IV. THE NEW GRAVITY
This section will add gravity to the universe. The provisions of section III(A) apply with the addition of the Newtonian gravitational (universal) constant (\(G = 6.67430x{10^{ – 11}}{\rm{ }}{{\rm{M}}^3}{{\rm{K}}^{ – 1}}{{\rm{S}}^{ – 2}}\) ). There is one kind of graviton (a massless tensor-boson) in GR, but it is not a gauge boson.
In the new gravity (NG), gravitons are tensor gauge-bosons, and there are more than one kind.
Thus, a new universal constant is added to Appendix Z (\({N_q}\) defines the number of NG graviton kinds). There is no limit to \({N_q}\)—only sufficient to agree with observations. For reasons that will become clearer below (see V(L)), I have chosen \({N_q} = 5\), where each kind has a different (extremely tiny) mass.
There are eight (small mass) gauge-bosons for the strong interaction and six (massive) gauge-bosons for the weak interaction. Thus, NG joins the standard model of particle physics. The only outlier is the electro-magnetic force with one massless gauge-boson (photon). One is led to wonder if perhaps there should be two or more photons with different (extremely tiny) masses like NG?
A. The tensor gravity fields
As mentioned in Section II, there is one primary metric tensor (the rest frame for the apex field,
\( g_{00} = 1 \) and \( g_{jk} = -\delta_{jk} \)) for which the equations of NCNG are meant to be applied.
Each of the five gravity fields is a dimensionless second-order symmetric tensor,
\( v^\mu_{\ \nu}(q,x) \), where \( q = 1…N_q \). In each case, the four-coordinate (x) will be implicit and the usual contravariant, covariant, and contraction algebra is used.
One can define some useful scalars and vectors derived from \( v^\mu_{\ \nu}(q) \):
\( S(q) \equiv v^\lambda_{\ \lambda}(q) \),
\( J(q) \equiv v^\alpha_{\ \lambda}(q) v^\lambda_{\ \alpha}(q) \),
\( S_\mu(q) \equiv \partial_\mu S(q) \),
\( N_\mu(q) \equiv \partial_\lambda v^\lambda_{\ \mu}(q) \).
There are two pieces to the Lagrangian density (LD) for each of the five gravity fields, the “kinetic” density, \( \mathcal{L}_g^{T}(q,x) \), and a “potential” density, \( \mathcal{L}_g^{V}(q,x) \). The complete LD for each gravity field is then,
\( \mathcal{L}_g = \mathcal{L}_g^{T} – \mathcal{L}_g^{V} \). Each piece is defined as follows:
where \( \kappa_q = 8 \pi G Z_{gq} c^{-3} \) and \( Z_{gq} \) is a dimensionless universal constant related to the strength of each gravity field. \( L_{gq} \) is a characteristic length associated with each gravity field—universal constants. I have chosen the following lengths:
\[ L_{gq} = q L_{g1} \tag{4.3} \]
These lengths are vast—\( L_{g1} \) will be about one million light years. The strength multipliers will be discussed below.
The Euler-Lagrange equation for each field in the absence of sources yields:
\[
\begin{aligned}
0 &= \Omega^{gq}_{\mu \nu}
+ \tfrac{1}{2} L_{gq}^{-2} \Big( v_{\mu \nu}(q) – \tfrac{1}{2} g_{\mu \nu} v^\lambda_{\ \lambda}(q) \Big), \\[6pt]
\Omega^{gq}_{\mu \nu} &= \Xi^{gq}_{\mu \nu} – \tfrac{1}{2} g_{\mu \nu} g^{\beta \lambda} \Xi^{gq}_{\beta \lambda}, \\[6pt]
2 \, \Xi^{gq}_{\mu \nu} &= \partial^\varepsilon \big( T^{gq}_{\varepsilon \mu \nu} \big), \\[6pt]
T^{gq}_{\varepsilon \mu \nu} &= \partial_\varepsilon v_{\mu \nu}
+ \tfrac{1}{2} \big( g_{\varepsilon \mu} S_\nu + g_{\varepsilon \nu} S_\mu \big)
– \big( g_{\varepsilon \mu} N_\nu + g_{\varepsilon \nu} N_\mu \big).
\end{aligned} \tag{4.4}
\]
Eq. (4.1) and Eq. (4.4) are relative to the \({x^0}\) evolution parameter, i.e., the apex clock. These equations correspond to the primed LD of section III when \(A = 0\). The gravity fields need to interact with the apex field.
The central field will be studied below. In that case, polar coordinates are useful, so \({g_{00}} = 1\), \({g_{rr}} = – 1\), \({g_{\theta \theta }} = – {r^2}\), \({g_{\phi \phi }} = – {r^2}{\sin ^2}\theta \). In Eq. (4.1) and Eq. (4.4), \({\partial _\lambda } \to {D_\gamma }\) (\({D_\lambda }\) is the usual covariant derivative) and \(D = {r^2}\sin \theta \). The invariant volume element is \(d{V^4} \equiv D\,{d^4}x\), where \(D \equiv {\left( { – {\rm{det}}\left( g \right)} \right)^{1/2}}\).
B. Gravity and the apex field
The five gravity-fields just represent another form of matter, and they couple to the apex field using the same method that was developed in section III. In this case, the details of the method are different from the photon method, which was different from the fermion method as well. Furthermore, the gravity method is far more complex. The following set of new tensors are useful:
\[
\begin{array}{l}
\psi _\mu ^\alpha = \sigma _\lambda ^\alpha \Sigma _\mu ^\lambda ,\\
\omega _\mu ^\alpha = \psi _\lambda ^\alpha \psi _\mu ^\lambda ,\\
\tilde \psi _\lambda ^\alpha \psi _\mu ^\lambda = \delta _\mu ^\alpha ,\\
\tilde \omega _\lambda ^\alpha \omega _\mu ^\lambda = \delta _\mu ^\alpha ,\\
\psi _0^0 = 1 – c\tau [{\partial _0}\ln(\mathcal{P})],
\end{array}\tag{4.5}
\]
where \(\sigma \) and \(\Sigma \) are defined in Eq. (3.2) and Eq. (3.6). The five bare primed LDs are just Eq. (4.1) and Eq. (4.2) with fields and derivatives primed. The dressed LD is obtained using the following substitutions:
\[\begin{array}{l}{v_{\mu \nu }}{\left( q \right)^\prime }\mathop \to \limits_{dress} \psi _\mu ^\alpha \psi _\nu ^\beta {v_{\alpha \beta }}{\left( q \right)^\prime },\\{\partial _\mu }^\prime \mathop \to \limits_{dress} \sigma _\mu ^\alpha {\partial _\alpha }^\prime .\end{array}\tag{4.6}\]
Great care must be applied to constructing the dressed LD (\(\overline{\mathcal{L}}_g\))
since there will be many terms. There is a useful third rank tensor, \({\Lambda _{\mu \nu \lambda }}^\prime = {\partial _\mu }^\prime {v_{\nu \lambda }}^\prime \), that illustrates this problem. The dressed version of \(\Lambda ‘\)is
\[\begin{array}{l}{\Lambda _{\mu \nu \lambda }}^\prime \mathop \to \limits_{dress} F_{\mu \nu \lambda }^{\alpha \beta \gamma }{\Lambda _{\alpha \beta \gamma }}^\prime + E_{\mu \nu \lambda }^{\beta \gamma }{v_{\beta \lambda }}^\prime ,\\F_{\mu \nu \lambda }^{\alpha \beta \gamma } = \sigma _\mu ^\alpha \psi _\nu ^\beta \psi _\lambda ^\gamma ,\\E_{\mu \nu \lambda }^{\beta \gamma } = \sigma _\mu ^\alpha {\partial _\alpha }^\prime (\psi _\nu ^\beta \psi _\lambda ^\gamma ).\end{array}\tag{4.7}\]
E is a tiny correction (of order \({L_{ax}}^{ – 1}\)) that must be included. The E factor of Eq. (4.7) will cause an unusual contribution to the source-free Euler-Lagrange equation (first term),
\[
0 = \partial \overline{\mathcal{L}}_g^{\rm{T}} / \partial {v^{\mu \nu }}^\prime
– \partial \overline{\mathcal{L}}_g^V / \partial {v^{\mu \nu }}^\prime
– {\partial ^\varepsilon }^\prime \left( \partial \overline{\mathcal{L}}_g^T / \partial {\Lambda ^{\varepsilon \mu \nu }}^\prime \right).
\tag{4.8}
\]
The solution of Eq. (4.8) can be written as follows,
\[0 = \xi _{\mu \nu }^{\alpha \beta \gamma }{\Lambda _{\alpha \beta \gamma }}^\prime – {\textstyle{1 \over 2}}L_{gq}^{ – 2}\omega _\mu ^\alpha \omega _\nu ^\beta {v_{\alpha \beta }}^\prime – {\partial ^\varepsilon }^\prime \left( {G_{\varepsilon \mu \nu }^{\alpha \beta \gamma }{\Lambda _{\alpha \beta \gamma }}^\prime + M_{\varepsilon \mu \nu }^{\beta \gamma }{v_{\beta \gamma }}^\prime } \right)\tag{4.9}\]
where G is a series of five contractions of two F’s. M and \(\xi \) are each a series of five contractions of E and F. These are too lengthy to exhibit, and there is another step! The last step requires the following substitution to convert Eq. (4.9) into a now equation that involves only physical time,
\[\begin{array}{l}{v_{\mu \nu }}{\left( q \right)^\prime } = \tilde \omega _\mu ^\alpha \tilde \omega _\nu ^\beta {v_{\alpha \beta }}\left( q \right),\\{\partial _\mu }^\prime = \Sigma _\mu ^\alpha {\partial _\alpha }.\end{array}\tag{4.10}\]
The resulting equation can be written as follows,
\[0 = T_{\varepsilon \mu \nu }^{\alpha \beta \gamma }{\partial ^\varepsilon }{\Lambda _{\alpha \beta \gamma }} + L_{gq}^{ – 2}\{ {v_{\mu \nu }} – {\textstyle{1 \over 2}}{g_{\mu \nu }}v_\lambda ^\lambda \} + U_{\mu \nu }^{\alpha \beta \gamma }{\Lambda _{\alpha \beta \gamma }} + W_{\varepsilon \mu \nu }^{\alpha \beta }{\partial ^\varepsilon }{v_{\alpha \beta }},\tag{4.11}\]
where U and W are tiny corrections (of order \({L_{ax}}^{ – 1}\)). T is a series of seven terms involving both \(\omega \), \(\tilde \omega \), and the metric tensor. U involves a derivative of T and a contraction of \(\xi \). W involves a contraction of M and of G.
The now equations for the five fields are analogous to Eq. (3.11) for fermions or Eq. (3.19) for photons and are obtained in the same manner—the limit as \(c\tau \to 0\) in Eq. (4.5). The now limit of psi is \(\psi _\mu ^\alpha \to \delta _\mu ^\alpha \) and \(\partial_0 \psi_\mu^\alpha \to – \delta_0^\alpha \delta_\mu^0 [\partial_0 \ln(\mathcal{P})]\). Using these limits in Eq. (4.11) gives the source free form,
\[0 = {\Omega _{\mu \nu }}\left( q \right) + {\textstyle{1 \over 2}}L_{gq}^{ – 2}\{ {v_{\mu \nu }}\left( q \right) – {\textstyle{1 \over 2}}{g_{\mu \nu }}v_\lambda ^\lambda \left( q \right)\} + \delta {\kern 1pt} {\Omega _{\mu \nu }}\left( q \right),\tag{4.12}\]
where \(\Omega \) is given in Eq. (4.4) with \({\partial _0} = \partial /\partial c\tau \), and \(2\delta {\kern 1pt} {\Omega _{\mu \nu }}\) is the U and W parts of Eq. (4.11). The tiny correction term has too many terms to display in general, so I will limit the analysis to three cases of general interest.
Case 1 corresponds to a static-weak field without symmetry. The non-zero fields will be \({v_{00}}\left( {q,{x^1},{x^2},{x^3}} \right)\), and \({\Lambda _{k00}}\) will be non-zero. Only \(U_{00}^{k00}\) and \(W_{k00}^{00}\) can contribute, but they are both zero. Thus, \(\delta {\kern 1pt} {\Omega _{\mu \nu }} = 0\).
Case 2 corresponds to a static central field of any strength. The non-zero fields will be \({v_{00}}\left( {q,{x^1}} \right)\) and \({v_{11}}\left( {q,{x^1}} \right)\), and both \({\Lambda _{100}}\) and \({\Lambda _{111}}\) will be non-zero (\({x^1} = r\) ). Only \(U_{00}^{100}\), \(U_{00}^{111}\), \(U_{11}^{100}\), \(U_{11}^{111}\), \(W_{100}^{00}\), \(W_{100}^{11}\), \(W_{111}^{00}\), \(W_{111}^{11}\) can contribute, but they are all zero. Thus, \(\delta {\kern 1pt} {\Omega _{\mu \nu }} = 0\).
Case 3 corresponds to a plane-transverse gravitational wave. The non-zero fields will be \({v_{23}}\left( {q,{x^0} = c\tau ,{x^1}} \right)\), and both \({\Lambda _{023}}\) and \({\Lambda _{123}}\) will be non-zero. Only \(U_{23}^{023}\), \(U_{23}^{123}\), \(W_{023}^{23}\), \(W_{123}^{23}\) can contribute, but only one (\(U_{23}^{023}\)) is non-zero. Since \(U_{23}^{023} = ({\partial _0}\omega _0^0)_{now} = – 2[{\partial _0}\ln(\mathcal{P})]\), the correction will be \(2\delta \, \Omega_{23} = U_{23}^{023}\Lambda_{023} = – 2({\partial _0}\ln(\mathcal{P})){\partial _0}v_{23}\). If one substitutes the correction into Eq. (4.12) and neglects the infinitesimal \({L_{gq}}^{ – 2}\)term, the equation of motion for a gravitational wave is the following,
\[
0 = {\partial _0}^2 v_{23} – {\partial _1}^2 v_{23} – 2({\partial _0} v_{23})[{\partial _0} \ln(\mathcal{P})]. \tag{4.13}
\]
Eq. (4.13) is the same as Eq. (3.20) for an electromagnetic wave. Thus, the apex field treats gravitons in flight in the same manner as photons. A gravitational wave train will appear to move with the same velocity (= c) for observers regardless of distance from the source, and the wavelength of the train will be red shifted (in early red summer) to the same degree as a photon. The apex field can directly add or subtract energy from a graviton in flight.
C. Coupling matter to the new gravity
The second pathology of GR is due to the strong field limit of the Schwarzschild solution. In the new gravity, one demands that the orbit equation for a gravitational source of finite mass should be well-behaved—no infinities or zeroes. That constraint will require a new set of dimensionless coupling tensors. I have chosen the following form,
\[\begin{array}{l}{V_{\mu \nu }}\left( x \right) \equiv \sum\limits_{q = 1}^{{N_q}} {{Q_{GC}}(q,E){v_{\mu \nu }}\left( {q,x} \right)} ,\\{{\tilde G}_{\mu \nu }} = {g_{\mu \nu }} + {H_{\mu \nu }}\left( {{V_{\alpha \beta }}} \right),\\{G^{\mu \lambda }}{{\tilde G}_{\lambda \nu }} = \delta _\nu ^\mu ,\end{array}\tag{4.14}\]
where \({Q_{GC}}(q,E)\) is the gravitational charge (defined in the following section). The H tensor is a power series in V (which takes a simple form when V is diagonal):
\[\begin{array}{l}
{H_{\mu \nu }} = \sum\limits_{n = 1}^\infty {{a_n}} {h_{\mu \nu }}(n),\\
{h_{\mu \nu }}(n + 1) = V_\mu ^\lambda {h_{\lambda \nu }}(n),\\
{h_{\mu \nu }}(1) = {V_{\mu \nu }},\\
H_\mu ^\nu = \delta _\mu ^\nu \sum\limits_{n = 1}^\infty {{a_n}} (V_\mu ^\mu )^n
\equiv \delta _\mu ^\nu H(V_\mu ^\mu ), \quad \text{if $V$ is diagonal.}
\end{array}\tag{4.15}\]
The leading coefficient must be \({a_1} = 1\) to give the correct weak field limit. If V is diagonal, H can be an arbitrary continuous-function—subject to the constraint \(\partial H(V)/\partial V = 1\) when \(V = 0\). The form for this function will be developed in section VI(D).
The method of coupling will be illustrated for three cases. First, consider a (fictitious) scalar boson (possibly having large mass). The LDM is modified, \({g^{\mu \nu }}{\partial _\mu }{\phi ^ * }{\partial _\nu }\phi \to {G^{\mu \nu }}{\partial _\mu }{\phi ^ * }{\partial _\nu }\phi \), and the Klein-Gordon equation of motion follows,
\[0 = {\partial _\mu }({G^{\mu \nu }}{\partial _\nu }\phi ) + {({m_B}c/\hbar )^2}\phi .\tag{4.16}\]
Eq. (4.16) is very useful because it yields the correct relativistic version of Hamilton’s principle function. Thus, Hamilton-Jacobi theory is available for the classical orbit of a featureless object of mass m moving in a static gravitational field. This methodology will also work for \(m = 0\) giving a simple method for photon ray tracing, all valid for the new gravity. Refer to Appendix E for central field details.
Next, consider a fermion. The modification is indirect,
\[\{ {\gamma ^\mu },{\gamma ^\nu }\} = 2{g^{\mu \nu }} \to \{ {\Gamma ^\mu }(x),{\Gamma ^\nu }(x)\} = 2{G^{\mu \nu }}(x),\tag{4.17}\]
and the Dirac matrixes are no longer constants. The Dirac equation follows,
\[0 = i{\Gamma ^\mu }{\partial _\mu }\psi + (i/2)[{\partial _\mu }({\Gamma ^\mu })]\psi – ({m_F}c/\hbar )\psi .\tag{4.18}\]
Eq. (4.17) demonstrates a clear necessity for G to be well-behaved.
Finally, consider electromagnetism in the absence of a source. The modification of \(\mathcal{L}_{EM}\)
is \({F_{\mu \nu }}{g^{\mu \alpha }}{g^{\nu \beta }}{F_{\alpha \beta }} \to {F_{\mu \nu }}{G^{\mu \alpha }}{G^{\nu \beta }}{F_{\alpha \beta }},\) and the equation of motion follows,
\[0 = {\partial ^\varepsilon }[\left( {G_\varepsilon ^\alpha G_\mu ^\beta – G_\mu ^\alpha G_\varepsilon ^\beta } \right){F_{\alpha \beta }}].\tag{4.19}\]
Eq. (4.19) is required to obtain the correct gravitational red shift for weak fields.
The sources for Eq. (4.12) for gravitational systems of interest are as follows,
\[{J_{\mu \nu }}\left( q \right) = {\Omega _{\mu \nu }}\left( q \right) + {\textstyle{1 \over 2}}L_{gq}^{ – 2}\{ {v_{\mu \nu }}\left( q \right) – {\textstyle{1 \over 2}}{g_{\mu \nu }}v_\lambda ^\lambda \left( q \right)\} = {\kappa _q}\,\partial \mathcal{L}_M / \partial v^{\mu \nu}(q), \tag{4.20}
\]
where \(\delta {\kern 1pt} \Omega \) is neglected for these systems, and Eq. (4.20) is for physical time (\({x^0} = c\tau \)).
Eq. (4.20) can be rewritten in a more transparent form using the unprimed version of Eq. (4.4),
\[
\begin{array}{l}
\Theta_{\mu \nu} \equiv 2 \, \Xi_{\mu \nu} + L_{gq}^{-2} v_{\mu \nu}, \\
\Theta_{\mu \nu} = 2 \left( J_{\mu \nu} – \tfrac{1}{2} g_{\mu \nu} J_\lambda^{\ \lambda} \right), \\
J_{\mu \nu}(q,x) = \kappa_q \, \partial \mathcal{L}_M / \partial v^{\mu \nu}(q,x).
\end{array} \tag{4.21}
\]
For each q one has the following:
\[{\partial ^\varepsilon }{\partial _\varepsilon }{v_\mu }_\nu + {\textstyle{1 \over 2}}\left( {{\partial _\mu }{S_\nu } + {\partial _\nu }{S_\mu }} \right) – \left( {{\partial _\mu }{N_\nu } + {\partial _\nu }{N_\mu }} \right) + L_{gq}^{ – 2}{v_{\mu \nu }} = 2\{ {J_{\mu \nu }} – {\textstyle{1 \over 2}}{g_{\mu \nu }}J_\lambda ^\lambda \} .\tag{4.22}\]
The systems of interest can be characterized by three qualifiers: weak/strong field, local/distal, and Newtonian/relativistic dynamics. Weak fields have \({V_{00}} < {10^{ - 3}}\). Local means that the range term in Eq. (4.22) can be ignored. Strong field systems are always local. In the next sections a wide range of systems will be examined.
The source term of Eq. (4.22) is a single particle defined by Eq. (4.16) or Eq. (4.17). However, the left-hand side of Eq. (4.22) is linear in \({v_{\mu \nu }}(q,x)\) so the superposition principle is applicable (unlike GR). The gravitational field of a group of particles is the sum of the individual fields generated by each member of the group—each (atom-sized) particle has a tiny contribution despite a possibly large G factor on the right-hand-side. Appendix J outlines the application of superposition for some problems of interest.
D. Gravitational Charge
For each of the five gravitons there are two universal constants, \({Z_{gq}}\) and \({L_{gq}}\). These two are independent of seasons. However, it will be necessary for gravity to be different in winter than in red or blue summer for all types of matter. That chore is handled by the gravitational charge in Eq. (4.14) where E is defined in Eq. (2.17).
In summer, \(E = 0\), and \({Q_{GC}}(q,0) = 1\). In winter, \(E = 1\), and the five charges (\({Q_{GC}}(q,1)\)) are universal constants tabulated in Appendix Z. The change of season is found in Eq. (7.5). I believe that it is prudent for the epoch and Higgs fields to be oblivious of gravity (\({Q_{GC}}(q,E) = 0\)).
E. Gravitons
Since Eq. (4.13) looks the same as Eq. (3.20) for an electromagnetic wave, the discussion of waves versus particles in electromagnetism should apply to gravity as well. There are five gravitons with different, extremely tiny masses.
For most applications, one can treat each graviton as a massless spin 2 particle with energy \(h\nu \) and momentum \(h\nu /c\). The only difference between electromagnetism studies and (the much weaker) gravity studies would be smaller numbers of gravitons. The importance of these observations is that there can be a significant graviton background radiation that we are completely unaware of. Unlike CMB, this radiation will be a function of \(\tau \) and \({\bf{x}}\).