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\begin{document}
\title{A Derivation of the Hubble- Lema\selectlanguage{ngerman}ître Law that Identifies Dark Energy and
Dark Matter\selectlanguage{english}}
\author[1]{Clive Anthony Redwood}%
\affil[1]{Affiliation not available}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
A generalized Friedmann-Lema\selectlanguage{ngerman}ître-Robertson-Walker metric specifies
gravitational fields with equations of state of \emph{w} = \selectlanguage{english}-1. In the
spacetime of the perturbed Robertson-Walker metric, the gravitational
pressure is negative. A gravitational positive feedback cycle,
recognized in Einstein's seminal work --~\emph{The Foundation of the
Theory of General Relativity}~- gives rise to the self-induction of
gravity that, under conservation of gravitational energy density, is
proposed to lead to continuous growth of the field, even in the absence
of changes in gravitating bodies. Physical space is proposed to be the
form of the field. This auto-induced growth of the field is, here,
proposed to manifest as the expansion of space -- elsewhere attributed
to dark energy. A gravitational expansion, by virtue of the conservation
of gravitational energy density and the equation of state, is locally
isobaric. The Hubble-Lemaitre law is derived from the isobaric nature of
the expansion. Data from cosmological probes
across\(\)\emph{\textbf{~}z =} 0 \emph{} to\emph{~z
\textgreater{}} 9\emph{~}support the linear redshift-distance relation
that demands a cosmic Hubble constant. The Hubble constant references a
universal maximum gravitational energy density \textasciitilde{}
1.52E-27 Kg m-3 @ \emph{H\textsuperscript{}\textasciitilde{}~}2.26E-18
s\textsuperscript{-1}. It is proposed that geodesics are dependent on
the conserved gravitational energy density. Thereby, the expansion of
the field in the vicinities of bodies leads to extended flat, or rising,
orbital velocity profiles and extensive regions of gravitational lensing
-- elsewhere attributed to dark matter. A gravitational expansion would
impart a historical character to the laws of gravitation. In this
spacetime, continuous gravitational expansion is essential in preventing
spatial collapse.~~\emph{}%
\end{abstract}\selectlanguage{ngerman}%
\sloppy
`\ldots{} I do not understand why only quantities with the
transformation characteristics of tensors should be granted physical
meaning.'
Albert Einstein, "\emph{On Gravitational Waves".}
\emph{\url{https://einsteinpapers.press.princeton.edu/vol7-trans/38}}
\section*{1.0~ ~INTRODUCTION}
{\label{169439}}
There is no widely accepted single theory that explains the sets of
gravitational phenomena separately attributed to dark energy (DE) and
dark matter (DM). So, there occurs a dichotomy within gravitational
science. The Friedmann-Lema\selectlanguage{ngerman}ître-Robertson-Walker (FLRW) theory of an
expanding universe without spatial curvature at very large scales,
through its determinations of spatial expansion, continues to be
affirmed by observations. However, at small (galactic) scales, it is not
applicable.
\par\null
Theoretical explorations~of the relations of spacetimes applicable to
galaxies, on one hand, and, on the other, to the FLRW spacetime have not
produced very useful results. In fact, in 1945, Einstein and Straus
established the conditions for the coexistence of the spherically
symmetric curvilinear static Schwarzschild spacetime -- that explained
significant phenomena at the scale of our solar system --~ the FLRW
spacetime~\cite{Einstein_1945}. However, the two had to be mutually
excluded in space and only had external relations, with the small-scale
Schwarzschild spacetime simply existing within the FLRW spacetime. It
has been shown that such coexistence was very fragile for the
Schwarzschild spacetime, with instability under isotropic radial changes
and vulnerability to non-spherical perturbations~\cite{Mars_2013}. So,
these spacetimes had to be separated by having the small-scale one
existing in a spherical vacuole within the FLRW spacetime
\cite{Mars_2013}.
Rich media available at \url{}
In the absence of a robust coherent connection between the spacetimes of
large- and small-scale regions, there has been a reassertion, in the
latter, of Newtonian science of gravity in cosmology and
astrophysics~\cite{Chisari_2011,Chisari_2011a}. For, at small scales, it is apparent
that Newtonian science of gravity is normatively applied in analysis and
so in simulations. So, though based on Einstein's theory of gravity,
modern cosmology was, in small-scale regions, mainly applying Newton's
methods against the background of a flat spacetime. This occurs even
though Newtonian physics had already been proven incapable of explaining
certain subtle gravitational effects such as the precession of the
orbits of planets and the deflection of light around massive bodies that
were explained by general relativity (GR) in~\cite{einstein1952}.
\par\null
Yet, however contradictory the situation appears, it is quite
comprehensible. For, in GR, orbits require inherently curvilinear
spacetimes \cite{einstein1952}. So, in GR, orbits are inexplicable in the
currently applied FLRW spacetime since the latter may, simply by means
of a time coordinate transformation, be shown to become, thereby, a
Minkowskian flat spacetime. So, orbits cannot be explained in the
GR-derived FLRW theory of a spatially flat universe. However, since
Newton's science of gravity applies only to flat space and, moreover, is
the only full-fledged alternative to Einstein's, then its reassertion is
fundamentally unavoidable in FLRW cosmology.
So, at these small scales, the current concordance model of cosmology
(\selectlanguage{greek}Λ\selectlanguage{english}CDM), in effect, pairs Newton's science of gravity with the
hypothetical cold dark matter (\selectlanguage{greek}Λ\selectlanguage{english}CDM). Even the application of
gravitational lensing techniques, based on GR, serves mainly Newtonian
objectives of determining mass distributions and potential functions.
Here, Newtonian methods are said to yield similar outcomes as
Einstein's.
\par\null
At large-scale, \selectlanguage{greek}Λ\selectlanguage{english}CDM proposes the cosmological constant \selectlanguage{greek}Λ
\selectlanguage{english}\cite{Riess_2001}~as the DE of the expansion. It is proposed to be a
vacuum quantum field of a nature that, via a reflex of the gravitational
field, expands space.
\par\null
So, at small scales, the field is absent and at large scales, its role,
in \selectlanguage{greek}Λ\selectlanguage{english}CDM cosmology, is secondary. Yet, classical -- that is, pre-FLRW --
GR, as a theory of gravitation, is distinguished by the central role of
a metric gravitational field -- as the only entity that couples both
with space and matter \cite{einstein1952}. So, the absence of the
gravitational field, or its secondary role, in \selectlanguage{greek}Λ\selectlanguage{english}CDM appears as
peculiarities of potentially fundamental impact within a GR-derived
theory. It is here proposed that the passivity, in \selectlanguage{greek}Λ\selectlanguage{english}CDM cosmology, of
the gravitational field is the basis of the persistence of the mysteries
of DE and DM.
\par\null
So, the endeavour, here undertaken, is to develop a unitary explanation,
within the paradigm of the gravitational field theory of GR, that
explains both sets of phenomena. The intent is to include the universal
empirical element of the Hubble expansion of space~\cite{Riess_2001}
within a strictly classical GR explanation that treats all matter
generically as energy tensors, while generally ignoring other aspects of
the various species of matter and radiation, some of which become so
important in \selectlanguage{greek}Λ\selectlanguage{english}CDM astrophysical cosmology.
\section*{}
{\label{303678}}
In \selectlanguage{ngerman}§2, a generalized FLRW metric determines certain common dynamic
attributes of the associated class of fields. In §3, a gravitationally
perturbed Robertson-Walker (gpRW) metric is used to develop descriptions
of initial states of certain spherically symmetric gravitational
systems. Then, in §4 a physical derivation of the Hubble-Lemaître law is
presented. Phenomena associated with DM are explained by the Hubble
expansion in §5. The recession in cosmic voids is explored in §6. In §7,
the Hubble expansion across all scales is explored. In §8, the
redshift-distance relation is explored. In §9, a gravitational process
proposed to energize the spatial expansion is presented. In §10, there
is a discussion of the views presented here and certain aspects of \selectlanguage{greek}Λ\selectlanguage{english}CDM.
The explanations given here are summarized in \selectlanguage{ngerman}§11.
\par\null\par\null
{ ~\cite{Demianski_2017}.~\cite{Harrison_1993}}
~\cite{Plaga_2005}.
\par\null
~\citep*{Karachentsev_2009}
\cite{Hu_2003}
\par\null
\section*{2.0\ \ \ GRAVITY IN A GENERALIZED FLRW SPACETIME}
\section*{2.1\ \ \ A Generalized FLRW Line Element}
It is clear that the assumed spatial homogeneity in FLRW cosmology is only approximated at very large scales in the universe. So, in order to accommodate small-scale $<$ 3 Mpc phenomena [12], the principle of homogeneity has to be surrendered.
However, the universe still appears fairly isotropic [11] as observed from within our solar system and since galaxies display galactocentric kinematics and distributions of matter, then a spherically symmetric metric seems appropriate. Most important in the choice of a spacetime, however, is the fact that the cosmic expansion is isotropic.
So, in observance of the empirically supported universal isotropy, consider a universe with a spherically symmetric cosmic line element of the form:
\begin{equation}
\label{eq1}
(ds)^2=g_{tt}(dx^0)^2+g_{rr}[(dx^1)^2+(dx^2)^2+(dx^3)^2]
\end{equation}
Rewrite as:
\begin{equation}
(ds)^2=g_{tt}(dx^t)^2+g_{rr}(dx^r)^2
\end{equation}
Where:
\begin{equation}
dx^r=[(dx^1)^2+(dx^2)^2+(dx^3)^2]^{1/2}
\end{equation}
The line element - $ds$ - is the invariant locally determined distance, or duration, corresponding to the infinitesimal coordinate vector components $dx^{\nu}$ by means of the $g_{\mu\nu}$ that are components of the metric, with tensorial indices $\mu, \nu= 0, 1, 2, 3.$ Equation (1) generalizes the curvature-free generic FLRW equation of the linear element that has $g_{tt}=1$ and $g_{rr}=-a^2,$ where $a=a(x^t)$ is the scale factor.
Equation (2) is the line element in an isotropic coordinate frame that simplifies descriptions in the isotropic and, effectively, irrotational applications that ensue.
\section*{2.2\ \ \ Barotropic Nature of Gravitational Fields in Generalized FLRW Spacetimes}
The gravitational field components are, here, defined by the negative of the Christoffel symbol of the second kind:
\begin{equation}
\label{eq4}
\Gamma^{\tau}_{\mu\nu}=-\frac{g^{\sigma\tau}}{2}(\frac{\partial{g_{\mu\sigma}}}{\partial{x^{\nu}}}+\frac{\partial{g_{\nu\sigma}}}{{\partial{x^{\mu}}}}-\frac{\partial{g_{\mu\nu}}}{{\partial{x^{\sigma}}}})
\end{equation}
(I use the original definition of the gravitational field component given in \cite{einstein1952}. It is the negative of the formula given in \cite*{a1973}.)
Substitutions of the tensorial Greek indices by $r$ yield:
\begin{equation}
\label{gtrr1}
\Gamma^r_{rr}=-\frac{g^{rr}}{2}(\frac{\partial{g_{rr}}}{\partial{x^r}}+\frac{\partial{g^{rr}}}{\partial{x^r}}-\frac{\partial{g_{rr}}}{\partial{x^r}})=-\frac{g^{rr}}{2}\frac{\partial{g_{rr}}}{\partial{x^r}}
\end{equation}
Similarly obtained, the other field components are given by:
\begin{equation}
\label{gttt}
\Gamma^t_{tt}=-\frac{g^{tt}}{2}\frac{\partial{g_{tt}}}{\partial{x^t}}
\end{equation}
\begin{equation}
\label{grtr}
\Gamma^r_{tr}=\Gamma^r_{rt}=-\frac{g^{rr}}{2}\frac{\partial{g_{rr}}}{\partial{x^t}}
\end{equation}
\begin{equation}
\label{gtrt}
\Gamma^t_{rt}=\Gamma^t_{tr}=-\frac{g^{tt}}{2}\frac{\partial{g_{tt}}}{\partial{x^r}}
\end{equation}
\begin{equation}
\label{grtt}
\Gamma^r_{tt}=\frac{g^{rr}}{2}\frac{\partial{g_{tt}}}{\partial{x^r}}
\end{equation}
\begin{equation}
\label{gtrr}
\Gamma^t_{rr}=\frac{g^{tt}}{2}\frac{\partial{g_{rr}}}{\partial{x^t}}
\end{equation}
Within the coordinate constraint of $\surd{(-g )}=1,$ where $g$ is the determinant of the metric, the energetic flux densities of gravity are given in \cite{einstein1952} as:
\begin{equation}
\label{pseud}
t^{\alpha}_{\sigma}=\frac{1}{\kappa}(\frac{\delta^{\alpha}_{\sigma}}{2}g^{\mu\nu}\Gamma^{\lambda}_{\mu\beta}\Gamma^{\beta}_{\nu\lambda}
-g^{\mu\nu}\Gamma^{\alpha}_{\mu\beta}\Gamma^{\beta}_{\nu\sigma})
\end{equation}
Where, $\kappa$ is the Einstein constant.
The expansion of space has usefully been ascribed to barotropic agents. [8, 24] Here, the gravitational field will be examined to determine its suitability for such a role. So, at first, setting $\alpha=\sigma=r$ in equation (11) and applying the Einstein summation rule yields:
\begin{equation}
\label{ktrr1}
\kappa t^r_r=\frac{g^{rr}}{ 2}\Gamma^{\lambda}_{r\beta}\Gamma^{\beta}_{r\lambda}
-g^{rr}\Gamma^r_{r\beta}\Gamma^{\beta}_{rr}+\frac{g^{tt}}{2}\Gamma^{\lambda}_{t\beta}\Gamma^{\beta}_{t\lambda}-g^{tt}\Gamma^r_{t\beta}\Gamma^{\beta}_{tr}
\end{equation}
Then, with further applications of the summation rule and simplifications, the above equation becomes:
\begin{equation}
\label{ktrr2}
\kappa \hat{t}^r_r=-\frac{g^{rr}}{2}\Gamma^r_{rr}\Gamma^r_{rr}+\frac{g^{rr}}{2}\Gamma^t_{rt}\Gamma^t_{rt}-\frac{g^{tt}}{2}\Gamma^r_{tr}\Gamma^r_{tr}+\frac{g^{tt}}{2}\Gamma^t_{tt}\Gamma^t_{tt}
\end{equation}
Proceeding similarly, by setting $\alpha=\sigma=t,$ or simply by interchanging the indices $r$ and $t$ in the above equation, then the other energetic flux density of interest here is given in:
\begin{equation}
\label{kttt}
\kappa \hat{t}^t_t= -\frac{g^{tt}}{2}\Gamma^t_{tt}\Gamma^t_{tt}+\frac{g^{tt}}{2}\Gamma^r_{rt}\Gamma^r_{rt}-\frac{g^{rr}}{2}\Gamma^t_{rt}\Gamma^t_{rt}+\frac{g^{rr}}{2}\Gamma^r_{rr}\Gamma^r_r
\end{equation}
Therefore:
\begin{equation}
\label{eos}
\hat{t}^t_t=-\hat{t}^r_r
\end{equation}
Recall:
\begin{equation}
\label{diag}
diag\{\hat{t}^{\alpha}_{\sigma}\}=[\hat{t}^0_0, \hat{t}^1_1, \hat{t}^2_2, \hat{t}^3_3]
\end{equation}
Where, the first element in the square brackets is the energy density (00 component of the energetic fluxes), here equal to $\hat{t}_t^t.$ The other elements are normal stresses that, within the isotropic universe, are all equal to the pressure $\hat{t}_r^r.$ Therefore, considering gravity as a fluid, equation (\ref{eos}) implies that it has an equation of state (EoS) - the ratio of the pressure to the energy density - of $w=-1.$
\section*{4.0\ \ \ HUBBLE EXPANSION IN A MODIFIED ROBERTSON-WALKER SPACETIME: IDENTIFICATION OF DARK ENERGY}
\label{4}
The \emph{EoS} of the gravitational field motivates exploration for a gravitational mechanism that may drive the spatial expansion of the universe. These considerations require a suitable spacetime and appropriate models of gravitational systems.
In considerations of the static central field, the (Exterior) Schwarzschild metric has been very successful. It is curvilinear in the vicinities of bodies and asymptotically Minkowskian in the remote voids. Fixing its transverse coordinates so that $d\theta=d\phi=0,$ then the terms in which these infinitesimal coordinate vectors appear in Schwarzschild\selectlanguage{english}'s equation of the linear element vanish and the spherically symmetric Schwarzschild metric is restricted to radial descriptions.
In order to provide for a gravitationally induced spatial expansion across the cosmos, the static Schwarzschild metric must be modified to include the Hubble scale factor.
So, by applying the Hubble scale factor $a$ in its $g_{rr}$ metric component, a version of the gpRW metric \cite{Bertschinger_1994} is produced. The scale factor is a function of coordinate time. The resulting metric and its inverse have the following components:
Where,\Phi$(r)$ is the gravitational potential. This metric simultaneously expresses gravitational attraction and the cosmic expansion.
At any current instant, the scale factor is usually normalized to unity. So, in general, the scale factor departs from unity across an interval of time.
The presence of the time dependent scale factor in the metric implies the evolution of the spacetime and the gravitational field that are functions of the metric.
(In SS5.1, it is shown how the constraint on pseudo-tensors, that arose in SS2.2, here expressed as \selectlanguage{english}[?]$(-g)=a=1,$ is not violated during this evolution.)
The gpRW metric is used in descriptions of the dynamic process of gravitational collapse that express and enhance cosmological inhomogeneity [1, 8]. In describing gravitational collapse, the potential term usually reflects baryonic mass-energy density fluctuations from an average background value.
However, here, this metric is applied to systems associated with spherical gravitating bodies including stars, black holes, and planets, as well as to spherical gravitational systems including solar systems, stellar clusters, spheroidal galaxies, and much larger spheroidal agglomerations in the forms of galactic clusters. All these systems form from gravitational collapse, exhibit gravitational attraction, and are, here, proposed to remain in the cosmic recession.
The explorations, here, will include a model of spherically symmetric gravitating bodies and an Hernquist analytical model representing spheroidal agglomerations such as galaxies and their clusters.
The systems' initial states are conveniently taken to be those existing at the times of completion of collapse to form bodies or the times of virialization to form galaxies or clusters. The state of the gravitational field at such times is assumed to be determined by the metric given here, with $a=1.$
\section*{3.2\ \ \ Initial Gravitational Pressures Around a Spherical Body}
The gravitational potential of a spherically-symmetric body is given by \ \Phi$(r)=-GM/r.$ Substituting this, along with the Schwarzschild radius $r_s=2GM/c^2,$ into equations (17), (18), (19), and (20) yields:
\begin{equation}
\label{grr1}
g_{rr}=-a^2(1-\frac{r_s}{r})^{-1}
\end{equation}
\begin{equation}
\label{grr2}
g^{rr}=-a^{-2}(1-\frac{r_s}{r})
\end{equation}
\begin{equation}
\label{gtt1}
g_{tt}=(1-\frac{r_s}{r})
\end{equation}
\begin{equation}
\label{gtt2}
g^{tt}=(1-\frac{r_s}{r})^{-1}
\end{equation}
The coordinate first derivatives of the components of the metric are:
\begin{equation}
\label{dgrrr}
\frac{\partial{g_{rr}}}{\partial{x^r}}=\frac{a^2r_s}{r^2}(1-\frac{r_s}{r})^{-2}
\end{equation}
\begin{equation}
\label{dgrrt}
\frac{\partial{g_{rr}}}{\partial{x^t}}=-2a\dot{a}(1-\frac{r_s}{r})^{-1}+\frac{a^2\dot{r}r_s}{r^2}(1-\frac{r_s}{r})^{-2}
\end{equation}
\begin{equation}
\label{dgttr}
\frac{\partial{g_{tt}}}{\partial{x^r}}=\frac{r_s}{r^2}
\end{equation}
\begin{equation}
\label{dgttt}
\frac{\partial{g_{tt}}}{\partial{x^t}}=\frac{r_s\dot{r}}{r^2}
\end{equation}
Substituting from equations (\ref{grr1}) to (\ref{gtt2}) into equations (\ref{gtrr1}) to (\ref{gtrr}) and, subsequently, from the latter set and into equation (\ref{ktrr2}), leads directly to:
\begin{equation}
\label{ktrr3}
\kappa t^r_r=-\frac{\dot{a}^2}{2a^2}(1-\frac{r_s}{r})^{-1}+\frac{\dot{a}\dot{r}r_s}{2ar^2}(1-\frac{r_s}{r})^{-2}
\end{equation}
By means of the Hubble-Lemaitre law, we may replace the quotient $\dot{a}/a$ by the Hubble parameter \selectlanguage{english}-- $H.$ Then, so that the energy tensors of the gravitating body determining the Schwarzschild radius - $r_s=2GM/c^2$ - do not change due to Hubble flow, we ensure that the observer moves with it away from the galactic centre - that serves as the origin of the coordinate chart here applied - at the radial speed of:
\begin{equation}
\label{dotr}
\dot{r}=\frac{dx^r}{dx^t}=\frac{1}{c}\frac{dr}{dt}=Hr
\end{equation}
Where:
\begin{equation}
\label{xt}
x^t=ct
\end{equation}
So, with these substitutions, equation (\ref{ktrr3}) becomes:
\begin{equation}
\label{ktrr4}
\kappa t^r_r=-\frac{\MakeUppercase{h}^2}{2}(1-\frac{r_s}{r})^{-1}+\frac{\MakeUppercase{h}^2r_s}{2r}(1-\frac{r_s}{r})^{-2}
\end{equation}
Figure 1 shows the radial profile of the gravitational pressure.
For $r>r_s$ the pressure turns out to be negative and directly proportional to the square of the Hubble parameter.
This negative pressure and the \emph{EoS} may be closely related to the Hubble expansion. They are signal features associated with the hypothetical DE of the cosmological constant \selectlanguage{english}-- \selectlanguage{greek}Λ
\selectlanguage{english}FigureFigure 1 illustrates the initially rapid rise of the magnitude of the negative pressure as the distance from the spherical body increases and its fairly rapid flattening thereafter. At very large radii, the negative pressure approaches its maximum magnitude of $H^2/2${\kappa}. This maximum magnitude of pressure is attained irrespective of the value of $r_s$ and so also of the mass-energy of the gravitating body. So, there is a saturation of the gravitational pressure (and energy density) occurring remotely from gravitating bodies of all scales of mass-energies and densities.
So, beyond the event horizon of the super massive black hole (SMBH), where $r > r_s$ defines the domain of application of the Exterior Schwarzschild metric, the pressure turns out to be negative and directly proportional to the square of the Hubble parameter. Clearly, this negative pressure is intimately related to the Hubble expansion. Both are signal features associated with the hypothetical dark energy of the cosmological constant - $\Lambda.$ Figure 1 illustrates the initially rapid rise of the magnitude of the negative pressu
The EoS of gravity revealed in equations (\ref{ktrr2}), (\ref{kttt}) and equation (\ref{eos}) imply that, without this continuous negative gravitational pressure and spatial expansion, the gravitational field and so space also would collapse. This suggests that gravitational pressure causes the spatial expansion.
\selectlanguage{ngerman}\section*{4.2\ \ \ Conservation of Energy in an Expanding Universe: a Derivation of the Hubble-Lemaître law}
Therefore, I propose that a growing gravitational field causes the expansion of space in a cooperation that preserves adherence to the energy conservation laws of gravity that apply at infinitesimal scales in the vacuum and of the total system - the latter including the gravitating matter and the gravitational field \selectlanguage{english}-- expressed by \cite{einstein1952}, respectively, as:
\begin{equation}
\label{consrv1}
\frac{\partial{t^{\alpha}_{\sigma}}}{\partial{x^{\alpha}}}=0
\end{equation}
\begin{equation}
\label{consrv2}
\frac{\partial}{\partial{x^{\alpha}}}(\MakeUppercase{t}^{\alpha}_{\sigma}+t^{\alpha}_{\sigma})
\end{equation}
Where, $T_{\sigma}^{\alpha}$ is the energy tensor of the gravitating body.
This cooperation may only occur if the incrementing energy only develops in emergent fields that come into being in infinitesimal emergent spaces and at the same energy density as that of the infinitesimal regions of the pre-existent field contiguous to the infinitesimal emergent spaces. Since, such a process of spatial expansion occurs at all radii greater than $r_s,$ then the expansion of space occurs as a smooth interstitial emergence of new spaces throughout the spatial manifold. This may only occur if space is the form of the gravitational field. This notion is buttressed by the fundamental metrical coupling of the field and space expressed by equations (\ref{eq1}) and (\ref{eq4}).
So, since the gravitational energy density \selectlanguage{english}-- with its magnitude, everywhere, being equal to that of the gravitational pressure - is conserved, then a gravitationally driven expansion would be locally isobaric.
All isobaric expansions, thermodynamic or cold, require the injection of energy. In thermodynamics, this energy is in the form of heat. In the cold gravitational isobaric expansion, emergent gravitational energy - that by virtue of the constraint of the conservation of energy density, may only occur in emergent spaces - performs the same role as heat does in isobaric thermodynamic expansions.
Consider the infinitesimal isobaric expansion of space in the form of a spherical shell of inner radius $r$ and thickness $dr.$ We propose that this expansion of space is due to the work that is done by the gravitational pressure within the infinitesimal region of the shell of volume $dV.$ Conservation of energy implies that $dW+dU=0,$ where $dW$ is the work done by the gravitational pressure and $dU$ is gravitational energy emerging in the simultaneously emergent space, solely as a result of the gravitational pressure.
The work $dW$ is the product of the pressure,
the area of the inner surface of the shell - $4{\pi}r^2$ and the displacement and is expressed here as:
\begin{equation}
\label{dW}
d\MakeUppercase{w}=4{\pi}r^2t^r_rdr
\end{equation}
Recall the volume of the sphere:
\begin{equation}
\label{v}
\MakeUppercase{v}=\frac{4}{3}\pi{r^3}
\end{equation}
And, by way of the radial derivative of the volume, the volume of the shell is:
\begin{equation}
\label{dv}
d\MakeUppercase{v}=4\pi{r^2}dr
\end{equation}
Conservation of energy implies:
\begin{equation}
\label{dU}
dU=-dW=-4{\pi}r^2t_r^rdr
\end{equation}
Therefore, with a substitution from (\ref{dv}), the energy density within the shell, by virtue of equation (\ref{eos}), is given by:
\begin{equation}
\label{rho}
\rho=\frac{d\MakeUppercase{u}}{d\MakeUppercase{v}}=-\frac{4{\pi}r^2t^r_rdr}{4{\pi}r^2dr}=-t^r_r=t^t_t
\end{equation}
This shows how the emergent space is created by gravitational pressure and simultaneously occupied by the incrementing gravitational energy at the pre-existent density. In this process, the incremental gravitational energy is created purely by the work constantly being performed by the gravitational pressure in creating emergent spaces. In this way, the expansion of space and the growth of the gravitational field combine to yield the conservation of gravitational energy, at infinitesimal scales, as first enunciated by \cite{einstein1952}.
Consider this expansion of space as it occurs in two separated spherical regions, each of arbitrary radius that comoves with the Hubble flow, located in gravitational fields possibly of different gravitational pressures. We desire, firstly, to compare the rates of fractional volumetric increases that result from these pressures.
The power per unit volume, isobarically applied, is directly proportional to the constant local pressure. So, we may write:
\begin{equation}
\label{prop}
\frac{1}{\MakeUppercase{v}}\frac{d{\MakeUppercase{w}}}{dt}\ {\propto}\ t^r_r
\end{equation}
Conservation of energy implies that the power - per unit volume - delivered by the gravitational pressure, in its locally isobaric expansion of space, is given by:
\begin{equation}
\label{iso}
\frac{1}{\MakeUppercase{v}}\frac{d\MakeUppercase{w}}{dt}=\frac{t^r_r}{\MakeUppercase{v}}\frac{d\MakeUppercase{v}}{dt}
\end{equation}
Denoting the two scenarios of the expanding spheres by the indices 1 and 2, respectively, the two relationships above allow us to write:
\begin{equation}
\label{max}
\frac{t^r_{r,2}}{t^r_{r,1}}=\frac{\frac{1}{\MakeUppercase{v}_2}\frac{d\MakeUppercase{w}_2}{dt}}{\frac{1}{\MakeUppercase{v}_1}\frac{d\MakeUppercase{w}_1}{dt}}=\frac{\frac{t^r_{r,2}}{\MakeUppercase{v}_2}\frac{d\MakeUppercase{v}_2}{dt}}{ \frac{t^r_{r,1}}{\MakeUppercase{v}_1}\frac{d\MakeUppercase{v}_1}{dt}}
\end{equation}
Which first yields:
\begin{equation}
\frac{1}{\MakeUppercase{v}_1}\frac{d\MakeUppercase{v}_1}{dt}=\frac{1}{\MakeUppercase{v}_2}\frac{d\MakeUppercase{v}_2}{dt}
\end{equation}
So, the fractional volumetric expansion rates are equivalent. With substitutions from (\ref{v}) and (\ref{dv}), we get:
\begin{equation}
\frac{1}{r_1}\frac{dr_1}{dt}=\frac{1}{r_2}\frac{dr_2
}{dt}=\MakeUppercase{h}
\end{equation}
Therefore, $H$ remains the same - irrespective of the strength of the gravitational field and the separation involved - and, thereby, also of location and time, due solely to the isobaric nature of the expansion of the invisible gravitational field underlying the observable spatial expansion. This affirms the uniformity and isotropy of the Hubble expansion across all distance and time scales and, thereby, confirms the nature of the Hubble parameter as a cosmic constant.
Thereby, this account identifies dark energy as the expanding gravitational field.
\section*{5.0\ \ \ INFLUENCES OF THE EXPANDING ENERGY FIELDS OF GRAVITY}
\section*{5.1\ \ \ Galactic Orbits and the Gravitational Field}
In this theoretical framework, the acceleration of bodies in free fall is directly due to the gravitational field, (as opposed to the direct attraction of the gravitating body, as in Newtonian physics). So, here the satellite's acceleration is given by the geodesic equation:
\begin{equation}
\label{geo}
\frac{d^2x^{\tau}}{ds^2}=\Gamma^{\tau}_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}
\end{equation}
In the absence of information on 3-velocities, we may first extract the acceleration by ensuring that the observer comoves with the test body by setting:
\begin{equation} \frac{dx^{\mu}}{ds}=\frac{dx^{\nu}}{ds}=0\ \ \ \ :\mu, \nu=1, 2, 3
\end{equation}
With these conditions, equations (1) and (19) yield:
\begin{equation}
\frac{dx^{\mu}}{ds}=\frac{dx^{\nu}}{ds}=\surd{(g_{tt})}\approx 1\ \ \ \ :r{\gg}r_s, \mu, \nu=0
\end{equation}
So, equation (31) yields:
\begin{equation}
\frac{d^2x^{\tau}}{ds^2}\approx\Gamma^{\tau}_{tt}\ \ \ \ :\tau=1, 2, 3
\end{equation}
These components of the 3-acceleration may be vectorially summed as:
\begin{equation}
\frac{d^2x^r}{ds^2} \approx \Gamma^r_{tt}
\end{equation}
With the acceleration so determined, then substituting from equations (\ref{grr2}), (\ref{dgrrt}), and (\ref{grtt}) into the above equation, the radial acceleration may be expressed as:
\begin{equation}
\frac{d^2x^r}{ds^2}\approx\Gamma^r_{tt}=-\frac{r_s}{2a^2r^2}(1-\frac{r_s}{r})
\end{equation}
Now, the orbits of the satellites of many galaxies are approximately circular, so the (averaged) speed of rotation is given by the square root of the product of the acceleration and the average radius, expressed here as:
\begin{equation}
\label{vel}
v_{rot}\approx |r\Gamma^r_{tt}|^{1/2}=[\frac{c^2r_s}{2a^2r}(1-\frac{r_s}{r})]^{1/2}
\end{equation}
Where $c$ is the speed of light that appears here, by virture of equation (\ref{xt}), in order to convert the natural time unit of measurement - equivalent to $c$ seconds - to the time unit of the second. For $a=1,$ Figure 2 shows the profile of a rotation curve in the vicinity of the galactic centre obtained by means of equation (\ref{vel}).
(Orbits of solar and planetary systems do not approach the event horizons - that are greatly exceeded by the extents of the stars and planets as closely as those of galaxies with their SMBHs enclosed by their event horizons, so their declining rotation curves appear Kleperian.)
However, exceptions abound. We propose that certain exceptions are due to the growth of the energy fields of gravity in the vicinities of galaxies. These include unexpected gravitational effects that are currently attributed to dark matter.
So, instead of directly considering the mass-energy of the gravitating bodies in order to understand the strange kinematics of their satellites, we now seek to understand the effects of the growth of the gravitational field on these orbits. For this purpose, we consider the gravitational energy density field obtained from equations (\ref{eos}) and (\ref{ktrr4}) as:
\begin{equation}
\label{kttt3}
\kappa t^t_t=\frac{\MakeUppercase{h}^2}{2}[(1- \frac{r_s}{r})^{-1}-\frac{r_s}{r}(1-\frac{r_s}{r})^{-2}]
\end{equation}
The profile of the energy density field is shown in Figure 3. The gravitational energy density is positive beyond $r_s.$ Note the high and flat energy density of regions remote from the centre. The strength of the energy density field dramatically declines as the galactic centre is approached.
As the gravitational field evolves, the resultant motions of the satellites progressively become more remotely connected to the gravitating bodies that initially gave rise to the field. Therefore, unexpected velocities may be due to unexpected behaviour of the gravitational field in galactic regions.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Rotation-Curve/Density-I}
\caption{{Radial Velocity Profile Around a Black Hole~
{\label{288764}}%
}}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Density/Vel-1}
\caption{{. Gravitational Energy Density Profile Around a Black Hole
{\label{988472}}%
}}
\end{center}
\end{figure}
\section*{5.2\ \ \ The Accelerated Expansions of Space and of the Gravitational Field}
The acceleration of the expansions of space and the gravitational field may be arrived at by differentiating equation (\ref{dotr}) to yield:
\begin{equation}
\label{accexp}
\ddot{r}=\MakeUppercase{h}\dot{r}=\MakeUppercase{h}^2r \end{equation}
Which we get since $H$ is a constant. So, two observer, while being separated only by the Hubble flow, will see each other in an accelerating recession.
Now, substitute from equation (\ref{xt}) into equation (\ref{dotr}), rearrange, and then integrate:
\begin{equation}
\label{expo}
\int_{r_0}^r\frac{dr}{r}=ln(\frac{r}{r_0})=c\MakeUppercase{h}\int^t_0dt=c\MakeUppercase{h}t
\end{equation}
So, we arrive at:
\begin{equation}
\label{expo2}
r=r_0e^{c\MakeUppercase{h}t}
\end{equation}
This equation describes the exponential expansion of space and the gravitational field over time (measured in seconds) corresponding to a constant Hubble parameter, that is, to a Hubble constant. The exponential term given in the equation above is the cosmic scale factor:
\begin{equation}
\label{scale}
a(t)=e^{c\MakeUppercase{h}t}
\end{equation}
So, the Hubble constant remains $H=(da/dx^t)/a=\dot{a}/a$ and $a(t_0)=1$ at time $t_0=0.$
\section*{5.3\ \ \ Historical Forms of the Laws of Curvilinear Gravity: Identification of Dark Matter}
Equation (\ref{expo2}) also constitutes a coordinate transformation of the radius across time and space due only to Hubble flow. Furthermore, since space is the form of the gravitational field and the latter determines the velocities of rotation of orbits, then equation (\ref{expo2}) specifies the space-time coordinate transformation that occurs due to Hubble flow and, thereby, applicable to the radial variable in equations (\ref{vel}) and (\ref{kttt3}).
Consider an observer, initially at a distance $r_0$, who moves - in Hubble flow - away from the galactic centre so that his distance changes according to the historical relationship given in the advanced-time scaled coordinate transformation given in equation (\ref{expo}). So, over time, his distance from the galactic centre increases according to equation (\ref{expo}).
However, the gravitational energy density would not change as the field, at his comoving locality, conserves its energy density. So, the retention of the value of the energy density as the radius extends would seem irregular from the ahistorical point of view taken in equation (\ref{kttt3})
Specifically, in this equation, only the radial variable would have changed in value, thereby rendering the result of its application incorrect.
However, he may restore the correct outcome to equation (\ref{kttt3}) by means of a coordinate transformation that, in this equation, substitutes the current radius r by its reversed-time scaled - and, thereby, initial - radius.
He may, then, also realize that with this coordinate transformation, equation (\ref{vel}) also yields the observed magnitude of the velocity of a satellite in a circular orbit at his radius that, previously puzzlingly, has also remained unchanged. So, at any radius $r>r_s:$
\begin{equation}
\label{kgt}
\begin{aligned}
&&-\kappa t^r_r=\kappa t^t_t= \frac{\MakeUppercase{h}^2}{2}[(1-\frac{r_s}{r'})^{-1}-\frac{r_s}{r'}(1 -\frac{r_s}{r'})^{-2}] \\ && :r'=(r-r_s)e^{-\MakeUppercase{h}t}-r_s
\end{aligned}
\end{equation}
Where, use is made of equation (40). The transformation preserves, within a changing galactocentric radius r, a constant Schwarzschild radius as this latter quantity does not undergo the Hubble expansion as it is not a function of the metric, but only of the mass of the SMBH, here considered constant.
This retention of the value of the gravitational energy density, at a comoving location, is the outcome of the conservation laws of gravity and the metric Hubble expansion associated with it.
Similarly:
\begin{equation}
\label{vrotg2}
v_{rot}=[\frac{c^2r_s}{2r'}(1-\frac{r_s}{r'})]^{1/2}
\end{equation}
Figure 4 shows, in a fixed galactocentric frame, the time-lapsed progression of a single galaxy. This series was produced by means of the applications of equations (53) and (54). Note the resemblance of the rotation curves to those of different galaxies. So, such galaxies may just be at different stages in a similar pattern of development.
The flattening of the rotation curves in the outer regions of numerous galaxies - resembling those shown in panels (e), (g), and (i) of Figure 4 - have been attributed to a clustering dark matter field. However, these profiles have been produced by the expansion of the gravitational field
In general, the observer may also calculate the velocities and gravitational energy densities at different radii at any time since the birth of the galaxy by the application of equations (53) and (54).
So, by means of equations (53) and (54), he may obtain time profiles of development at fixed radii, as well as radial characteristics at fixed times - such as those of Fig. 4 - of the expansion of the rotation curves and of the energetic fields of gravity.
Therefore, the expansion of the gravitational field yields the same results - flattening rotation curves - as does the hypothetical dark matter, without the latter's challenges of identification and the 'cuspy' central gradients of its mass-energy density profiles as described by its prominent models [18] -- as may be seen in Fig. 4. Here, the energy density profiles are either 'cored' or flattened, according to the age of the galaxy.
The flattening of the rotation curves in the outer regions of numerous galaxies - resembling those shown in panels (e), (g), and (i) of Figure 4 - have been attributed to a clustering dark matter field. However, these profiles have been produced by the expansion of the gravitational field.
In general, the observer may also calculate the velocities and gravitational energy densities at different radii at any time since the birth of the galaxy by the application of equations (53) and (54).
So, by means of equations (53) and (54), he may obtain time profiles of development at fixed radii, as well as radial characteristics at fixed times - such as those of Fig. 4 - of the expansion of the rotation curves and of the energetic fields of gravity.
Therefore, the expansion of the gravitational field yields the same results - flattening rotation curves - as does the hypothetical dark matter, without the latter's challenges of identification and the 'cuspy' central gradients of its mass-energy density profiles as described by its prominent models [18] -- as may be seen in Fig. 4. Here, the energy density profiles are either 'cored' or flattened, according to the age of the galaxy.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Multiple-Graphs/Evolution-BH}
\caption{{Evolution of Velocity and Gravitational Energy Density Profile Around a
Black Hole
{\label{957027}}%
}}
\end{center}
\end{figure}
Now, orbits only occur in inherently curvilinear regions of the gravitational field [4]. So, so-called 'dark matter haloes' are regions of curvilinear gravity in the vicinities of matter fields. However, such regions extend well beyond the galactic orbits of most of 'dark matter's' baryonic tracers. Evidence of the gravitational curvilinearity beyond the orbits is obtained by means of its optical effects.
The 4-D trajectories of radiation are null geodesics - mathematically obtained by setting the proper duration to zero in an equation of the linear element -- that often are expressed as 3-velocities, with directions that vary according to Huyghen's Principle. So, generally, null geodesics are defined by the condition:\selectlanguage{greek} $g_{μν} \selectlanguage{english}dx^\selectlanguage{greek}μ\selectlanguage{english}dx^\selectlanguage{greek}ν\selectlanguage{english}=0,$ where, the Einstein summation rule is implied.
In flat space-time, where the\selectlanguage{greek} $g_{μν}\selectlanguage{english}$ are constants, the trajectories are rectilinear. However, where the components of the metric inherently are curvilinear functions of the coordinates, the null geodesics are also curvilinear. Therefore, the occurrence of gravitational lensing solely depends on the presence of inherently curvilinear space-times, as in the case of orbits. However, unlike most galactic orbits that are intragalactic phenomena, gravitational optics is evidenced throughout the 'dark matter haloes' enclosing galaxies.
The use of gravitational lensing is the only technique presently available to determine the full extent of regions of curvilinear gravity that are termed dark matter haloes [12] and the variability of the metric in those regions. (Bound hypervelocity stars show great promise to assist in determining the extents of gravitational curvilinearity. However, the low densities of such stars in the haloes limit their usefulness.)
Gravitational lensing, thereby, provides a straightforward confirmation of the proposal that the regions of 'dark matter' are zones of curvilinear space-time -- the latter with line elements characterized by curvilinear metric components - and, thereby, also of curvilinear gravitational field and energy components: curvilinear gravity.
\section*{5.4\ \ Cosmic Voids and an Appproximately Flat Universe}
The regions of gravitational curvilinearity - the 'dark matter halos' of enclosed matter fields - are embedded in a spatially dominant and approximately flat domain of the gravitational manifold. In these parts, in the limit - $r_s/r\rightarrow \ 0,$ the equation of the line element obtained by substituting equations (\ref{gtt1}), (\ref{grr1}), and (\ref{scale}) into equation (\ref{eq1}) turns out to be that of \emph{de} Sitter's:
\begin{equation}
\label{deS}
(ds)^2=(dx^0)^2-e^{2c\MakeUppercase{h}t}[(dx^1)^2+(dx^2)^2+(dx^3)^2]
\end{equation}
So, in these vast spaces, the metric approximates that of the flat spherically symmetrical expanding universe. Yet, it is curvilinear in galactic neighbourhoods. Therefore, the same metric applies across all distance scales. So, it avoids the challenges of integrating the curvilinear space-time of galaxies with a spatially flat space-time of the universe at large scales that confronted Einstein and Straus.
The flatness of space occurs beyond the 'dark matter halos' of galaxies, galactic clusters, and cosmic walls and in the great voids of gravitational flatness. So, the maximal intensity and flatness of the field, and not its Newtonian absence, prevail over the significantly greater portion of the observable universe consisting of large empty spaces. (This resolves the 'flatness problem' of the \selectlanguage{greek}Λ\selectlanguage{english}CDM model.)
Equation (\ref{eq4}) shows that the field components of gravity are functions of the coordinate derivatives of the components of the metric. So, an approximately constant metric in the void generally implies mostly vanishing gravitational field components. However, in this space-time, there is one space component of the gravitational field - $\Gamma_{tr}^r$ -- that, at large radii, approaches a constant non-zero value.
Equations (\ref{gtt1}), (\ref{grr1} ), and (\ref{grtr}) yield:
\begin{equation}
\Gamma^r_{tr}=-\frac{g^{rr}}{2}\frac{\partial{g_{rr}}}{dx^t}=-\MakeUppercase{h} +\frac{\MakeUppercase{h}r_s}{2r}(1-\frac{r_s}{r})^{-1}
\end{equation}
So:
\begin{equation}
\label{goo}
\Gamma^r_{tr}=-\MakeUppercase{h}\ \ \
:r_s/r\ \rightarrow\ 0
\end{equation}
It is this field component that gives rise to the approximately maximum value of gravitational energy density and pressure in these remote regions, as shown by equations (\ref{ktrr4}) and (\ref{kttt3}).
Now, consider the geodesic 3-acceleration associated with this field component that is given by:
\begin{equation}
\label{geod1}
\frac{d^2x^r}{ds^2}=\Gamma^r_{rt}\frac{dx^r}{ds}\frac{dx^t}{ds}
\end{equation}
Substitute from equations (\ref{dotr}), (\ref{xt}), (\ref{accexp}), (\ref{goo}), and $ds=cd\tau,$ and rewrite (\ref{geod1}) as:
\begin{equation}
\label{geod2}
\frac{d^2r}{d\tau^2}=-c\MakeUppercase{h}\frac{dr}{d\tau}\frac{dt}{d\tau}
\end{equation}
Since, up to this point, the constant $H$ has units of per $c$ seconds, then converting to per second absorbs the constant $c.$ So, substituting from (\ref{accexp}):
\begin{equation}
\label{tree}
\frac{d^2r}{d\tau^2}= -\MakeUppercase{h}\frac{dr}{dt}(\frac{dt}{d\tau})^2=-\frac{d^2r}{dt^2}(\frac{dt}{d\tau})^2
\end{equation}
On the left of this equation is the acceleration, measured locally, due to the Hubble expansion. Locally, the coordinate radius $r$ is equivalent to the proper distance $-D.$ This is obtained in the time slice $dt=0$ taken at the time of emission, $t=0,$ in the equation of the line element (\ref{deS}). So, making this substitution yields:
\begin{equation}
\frac{d^2\MakeUppercase{d}}{d\tau ^2}=\frac{d^2r}{dt^2}(\frac{dt}{d\tau})^2
\end{equation}
This arranges the geodesic equation so that on the left is the local expansion, measured locally, and on the right is the same expansion measured from afar, as adjusted by a linear and homogenous (tensorial) transformation across time.
Consider the proper speed of recession measurable at the target and that is to be determined at the observer given as:
\begin{equation}
\frac{d\MakeUppercase{d}}{d\tau}=\frac{dr}{d\tau}=\frac{dr}{dt}\frac{dt}{d\tau}
\end{equation}
In order to obtain the values of the proper separation's proper time derivatives, on the left of the two equations above, from the observed coordinate time derivatives of the radius on the right, the factor $dt/d\tau$ must be determined. This factor reflects the changes of the infinitesimal time ('tick') rate of the target's oscillator (source of electromagnetic radiation) as measured, usually from a distant, by an observer with an oscillator (clock) in his vicinity. This occurs because of the expansion of the wavelength as the radiation travels from the emitter to the observer. Since the wavelength lengthens, then the frequency reduces; redshifts.
Denote the parameters of the wave produced by the emitting oscillator by the subscript $e$ and measurements of them by means of the observer's oscillator by $o.$ Now, since the time rate is proportional to the period and thereby to the wavelength, then:
\begin{equation}
\frac{dt}{d\tau}=\frac{\lambda_o}{\lambda_e}=\frac{f_e}{f_o}=\frac{\lambda_o-\lambda_e}{\lambda_e}+1=z+1
\end{equation}
Where $f$ is the frequency and $z$ is the fractional change in wavelength that occurs in transmission - the redshift.
Furthermore, as the wavelength changes due to the Hubble expansion, then:
\begin{equation}
1+z=\frac{\lambda_o}{\lambda_e}=\frac{a(t_o)}{a(t_e)}=e^{c\MakeUppercase{h}(t_o-t_e)}
\end{equation}
Where, equation (\ref{scale}) has been used to produce the last term above.
(In practice, the redshift $z$ is determined spectroscopically. The separation is determined optically by relative luminosity of certain types of stars - 'standard candles' - or by the light curves of SN1a supernovae. The proper speed of the Hubble recession is obtained by means of the redshift and a cosmological model.\cite{Guth_1981})
So, it is the field component $\Gamma^r_{tr}$ that, in the vast voids, governs the isotropic geodesic metric acceleration of the Hubble recession.
It must be kept in mind that, although being made clear here due to simplification of the metric, this field component acts in the same manner to drive the Hubble recession at all radii greater than $r_s.$ However, within the regions of curvilinear gravity, its appearance is complexified by the gravitationally induced kinematical - as opposed to metrical - redshifting associated with the factor $(1-r_s/r),$ that occurs to radiation emitted from gravitational wells. But, its own effects remain the same, in terms of the recessional rate it gives rise to. This is due to the invariance of the relative recessional rate due to its isobaric origins.
The attainment - in these universally volumetrically dominant voids \cite{van_de_Weygaert_2014}- of maximal gravitational energy density, pressure, and, simultaneously, of flat space-time reflects the complexity and the importance of such regions in comprehending cosmic phenomena.
\section*{6.0\ \ \ AUTO-INDUCTION: THE ENGINE OF GRAVITATIONAL EXPANSION }
In presenting his theory of gravitation, Einstein developed his general field equations that represent the 'total system' - consisting, for example, of the massive gravitating body and its gravitational field - from his field equations of gravity 'in the absence of matter' - $R_{\mu\nu}=0$ - and equation (\ref{eq4}). So, he first arrives at the following relationship:
\begin{equation}
\label{induct}
\frac{\partial}{\partial{x^{\alpha}}}(g^{\beta\sigma}\Gamma^{\alpha}_{\mu\beta})=-\kappa(t^\sigma_\mu-\frac{\delta^{\sigma}_{\mu}}{2}t)
\end{equation}
Then at this point, Einstein made the remarkable substitutions:
\begin{equation}
\label{sub}
t\rightarrow t+T \ \ \ \ \ t^\sigma_\mu \rightarrow t^\sigma_\mu+T^\sigma_\mu \end{equation}
Where, $t=t_\mu^\mu=trace\{t^{\sigma}_{\mu}\}$. Here, the trace is the sum of the elements along the diagonal of the pseudo-tensor as given in (\ref{diag}). So, as a result of its EoS, the trace of the energetic gravitational fluxes of gravity is given by:
\begin{equation}
\label{trace}
t=2t_r^r=-2t_t^t
\end{equation}
By way of justification, Einstein offered the following:
'It must be admitted that this introduction of the energy-tensor of matter is not justified by the relativity postulate alone. For this reason, we have deduced it from the requirement that the energy of the gravitational field acts in the same way as other kinds of energy.' [Emphases added.]
So, by means of these substitutions, Einstein first arrives at:
\begin{equation}
\label{induce}
\frac{\partial}{\partial{x^{\alpha}}}(g^{\beta\sigma}\Gamma^{\alpha}_{\mu\beta})=-\kappa{[( t^\sigma_\mu-\MakeUppercase{t}^\sigma_\mu)-\frac{\delta^\sigma_\mu}{2}(t-\MakeUppercase{t})]}
\end{equation}
And by such a pathway, Einstein arrived at his general field equations of gravitation.
With regards to the components of gravity, Einstein's substitutions amount to a feedback mechanism that drives continuous changes in the field components and, correspondingly, in the gravitational energy, even in the absence of changes in the energetic fluxes of matter. This is auto-induction of gravity.
In the exploration of the process of gravitational auto-induction, it is facilitative to ensure the exclusion of external factors by setting the energy tensors to zero, as they are in the vacuum. We may further simplify considerations by setting $\sigma=\mu$
in equation (\ref{induct}), whereby the Kronecker delta becomes $\delta_\selectlanguage{greek}μ\selectlanguage{english}^\selectlanguage{greek}μ\selectlanguage{english}=4.$ This yields:
\begin{equation}
\label{ind}
\frac{\partial}{\partial{x^\alpha}}(g^{\beta\mu}\Gamma^{\alpha}_{\mu\beta})=\kappa{t}
\end{equation}
Rewrite as:
\begin{equation}
d(g^{\mu\beta}\Gamma^{\alpha}_{\mu\beta})=\kappa{t}dx^{\alpha}
\end{equation}
So, the components of the gravitational field that participate in time evolution at any - even at a fixed - radius r, are here identified by substituting $\selectlanguage{greek}α\selectlanguage{english}$ by $t$ in the above equation. Then, integrating, we get:
\begin{equation}
\Delta(g^{\mu\beta}\Gamma^t_{\mu\beta})|^{ct}_{ct_0}=\kappa\int^{ct}_{ct_0}tdx^t
\end{equation}
Where $c$ is the speed of light and $x^t=ct.$ Rewrite:
\begin{equation}
g^{\mu\beta}(r,t)\Gamma^t_{\mu\beta}(r,t)=g^{\mu\beta}(r_0,t_0)\Gamma^t_{\mu\beta}(r_0,t_0)+\kappa\int^{ct}_{ct_0}tdx^{\alpha}
\end{equation} Multiplying by the metric and, as $g_{\mu\beta}g
^{\mu\beta}=\delta^\mu_\mu=4,$ we arrive at the auto-induced augmented field components:
\begin{equation}
\begin{aligned}
\label{xyz}
&&\Gamma^t_{\mu\beta}(r,t)=\frac{1}{4}g_{\mu\beta}(r,t)g^{\mu\beta}(r_0,t_0)\Gamma^t_{\mu\beta}(r_0,t_0)
\\
&&+\frac{1}{4}g_{\mu\beta}(r,t)\int^{ct}_{ct_0}tdx^{\alpha}
\end{aligned}
\end{equation}
This equation, that increments the field, is the critical regenerative phase of a positive gravitational feedback cycle: it describes the auto-induction of the time components of the gravitational field. However, such incremented field components have the same regenerative impact in the space components $t_r^r$ of the energy fluxes as in their time components $t_t^t,$ as is shown in equations (\ref{ktrr2}) and (\ref{kttt}). The updated field component above updates the energy density of equation (\ref{kttt}) in its historical form given by equation (\ref{ktttg}), then the trace of the gravitational pseudo-tensor is updated by means of equation (\ref{trace}), whereby the cycle returns to the equation above to recommence.
So, as these field components evolve, so do the energetic fluxes, and - since the gravitational energy density is conserved - the field expands and, thereby, so does its form - space. By such a pathway, the auto-induced gravity results in an unconditional and continuous growth of the field, once it is seeded by the gravitating body.
\section*{6.1\ \ \ Relative Mass-energy Density Weightings in the Induction of Gravity}
Consider a spherical region of space undergoing the Hubble expansion. Denoting its initial conditions by the subscript 0, then with equation (\ref{expo2}) and with Einstein's substitutions (\ref{sub}) into equation (\ref{ind}), there results the historical form of the equation of gravitational induction:
\begin{equation}
\label{induct2}
\frac{\partial}{\partial{x^{t}}}(g^{\mu\beta}\Gamma^t_ {\mu\beta})=\kappa [t+\MakeUppercase{t}_{\MakeUppercase{b},0}e^{-3\MakeUppercase{h}x^t}+\MakeUppercase{t}_{\MakeUppercase{r},0}e^{-4\MakeUppercase{h}x^t}]
\end{equation}
Where, the subscripts B and R indicate baryonic matter and radiation, respectively. The Hubble expansion dilutes the mass-energy fields of radiation and baryons. So their induction of the gravitational field will be progressively reduced. For baryons, the dilution of energy density is directly proportional to the volume. For radiation, dilution occurs due to volumetric expansion as well as to its accompanying redshifting of the frequencies. However, the gravitational field preserves its density during spatial expansion.
So, over time, gravitational energy increasingly dominates gravitational induction. Present estimates of 'dark energy' and 'dark matter' place the relative weight of gravitational energy density at approximately ninety-five percent.
\section*{7.0 EINSTEIN'S ENERGY MOMENTUM PSEUDO-TENSOR} This quantity describes the energy components of gravity according to \cite{einstein1952}. There, it is revealed that this energy is similar to that of baryons and radiation in the induction of gravity. However, beyond their common induction of gravity, here it is shown that the pseudo-tensor is qualitatively different from the energy tensor. The first grows to create more space, the second drives interactions of bodies and radiation. Both processes are energetic.
Everywhere, energy tensors of bodies and radiation interact locally, from time to time, under the constraint of the conservation of energy, the latter both at infinitesimal and finite scales. The pseudo-tensors of gravity, under the constraint of the conservation of energy, only at infinitesimal scales, act simultaneously everywhere to maintain and expand the field, and thereby, its form - space - that accommodates all baryons and radiation.
So, the scopes of the interactions are at vastly different. Tensors interact locally, regardless of the frame of reference applied, and therefore are covariant. Pseudo-tensors act cosmically that is, non-locally, and may appear to disappear as an artefact of a coordinate system. This, for example, occurs in charts riding on geodesics, because the pseudo-tensors, as shown in equation (\ref{eq4}), is a function of coordinate derivatives of components of the metric, the latter being constants in the locally flat space-time of a geodesic.
However, geodesics are not energetic manifestations of the gravitational field, but are influences to the field on baryons and radiation essentially related only to the growth and curvilinearity of these field components and not directly to the actions of the pseudo-tensors that may also be curvilinear due to the same cause. The energetic fluxes of gravity, energy pseudo-tensors, act only to grow the field.
\section*{~}
{\label{464339}}
\section*{\texorpdfstring{{8.0
DISCUSSION~}}{8.0 DISCUSSION~}}
{\label{698828}}
The theoretical results obtained here strongly motivate further
endeavours to advance empirical investigations to determine the
linearity of the Hubble-Lema\selectlanguage{ngerman}ître law at small scales less than 5Mpc
\selectlanguage{english}\cite{Sandage_2010}{~below 0.7 Mpc {[}9{]}~\citep*{Karachentsev_2009}. At
galactic and solar scales [?] 100 pc, innovative techniques will have to
be developed. This will test the proposal that the Hubble parameter is
invariant.~}
{}
The historical forms of the laws of the field and their influences,
given here, may assist in providing the bases for the development of
techniques applicable to the determination of the expansion, especially
in regions of curvilinear gravity.
\par\null
Due to its EoS, gravity's energetic fluxes - pressure (`dark energy')
and energy density (`dark matter') - are equal in magnitude everywhere.
This avoids the `cosmic coincidence problem' confronting the
\(\)\selectlanguage{greek}Λ\selectlanguage{english}CDM model, of infinitesimal likelihood, in any single
epoch, of the occurrence of comparable magnitudes of a continuously
decreasing dark matter density and that of an unchanging dark energy
density.
\par\null
`This ratio changes from almost infinity to zero in the \selectlanguage{greek}Λ\selectlanguage{english}CDM
model'~\cite{Velten_2014}.
\par\null
Given this latitude, the question is: why is it that, in this epoch,
this ratio is of an order of magnitude of one.
`Since the missing energy density and the matter density decrease at
different rates as the universe expands, it appears that their ratio
must be set to a specific, infinitesimal value in the very early
universe in order for the two densities to nearly coincide today, some
15 billion years later'~\protect\hyperlink{Zlatev_1999}{(Zlatev 1999)}.
\par\null
{The volumetrically dominant cosmic voids and empty intergalactic spaces
{[}16{]} are, here, described as domains of maximal gravitational energy
density and of approximately flat space-time. This makes for an overall
proximate flatness of the field across the cosmos. Furthermore, the
growth of the field increases the volumes of these flat regions as it
reduces its gradients in the gravitationally curvilinear vicinities of
galaxies and, thereby, increases the overall cosmic flatness of the
field.~}
\par\null
This avoids the 'flatness problem' of \selectlanguage{greek}Λ\selectlanguage{english}CDM presented by its requirement
that:
``the initial value of the Hubble constant must be fine tuned to
extraordinary accuracy to produce a universe as flat (i.e., near
critical mass density) as `the one we see today (flatness problem)'
``~\cite{Guth_1981}
\par\null
In \selectlanguage{greek}Λ\selectlanguage{english}CDM, the geometry of the universe depends on the density parameter:
ratio of the density of its contents relative to a critical density that
is directly proportional to the square of the Hubble parameter. Now,
according to \selectlanguage{greek}Λ\selectlanguage{english}CDM if the initial value of the cosmic density parameter
was not within the range of 1.0\selectlanguage{ngerman}±1.0E-15 at the Big Bang, then there
would have been an exponentially fast departure from flatness. Since
\selectlanguage{greek}Λ\selectlanguage{english}CDM makes no prediction on the cosmic density at the beginning of the
universe and that it predicts widely varying rates of spatial expansion,
then, to itself, the initial closeness of the cosmic and critical
densities is a mere coincidence.
Here, the geometry of the universe is that of the gravitational field.
It is almost flat at large-scales, and curvilinear in the galactic
haloes.
\par\null
It is quite remarkable that the maximum, and approximately average
cosmic, value here predicted of the energy density of the gravitational
field, H\textsuperscript{2}/2\selectlanguage{greek}κ\selectlanguage{english}~\(\)[?] 1.52E-27 Kg
m\textsuperscript{-3~}for H = 2.26E-18 s\textsuperscript{-1}, is of the
same order of magnitude as \selectlanguage{greek}Λ\selectlanguage{english}CDM's current `observed' value of the
density of `dark energy' which approximately is 6.9E-27 Kg
m\textsuperscript{-3}. This is significant because `dark energy' is, in
\selectlanguage{greek}Λ\selectlanguage{english}CDM, the sole occupant in the cosmic voids. However, by the account
given here, the first occupant of the vacuum is the energetic
gravitational field that produces the vacuum as its form - space, as
opposed to resulting from it.~ This shared order of magnitude of energy
density of the field and `dark energy' supports the identification of
the latter as being the gravitational field.
This approximate value, as determined here, of the average cosmic energy
density of the gravitational field in the vacuum also avoids the
`cosmological constant problem' {[}19{]} confronting explanations of
dark energy based on particle physics. In the~ \selectlanguage{greek}Λ\selectlanguage{english}CDM description of dark
energy it is represented by a cosmological constant with a uniformly
constant energy density. This energy is proposed by particle physicists
to be due to energetic quantum fluctuations of the vacuum. The problem
is that the predicted minimum energy density of physical quantum
processes (electroweak) is of an order of magnitude 1.0E122 greater than
\selectlanguage{greek}Λ\selectlanguage{english}CDM estimates of the density of the hypothetical dark energy.
\par\null
It must be noted that the `model' of `dark matter' presented here --
that of the gravitational energy density field -- has a galactocentric
radial energy density profile that displays no `cuspy' gradient in the
vicinity of the galactic centre. In fact, as the centre is approached,
the density declines, as do the velocities. This decline of velocities
in the galactic core, that challenges prominent models of dark matter
{[}4{]}, is generally empirically confirmed.
\par\null
It is also clear that the flattened orbits in galaxies and clusters,
considered here, do not violate the law of conservation of angular
momentum and Einstein's laws of gravitation. The orbits are displaying
the gravitational effects of an unseen energetic field; the
gravitational field.
\par\null
Indeed, it is because all forms of energy are equivalent in inducing
gravity - as shown by equation (68) - that the auto-induced energetic
gravitational field will appear, purely through its gravitational
effects, just as it really is, that is, as an invisible and thereby
dark, `dissipationless' and thereby cold, growing and thereby
clustering, mass-energy field. So, it is understandable that for
Newtonian science - that presumes action-at-a-distance of a massive
gravitating body and does not make provisions for the agency, in such
matters, of a gravitational field -- that these apparently anomalous
orbits cannot be due to anything other than the attraction by some cold
dark matter.
\par\null
Subsequent to the failure to identify baryonic dark matter, the search
has focussed on non-baryonic dark matter (NBDM), but without success so
far \cite{Weinberg_1989}
\par\null
The NBDM is here identified as the gravitational field.
\par\null
Phenomena elsewhere attributed separately to `dark matter' and `dark
energy' are here attributed to the common cause of the qualitatively
unchanging gravitational field. This explanation, thereby, avoids
problematics associated with explanations that are based on
particularities, often exotic, of the natures of the variously
hypothesized species of matter that commonly occur in speculations
concerning both dark energy and dark matter.
\par\null
It should be clear, in view of the account given here, that dark matter
and dark energy are conceptions created to pre-configure explanations of
two sets of dynamical manifestations of one and the same thing -- the
auto-induced expansion of the gravitational field into new spaces that
it creates. That is, the expansion of the gravitational field appears,
through its different effects -- the isotropic recession of galaxies, on
one hand, and unexpected orbits in galaxies and in their clusters and
gravitational lensing in their haloes, on the other - as being two
different entities operating in mutually exclusive regions. So, the
auto-inductive gravitational field provides a unifying explanation of
phenomena attributed to dark energy and dark matter, without the
problematics associated with their models.
\par\null
The application of the gravitationally perturbed Robertson-Walker metric
has been quite productive in this work. It provides a single versatile
metric suitable for descriptions of phenomena both in large- and
small-scale regions. So, it avoids the challenges of integrating the
small-scale curvilinear space-time of galaxies with a spatially flat
space-time of the universe at large scales that confronted Einstein and
Straus.
\par\null
It is also interesting to note the appellation - gravitationally
perturbed Robertson-Walker metric. It seems to assign to curvilinear
gravity -- which is associated with such distinctly metric phenomena as
orbits, gravitational lensing, and, even with the spatial expansion -
the insignificance of merely being a perturbation of the flat cosmic
Robertson-Walker metric that underpins FLRW cosmology, as opposed to
being an essential structural aspect of the cosmos.
\par\null
Across the universe, the field appears as being, unintuitively, a high
and flat -- and, thereby, non-Minkowskian - plateau of energy density
pocked by innumerable narrow curvilinear declines into deep
gravitational wells - associated with widely isolated baryonic fields -
reaching down to zero energy density at event horizons.
The gravitational field manifests two modes of geodesic and metric
influences on bodies and radiation: (a) energetically in the inducement
of a universal and constant, uniform, isotropic recession, and (b)
non-energetically in the inducement of recession-influenced baryonic
kinematics and optics -- rectilinear in voids and curvilinear in the
vicinities of baryons.
The so-called `gravitationally bound' systems of galaxies and solar
systems do not resist the gravitational recession, but participate fully
in the cosmic Hubble flow that manifests itself, here, in strange
flattened orbits. The term - gravitationally bound - has traditionally
been used to infer orbiting bodies or those on course to collide with
the gravitating bodies. Such bodies are said to be able to escape their
fates if supplied with sufficient energy. However, from the point of
view here taken, this energy is that required to take the bound body
from the deep gravitational wells of relatively low energy density to
the remote regions of higher gravitational energy intensity which are
also flat and thereby devoid of orbits. So, they appear unbound.
However, these bodies still participate in the recession due to gravity.
So, in fact, all bodies are gravitationally bound, hence their universal
mutual recession. There is no space outside of gravity.
\par\null
If the explanation here given proves to be useful, then such an outcome
would be consistent with the conjecture that the operational absence of
the gravitational field in \selectlanguage{greek}Λ\selectlanguage{english}CDM cosmology has led to its failure to
identify the agents of dark energy and dark matter.
\par\null
Possibly due to the non-locality and non-covariance of its
pseudo-tensor, gravitational energy is not given a place in \selectlanguage{greek}Λ\selectlanguage{english}CDM's
cosmic density parameters. Here, the non-locality of the pseudo-tensor
is consistent with its cosmic application. Yet, inadvertently, \selectlanguage{greek}Λ\selectlanguage{english}CDM
grants to gravitational energy, through the proxies of dark energy and
dark matter, an allocation of about ninety-five percent of the cosmic
mass-energy density.
\par\null
This work is a proof of principles. As such, it considers an ideal
galaxy, one with a dominant central SMBH and infinitesimal baryonic
tracers. Applications to more realistic situations involving different
types of bound systems - galaxies, clusters, and walls - may not only
discern the influence of the cosmic expansion, but also contribute to
the comprehension of the formation of their structures.
\par\null
A useful application of the relationships, such as here developed, is in
the mapping of the gravitational field of regions of the observable
universe. This would include data from on-going and future sky surveys
of the distribution of matter fields; regions of curvilinear gravity
(dark matter haloes) as determined by orbital tracers and gravitational
lensing; and cosmic voids and walls. Simulations of past and future
baryonic trajectories within the developing gravitational field, both at
large- and small-scales, will enable a significantly greater
comprehension of the universe and its development.
\par\null
\section*{9.0~ ~CONCLUSIONS}
{\label{464339}}
So, here stands a derivation of the Hubble-Lema\selectlanguage{ngerman}ître law as a law of
gravity and a unitary identification of the dark energy and dark matter
of the cosmos as being the singular energetic gravitational field.
\par\null
It is the application of the Hubble-Lemaître law that leads to an
explanation of both sets of phenomena separately attributed to dark
matter and to dark energy.
\par\null
New and/or significant aspects in this explanation include:~
1.~ A single universal metric -- the radially symmetric gravitationally
perturbed Robertson-Walker metric - that:
\begin{itemize}
\tightlist
\item
is applicable at all scales across the universe, smoothly yielding
descriptions of both small- and large-scale phenomena.~
\item
at large scales, recovers an FLRW line element, the one without
curvature, that large-scale observations confirm. Here, in deep space,
the metric appears as that of an empty de~Sitter space-time.
\end{itemize}
2. The determination, in the space-time defined by this metric, of an
equation of state (EoS) of the gravitational field as being: w=-1.
\par\null
3. The identification of the mechanism of the auto-induction of gravity
in Einstein's theory of general relativity.
\par\null
4. The recognition that space is the form of the transparent
gravitational field embedding baryons and traversed by the radiation
emitted and absorbed by the latter.
\par\null
5. The recognition that the expansion of gravity and so of its form -
space - is due to the auto-induction of gravity and the nature of its
EoS, under the constraints of conservation of gravitational energy
density. The expansion is non-thermal and locally isobaric, with the
continuous creation of new gravitational fields in the form of spaces
occurring everywhere.
\par\null
6. A unitary explanation of the cosmic phenomena separately attributed
to dark energy and dark matter as being due to the actions of the
auto-induced expanding gravitational field on embedded baryonic matter
and radiation.
\par\null
7. The recognition that `dark matter haloes' essentially are regions of
curvilinear gravity (of its field components and energetic fluxes) and
curvilinear space-time - in the vicinities of baryonic matter fields -
with the curvilinear geodesics of orbits and lensing.
\par\null
8. The demonstration that:
\begin{itemize}
\tightlist
\item
the Hubble parameter, previously only empirically determined at large
scales, is a cosmic constant. This is confirmed through the derivation
of the Hubble-Lemaître law from the general conditions of
gravitational expansion outlined in paragraph 5 above.
\item
the Hubble constant references a universal maximum magnitude, both of
the gravitational pressure and of the gravitational energy density,
expressed as:~\(\)\emph{H\textsuperscript{2}}/2\selectlanguage{greek}κ \selectlanguage{english}[?]
\selectlanguage{english}1.52E-27 Kg m\textsuperscript{-3~}for \emph{H~}=2.26E-18
s\textsuperscript{-1}.
\end{itemize}
9. The scale factor - independently of the particular qualities of
material bodies and radiation, as well as of the curvilinearity of the
gravitational field, and so of time and location, thereby being cosmic -
is given by\(a(t)=e^{cHt}\).
\par\null
10. The recognition that the vast empty regions of flat space-time are
not those of vanishing fields of gravity, but of the field's maximal
gravitational intensity - pressure and energy density -- and are,
thereby, non-Minkowskian.~
\par\null
11. The recognition of the historical nature, and the development here,
of certain laws of nature including:
\begin{itemize}
\tightlist
\item
gravitational pressure and gravitational energy density.
\item
gravitational acceleration and the velocities of circular orbits.
\item
gravitational induction.~
\end{itemize}
12. The recognition of the prime position of gravity in the equation of
cosmic mass-energy density parameters.~\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Pressure/Pressure}
\caption{{This is a caption
{\label{515114}}%
}}
\end{center}
\end{figure}
\selectlanguage{english}
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