EDM, PNM, and NDM. A CCNS or CCBO can exist for billions of years during summer, so one must expect evolution.
There is a third NCNG type of interest, MBO (massive boundary object), which replaces the super massive black hole. An MBO is just a CCBO that has grown in a special manner to a very large mass. The evolution of a MBO and the fate of all cinder types in winter are discussed in sections VI(G) and VII(E).
Returning to the method of solution, one can cobble together an EOS that spans a large range of densities and corresponding pressures (providing \(\partial P/\partial \rho \)). The integrations start at the center (r = 0) with initial conditions, \[\begin{array}{l}U(0) = 0,\\V(0) = {V_0} < 0,\\\rho (0) = {\rho _0}.\end{array}\tag{6.10}\] The integrations using Eq. (6.7) continue until \(r = {r_{stop}}\), where \(\varepsilon = 1 - U({r_{stop}})/V({r_{stop}})\) and epsilon is positive and sufficiently small (e.g., \({10^{ - 7}}\)). If \(\rho ({r_{stop}})/{\rho _0}\) is sufficiently small, then one has a plausible solution for that choice of \({V_0}\) and \({\rho _0}\): \[\begin{array}{l}{R_{source}}({\rho _0}) = {r_{stop}},\\{M_I}({\rho _0}) = {M_{inertial}}({r_{stop}}),\\{M_G}({\rho _0}) = {M_{gravity}}({r_{stop}}),\end{array}\tag{6.11}\] otherwise, a different \({\rho _0}\) must be chosen (for the same \({V_0}\)) and the integration procedure repeated until a plausible solution is found. The inertial and gravitational masses are accumulated during the integration process—not necessarily the same due to the H factor on the right-hand side of Eq. (6.4). There is only one solution per \({V_0}\), but there could be two different solutions per \({\rho _0}\). Appendix H has details for the summer solutions of Eq. (6.7) for objects of interest.
Eq. (6.7) represents the NCNG version of the TOV (Tolman, Oppenheimer, Volkoff) equation which can be found as Eq. (11) of [9]. TOV is still in use today—use a modern EOS for nuclear matter to obtain \({M_I}({\rho _0})\). Even if GR could be a correct theory of gravity, TOV would still be a crude approximation. TOV is derived using a simple two parameter model of the source energy-momentum tensor (an ideal fluid). Source material is not an ideal fluid due to (second degree) phase changes and an inhomogeneous consistency. Furthermore, a “cold” temperature is used for the EOS in [9]. The assumption that the internal temperature is cold is suspect. The cooling time for a white dwarf cinder is measured in billions of years. There is no reason to believe that neutron star cinders would cool significantly faster despite some (zonal) neutrino cooling during the actual core collapse.
The only observable consequence of Eq. (6.7) is the gravitational mass of the source object (which determines the orbit of an object distant from the source). The fields for the distant orbiting object are \(U = V = – b/r\), where \(b = 2G{M_G}{\kern 1pt} {c^{ – 2}}\). The central density (\({\rho _0}\)) can only be inferred by using Eq. (6.7). The only (observable) information that can determine \(H(V)\) are the ranges of mass for certain classes of source and the rate of accretion during their formation.
The mass spectrums of two classes in summer (CCBO, CCNS) give rise to an important clue for the form of \(H(V)\). There appears to be a gap between the heaviest mass (observed) for a CCNS and the lightest mass for a CCBO. Within the source for each type, \(\partial H/\partial V > 0\) (an inward gravitational force). If this condition is met for all \(V < 0\), there would be no expectation for a mass gap. Therefore, one must expect that there is a (small) region where \(\partial H/\partial V < 0\) (a repulsive barrier). I have chosen to place this boundary at \(V = - 1\). In essence, NG replaces the event horizon of GR with a (finite) repulsive boundary. The boundary form for \(H(V)\) is as follows: \[\begin{array}{l}H(V) = {\theta _{SL}}(V){h_ > }(V) + (1 – {\theta _{SL}}(V)){h_ < }(V),\\{\theta _{SL}} = 0.5\{ 1 + \tanh [{\Gamma _{SL}}(V + 1)]\} ,\end{array}\tag{6.12}\] where \({\Gamma _{SL}}\) is a universal constant (\({\Gamma _{SL}} > > 1\)). Both \({h_ > }\) and \({h_ < }\) are continuous functions defined for all \(V\), and H is continuous as well.
A cinder cannot be stable if there is a repulsive boundary within the source. Therefore, there will be two classes separated by the boundary. The heaviest CCNS will have \(V(0) = – 1\), while the lightest CCBO will have \(V({R_{source}}) = – 1\). There are no reliable observations of low candidate CCBO masses (largely due to controversy and uncertainty about their distances). The TOV models [9] give maximum CCNS masses in the range (1.8-2.25) solar masses, but there are observations as high as 2.4 solar masses (also with controversy and uncertainty). It is more convenient to use the following definition: \[\begin{array}{l}Q(V) \equiv 1 + H(V),\\Q(V) = {\theta _{SL}}(V){q_ > }(V) + (1 – {\theta _{SL}}(V)){q_ < }(V),\end{array}\tag{6.13}\] where \(0 < Q(V) < 1\) for \(V < 0\), \(Q(0) = 1\), and \(\partial Q/\partial V = 1\) at \(V = 0\). The following functions satisfy these conditions: \[\begin{array}{l}{q_ > }(V) = (1 + V + {\varepsilon _{SL}}{V^2}){(1 + 0.5{V^2} + {\gamma _{SL}}{V^4})^{ – 1}},\\{q_ < }{(V)^{ - 1}} = {\delta _{SL}}^{ - 1} - 1 - {\alpha _{SL}} - V + {\alpha _{SL}}{V^{ - 2}},\end{array}\tag{6.14}\] where \({\varepsilon _{SL}}\), \({\gamma _{SL}}\), \({\delta _{SL}}\), \({\alpha _{SL}}\), and \({\Gamma _{SL}}\) are universal constants. The universal constants are subject to constraints: \(0 < {\varepsilon _{SL}} < 1\), \(0 < {\gamma _{SL}} < < 1\), \(0 < {\delta _{SL}} < 1\), \({\alpha _{SL}} < 0\), and \({\Gamma _{SL}} > > 1\), Furthermore, these constants control different observable aspects. \({\Gamma _{SL}}\) and \({\delta _{SL}}\) control the flow of matter and photons across the boundary at \(V = – 1\). \({\varepsilon _{SL}}\) controls the mass of the heaviest neutron star. \({\alpha _{SL}}\) controls the mass of the lightest CCBO, \({\gamma _{SL}}\) controls the maximum size of a MBO source. The universal constants are determined in Appendix H for a set of observable aspects (gravitational mass spectrum). The mass spectrum is not well known at present, so I have chosen plausible values to establish the strong-field-limit component of a standard NCNG model (Appendix Z), \[\begin{array}{l}{\varepsilon _{SL}} = 0.487,\\{\gamma _{SL}} = 2.8{\rm{x}}{10^{ – 7}},\\{\alpha _{SL}} = – 1.9,\\{\delta _{SL}} = 0.95{\rm{ }}{\rm{.}}\end{array}\tag{6.15}\] The limits are based on a maximum CCNS mass =2.406 solar masses, minimum CCBO mass = 4.093 solar masses, and maximum MBO source size \( = 1.597{\rm{x}}{10^7}\) km. The required rate of flow across the boundary is an extremely complex subject since it involves the details of core collapse—visited in a subsection below. Therefore, my choice for \({\delta _{SL}}\) represents a starting assumption. Appendix H has Figures showing size and gravitational mass as functions of the central density for the CCNS, CCBO, and MBO.
There are situations of interest where time dependence and thermodynamics are important—formation and seasonal changes in physics. These situations will be discussed in various sections.

E. Core collapse and cinder formation

Core collapse and the resultant formation of central dense objects (cinders to be) is the most challenging problem in astrophysics. NG (with the new strong field limit) does not simplify this problem. Instead, NG significantly complicates core collapse because one must carry the calculations through any boundary all the way to the center. Furthermore, composition and thermodynamics of the forming central object within any boundary must be considered as well. In NC, core collapse can only occur during summer—no stars in winter.
How will NG change the current model of core collapse evolution? A detailed study of this subject is beyond the scope of this document, but I can make the following observations. Refer to Appendix M for what follows. Using Eq. (M.2), one obtains the velocities—\(c{\beta _ > }\) is the material velocity on the \(V > – 1\) side of the boundary, and \({\beta _ < }\) is the velocity on the \(V < - 1\) side. The velocity on each side has the same sign, positive for outbound particles and negative for inbound.
If the core is not spinning, the velocity profile of material will be radial (\(v(r,\tau )\)), where \(v(0,\tau ) = 0\). A boundary is only a few meters thick. The gravitational potential, \(V(r,\tau )\), will not be directly connected to the velocity profile. The radius of a boundary, \(V({R_B}(\tau ),\tau ) = – 1\), can move. A boundary can come into existence at any radius or vanish at any time. This evolution is very complicated—many moving parts.
The time evolution of gravitational systems has not been developed in this document. However, I believe that Eq. (M.2) can be applied to a special instant in time during core collapse (\({\tau _{cc}}\)). That instant occurs when there is a boundary, \(v({R_B}({\tau _{cc}}),{\tau _{cc}}) = 0\), and \({\partial _\tau }v({R_B}({\tau _{cc}}),{\tau _{cc}}) = 0\)—the region near the boundary is quasi-static (in the frame where the boundary is stationary). Thus, \(c{\beta _ < } = 0\) and \(c{\beta _ > } = \pm \,c\,{a^{1/2}}\). There can be a sudden reversal of material flow: \(c{\beta _ > } = – \,\,c{a^{1/2}}\) (inbound) when \(\tau = {\tau _{cc}} – \varepsilon \), and \(c{\beta _ > } = \,c\,{a^{1/2}}\) (outbound) when \(\tau = {\tau _{cc}} + \varepsilon \) (epsilon is a very small time). The inbound matter is bounced off the (moving) boundary. If the boundary is inbound \({v_B} < 0\), then the bounce can develop into a very robust outbound shockwave that will start with \({v_{SW}} = - 2{v_B}\) and move upstream into a nearby region where there is fusion fuel available.
The creation of an outbound shockwave is also a feature of present-day models. Once this shockwave enters a region where fusion is ongoing, it could radically increase the rate of fusion (if it is sufficiently robust)—create a new shockwave. The new shockwave would have two fronts, outbound and inbound. The outbound side could set off a runaway increase in fusion (a type II supernova). The inbound side could increase the mass of the source and perhaps create a new bounce.
I believe that a comprehensive study of the moving parts in NCNG will reproduce the basic elements of the scenario above. However, there will be a significant difference from present-day models. The NG version of the boundary is very dynamic. A boundary with a specific behavior is required to have a robust bounce, otherwise the (weaker) bounce will depend on the properties of the source EOS. Is the source a CCNS or a CCBO? The NG answer is simple. If there is a static boundary after the source becomes isolated, it is a CCBO, otherwise a CCNS. I also believe that the present-day models have used the incorrect strong field limit of GR to produce suspect results.
The composition profile of a newly formed CCBO or CCNS summer-cinder is governed by the environment of the center of the collapse at its end—iron at (\(T(r = 0) \sim 2.7 \times 10^9 \, ^\circ\mathrm{K}\)). \(T(0)\) is the burn-temperature of the last fuel to be exhausted (Silicon). The EOS of new cinder material causes the composition profile to have three or less spherical layers. An aging cinder may have extra layers due to growth. Each layer represents a different (second order) phase or composition of the EOS, but the EOS is continuous.
The number of layers in a new cinder (and their thickness) depends on the gravitational mass of the new cinder. The density, pressure, and temperature all decrease moving outward from the center for all possible sources that do not have fusion within. A small amount of fusion is possible only in summer for a growing cinder with a very specific temperature profile.
The outermost layer of a new cinder consists of iron EDM. This EDM layer is always present for all new cinders of any mass. In many cases it is the only layer (heavy CCBOs or light CCNSs). The amount of degeneracy and free-electron velocity decline moving outward in this layer. One should expect that the heat conductivity will decline as well.
A second layer of PNM (proton neutron matter) for a new cinder can form inside the EDM layer for lighter CCBOs or heavier CCNSs. The EOS undergoes a (second degree) phase change at the EDM-PNM boundary when the pressure rises above the maximum pressure for a fully degenerate electron gas. Furthermore, the protons in the iron nuclides that underly the electron gas of the EDM layer can undergo electron capture and turn into neutrons (and fleeing neutrinos). Thus, the matter transitions into an underlying solid of protons and neutrons and a depleted degenerate electron gas. As pressure increases toward the origin, the proton density and electron gas density decline and can vanish. The PNM material is a solid.
A third layer of NDM (neutron degenerate matter) for a new cinder can form inside the PNM layer for the lightest CCBOs or heaviest CCNSs. The EOS undergoes a (second degree) phase change at the NDM-PNM boundary. The NDM material is a liquid.
I have used the EOS of [9] to calculate the mass spectrums of Appendix H (despite questions of accuracy). The results of Appendix H should be viewed as a comparison of the strong field limit of NCNG versus GR, rather than the correctness of the EOS. The source-density at the EDM-PNM edge is \(3.6883{\rm{x}}{10^{14}}\) (MKS) and \({\rm{1}}{\rm{.3038x}}{10^{17}}\) (MKS) at the PNM-NDM edge. The figures of Appendix H give rise to summer-time observations. All CCNS cinders have all three layers. All CCBO objects that are heavier than \({10^5}\)solar have only one layer (EDM). All CCBO objects lighter that roughly 30000 solar have all three layers and the remainder have two layers (EDM and PNM).

F. Gravitational lensing

The path of photons through an NG gravitational field containing a source is an important subject today. This discussion will be limited to static central fields of all strengths—an excellent approximation for canisters and ellipsoidal galaxies.
The necessary equations are derived from the Hamilton-Jacobi procedure outlined in Eq. (6.1). One simply takes the limit where the mass of the orbiting object goes to zero. The essential difference between NG and GR is that a photon can never be trapped within a gravitational well. A photon can be created at the surface of a source, and it will always trace a path that will end up at infinite radius. A photon can come from infinite radius and trace a path that leads it to be absorbed by the surface of the source. A photon can come from infinite radius and trace a path that never intersects with the surface of the source and returns to infinite radius. All three possibilities have a single common identifier, the impact parameter of scattering theory.
The orbit equation for a photon is obtained from Eq. (6.3) by setting \(m = 0\), \[\begin{array}{l}Q(V) = 1 + H(V),\\dr/d\phi = \pm {\kern 1pt} \,{r^2}Q{( – U)^{ – 1/2}}{[Q{(V)^{ – 1}}({E^2}/{c^2}{A^2}) – {r^{ – 2}}]^{1/2}},\\{({E^2}/{c^2}{A^2})_{m = 0}} = {R_{impact}}^{ – 2}.\end{array}\tag{6.16}\] The conserved angular momentum (A) of the photon (relative to the origin) is \(A = {R_{impact}}E/c\), where \(E = h\nu (\infty )\). The impact parameter is given by the closest distance from the origin of the vector momentum of the photon at infinity extended as a straight line to the vicinity of the origin. There is a turning point (closest approach to the origin), \[{R_{TP}}Q{(V({R_{TP}}))^{ – 1/2}} = {R_{impact}}.\tag{6.17}\] There are two categories of interest. The first is for local opaque sources surrounded by a vacuum (stars, NSs, and BOs). In this case, orbits of interest have \(U = V = – b/r\), where \(b = 2G{M_{source}}{c^{ – 2}}.\)For stars, the potential is weak, and Eq. (6.14) yields Eq. (E.7) and the GR result. For a neutron star, the field is strong with more complex optics. For a boundary object, the field is even stronger, and the obits can cross the boundary with even more complex optics. The second category is for vast weak fields (galaxies and canisters). The most interesting problem is the optical characteristics of a canister. The primary object of study for category one is the MBO at the center of our galaxy, where the mass is \(4.1{\rm{x1}}{{\rm{0}}^6}\) solar masses, \(b = 1.2109{\rm{x1}}{{\rm{0}}^7}\) km, and \({R_{source}} = 1.3767{\rm{x1}}{{\rm{0}}^5}\) km. The numerical values that follow are based on these three source constants and Eq. (6.13). One can simplify Eq. (6.14) for photon orbits that do not intersect the source for this case, \[\begin{array}{l}dV/d\phi = \pm {\kern 1pt} \,Q{( – V)^{ – 1/2}}{[Q{(V)^{ – 1}}{\xi ^2} – {V^2}]^{1/2}},\\V = – b/r,\\\xi = b/{R_{impact}}{\rm{ }}{\rm{.}}\end{array}\tag{6.18}\] A (non-intersecting) photon can cross the boundary but cannot be trapped within. All solutions of Eq, (6.16) start and end with \(r = \infty \) (\(V = 0\)), and the orbit lies in a plane defined by the origin and the two locations at \(r = \infty \). The only observable is the angle of deflection (\(\Delta {\phi _D}\)), which is given by \({{\bf{n}}_{in}} \cdot {{\bf{n}}_{out}} = \cos (\Delta {\phi _D})\). The unit vectors correspond to the direction of motion of the photon at infinity inbound and outbound. A numerical solution of Eq. (6.16) starts at the turning point (\(V = {V_{TP}} < 0,{\rm{ }}\phi = \pi /2\)) and continues along the – branch (\(dV > 0,{\rm{ }}d\phi < 0\)) until (\(V = 0,{\rm{ }}\phi = {\phi _{end}} < 0\)). There is a discontinuity in \(dV/d\phi \) at \(V = - 1\), so it is necessary to break the process into two portions using \({Q_ < }\) for the inner part and \({Q_ > }\) for the outer. The orbit is symmetric about the turning point, so only the outbound part needs to be calculated. For large \({R_{impact}}\) (weak field), \(\Delta {\phi _D}\) is small and positive—the inbound photon is bending toward the source. For the photon to avoid intersecting the source, the turning point must be \({V_{TP}} > – 87.9556\). Note that \({\xi ^2} = {V_{TP}}^2Q({V_{TP}})\), and the boundary is assumed to have infinitesimal thickness.
There are three zones for solutions of Eq. (6.16), \[ \begin{array}{l} \text{external: } \xi < 0.56979 \ \text{ and } \ V_{TP} > -1, \\ \text{bounce: } 0.56979 < \xi < 0.97468 \ \text{ and } \ V_{TP} = -1, \\ \text{crossing: } 0.97468 < \xi \ \text{ and } \ -87.9556 < V_{TP} < -1.6734. \end{array} \tag{6.19}\] The external zone is familiar—small positive deflections. The bounce is new. Bounce photons have too much angular momentum to penetrate the boundary, so they bounce off it. Bounce calculations should use \({Q_ > }(V)\).
The crossing zone is also new. Inbound (crossing) photons have small enough angular momentum to cross the boundary. They spiral inward toward the turning point, then spiral outward and recross the boundary outbound toward infinity. The spiral nature of the inner photons can cause \(\Delta {\phi _D} > 2\pi~ \)and so require the observed deflection to be \({\rm{mo}}{{\rm{d}}_{2\pi }}(\Delta {\phi _D})\). \(\Delta {\phi _D}\) is a single-valued function of \({R_{impact}}\) for crossing photons. As \({R_{impact}}\) decreases, \(\Delta {\phi _D}\) increases steeply—there are shrinking slices of \({R_{impact}}\) within which \({\rm{0}} \le {\rm{mo}}{{\rm{d}}_{2\pi }}(\Delta {\phi _D}) < 2\pi \). There are 1025 slices for our MBO. Fig. 3 shows the first three slices.
It should be clear that photons from a distant star occulted by our MBO would have negligible probability of ending up in a telescope since they can be moving in any direction—invisible star. The apparent motion of a star moving toward occultation with our MBO would be a direct observation of the strong field limit of NG (or GR). All other possibilities have b too small or the MBO is too distant. Unfortunately, an infrared (2.2 micron) interferometer with a base of order 100 km would be required for accurate measurements of this best case!

Graph

Figure 3.Degree of spiraling of a photon that crossed the boundary of our MBO (then exited) versus impact parameter.

The second category (canister lensing) is developed in part of Appendix K. A solution of Eq. (K.12) gives the photon deflection angle (radians) as a function of impact parameter (million light-years) for a given MDGO. The geometry of a photon path through a canister and the impact parameter is explained as well.
Fig. 4 illustrates an example deflection profile for an MDGO having \(M = {10^{14}}{\rm{ solar}}\) within a radius (\({R_E}\)) of 2 million light-years. \({R_{impact}}\) is given in million light-years and \({\theta _D}\) in radians.

Graph

Figure 4.The deflection of a photon that passed through the example canister defined above.

The optical properties of a canister (isolated and containing a single MDGO) are completely determined by the function \({\theta _D}\). The resulting observations can be complex. A simple example will illustrate an observation. Suppose that a camera is aimed at the center of the canister of Fig. 5 which is located at a distance \({F_{camera}}\) (million light-years) from the camera. Suppose that there is a point photon-source (radiating uniformly in all directions) located on a line joining the canister-center and the camera-lens at a distance of \({F_{source}}\)(same units) from the canister-center (opposite to the camera). The only photons that can arrive at the camera lens must bend through a specific angle, \[\begin{array}{l}{R_{impact}} = {F_{source}}\sin \alpha ,\\{R_{impact}} = {F_{camera}}\sin \psi ,\\\alpha + \psi = {\theta _D}({R_{impact}}).\end{array}\tag{6.20}\] Due to the cylindrical symmetry of this arrangement, the image of the source on a photograph would be a circle with a radius corresponding to \(\psi \). A more realistic source would have size, shape, and a spectrum. The image would be an annular ring with average radius \(\psi \).
Fig. 5 shows the annular radius in radians versus \({\log _{10}}({F_{source}})\) when \({F_{camera}}\) is 700 million light-years (for this example canister). Each portion of the annulus would have the same spectral characteristics—a smeared-out image of a single object. In general, \({F_{source}}\) and \({F_{camera}}\) can be established through the cosmological red shift of the source and canister galaxies. The size and the location of the center of the canister MDGO can be determined by soft-Xray photography. Thus, a knowledge of \(\psi \) leads to a determination of the mass of the container for this type of arrangement.

Graph

Figure 5.The radius of the anulus image in radians versus log 10 of F_source defined above in an example.

The camera image is far more complex when the source is not in line with the canister and camera. As the angle of the source increases, the annular ring breaks up into a diminishing number of displaced partial segments. These blobs can be identified as images of the same object because they will have the same spectrum. Furthermore, the path-distance of the photons for each image will be slightly different (but calculable). In that case, there will be an arrival time difference of observable events between images of the same object, thus leading to a knowledge of the distance of the source that is independent of cosmological red shift. It should be clear that using NG to study canisters and lensing will lead to a more precise ladder of distance versus red shift, mass distribution of canisters, and the universal constants of Eq. (5.10).

G. Growth of a massive boundary object during summer

The formation and growth of a massive boundary object (MBO) is yet another NCNG topic that is extremely complex (beyond the scope of this document), but I can outline an evolution model. I will focus on the MW MBO—the best-known object.
NCNG requires that there will be no relic MBOs in any galaxy at the end of any winter (when local gravity is turned on). Otherwise, the universe could become filled with MBO debris.
Section VII(E) deals with the fate of a MBO at the end of any summer in any galaxy (when local gravity is turned off). NCNG requires that a MBO must disintegrate in such a manner that the debris can be efficiently recycled during the following winter—no buildup of MBO debris.
The evolution model must start (local gravity turned on) with no MBO, newly borne Kelvin contraction stars, and a stable (small) amount of relic iron debris, The evolution ends (local gravity turned off) when a MBO begins to disintegrate (in the right manner). The growth model will use information from the following sections: VI(C), VI(F), VII(B), Appendix H, and Appendix M.
The early evolution model will have an important event. Fusion will be turned on about one million years after the start—see VII(B). Some of the newly borne Kelvin contraction stars will be heavy and hot enough to ignite and form a stellar cinder. The newly created CCBO that is closest to the galaxy center is most likely to be the seed that will grow to become the MW MBO today (about 21 billion years after the start).
How does a boundary object grow? The MW MBO would have to gain an average of one solar mass per 5100 years—cross the MW MBO boundary. This robust diet can only be sustained by a bombardment of material objects (projectiles) at the growing MW MBO boundary (target). Thus, one needs to understand many applicable topics—a scattering theory problem.
What are the projectiles? The environment near the center of MW consists of stars, stellar cinders, gas and gravitational fields. Gas as a projectile can be neglected since gas crossing the target boundary is an inefficient equilibrium problem (Appendix M). Stars are the largest projectiles and easily observed. There is a zoo of stellar cinders: WDs, CCNSs, CCBOs, and even fledging MBOs. Cinder projectiles are much smaller than stars and not observable (a major problem). Gravitational fields are important but difficult to calculate, and they change as the MW MBO grows.
What happens when a projectile collides with the target boundary? There is no simple answer to this important question. The fate of the projectile depends strongly on the type and velocity of projectile and the mass of the target. None of the alternatives have been studied. Some examples will illustrate the problem. The target boundary is safely assumed to be stationary at the center of MW. The radius of the target boundary is \({R_{MBO}} = 2.9533\;\,{M_{MBO}}/{M_ \odot }\,{\rm{ km}}\). Suppose the projectile is the sun moving at 1200 km/s—a typical star near the center of MW today. Furthermore, assume that the projectile is aimed at the center of the target.
The first example has 1000 solar target mass (5906 km diameter) in the early red summer. The diameter of the projectile is 1.39 million km, and the target is within the projectile for 19.3 minutes. Although the gravitational force of the target is large compared to that of the projectile, the force is an impulse that reverses as the center of the projectile crosses the center of the target. It should be clear that the collision will have negligible effect on the projectile, and the target will be left with a (small) fraction of the projectile mass in an accretion halo on the target boundary. The mass of the accretion halo, and the amount of that mass that stays inside the target boundary needs to be calculated—extremely difficult.
The second example has 235600 solar target mass in later summer. Then the target is the same size as the projectile, and the projectile is completely destroyed. The target will be left with a (larger) fraction of the projectile mass in an accretion halo, and the rest of the projectile mass will be gas released into the environment (where it can contribute to the creation of new projectiles). This calculation is much more difficult.
When the projectile is a cinder, the result is orders of magnitude more complex than a star because one must calculate the effects of the collision on the projectile source. For example, a CCBO projectile can cross the target boundary in a special manner because the target and projectile boundaries can merge, and the projectile will not actually cross a boundary. However, the source of the CCBO will be subject to severe tidal forces and become distorted or even dispersed. If that is not difficult enough, the projectile can be in an orbit that will take the (wrecked) projectile back outside of the target boundary! This topic requires a huge amount of study.
What happens to the source of the target after a collision? This topic requires the thermodynamics of the target source and the projectiles that formed it—also requiring a huge amount of study. The source is all the matter that is trapped inside the boundary. The source is layered by phase. Innermost is the solid phase (EDM), surrounded by an ocean (liquid phase). Outermost is an atmosphere (gas phase). The EDM ball has most of the source mass.
EDM is well studied and has important features: the EOS is independent of temperature, the formative nuclides are largely retained, and the heat conductivity is dominated by the temperature and density of the electron gas. The layers are in thermodynamic equilibrium. I believe that the EDM nuclides are also organized so that the nuclide mass decreases as the radius increases. When a projectile permanently crosses the target boundary, the equilibrium is briefly unbalanced.
There will only be a few formative nuclides: iron, carbon, oxygen, helium, and hydrogen. A collision with a star will provide hydrogen and helium. A collision with a WD will most likely provide carbon and oxygen. A collision with a CCNS or CCBO will provide iron. A collision with a MBO can provide all five.
The formative nuclides will slowly diffuse through the EDM toward the center until they reach an equilibrium zone. The zones would be an iron zone at the center (then moving outward), an oxygen zone, a carbon zone, a helium zone, and a hydrogen zone. Each zone would have a small flux of heavier nuclides moving inward.
The heat conductivity of each zone would be proportional to the mass of the zone nuclides and decline as the temperature declines. The temperature at the center (iron zone) would be extremely high (burn temperature of silicon) and would decline to a low value where the hydrogen EDM meets the liquid phase. Modest temperature hydrogen EDM is an excellent insulator. The high temperature at the center of the iron zone is because the original MBO to be CCBO was created in the environment where the temperature would be the burn temperature of the last available fuel element (silicon)—about \(3x{10^9}{\rm{ K}}\). The rapid early growth of this cinder would cover the center with layers of EMD that would act as insulation. The center would not have an opportunity to cool significantly.
Now, one comes upon the joker in this deck. It is summer and fusion is turned on. Fusion can occur in any zone except the iron zone. The topic of fusion within layered EDM needs to be developed, but I believe that the heat energy released into the EDM (\(E(r)\)) will take the familiar form, \(\log E(r) = \alpha (r,T(r))\log T(r)\), where r is the distance from the center of the EMD ball. NCNG requires that the thermal radiation from the MW MBO today (and at the end of summer) to be small enough to be difficult to observe. That would require \(T(r)\) to be small enough through all zones except the iron zone.
The answers to these questions will form the required model for growth after a huge effort of study. Furthermore, the resulting model will set the initial conditions for section VII(E).