Section V

V.THE WEAK FIELD LIMIT OF THE NEW GRAVITY

There are many systems in the universe that fall into the weak category. The first topic is the most basic.

The weak field limit of static-matter sources and their gravitational fields is basic to an understanding of both local and distal systems. The non-relativistic motion of traceable matter in these fields provides a measure of their form. Eq. (4.16) will be central to the quest.
The weak field limit has \({G^{00}} = 1 – {V^{00}}\left( {\bf{x}} \right),{\rm{ }}{G^{jk}} = {g^{jk}}\), then one can convert Eq. (4.16) to the Schrodinger equation by substituting \(\phi \left( {{x^0},{\bf{x}}} \right) = \psi \left( {\tau ,{\bf{x}}} \right)\,\exp \left( {i{x^0}mc/\hbar } \right)\) and keeping the leading terms in c (\({x^0} = c\tau \)). The connection between the Newtonian potential (\({V_N}\)) and \({V^{00}}\) results, \[\hbar i(\partial \psi /\partial \tau ) + ({\hbar ^2}/2m){\nabla ^2}\psi \equiv {V_N}\psi = (m{c^2}/2){V^{00}}\psi .\tag{5.1}\] Use the Ehrenfest theorem to obtain Newtonian dynamics for a (neutral) particle moving in the five gravitational fields, \[m({d^2}{\bf{x}}/d{\tau ^2}) = – \nabla {V_N}\left( {\bf{x}} \right),\tag{5.2}\] \[ V_N(\mathbf{x}) = m \sum_{q = 1}^{N_q} \mathcal{A}_N \left( q, \mathbf{x} \right), \tag{5.3} \] \[ \mathcal{A}_N \left( q, \mathbf{x} \right) = \tfrac{1}{2} c^2 v_{00}(q,\mathbf{x}), \tag{5.4} \] The next step is to adapt Eq. (J.5) in Appendix J to a situation where the source is a static collection of (atom-sized) particles that are defined by a mass density (\(\rho ({\bf{x}})\)). Eq. (5.2) requires only \({v_{00}}(q,{\bf{x}})\)—justifying the neglect of other gravitational components in Eq. (J.5). The static version of Eq. (J.5) has a right-hand side that is proportional to a sum of delta functions for the location of each member of the collection. Auto-correlate both sides of Eq. (J.5), i.e., integrate over volume elements that contain very many sources, but are small compared to the inverse of the logarithmic derivative of \({{\rm{v}}_0}_0\). Divide both sides by the volume of the element. The right-hand side will be proportional to the number of particles per unit volume (\(\rho ({\bf{x}})/m\)). Rearrange the terms to obtain the desired equations, \[ 4\pi G Z_{gq} \, Q_{GC}(q,E)\,\rho(\mathbf{x}) = \nabla^2 \mathcal{A}_N(q,\mathbf{x}) – L_{gq}^{-2} \mathcal{A}_N(q,\mathbf{x}),
\quad \dfrac{d^2 \mathbf{x}}{d\tau^2} = – \sum\limits_{q = 1}^{N_q} \nabla \mathcal{A}_N(q,\mathbf{x}). \tag{5.5} \] where \(\rho \) is the mass density of the source. Eq. (5.5) will be used primarily for the study of galaxy-sized objects or larger. For clarity, there is no dark matter, dark energy, or dark anything in the NCNG universe. Rho is the density of ordinary matter (stars, gas, dust, white dwarfs, neutron stars, and the NCNG version of black holes of all masses)—usually called visible matter.

The solution of Eq. (5.5) for a point source of mass M is given, \[ \mathcal{A}_N(r) \equiv \sum\limits_{q = 1}^{N_q} \mathcal{A}_N(q,r) = – \frac{MG}{r} \sum\limits_{q = 1}^{N_q} Z_{gq} \, Q_{GC}(q,E(\phi_{ep})) \, \exp\left( – \tfrac{r}{L_{gq}} \right), \tag{5.6} \] where I have included the full seasonal dependence of \({Q_{GC}}\). In summer, one requires that the acceleration potential \(\mathcal{A}_N(r)\) must reduce to Newton’s Law when \(r < < {L_{g1}}\). Since \({Q_{GC}}(q,0) = 1\) for matter in summer, there is a constraint, \[\sum\limits_{q = 1}^{{N_q}} {{Z_{gq}}{\kern 1pt} } = 1\tag{5.7}\] There is another constraint for all seasons. The gravitons can represent either attractive or repulsive forces. Therefore, I choose the following: \[\begin{array}{c}{{\rm{Z}}_{g1}}{\rm{ < 0 (repulsive),}}\\{Z_{g2}}{\rm{ > 0 (attractive),}}\\{Z_{g3}}{\rm{ < 0 (repulsive),}}\\{Z_{g4}}{\rm{ > 0 (attractive),}}\\{Z_{g5}}{\rm{ < 0 (repulsive),}}\end{array}\tag{5.8}\] where the current values of these universal constants \({Z_{gq}}\) are listed in Appendix Z and used throughout NCNG. Appendix Z tabulates all the universal constants that define NCNG. Since NCNG is a work in progress, Appendix Z will be periodically updated as more accurate observations constrain the predictions of NCNG.
Each \({Q_{GC}}(q,0)\) can be different from 1 during winter, and they are also universal constants tabulated in Appendix Z. The analog of Eq. (5.8) follows, \[\sum\limits_{q = 1}^{{N_q}} {{Z_{gq}}{\kern 1pt} {Q_{GC}}(q,1)} = 0{\rm{ (winter)}}.\tag{5.9}\] There is no local gravity in winter, so there can be no stars, no white dwarfs, no core-collapse cinders, and no NCNG versions of supermassive black holes in winter. This subject will be revisited in the section about physics in winter.
The Yukawa form of Eq. (5.6) has far-reaching consequences in all seasons. The most obvious consequence is that \(r \, \mathcal{A}_N(r) \to 0\) for finite r. Furthermore, the point-source gravitational force becomes weakly repulsive at large distances. The Green’s function for Eq. (5.5) has a structure that requires a knowledge of \(\rho \) for all values of r to calculate \(\mathcal{A}_N(r)\) for small r—a serious complication. Since Eq. (5.6) does not represent an inverse-square force law, the virial theorem (a frequently used tool in astrophysics) is only useful for stars and unreliable for galaxies and larger objects.

The fact that NG gravitation has finite range brings a new perspective of scale for matter organization. The five graviton types defined in Appendix Z give rise to a specific point source gravitational field. Using Eq. (5.6), one finds that the summer force is attractive for \(r < {R_1}\), repulsive for \({R_1} < r < {R_2}\), and attractive for \(r > {R_2}\), where \({R_1} = 11.6\) MLY and \({R_2} = 20\) MLY. Furthermore, \({R_2}\) is the same in winter. \({R_1}\) and \({R_2}\) are independent of the source mass. There is a small repulsive bump centered on \({R_1}\) and a very shallow gravitational well centered on \({R_2}\). This shallow well slightly favors two sources to be 20 MLY apart. The consequences of this result will be examined in section V(L).
The source of an object of interest is a collection of matter located near the center of the object. The resulting gravitational field at large distances is well approximated by the point source above. The gravitational field contains the matter, so I call the object a canister. The sources for a canister are gas molecules, stars, stellar cinders, galaxies, and clusters of galaxies. The size of a canister is finite, but two canisters will be oblivious of each other only when separated well beyond \({R_2}\). If so separated, the canister source matter within will evolve undisturbed.
Canisters can move and they can (rarely) collide. When two canisters collide, the final state depends on the mass and matter distribution of each. The collision will be inelastic. The final state will be one or more canisters and possibly a splattering of unconfined matter contributed to the local inter-canister space. It is assumed that the matter within a canister is concentrated near the center, then the impact parameter for the collision would have to be smaller than \(2{R_1}\) for a significant disruption of the sources.
The second level of sequestration is the division between gas, stellar, galactic, or cluster mass objects within a canister. For some (massive) canisters the intergalactic gas is thought to contribute much more mass to the canisters than the galaxies.
The intergalactic gas is a mixture of hydrogen, helium, and fusion nuclides (stirred in from galaxies moving through the gas), and it is not visible using optical photography. This extremely diffuse gas is very hot (\(10^6 \, K\)) and clearly seen using soft Xray photography (0.4 Kev). Furthermore, gas is found to be concentrated in a spherically symmetric massive diffuse gas object (MDGO). The radius of a MDGO can be several million light years, and the mass can be as small as a dwarf galaxy or as great as 100 heavy galaxies. Appendix F has information about MDGOs and their gravitational fields.
After a collision, a canister could have two or more MDGOs as well as galaxies. This N body orbit problem would be extremely complex. The simplest canister would have one galaxy and one much heavier MDGO. The center of the canister would be at the center of the MDGO, and the galaxy would orbit around the center. Appendix K has information about galaxy orbits within a MDGO, and Section V(L) has a study of our own milky way (MW) canister.
Even a huge mass MDGO still has a weak-field gravitational potential and an extremely tiny density. Thus, a MDGO cannot support any pressure differential—e.g., order one millionth atmospheres between the center and the outer edge two million light years above! This fact supports the model of Appendix F, where the gas pressure, temperature, and density are constant out to a radius (and zero beyond).
There are two examples of two MDGOs in a canister. Chandra Xray photography shows that Abell 399 and Abell 401 are headed for a collision in the future. The centers of the MDGOs for these two clusters are currently 10 MLY apart—clearly within the same canister. NG would consider these two clusters as one object (accounting by canisters). A Chandra photo of Abell 2744 shows two co-orbiting MDGOs (unequal size) close enough to be in the same canister.
It is likely that there is an NG version of a super-massive black hole near the center of each MDGO. When such an object is active, it becomes a very bright source of high energy (10 Kev) photons. The faint cosmic Xray background is thought to come from such distant sources.
Many Xray studies are for containers that have MDGOs in collision. Such collisions cause compression shock waves moving at sub-light velocities that are much hotter and brighter than an unperturbed MDGO. Furthermore, the photons from the compressed zone move ahead of the shock waves and heat the unperturbed gas (a chaotic evolving temperature map).
The past or future of canisters with two or more co-orbiting MDGOs is very difficult to predict from present day observations. However, viscosity shear forces are independent of gas density, so one should expect that viscosity would play a big role with the tiny-density MDGOs. I believe that the eventual fate of a multi MDGO canister is a one MDGO canister. The mass of the MDGO would be conserved, but the size, density, and temperature would change. There is an important fact that will end this subsection. The soft Xray emissions of a MDGO are powered by Kelvin (and Helmholtz) contraction (Appendix F). Therefore, soft Xray emissions from a canister in winter will also be observable. The only direct information about winter must come from soft Xray astronomy.

There is a significant difference between the NCNG explanation and the GRLCDM explanation for large-scale structure (LSS). With GRLCDM, LSS must be created in the earliest stage of the Big Bang. NCNG does not require any explanation for LSS—it is what it is. Canisters eternally move about. Their contents evolve independently. Rarely, there are canister collisions—new canisters moving in different directions replace the two colliders.
Canisters can only be identified using high resolution soft Xray telescopes to see the MDGO sources (none exist now). Thus, NCNG has a hidden order that cannot be observed using (the many) longer wavelength telescopes. What one sees is a snapshot of a slowly changing vista that might make more sense using soft Xray astronomy.
There is another important difference. The GRLCDM version of structure is created at the same time, and the statistical nature of the proposed creation mechanism makes correlations universal. For example, the correlation-coefficient for the relative positions of galaxies in GRLCDM is thought to be a function only of the (scalar) distance between any two galaxies—independent of their location or orientation. In NCNG, the correlation-coefficient is not universal (a pathological function instead) and has no useful application for LSS. This difference cannot be tested directly since that would require two observers communicating over a vast distance. There is another important scale structure developed in section V(L).

For purposes of comparison, the general relativity summer version of Eq. (5.5) follows, \[\begin{array}{l} 4\pi G[{\rho _{VM}}\left( {\bf{x}} \right) + {\rho _{DM}}\left( {\bf{x}} \right)] = {\nabla ^2}\mathcal{A}_{GR}\left( {\bf{x}} \right),\\ {d^2}{\bf{x}}/d{\tau ^2} = – \nabla \mathcal{A}_{GR}\left( {\bf{x}} \right). \end{array}\tag{5.10}\] The \({\rho _{VM}}\) factor is the density profile for visible matter. The \({\rho _{DM}}\) factor represents the density profile of what is called dark matter—an unknown gaseous substance that is thought to surround gravitating visible masses in a halo.
For spiral galaxies, there are various shapes for \({\rho _{DM}}\) in the literature. A common choice is a three-parameter spherically symmetric profile, \[{\rho _{DM}}(r) = \rho _{DM}^0{[1 + {(r/{r_{DM}})^2} + {\lambda _{DM}}{(r/{r_{DM}})^4}]^{ – 1}}.\tag{5.11}\] Note that \({\lambda _{DM}} > 0\), otherwise the volume integral of \({\rho _{DM}}\)will be infinite—not acceptable. Other shapes also require three parameters for finiteness [6].
There are important consequences of Eq. (5.10) and Eq. (5.11). If \({\rho _{DM}} = 0\), then there is no choice for \({\rho _{VM}}\) that will come even remotely close to providing a correct rotation velocity profile for any spiral galaxy. GR can be a candidate for a correct gravitational theory only if dark matter exists.
The GR solution for rotation always requires using the three extra parameters of Eq. (5.11) compared to Eq. (5.5). Worse yet, the dark matter parameters are adjustable (different for each galaxy), and they cannot be predicted by using a knowledge of the shape of \({\rho _{VM}}\). In contrast, the constants that define all aspects of NG gravity are universal constants—the same for all galaxies, for all times, and for all locations in the universe.
It should be clear that dark matter is a fudge-factor that allows one to pretend that GR is a correct gravitational theory, not an actuality. The correctness of NG or GR can be determined by (existing and new) observations.

While all galaxies exhibit some degree of angular momentum and a rotational velocity distribution, very few are suitable objects for study. References [7] and [8] detail part of the selection process, the velocity measurement process, and the velocity profiles of a set of candidate suitable spiral galaxies. Appendix D has all the information of interest for the NCNG version of this topic. The following is a brief review of what I believe are the requirements for a suitable study to compare NCNG with GRLCDM predictions.
The observatory must be an array of numerous radio telescopes (there are only four). The observation is the (relative) doppler shift of the 21 cm line of H1. This is the most accurate measure for velocities at large distances from the galactic center where there are few stars, and where the deviations from Newtonian rotation are expected to be greatest.
The distance to the candidate spiral galaxy must be accurately known. I consider observations of cepheid variables within the candidate as the only reliable method. The ladder of distances is unreliable and subject to considerable disagreement. There are very few such observations.
The candidate galaxy must have a flat and thin circular disk oriented between face-on and edge-on. Bars should be faint, and arms should be faint and confined within the circular disk. A bulge at the center is ok, but significant lumps or imbedded dwarf galaxies are not ok. It is also necessary that there are no oddities in the observed rotation velocity profile, e.g., a band which has negative angular momentum compared to the remainder of the profile. Once again, there are few galaxies that satisfy this set of constraints.
There is only one galaxy that I know about that satisfied all these conditions, NGC 3198. In Appendix D, A hypothetical galaxy which is similar to NGC 3198 was used to calculate an apples to apples set of NCNG and GRLCDM rotation velocity profiles. FIG. D2 and FIG. D3 illustrate these two profiles. The NCNG profile was chosen to resemble the observed profile of NGC 3198. The figures show that the GRLCDM profile is not even remotely close to the NCNG profile—a dismal failure.
There are many rotation profiles that are based on visible wavelengths instead of the 21 cm wavelength. These profiles are not as accurate nor extensive as the 21 cm profiles, but they may extend beyond the distance from the galactic center where the NCNG and GRLCDM profiles start to diverge. If some of these galaxies also happen to have cepheid variable distances tabulated, then one can generate figures like those in Appendix D. This method resembles a catalogue search. I believe that any additional information will end up destroying GRLCDM as a viable theory.

In this subsection, the gravitational field will be local, static, and spherically symmetric. Eq. (4.20) will be solved in polar coordinates within the region where the spherically symmetric source is zero—a point source. The purpose of this study is to develop the relativistic corrections to weak field orbits. The distances will be local, and the season will be red summer. Omega is a linear function of \({v_{\mu \nu }}\left( {q,r} \right)\). Local and summer mean that one can rewrite (unprimed) Eq. (4.4) in terms of the compound five gravitational fields, \[\begin{array}{l}{V_{\mu \nu }} = \sum\limits_{q = 1}^{{N_q}} {{v_{\mu \nu }}\left( {q,r} \right)} ,\\2{\Xi _\mu }_\nu = {D^\varepsilon }\left( {{T_{\varepsilon \mu \nu }}} \right),\\0 = {\Omega _\mu }_\nu = {\Xi _\mu }_\nu – {\textstyle{1 \over 2}}{g_\mu }_\nu {g^{\beta \lambda }}{\Xi _{\beta \lambda }}{\rm{,}}\\{T_{\varepsilon \mu \nu }} = {D_\varepsilon }{V_\mu }_\nu + {\textstyle{1 \over 2}}\left( {{g_{\varepsilon \mu }}{S_\nu } + {g_{\varepsilon \nu }}{S_\mu }} \right) – \left( {{g_{\varepsilon \mu }}{N_\nu } + {g_{\varepsilon \nu }}{N_\mu }} \right),\\{S_\mu } = {\partial _\mu }V_\lambda ^\lambda ,\\{N_\mu } = {D_\lambda }V_\mu ^\lambda .\end{array}\tag{5.12}\] Eq. (5.12) is solved by using two potentials and careful application of the covariant derivatives for polar coordinates, \[\begin{array}{l}{V_{00}} = V(r)\\{V_{rr}} = U(r)\\0 = {\Omega _{00}} = ({\partial _r}U + U{r^{ – 1}}){r^{ – 1}},\\0 = {\Omega _{rr}} = – ({\partial _r}V + U{r^{ – 1}}){r^{ – 1}},\\0 = {\Omega _{\theta \theta }} = – (r/2)(r\partial _r^2V + {\partial _r}V + {\partial _r}U),\\0 = {\Omega _{\phi \phi }} = {\Omega _{\theta \theta }}{\sin ^2}\theta .\end{array}\tag{5.13}\] Eq. (5.13) is solved as follows, \[\begin{array}{l}U = V = – b/r,\\b = 2GM{c^{ – 2}},\end{array}\tag{5.14}\] where M is the mass of the source and G is the universal gravitation constant. The b length factor comes from Eq. (5.2) and Eq. (5.6)—known in GR as the Schwarzschild radius. Eq. (5.14) is correct for strong fields as well, and it will be revisited in the strong-field limit section. Eq. (4.14) and Eq. (4.15) can be solved using Eq. (5.14), \[\begin{array}{l}{G^{00}} = {[1 + H(V)]^{ – 1}},\\{G^{rr}} = – {[1 + H( – U)]^{ – 1}},\\{G^{\theta \theta }} = – {r^{ – 2}},\\{G^{\phi \phi }} = – {r^{ – 2}}{\sin ^{ – 2}}\theta ,\end{array}\tag{5.15}\] where H is an arbitrary function of V except for the following constraints. The physical region is for \( – \infty < V < 0\), and in that region \( - 1 < H < 0\). Furthermore, \(\partial H/\partial V = 1\) at \(V = 0\). The orbits for the potentials of Eq. (5.14) are developed in Appendix E using the weak-field limit of Eq. (5.15).
This problem is the first instance of the need for gravitational components beyond \({v_{00}}(q,x)\). The orbit solutions of Appendix E illustrate an important fact; even though \({V_{00}}\) and \({V_{rr}}\) have the same strength, \({V_{00}}\) causes the Newtonian orbit for a non-relativistic object, but \({V_{rr}}\) causes only the tiny advance of perihelion. This fact is the rationale for neglecting the nine gravitational components (beside \({V_{00}}\)) for non-relativistic objects. The story for photons (or relativistic objects) is very different—\({V_{rr}}\) is responsible alone for the deflection of photons. Thus, one must have more than one gravitational component to describe the path of a photon in a gravitational field.

If the mass of a canister is dominated by a MDGO, the galaxies can be ignored in this subsection. The goal is to develop an equation for the potential well using Eq. (5.5) for a canister with one MDGO. Refer to Appendix F for details of the solution for Eq. (5.5) and other important information. A MDGO is like the elephant in the living room—vitally important but often ignored.
Since the MDGOs and galaxies are made up of non-relativistic matter, Eq. (5.5) is sufficiently accurate as noted in the preceding subsection. The path of photons through a canister is much more complex (see VI(F)).
Figure 1 shows the dimensionless (red summer) NG gravity field (\({V_{00}} = 2{A_N}(r){c^{ – 2}}\)) for an example constant-density MDGO having a size \({R_{MDGO}} = 2x{10^6}\) LY and a mass \(M = {10^{14}}\) solar masses. The NG gravitational constants are found in Appendix Z. This gravity well is very deep. The escape velocity for an object at \({R_{MDGO}}\) is 15013 km/s, and an object must have a radial velocity greater than 1480 km/s to get over the repulsive barrier (enter the canister during summer).

Figure 1. The dimensionless NG gravitational potential (for the example canister described above) versus distance from the canister center. The peak of the weak repulsive barrier is at the right side of the figure.

No compact object is likely to enter or leave a canister of this mass, but another canister can enter. Given the gravitational constants, the dimensionless field (\({V_{00}}(r)\)) is a function of M and \({R_{MDGO}}\).

The orbital motions of individual galaxies within a canister are determined in a stepwise manner. The mass of a contained MDGO is assumed to be significantly greater than the mass of the galaxies associated with that MDGO. Thus, the galaxies can be considered as perturbation.
The first step is to obtain the orbit for an object in the radially symmetric field of a single MDGO using the equations of Appendix F and the techniques of Appendix E. This step will provide useful information (the subsequent steps are beyond the scope of this document).
One can use the example model MDGO of section V(H) to calculate the example orbit of a galaxy. The orbit will lie in a plane containing the center of the MDGO (with arbitrary orientation). The orbital information is summarized by studying two extremes of angular momentum. Case one is for a circular orbit of radius R (maximum angular momentum), and case two is for a linear orbit (angular momentum = 0) with turning points at R and -R. Figure 2 shows the orbital periods for the two cases as a function of R.

Figure 2.Periods of a galaxy orbiting within an example MDGO. Circular orbit (green) and zero angular momentum (red).

The period for any angular momentum will lie between the two curves of Figure 2. The primary observation for this subsection is that the orbits for this model will have periods of order 350 million years—small compared to the average canister collision frequency. Thus, considerable evolution of the relative positions of the MDGO galaxies will occur between potentially disruptive collisions.

Ultra-diffuse galaxies (UDGs) can have significant size (like the milky way) but faint luminosity (like a dwarf galaxy). UDGs of large size are found in distant clusters of galaxies, generally nearer to the center. There could be smaller UDG’s in a distant cluster that are too small and faint to be observed. The UDG velocity dispersion suggests a robust gravitational force, but the faint luminosity suggests a small visible mass. Using GRLCDM to explain their structure via Eq. (5.10) yields absurdly high ratios of dark-matter mass / visible mass—not acceptable. The NG explanation is plausible but extremely complex.
The most dramatic difference between GR and NG is illustrated by a simple example distribution of (visible) matter. Consider a thin radial shell of radius \({R_S}\) having constant density, \[\begin{array}{l}\rho (r) = \sigma \,\delta (r – {R_S}),\\{M_S} = 4\pi \sigma {R_S}^2,\end{array}\tag{5.16}\] where \(\delta \) is the Dirac delta function and \({M_S}\) is the shell mass. The shell is an approximation for a group of galaxies distributed at a constant distance from a center. For GR, there is no gravitational force within the shell (constant potential). For NG inside the shell vacuum, there is a gravitational potential bulge (repulsive force supported by no matter inside the shell). Using Eq. (5.16) in Eq. (F.1) yields the NG potential: \[\begin{array}{l}{V_{00}}(r) = – 2{M_S}G\,{({R_S}{c^2})^{ – 1}}F(r),\\F(r) = \sum\limits_{q = 1}^{{N_q}} {{Z_q}} {L_{gq}}{r^{ – 1}}\sinh (r{L_{gq}}^{ – 1})\exp ( – {R_S}{L_{gq}}^{ – 1}),\end{array}\tag{5.17}\] where the constants are tabulated in Appendix Z. Eq. (5.17) is presented as an example comparison of NCNG and GRLCDM. A real canister can have many galaxies orbiting about the MDGO center with perturbations between each other. At any point where the vector NG force of all the orbiting galaxies is zero, there will be the maximum of a vacuum repulsive gravitational potential (a peak). One should expect that there will be many peaks. Since the galaxies are moving, the peaks will move, wax or wane, appear or disappear. I believe that the force from the MDGO can be neglected since it is canceled by the acceleration of each orbiting galaxy. If a peak lies within a galaxy, there will be a consequence depending on the size and strength of the repulsive volume and the size and mass of the galaxy (if it was outside the repulsive volume). The attractive force of the NG galactic matter is countered by the repulsive volume.
A UDG results only if the repulsive volume is large enough and strong enough to hold all the mass of the galaxy with significantly smaller density. Otherwise, the galaxy would be largely intact, but it would contain a small lower density volume within (difficult to observe).
All of this represents a very complex evolution. There are peaks wandering about and galaxies moving as well. A UDG can only exist when a peak and a galaxy coincide (their paths cross). Thus, one should expect that a specific galaxy could have a UDG appearance during a brief period (likely less than 100000 years), and after the appearance the galaxy would be largely the same as before the appearance. The NCNG version of a UDG is radically different compared to the (failed) GRLCDM version. The complexities of this topic are beyond the scope of this document.

Our own canister (MWC) has the virtue of relative simplicity. What is known as the local group consists of two massive galaxies (MW and M31—Andromeda) and many lesser objects (negligible mass). These objects are associated with a MDGO of unknown mass and location. There are not any other candidate MDGOs within ten million light years, so one may assume that MWC has only one MDGO. In that case, gravitation within the canister devolves into a three-body problem (MDGO, M31, MW). This is an example of the elephant in the living room problem since studies of interactions between M31 and MW have been treated as a two-body problem, e.g., the erroneous notion that MW and M31 will collide in the future.
Direct evidence about our MDGO is problematic. The gas associated with our MDGO has been observed but misidentified as a wisp of warm-hot intergalactic matter (WHIM) —the defunct dark cosmology. These observations indicate that both M31 and the MW are within the boundary of our MDGO. Therefore, the center of our MDGO is within (roughly) 2 MLY of both massive galaxies. It is likely that MDGOs have a NCNG version of a super-massive black hole near their center, but that object would only be observable when ingesting an occasional star—not the case for MWC now.
The distance between MW and M31 is accurately known now (0.770 Mpc—cepheid), but the GRLCDM mass (including dark matter) of M31 is very poorly known. The rotation velocity profile of M31 has been established [17], so an accurate NCNG mass can be established using the methods of Appendix D. MW rotates, but the velocity profile is not accurate enough to get a good NCNG mass.
I believe that the warping of the rotating discs of MW and M31 are both caused by our MDGO instead of each other. Therefore, it could be possible to establish a direction toward our MDGO—an exercise for the reader, A good MDGO direction would simplify the search for a NG version of a super massive black hole near the center of the MDGO. Such an object (even if inactive) would likely emit radiation observable by the JWST.
MWC gravitation includes the three massive objects plus many dwarf satellite galaxies. There could be several (very small) UDGs near MW and M31 (as described in section V(J)). If there are genuine UDGs, their locations would constrain the mass and locations of the three massive objects. A possible candidate UDG would be And II [Ra(1,16,29.8), Dec(33,25,9)]—a satellite of M31.
Spectral red shifts are another source of information about the members of the MWC. However, red shift has three components (cosmologic, doppler, gravitational) and then only for the line-of-sight portion of information. Gravitational red shift in NG is a larger factor than in GR, thus making doppler determinations less reliable. One does not know where the center of our MDGO is located. For a plausible set of parameters, the extremes of the gravitational contribution to the apparent doppler velocity are \( \pm 41\) km/s (either MW or M31 is at the center of our MDGO). M31 is thought to be moving toward Milky Way at 110 km/s. The actual receding velocity could be \( – 110 \pm 41\) km/s—a major uncertainty due to NG. Furthermore, MW and M31 have different orbits about the MDGO center (see Appendix K). The risible notion of a near-future collision is not supported.
NCNG can yield an impressive crop of accurate information about MWC, whereas GRLCDM can yield only question marks.

This topic is by far the most complex application of both NC and NG. NG astronomy up to this point has been all about canisters. Canisters are all about MDGOs, and MDGOs are all about soft Xray astronomy. There is no soft Xray astronomy now, and the two toy observatories of the past were hopelessly inadequate. Conventional astronomy of interest here is all about galaxies (Their orbits are influenced by MDGOs that cannot be located). The next step-up is the organization of canisters on a larger scale—filaments of canisters.
This subsection is also the most important one of section V because this is the first case where knowledge of the winter seasons is important. Filaments of galaxies (residing in canisters) are considered as the largest structures in the universe. They are billions of light years in size and appear as long kinky lines in the sky. It is difficult to understand how filaments had enough time to organize a form in a GRLCDM universe. This problem does not exist in NCNG since thousands of cycles are available—evolution during all seasons. What one sees today is just a snapshot of an evolution that has continued for eons.
The rationale for my choice of five gravitons in NG will now become clear. A minimum of five gravitons is required so that the NG gravitational potential of a point source (Eq. (5.6) where \({V_{00}} = 2{c^{ – 2}}{A_N}\)) will have a single shallow attractive gravitational well centered at a distance \({D_W}\)from the source. \({D_W}\) (\({R_2}\) of subsection V(C)) is independent of the mass of the source. Thus, point sources of any mass can become trapped when they are about \({D_W}\) apart. \({D_W}\) is a universal constant since it is calculated using the universal constants of Appendix Z. Can one find a value for \({D_W}\)?
A study of four nearby major galaxies and their satellites (M81, M94, NGC5128, and Sculptor) lead to an interesting result. The distance from earth of the four major galaxies are known and in the same distance ladder slot. The directions are very accurate, so the distance between any pair of the four major galaxies is known. They are far enough apart so that there are four separate single major-galaxy and single MDGO canisters. The location of each canister center is unknown but likely to be within a million light years of each major galaxy. Of the six possible pairs, four have nearly the same separation (\(19.95 \pm 0.77\) million light years), and the other two are significantly different. The four major galaxies do not lie in a plane, so one has four possible equilateral triangles (defining a plane) and one outlier not in the plane. I have chosen the equilateral triangle (M81, NGC5128, and Sculptor) and M94 as the outlier. In that case, \({D_W} = 20.081\). There is another canister of interest (MWC is a two major-galaxy and one MDGO canister). MWC is too close to the triangle to be part of the connectivity.
The gravitational constants of Appendix Z are constrained by this choice of distance. If future data requires a different distance, the reasoning below will not be changed, but details will change (for every topic in NG). I also believe that smooth evolution is best served by having the same distance \({D_W}\) in winters—a constraint on winter NG constants in Appendix Z. Using point sources for this study is a good approximation at the distances of interest. The point source coincides with the center of mass for a canister. The gravitational well is shallow but large enough to hold a MDGO and all the galaxies of a canister. It will be important to keep in mind that a canister source in the well can orbit about the center of the well. An observed galaxy in the well will orbit within the canister (which in turn will orbit within the well). The lack of information about canister MDGOs will be a persistent problem—no soft Xray astronomy.
It should be obvious that a chain of sources separated by \({D_W}\) could form the frame for a filament. There will be individual galaxies and canisters that are not part of the chain-frame connectivity but move in orbits within the gravitation of the chain-frame (I call them rogues). For example, MWC could be a rogue. There are also rare interlopers that pass through the chain-frame gravitation without being trapped. MWC could also be an interloper—a question of unknown velocities.
Rogues can (rarely) interact with frame sources and possibly dislodge them, thus breaking the chain. Rogues can also conspire to replace a missing frame source, thus repairing the chain. An interloper can also knock out a frame source. The replacement frame source can be a newly formed canister or the old, displaced canister. Once again, one sees an extremely complex evolution with details beyond the scope of this document. However, some examples will give one an insight into the stability of a filament.
Consider a very long linear-filament consisting of equal mass canister-sources separated by \({D_W}\). The NG dimensionless gravitational potential surrounding the filament is \({V_{00}}(r,z)\), where z the position along the line joining the centers of the sources, and r is a radial distance perpendicular to z. \({V_{00}}\) is independent of the radial angle, \(\phi \)—cylindrical symmetry. There is another symmetry, \({V_{00}}(r,z + {D_W}) = {V_{00}}(r,z)\).
The origin (\(r = z = 0\)) is at the center of an arbitrary source in the linear filament. There are two separate orbits of interest. The orbit of a rogue is determined by the sum of the NG potentials (Eq. (5.6)) of all the filament canister sources. The orbit of a filament-source (near the origin) is determined by the total sum excluding the potential of the source at the origin. \({V_{00}}\) is well approximated by summing the two or three source-potentials of the sources near the origin. Refer to Appendix G for some details of this study.
The main discovered factor in this study is that the period of an orbiting canister-source is measured in tens of billions of years (up to trillions). Thus, canister sources in a linear frame are very weakly bound, and an observed filament is favored by having the heaviest possible canister sources. Furthermore, rouges are the only candidates for dislodgement, so the lightest possible rogues are favored. In that case, a single pass of a rouge close to a canister source will provide an impulse that is small enough so that the source is still bound in the future when another pass occurs (providing an impulse in a different direction). The dislodgement mechanism is significantly slowed by the randomness of the impulse directions. I believe that a linear filament can last for tens of billion years—very slow evolution.
An observed filament has galaxies closer together than \({D_W}\). The extra galaxies are rogues with orbits favoring the z axis between canister sources. Motion along that axis is agonizingly slow—hundreds of million years. This topic deserves much study.