II.THE NEW COSMOLOGY IN AN EMPTY UNIVERSE
Consider a (vanilla) universe that contains only two (classical) isotropic fields. The universe has four dimensions
(\(x^0, x^1, x^2, x^3\)) measured in meters and a dimensionless (vanilla) metric tensor
(\(g_{00} = 1\) and \(g_{jk} = – \delta_{jk}\), where \(\delta_{JJ} = 1\) else zero).
Both fields are self-supporting, i.e., either one can pervade all the universe without a source. Furthermore,
the fields can interact with each other in a very specific manner. Each field will be dimensionless but
characterized by a length.
There is no matter, so there are no observers (physicists or astronomers). There are no laws of physics—no matter
and no physicist to perform experiments. There is no relativity—no pair of observers. If one finds that each field
is a function only of \({x^0}\) when there is no matter, then there is no time involved in evolution (only length).
Time can only result from the interaction of the two fields with matter—matter can be thought to create the need for time.
How does one discover the evolution of the fields? Our toolbox is nearly empty, but the principle of least action can be used,
\[0 = \delta \int d{x^0}d{x^1}d{x^2}d{x^3} \, \mathcal{L}\tag{2.1}\]
\(\mathcal{L}\) is the Lagrangian density (LD) for each field, and it must only be a function of universal constants,
fields, and first derivatives of fields. The principle was created by a material physicist, so it uses (familiar)
material units (MKS). The LD has units of \({\rm{K}}{{\rm{S}}^{ – 1}}{{\rm{M}}^{ – 2}}\), but the empty universe only has M—no S or K.
One needs two universal constants to generate S and K from M.
I have chosen to define the needed constants as a velocity
(\(c = 2.99799 \times 10^8 \, {\rm{M}}{\rm{S}}^{-1}\)) and a unit of action
(\(h = 6.62607015 \times 10^{-34} \, {\rm{M}}^2 {\rm{K}} {\rm{S}}^{-1}\)).
There are no photons in an empty universe, so any connection between photons and \(c\) will be deferred to subsequent sections.
A constant is universal if \({\partial _\mu }{\rm{constant = 0}}\), where \({\partial _\mu } \equiv d/d{x^\mu }\).
In what follows, the usual conventions of tensor algebra and calculus are used. Covariant and contravariant indexes are
raised and lowered using the vanilla metric tensor defined above. There are no meaningful transformations of coordinates
in the empty universe.
The LD for the fields is a scalar (zero rank tensor). The apex field is a rank two tensor, and the epoch field is a scalar.
The (more robust) apex field will be studied first.
A. The tensor apex field
The apex field is a dimensionless second-order symmetric tensor,
\( v^{ax}_{\mu\nu}(x) \). The apex label (ax) and the four-coordinate (x)
are implicit in what follows. One can define some useful scalars and vectors
(first order tensors) derived from \( v_{\mu\nu} \):
$
S \equiv v^{\lambda}_{\lambda}, \quad
J \equiv v^{\alpha}_{\lambda} v^{\lambda}_{\alpha}, \quad
S_{\mu} \equiv \partial_{\mu} S, \quad
N_{\mu} \equiv \partial_{\lambda} v^{\lambda}_{\mu}.
$
There are two pieces to the apex LD, the “kinetic” density,
\(\mathcal{L}_{{ax}}^{T}(x)\), and a “potential” density,
\(\mathcal{L}_{{ax}}^{V}(x)\).
The complete LD is then,
\(\mathcal{L}_{{ax}} = \mathcal{L}_{{ax}}^{T} – \mathcal{L}_{{ax}}^{V}\).
Each piece is defined as follows:
\[
\begin{array}{c}
4\kappa_{ax}\,\mathcal{L}_{{ax}}^{T}(x)
= g^{\mu\nu} \{ \partial_\mu v_\beta^\alpha \partial_\nu v_\alpha^\beta
– S_\mu S_\nu
+ 2N_\mu (S_\nu – N_\nu) \},
\end{array}
\tag{2.2}\]
\[
4\kappa_{ax}\,\mathcal{L}_{{ax}}^{V}(x)
= L_{ax}^{-2} F_{ax}(J),
\tag{2.3}
\]
where
$
\kappa_{ax} = L_{ax}^2 \hbar^{-1} \mathcal{N}_{ax}^{-1}
$
and $\mathcal{N}_{ax}$ is a dimensionless universal constant related to the energy density of the apex field.
\( L_{ax} \) is a characteristic length associated with the apex field—a universal constant.
Now vary the field, $ \delta v^{\mu}_{\nu} $, within the action integral to obtain the apex field equation in the absence of sources,
\[
0 = \Omega^{ax}_{\mu\nu} + \tfrac{1}{2} L_{ax}^{-2}
\left( v_{\mu\nu} \frac{\partial F_{ax}}{\partial J} \right). \tag{2.4}
\]
where the coordinate \( x \) is implicit. Furthermore:
\[
\Omega^{ax}_{\mu\nu} = \Xi^{ax}_{\mu\nu}
– \tfrac{1}{2} g_{\mu\nu} g^{\beta\lambda} \Xi^{ax}_{\beta\lambda}, \tag{2.5}
\]
\[
2 \Xi^{ax}_{\mu\nu} = g^{\alpha\beta} \partial_{\alpha} \partial_{\beta} v_{\mu\nu}
+ \partial_{\mu} \partial_{\nu} S
– \partial_{\mu} N_{\nu} – \partial_{\nu} N_{\mu}. \tag{2.6}
\]
Solve Eq. (2.4) by using an isotropic apex field that depends only on the evolution parameter,
\( v^{ax}_{jk}(x) = A(x^{0}) \delta_{jk} \) (the other elements are zero).
Furthermore, a specific three-term form for the potential is chosen:
\[
F_{ax}(J) = – \sum_{n=1}^3 6 d_{n}^{ax} \, 3^{-n} J^n. \tag{2.7}
\]
where \( J = 3A^2 \), and \( d_{n}^{ax} \) are a set of three dimensionless universal constants
to be determined from astronomical observations. Using this potential in Eq. (2.4) leads to the following result,
\[
\frac{d^2 A}{(d x^0)^2} = – L_{ax}^{-2} A \sum_{n=1}^3 n d_{n}^{ax} \left( A^2 \right)^{n-1}. \tag{2.8}
\]
A first integral of Eq. (2.8) is as follows:
\[
\frac{dA}{dx^0} = \pm L_{ax}^{-1}
\left[ \varsigma_0^2 – \sum_{n=1}^3 d_{n}^{ax} (A^2)^n \right]^{1/2}. \tag{2.9}
\]
where \( \varsigma_0 \) is a constant of integration. Eq. (2.9) is valid only for values of \( A \)
for which the term in brackets is not negative. This solution gives cyclic behavior with \( A \) endlessly oscillating
between two turning points (where \( dA/dx^0 = 0 \)). There are two branches of Eq. (2.9).
For reasons that will become clear in section III, the minus branch (decreasing A) is labeled as the red branch.
Increasing A is the blue branch.
It should be clear that the apex field is a clock, but it does not measure time.
Furthermore, one should expect that interactions between \( A \) and material particles will cause any material clock
to have a complex relation to the apex clock.
Since \( J = 3A^2 \), it follows that one would expect the scalar \( J \) to be invariant under a Lorentz transformation of coordinates
(thus likewise for \( A \)). The right-hand side of Eq. (2.9) would be expected to be invariant, but the left-hand side
would transform like the time-like component of a vector. This dichotomy would go away if \( L_{ax} \to \infty \).
\( L_{ax} \) will be very large (9 billion light-years) but finite. The apex field breaks Lorentz invariance to a tiny degree
and renders special relativity incorrect (but an extremely good approximation to reality).
This result should be expected since the apex field embodies a special frame of reference for the universe.
General relativity is seriously wrong because special relativity is a little bit wrong.
In all that follows, it is necessary to keep in mind that there will be no automatically invariant coordinate transformations in NG (unlike GR).
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. A coordinate transformation may introduce new physics,
e.g., a rotating coordinate system plus Newtonian physics generates a Coriolis force.
Furthermore, the label of scalar, vector, etc. in NCNG means the transformation label in the limit \( L_{ax} \to \infty \).
The energy density of the apex field (in the absence of the epoch field) is proportional to the potential density of Eq. (2.3) evaluated at a turning point,
\[
U_{ax} = – \tfrac{3}{2} \hbar c \mathcal{N}_{ax} L_{ax}^{-4} (\varsigma_0)^2. \tag{2.10}
\]
Eq. (2.9) is correct if there is no source term in Eq. (2.4).
If the LD for the epoch field contains \( A \), then there will be an interaction term.
The epoch field must be studied before finishing with the apex field.
The idea of a self-supporting all-pervading scalar field with two possible vacuum states (true and false) is largely due to Sidney Coleman [3]. I have taken this idea into a different arena. In NC, the epoch field will couple with the apex field, and each vacuum state will alternate between true and false as the apex field evolves. In contrast to the situation in [3], changes will proceed with cosmologic slowness.
The epoch field (\({\phi _{ep}}\)) is dimensionless and real. The epoch field LD (\({\mathcal{L}_{ep}}\)) has the following form:
\[\kappa_{ep}\,\mathcal{L}_{ep}(x) = \tfrac{1}{2} g^{\mu \nu} \partial_\mu \phi_{ep} \partial_\nu \phi_{ep} – L_{ep}^{-2} V_{ep}(\phi_{ep}),\tag{2.11}\]
\[
V_{ep}(\phi_{ep}) = -\tfrac{1}{2}\phi_{ep}^2 + \tfrac{1}{3}\lambda(A)\phi_{ep}^3 + \tfrac{1}{4}a_{ep}^2\phi_{ep}^4
\tag{2.12}
\]
where \(\kappa_{ep} = L_{ep}^2 \hbar^{-1} \mathcal{N}_{ep}^{-1} \quad \text{and} \quad \mathcal{N}_{ep}\) is a dimensionless universal constant related to the energy density of the epoch field. \(L_{ep}\) is a characteristic length and \(a_{ep}\) is a positive dimensionless factor—both universal constants associated with the epoch field. \(\lambda(A)\) is a dimensionless function of the apex field,
\[
\lambda(A) = -\lambda_{ep} + \beta_{ep}\tanh \big( \Gamma_{ep}\{ A(x^0)^2 – A_{ep}^2 \} \big)
\tag{2.13}
\]
where \(\lambda_{ep}, \; \beta_{ep}, \; \Gamma_{ep}, \; \text{and } A_{ep} > 0\) are dimensionless universal constants. Eq. (2.13) is legal because \(A\) is a scalar, and the resulting LD does not have explicit dependence on \(x^0\). There are seven universal constants for the epoch field (tabulated in Appendix Z), but all of them can be determined (in principle) from observations. Note that \(\lambda(-A) = \lambda(A)\), so the resulting cosmology will be symmetric as well.
There are two possible vacuum states that correspond to local minima of Eq. (2.12), and they occur at the following values of \(\phi_{ep}\):
\[
\phi_{ep}^{\pm} = \frac{-\lambda(A) \pm \sqrt{a_{ep}^2 + \lambda(A)^2}}{2a_{ep}^2}
\tag{2.14}
\]
If \(V_{ep}(\phi_{ep}^+) < V_{ep}(\phi_{ep}^-)\) then \(\phi_{ep}^+\) is the true vacuum state and \(\phi_{ep}^-\) is the false vacuum, otherwise the reverse is true. When \(V_{ep}(\phi_{ep}^+) = V_{ep}(\phi_{ep}^-)\) the vacuum states are degenerate, and that occurs when \(\lambda(A_D) = 0\). As \(x^0\) increases, \(A\) changes (very slowly), and it is possible that when \(A\) crosses \(A_D\), what was previously a true vacuum has become a false vacuum. In that case it is possible for the false vacuum to decay into a true vacuum. The decay process is explained in [3] and [4] and in Appendix B. This process will be important in NC and described in greater detail below. The goal in this sub-section is to develop the energy density of the epoch field and its broad behavior.
The study will start with the assumption that \({\Gamma _{ep}} \gg 1\). Eq. (2.13) shows that the epoch field will be independent of the apex field except for a tiny region around \(A(x^0)^2 = A_{ep}^2\) (where vacuum decay will play a role). In that case, Eq. (2.14) takes the simplified form:
\[
\begin{array}{l}
\phi_{ep}^{\pm} = \{ \lambda_{ep} – \beta_{ep} \pm \sqrt{a_{ep}^2 + (\lambda_{ep} – \beta_{ep})^2} \} / 2a_{ep}^2, \quad A^2 > A_{ep}^2, \\
\phi_{ep}^{\pm} = \{ \lambda_{ep} + \beta_{ep} \pm \sqrt{a_{ep}^2 + (\lambda_{ep} + \beta_{ep})^2} \} / 2a_{ep}^2, \quad A^2 < A_{ep}^2.
\end{array}
\tag{2.15}
\]
There will be constraints for \(\lambda_{ep}\), \(\beta_{ep}\), \(a_{ep}\), but \(\beta_{ep} > |\lambda_{ep}|\) will suffice here. If \(A^2 > A_{ep}^2\), the true (surviving) vacuum will be \(\phi_{ep}^- (< 0)\), and for \(A^2 < A_{ep}^2\) the true vacuum will be \(\phi_{ep}^+ (> 0)\).
\(A_{ep}^2\) will lie between the turning points of A, so each branch of A will have two distinct epochs with different vacuums. I choose to name the branch-epochs after seasons—red or blue summer for \(A^2 < A_{ep}^2\) and red or blue winter for \(A^2 > A_{ep}^2\). There are also two (very brief) transition periods for each branch when A is very nearly equal to \(A_{ep}\) also named after seasons—spring when \(A^2\) decreases through \(A_{ep}^2\), and fall when \(A^2\) increases through \(A_{ep}^2\). In the new cosmology the universe will endlessly pass in order through the eight seasons—red spring, red summer, red fall, red-blue winter, blue spring, blue summer, blue fall, blue-red winter, red spring, etc., (a cyclic cosmology). We are in early red summer. The color tags for the seasons will be explained in section III(D).
It is helpful to relabel the vacuums as follows:
\[
\begin{array}{l}
\phi_W = \{ \lambda_{ep} – \beta_{ep} – \sqrt{a_{ep}^2 + (\lambda_{ep} – \beta_{ep})^2} \} / 2a_{ep}^2 \quad \text{winter}, \\
\phi_S = \{ \lambda_{ep} + \beta_{ep} + \sqrt{a_{ep}^2 + (\lambda_{ep} + \beta_{ep})^2} \} / 2a_{ep}^2 \quad \text{summer}.
\end{array}
\tag{2.16}
\]
Furthermore, it will be useful to define a linear scalar-function of \(\phi_{ep}\) as follows:
\[
E(\phi_{ep}) = \frac{\phi_{ep}(x^0) – \phi_S}{\phi_W – \phi_S},
\tag{2.17}
\]
where \(E = 1\) in winter and \(E = 0\) in summer.
When matter is added to the universe, \(E\) (or any function of \(E\) and/or \(A\)) can be inserted anywhere in the LD for constituents. Therefore, one can have different physics in winter versus summer. Fig. C1 in Appendix C shows (red spring) \(E\) as a function of \(A\) for the universal constants tabulated in Appendix Z. The gap in the figure corresponds to the very brief period of active vacuum decay (what I will call the bubble era). The continuous portion is the more-lengthy “classical” part of red spring.
The epoch field energy density is given by the following forms:
\[
\begin{array}{l}
U_{ep}^W = \hbar c \mathcal{N}_{ep} L_{ep}^{-4} V_{ep}(\phi_W) \quad \text{winter}, \\
U_{ep}^S = \hbar c \mathcal{N}_{ep} L_{ep}^{-4} V_{ep}(\phi_S) \quad \text{summer}.
\end{array}
\tag{2.18}
\]
Furthermore, \(U_{ep}^S < U_{ep}^W < 0\).
I must state the most basic aspect of NCNG at this point—the hierarchy of energy densities and ranges,
\[
\begin{array}{l}
|U_{ax}| > |U_{ep}| \gg U_{matter} \quad \text{all seasons}, \\
L_{ax} \gg L_{ep} \gg \hbar / m_{proton} c.
\end{array}
\tag{2.19}
\]
The average energy density of matter in the universe is thought to be about 250 MeV per cubic meter (a suspect value). The magnitude of the epoch field energy density will be at least eight orders of magnitude larger than average matter. The magnitude of apex energy density will be about two orders of magnitude greater than the epoch field. Thus, matter can be considered as an insignificant froth having negligible impact on the cosmology generated by the apex and epoch fields (despite the interaction of these fields with matter).
Neglecting matter leads to the following relations between the seasonal apex and epoch energy densities:
\[
\begin{array}{l}
U_{ax}(x^0) + U_{ep}(x^0) = \text{constant}, \\
U_{ep}^S < U_{ep}^W < 0 \Rightarrow 0 > U_{ax}^S > U_{ax}^W, \\
\alpha_W^{ax} \equiv \frac{U_{ep}^S – U_{ep}^W}{U_{ax}^S}, \\
1 \gg \alpha_W^{ax} > 0, \\
U_{ax}^S = -1.5 \, c L_{ax}^{-4} \hbar \mathcal{N}_{ax},
\end{array}
\tag{2.20}
\]
where \(\alpha_W^{ax}\) is a universal (apex) constant tabulated in Appendix Z. The apex energy density in summer is c times the kinetic LD (evaluated where the potential LD is zero).
The final step in this section is the inclusion of the epoch field into a source term for Eq. (2.8). The Euler-Lagrange equation is as follows:
\[
\begin{array}{l}
0 = \partial \mathcal{L}_{ep} / \partial v_{ax}^{\mu \nu }
– \partial \mathcal{L}_{ax}^V / \partial v_{ax}^{\mu \nu }
– \partial^\beta \left[ \partial \mathcal{L}_{ax}^T / \partial (\partial^\beta v_{ax}^{\mu \nu }) \right],\\
\frac{d^2 A}{(d x^0)^2} = – L_{ax}^{-2} A \sum_{n=1}^3 n d_n^{ax} (A^2)^{n-1} + J(x^0).
\end{array}
\tag{2.21}
\]
A tedious calculation produces J,
\[
J(A) = \frac{2 \kappa_{ax}}{9 \kappa_{ep} L_{ep}^2} \, \phi_{ep}(A)^3 \, \beta_{ep} \Gamma_{ep} A \, \{ \cosh [\Gamma_{ep}(A^2 – A_{ep}^2)] \}^{-2}.
\tag{2.22}
\]
Reference to Appendix Z shows that \(\Gamma_{ep}\) is indeed very large and \(A_{ep}\) is very close to 1, so J is sharply peaked at \(A = A_{ep}\). The following is a very good approximation for J:
\[
\begin{array}{l}
J = \Lambda \{ \cosh [2 A_{ep} \Gamma_{ep} (A – A_{ep})] \}^{-2}, \\
\Lambda = \frac{2 \kappa_{ax}}{9 \kappa_{ep} L_{ep}^2} \, \phi_{ep}(A_{ep})^3 \, \beta_{ep} \Gamma_{ep} A_{ep},
\end{array}
\tag{2.23}
\]
where \(\Lambda\) is a constant (made up from universal epoch-constants). One can make a closed form solution to Eq. (2.21) by using Eq. (2.23):
\[
\begin{array}{l}
G_{ax}(A) = d_1^{ax} A^2 + d_2^{ax} A^4 + d_3^{ax} A^6, \\
(dA/dx^0) = \pm L_{ax}^{-1} \mathcal{F}(A), \\
\mathcal{F}(A) = \left[ 1 + \xi(A) – G_{ax}(A) \right]^{1/2}, \\
\xi(A) = (\alpha_W^{ax}/2) \{ 1 + \tanh [2 A_{ep} \Gamma_{ep} (A – A_{ep})] \}, \\
\alpha_W^{ax} = 2 \Lambda L_{ax}^2 / (A_{ep} \Gamma_{ep}).
\end{array}
\tag{2.24}
\]
It is convenient to feature \(\alpha_W^{ax}\) (a tabulated universal constant) since it appears in the equation for A in winter. However, \(\alpha_W^{ax}\) is also defined in two equations—the bottom element of Eq. (2.24) and the third element of Eq. (2.20). Each of the elements are constructed from some universal apex and epoch constants. Given \(\alpha_W^{ax}\), there are two constraints on the construction constants—the tabulated Appendix Z constants are not all independent. In the case above, the relations are as follows:
\[
\begin{array}{l}
\alpha_W^{ax} = \frac{4}{9} \, \beta_{ep} \, \phi_{ep}(A_{ep})^3 \, (\mathcal{N}_{ep}/\mathcal{N}_{ax}) \, (L_{ax}/L_{ep})^4, \\
V_{ep}(\phi_W) – V_{ep}(\phi_S) = \frac{2}{3} \, \beta_{ep} \, \phi_{ep}(A_{ep})^3.
\end{array}
\tag{2.25}
\]
The bottom element of Eq. (2.25) is very interesting since it involves only \(a_{ep}, \lambda_{ep}, \text{ and } \beta_{ep}\).
C.The spring and fall transitions
The duration of a transition is somewhat arbitrary since it depends on the asymptotic behavior of Eq. (2.13).
I choose to define the start of spring when $E(\phi_{ep}) = 0.999$
and ending when
$E(\phi_{ep}) = 0.001$
The fall transition has the reverse order.
The epoch field vacuum becomes degenerate when
$A = A_D, \quad A_D = \pm \left[ A_{ep}^2 + \Gamma_{ep}^{-1} \, \text{arctanh}\left(\frac{\lambda_{ep}}{\beta_{ep}}\right) \right]^{1/2}$
\(A_D\) lies between the start and end of a transition since \(\beta_{ep} > | \lambda_{ep} |\).
Furthermore, \(A = A_D\) marks the beginning of a period where active vacuum decay can occur.
The active vacuum decay period can be divided into two segments. The first segment is quiescent (included in the continuous portion of Fig. C1).
The following very brief segment is explosive (the discontinuous portion of Fig. C1).
It is shown in Appendix B that vacuum decay corresponds to the creation of expanding bubbles throughout false vacuum space where the true vacuum resides inside each bubble.
The probability of a bubble being formed is a steep nonlinear function of \(\left| A – A_D \right|\).
During the lengthy quiescent period, the average probability of bubble formation is negligible.
Once the first average bubble forms in a volume of space, the volume rapidly becomes filled with expanding bubbles—a true vacuum volume.
In spring, the quiescent period extends from \(A = A_D\) to \(A = A_D – x_{Bera}\), where \(x_{Bera}\) is treated as a dimensionless universal constant tabulated in Appendix Z.
The quiescent period can be three orders of magnitude longer than the explosive period.
Refer to Appendix B for a review of spring bubble-formation.
One can see that the apex field is continuous during the classical period in red spring (before the red-spring bubble era).
Furthermore, \(dA/dx^0\) is continuous and only changes significantly during a very brief period around \(A = A_{ep}\) \(\tag{2.30}\) (well before the chaotic bubble era).
Consequently, \(A\) and \(dA/dx^0\) (during the bubble era) will be very nearly the same on each side of a bubble wall.
Thus, the bubble era will have very little effect on the apex field—the bubble era becomes marginalized.
The same is true for the four bubble eras during an apex field cycle.
One will learn below that I have chosen to marginalize the bubble era for summer/winter physics transitions as well (Eq. (7.4) and Eq. (7.5)).
In the empty universe there is only one interesting variable, the epoch parameter \(E(\phi_{ep})\) \(\tag{2.31}\) defined in Eq. (2.17).
The universe is ancient and vast in NC—at the limits of comprehension.
One must expect that the universe has gone through thousands of cycles (each about 731 billion years).
I have chosen to avoid the creation question. Was the universe created many cycles ago, or is it eternal?
These questions can never be answered since any evidence of a past creation is lost beyond our ability to observe.