Re-examination of globally flat space-time

Michael R. Feldman1,∗

1 Michael R. Feldman Private researcher, New York, NY, United States of
America
$\ast$ E-mail<EMAIL_ADDRESS>
## Abstract

In the following, we offer a novel approach to modeling the observed effects
currently attributed to the theoretical concepts of ‘dark energy’, ‘dark
matter’, and ‘dark flow’. Instead of assuming the existence of these
theoretical concepts, we take an alternative route and choose to redefine what
we consider to be inertial motion as well as what constitutes an inertial
frame of reference in flat space-time. We adopt none of the features of our
current cosmological models except for the requirement that special and
general relativity be local approximations within our revised definition of
inertial systems. Implicit in our ideas is the assumption that at “large
enough” scales one can treat objects within these inertial systems as point-
particles having an insignificant effect on the curvature of space-time. We
then proceed under the assumption that time and space are fundamentally
intertwined such that time- and spatial-translational invariance are not
inherent symmetries of flat space-time (i.e. observable clock rates depend
upon both relative velocity and spatial position within these inertial
systems) and take the geodesics of this theory in the radial Rindler chart as
the proper characterization of inertial motion. With this commitment, we are
able to model solely with inertial motion the observed effects expected to be
the result of ‘dark energy’, ‘dark matter’, and ‘dark flow’. In addition, we
examine the potential observable implications of our theory in a gravitational
system located within a confined region of an inertial reference frame,
subsequently interpreting the Pioneer anomaly as support for our redefinition
of inertial motion. As well, we extend our analysis into quantum mechanics by
quantizing for a real scalar field and find a possible explanation for the
asymmetry between matter and antimatter within the framework of these
redefined inertial systems.

## Introduction

The purpose of this paper is to present the foundational groundwork for a new
metric theory of flat space-time which takes into account the observed effects
currently expected to be the result of ‘dark energy’[1], ‘dark matter’[2], and
‘dark flow’ [3] without resorting to these theoretical concepts that we have
yet to observe in the laboratory. We emphasize above the fact that we are
working in flat space-time as this paper is not concerned with reformulating
gravity. Meaning, we assume gravity is the consequence of local curvature in
space-time resulting from the energy-momentum content associated with an
object as formulated by Einstein in his theory of general relativity. However,
for our discussion, we operate under the assumption that at “large enough”
scales we may treat massive objects in our proposed inertial reference frames
as point-particles having an insignificant effect on the curvature of space-
time for the purpose of examining the motion of said objects within the
context of these larger scales. Consequently, we assume that space-time is
essentially flat at these scales, and therefore, the energy density throughout
our inertial systems is taken to be approximately zero. Thus, we assume that
the deviation away from flat space-time inertial paths due to curvature in
space-time is insignificant in our analysis. Furthermore, it is assumed that
the observed motion of these large-scale objects about central points (e.g.
stars orbiting the center of a galaxy, galaxies orbiting the center of a
group/cluster, groups/clusters orbiting the center of a supercluster, etc.) is
not due to the presence of gravitational sources at these centers but is
instead a manifestation of the way in which objects move when no net external
forces are acting upon them. In other words, the following work is concerned
with reformulating our understanding of inertial motion. Furthermore, we focus
on reformulating the global properties of an inertial reference frame while
disregarding the potential local effects that objects moving within this
global inertial system may have on the curvature of space-time. To begin with
our reformulation, we explicitly state for the reader the assumptions of flat
space-time as given by Einstein’s special relativity [4]:

1\. An object will travel in a straight line at a constant speed when no net
external forces are acting upon this object (inertial motion adopted from
Newton; see section titled “Definitions” in [5]).

2\. An observer undergoing inertial motion has the freedom to describe events
by “carrying rulers” in any three arbitrarily chosen spatial directions
(perpendicular to one another) and calibrating clocks according to Einstein’s
prescription for synchronization (an inertial frame of reference). As well,
inertial reference frames moving with uniform (constant velocity) rectilinear
motion relative to one another are treated equally (i.e. there are no
preferred inertial frames of reference in flat space-time).

3\. The speed of light remains constant in all of these observer dependent
inertial frames.

While operating under these assumptions in addition to those of general
relativity [6], our cosmological models (e.g. $\Lambda$CDM [7]) then require a
‘Big-Bang’ event[8][9][10], ‘inflation’ [11], ‘dark energy’, ‘dark matter’,
and ‘dark flow’ as explanations for observed phenomena on cosmological scales
given our assumed understanding of inertial motion and inertial reference
frames as stated above. In contrast, our claims in this paper are that in
order to reproduce the observed behavior attributed to the theoretical
concepts of ‘dark energy’, ‘dark matter’ and ‘dark flow’, it is not necessary
to assume that these supplements must exist. Instead, it is possible to
reproduce this behavior by simply incorporating it into a revised
understanding of inertial motion and inertial reference frames in empty flat
space-time, thereby no longer assuming the three pillars of theoretical
physics as listed above and no longer requiring the occurrence of a ‘Big-
Bang’, ‘inflation’, and expansion of space. While seemingly rash at first
glance, we claim that in what we term as our “Theory of Inertial Centers”, as
laid out in the following work, one can reproduce with inertial motion in our
redefined inertial reference frames the following observed features:

1\. Accelerated redshifts [12] and the Hubble relation [13].

2\. Plateauing orbital velocity curves at large distances from a central point
about which objects orbit [14].

3\. Consistent velocity “flow” of objects toward a central point [3] [15].

4\. An orientation associated with a particular frame of reference [16] (i.e.
we do not take the cosmological principle to be a valid assumption as can be
seen from experimental evidence such as [17]).

In our theory of flat space-time, inertial motion remains defined to be the
motion of an object when it is subjected to no net external forces. In
addition, an inertial reference frame is defined to be a system within which
objects move along inertial trajectories when no net external forces are
acting upon them. We then make the following assumptions and requirements in
our theory:

1\. Inertial motion is not characterized by an object moving in a straight
line at a constant speed. Instead, inertial motion is characterized by
geodesics about “inertial center points” in the radial Rindler chart as
examined in the following discussion (the radial Rindler chart has been
mentioned in other contexts such as [18] and [19]). Note that implicit in this
assumption is the idea that time and space are fundamentally intertwined such
that time-translational invariance and spatial-translational invariance are
not inherent symmetries of flat space-time. Mathematically, this notion
reduces to incorporating both time and spatial distance into the invariant
interval associated with our metric. Meaning, the physically observable
elapsed time as measured by a clock carried along a given curve, denoted as
“proper time” $\tau$, is not our affine parameter and thus is not invariant.
Therefore, observable clock rates depend upon both spatial position in a
particular inertial frame as well as in which inertial frame the observer is
observing. Our affine parameter $\chi$ in the theory of inertial centers is
then taken as a function of proper time in a particular inertial frame to be

$\chi=\sqrt{\Lambda}\cdot\int r(\tau)d\tau$

where $r=r(\tau)$ represents the physical distance to the inertial center
about which the observer moves at a particular observable clock time $\tau$ in
the inertial system and $\sqrt{\Lambda}$ is taken to be the Hubble
constant[13][20]. In addition, these inertial center points define the centers
of our inertial reference frames.

2\. An observer does not have the freedom to describe an inertial reference
frame in whichever way he/she chooses as in special relativity. We, as
observers, are forced to adopt the orientation of the inertial reference frame
that nature provides for us at the particular scale in which we are describing
phenomena. As well, the inertial motion of an object must be thought of
relative to the inertial center point about which said object orbits
(throughout this paper, we will use the term “orbit” to refer to the inertial
motion of an object about an inertial center point).

3\. The speed of light is not constant throughout these inertial reference
frames.

4\. Locally within a confined region of each of these newly defined inertial
reference frames, our theory reduces to and abides by the axioms of special
relativity and general relativity.

Our analysis is organized in the following manner. First, we explore the
limiting behavior of our equations of motion with the radial Rindler chart in
flat space-time. Out of this, we come upon the ability to model the observed
features as listed above. Second, we determine the limit in which our theory
reduces to special relativity, while also proposing the form of our invariant
interval in terms of both time and distance to an inertial center. We have
stated the form of our affine parameter earlier in this introduction as a
preface to the logic used in this proposition. Third, we examine the potential
observable effects of this theory within our solar system and interpret the
Pioneer anomaly [21] as support for our ideas. Fourth, we extend our analysis
by quantizing our theory for a real massive scalar field. Within the context
of this extension, we find a potential explanation for the asymmetry between
matter and antimatter in our observable universe through the possibility of a
parallel region to each inertial system embodied mathematically by the “other”
radial Rindler wedge. We conclude by proposing future work including
addressing the source of the cosmic microwave background [22] in this theory,
attempting to explain other astrometric anomalies within our solar system
besides Pioneer [23], and extending our quantum mechanical analysis to complex
fields with spin.

## Discussion

### Geodesic paths

Adopting the signature $(-,+,+,+)$ and employing abstract index notation
throughout our analysis (see Chapter 2.4 of [24]), we work in the following
metric:

$-d\chi^{2}=-{\Lambda}r^{2}dt^{2}+dr^{2}+r^{2}\cosh^{2}(\sqrt{\Lambda}t)d{\Omega}^{2}$
(1)

where $d{\Omega}^{2}=d{\theta}^{2}+d{\phi}^{2}\sin^{2}{\theta}$;
$0\leq\theta\leq\pi$, $0\leq\phi<2\pi$, $-\infty<t<\infty$, $0<r^{2}<\infty$,
and $\Lambda$ is a positive constant. In a subsequent section, we’ll deduce
that $\Lambda$ must be the square of the Hubble constant. $d\chi^{2}$ denotes
the invariant interval associated with this metric where $d\chi^{2}\neq
c^{2}d\tau^{2}$ assuming $\tau$ denotes proper time, defined as the physically
observable elapsed time between two events as measured by a clock passing
through both events carried along a particular curve, and $c$ denotes the
constant associated with the speed of light in special relativity. Therefore,
in contrast with special relativity, our proper time interval is not assumed
to be invariant, and the speed of light in flat space-time is not assumed to
be constant. However, in subsequent sections, we shall show how special
relativity can be treated as a local approximation to our theory of inertial
centers. As in special and general relativity, massless particles travel along
null geodesics. Thus, with this radial Rindler chart as the description of our
inertial frame of reference and our redefinition of the invariant interval
associated with the metric, we implicitly assume that time and space are
fundamentally intertwined such that time-translational invariance and spatial-
translational invariance are not inherent symmetries of flat space-time. In
other words, one cannot progress coordinate time $t$ forward (i.e. replace
$t\rightarrow t+t_{0}$ where $t_{0}$ is a constant) without considering the
effect of this action on space and vice versa. As well, this concept requires
that we incorporate into the invariant interval associated with our metric
both distance to inertial centers as well as proper time. Later in our
analysis, we will express $d\chi^{2}$ for this theory of inertial centers in
terms of the proper time interval in a particular inertial frame.

For the affine connection terms, Ricci tensor elements, curvature scalar and
square Riemann tensor, we refer to Appendix A. From these calculations it is
clear that this space-time geometry is indeed flat. Taking the Rindler
transformation equations, $cT=r\sinh(\sqrt{\Lambda}t)$,
$R=r\cosh(\sqrt{\Lambda}t)$, we find our metric equation becomes

$-d\chi^{2}=-c^{2}dT^{2}+dR^{2}+R^{2}d\Omega^{2},\indent\forall
R,c^{2}T^{2}<R^{2}$

where $c=$ speed of light in the local Minkowski reference frame [25]. If one
operates under the assumptions of special relativity, $d\chi^{2}$ would in
fact equal $c^{2}d\tau^{2}$, and then the metric in (1) can be used to model
uniformly radially accelerated motion with respect to Minkowski space-time
confined to either of the Rindler wedges: left wedge for $|T|<-R/c$ and right
wedge for $|T|<R/c$ [26]. For the rest of our analysis, however, we no longer
assume that special relativity is valid throughout globally flat space-time
(again, $d\chi^{2}\neq c^{2}d\tau^{2}$) and instead examine the geodesic
motion of point-particles in this radial Rindler coordinate system with time
and radial distance from our inertial center point corresponding to the
coordinate labels $t$ and $r$, respectively. Additionally, as $d\chi^{2}\neq
c^{2}d\tau^{2}$, we do not assume that the reference frame itself is radially
accelerating. Instead, we are re-examining inertial motion under the
guidelines presented in our introduction keeping in mind that the form of our
invariant interval is different from that of special and general relativity.
And since our affine parameter is different from that of special and general
relativity, the geodesics of our theory will also be different. Consequently,
our employment of the radial Rindler chart in the following analysis is our
way of establishing that this coordinate system is the “natural” one for
describing an inertial system in the theory of inertial centers (i.e.
coordinate time in the radial Rindler chart progresses at the same rate as the
physical clock of a stationary observer in the inertial system). Thus, in the
following work, we abandon the idea that Minkowski coordinates can cover all
of an inertial system in flat space-time. Furthermore, we propose that the
radial Rindler chart should be our “natural” coordinate system for describing
an inertial frame of reference in the theory of inertial centers.

Referring to Appendix B, we find for the equations of motion of a particle
within a particular inertial system ($U^{a}{\nabla}_{a}U^{b}=0$, where our
‘proper velocity’ in component form is $U^{\mu}=dx^{\mu}/d\sigma$):

$\displaystyle
0=\frac{d^{2}t}{d\sigma^{2}}+\frac{2}{r}\frac{dt}{d\sigma}\frac{dr}{d\sigma}+\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
(2) $\displaystyle 0=\frac{d^{2}r}{d\sigma^{2}}+\Lambda
r\bigg{(}\frac{dt}{d\sigma}\bigg{)}^{2}-r\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
(3) $\displaystyle
0=\frac{d^{2}\theta}{d\sigma^{2}}+2\frac{d\theta}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\frac{1}{r}\frac{dr}{d\sigma}\bigg{]}-\sin{\theta}\cos{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}$
(4) $\displaystyle
0=\frac{d^{2}\phi}{d\sigma^{2}}+2\frac{d\phi}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\frac{1}{r}\frac{dr}{d\sigma}+\cot{\theta}\frac{d\theta}{d\sigma}\bigg{]}$
(5)

And our norm for the ‘four-velocity’ is given by

$-k=g_{ab}U^{a}U^{b}=-\Lambda
r^{2}\bigg{(}\frac{dt}{d\sigma}\bigg{)}^{2}+\bigg{(}\frac{dr}{d\sigma}\bigg{)}^{2}+r^{2}\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
(6)

where

$k=\left\\{\begin{array}[]{l l}0&\quad\textrm{massless particle}\\\
1&\quad\textrm{massive particle}\\\ \end{array}\right.$

and $\sigma=\chi$ for massive particles. Notice that our ‘four-velocity’
$U^{a}$ in this theory is dimensionless for spatial components and has units
of [time]/[distance] for our time component since $\chi$ (and therefore
$\sigma$) has units of [distance]. Multiplying each term in our radial
equation of motion by $r$ and plugging in (6),

$0=r\frac{d^{2}r}{d\sigma^{2}}+\bigg{(}\frac{dr}{d\sigma}\bigg{)}^{2}+k$ (7)

But to remain at a constant radial distance away from our inertial center:
$d^{2}r/d\sigma^{2}$, $dr/d\sigma=0$. Therefore, only massless particles can
have circular orbits.

Possible geodesic paths obey the relation $U^{a}U_{a}\leq 0$ from (6), and
solving for $d\theta/dt$ and $d\phi/dt$, we find that

$\frac{1}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)}\bigg{[}\Lambda
r^{2}-\bigg{(}\frac{dr}{dt}\bigg{)}^{2}\bigg{]}\geq\bigg{(}\frac{d\theta}{dt}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{dt}\bigg{)}^{2}$

Examining our $\theta$ equation of motion (4), we see that a particle remains
at a constant value of $\theta$ for non-zero angular velocity in $\phi$ if and
only if $d\theta/d\sigma=0$ and $\theta=0$, $\pi/2$, $\pi$. Consequently, the
angular velocity of a particle traveling in the equatorial plane
($\theta=\pi/2$) of this inertial reference frame is bound by the range:

$-\frac{\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)}\leq\frac{d\phi}{dt}\bigg{|}_{\theta=\pi/2}\leq+\frac{\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)}$
(8)

Then, for a photon traveling in a circular orbit in the equatorial plane, we
find

$\frac{d\phi}{dt}\bigg{|}_{k=0,\theta=\pi/2,{\rm
circular}}=\pm\frac{\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)}$ (9)

Later, we’ll see that a massless particle can have circular orbits only for
$\theta=\pi/2$ (orbits with $\phi=\phi_{0}$ cannot be circular).

For massive particles nearly at rest with respect to the center of this
inertial system (i.e. spatial ‘velocity’ terms are much smaller than our
‘velocity’ term in time so that these spatial terms can be taken as nearly
zero), these four equations of motion (2), (3), (4), and (5) reduce to two:

$0=\frac{d^{2}t}{d\chi^{2}}\indent{\rm and}\indent
0=\frac{d^{2}r}{d\chi^{2}}+\Lambda r\bigg{(}\frac{dt}{d\chi}\bigg{)}^{2}$

And solving for the radial acceleration, we find that

$\frac{d^{2}r}{dt^{2}}=-\Lambda r$ (10)

In this limit, the inertial motion of our point-particle is described by a
spatial acceleration in $r$ pulling inward toward the center of this
particular reference frame scaled by the square of the time-scale constant.
Thus, slowly moving objects at large radial distances experience a large
radial acceleration pulling inward toward the center of the inertial system
about which the objects orbit.

Then, let us examine the case where the motion of particles far from an
inertial center (large $r$) is dominated by angular velocities with
approximately circular radial motion ($r\approx{\rm constant}$). Our equations
of motion reduce to

$\displaystyle
0=\frac{d^{2}t}{d\sigma^{2}}+\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
$\displaystyle 0=\Lambda
r\bigg{(}\frac{dt}{d\sigma}\bigg{)}^{2}-r\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\bigg{(}\frac{d\theta}{d\sigma}\bigg{)}^{2}+\sin^{2}{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}\bigg{]}$
$\displaystyle
0=\frac{d^{2}\theta}{d\sigma^{2}}+2\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{d\theta}{d\sigma}\frac{dt}{d\sigma}-\sin{\theta}\cos{\theta}\bigg{(}\frac{d\phi}{d\sigma}\bigg{)}^{2}$
$\displaystyle
0=\frac{d^{2}\phi}{d\sigma^{2}}+2\frac{d\phi}{d\sigma}\bigg{[}\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\cot{\theta}\frac{d\theta}{d\sigma}\bigg{]}$

Plugging $d^{2}t/d\sigma^{2}$ and $dt/d\sigma$ into our expressions for
$d^{2}\phi/d\sigma^{2}$ and $d^{2}\theta/d\sigma^{2}$:

$\displaystyle
0=\frac{d^{2}\phi}{dt^{2}}+\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{d\phi}{dt}+2\cot{\theta}\frac{d\theta}{dt}\frac{d\phi}{dt}$
$\displaystyle
0=\frac{d^{2}\theta}{dt^{2}}+\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\frac{d\theta}{dt}-\sin{\theta}\cos{\theta}\bigg{(}\frac{d\phi}{dt}\bigg{)}^{2}$

Now, if we assume $d\phi/dt\gg d\theta/dt$ and integrate:

$\frac{d\phi}{dt}=\frac{\phi_{0}}{\cosh(\sqrt{\Lambda}t)}$

where $\phi_{0}\rightarrow\pm\sqrt{\Lambda}$ for light taking a circular orbit
in the equatorial plane and thus $|\phi_{0}|\leq\sqrt{\Lambda}$ for
$\theta=\pi/2$. Plugging in for $d\phi/dt$, we have

$\frac{d^{2}\theta}{dt^{2}}=\sin{\theta}\cos{\theta}\bigg{(}\frac{\phi_{0}}{\cosh(\sqrt{\Lambda}t)}\bigg{)}^{2}$
(11)

where $\phi_{0}$ is a constant. In the large $d\phi/d\sigma$ limit:

$\displaystyle\frac{d^{2}\theta}{dt^{2}}>0\indent\textrm{for}\indent
0<\theta<\frac{\pi}{2}$ (12)
$\displaystyle\frac{d^{2}\theta}{dt^{2}}=0\indent\textrm{for}\indent\theta=\frac{\pi}{2}$
(13)
$\displaystyle\frac{d^{2}\theta}{dt^{2}}<0\indent\textrm{for}\indent\frac{\pi}{2}<\theta<\pi$
(14)

As long as the particle is not located at either of the poles ($\theta\neq 0$,
$\pi$), we see a sinusoidal spatial angular acceleration that decreases with
$t$ and moves the object toward $\theta=\pi/2$. One can then picture spiral
galaxy formation resulting from objects orbiting an inertial center with large
angular velocity in $\phi$.

If we refer back to our expression for $d\phi/dt$, we find for the orbital
velocity ($v=r\cdot d\phi/dt$) of a particle in this limit:

$v=\frac{\phi_{0}}{\cosh(\sqrt{\Lambda}t)}r$ (15)

And for $\sqrt{\Lambda}t\approx 0$, our particle’s speed is linearly
proportional to its radial distance away from the inertial center about which
it orbits. In this limit at large $r$, the relationship between orbital
velocity and radial distance mimics the relationship between orbital velocity
and radial distance found in our observed galaxy rotation curves[14] for
comparably small values of $\sqrt{\Lambda}$ and therefore $\phi_{0}$. However,
the analysis above will apply to the classical (in the sense that we are not
taking into account quantum mechanics) inertial motion of an object in any
particular inertial system (e.g. galaxies, groups, clusters, etc.). Later in
our analysis, we’ll provide an experimental scale for the time-scale constant
$\sqrt{\Lambda}$ by analyzing the inherent redshift that occurs in these
inertial frames (i.e. we’ll take $\sqrt{\Lambda}$ to be the Hubble constant).
Since $|\phi_{0}|\leq\sqrt{\Lambda}$, this value will also give us an upper
limit for the slope of our orbital velocity curves at large $r$. Thus, we
claim that the linear relationship found in (15) models the experimental
relationship found from our observed orbital velocity curves for objects far
from the center of the galaxy within which they orbit. We base this claim off
of the idea that the plateauing nature of our experimental curves would be
interpreted in our model to be the result of the small scale of
$\sqrt{\Lambda}$ relative to galactic distance and orbital velocity scales.

### Conservation laws

Since this metric is just a coordinate transformation away from Minkowski, we
expect to find ten linearly independent Killing vector fields as vector fields
are geometric objects independent of our coordinate parametrization. One could
obtain these using the radial Rindler transformation equations, but we find it
helpful to explicitly derive them. We refer to Appendix C for more detail as
well as a full list of all Killing vector fields given in the radial Rindler
chart. Rewriting here for reference the three we will be using:

$\displaystyle\rho^{\mu}\rightarrow\langle\frac{1}{\sqrt{\Lambda}r}\cosh(\sqrt{\Lambda}t),-\sinh(\sqrt{\Lambda}t),0,0\rangle$
(16)
$\displaystyle\Theta^{\mu}\rightarrow\langle\frac{1}{\sqrt{\Lambda}}\cos{\theta},0,-\sin{\theta}\tanh(\sqrt{\Lambda}t),0\rangle$
(17) $\displaystyle\psi^{\mu}\rightarrow\langle 0,0,0,1\rangle$ (18)

Applying Noether’s theorem ($U^{a}\xi_{a}={\rm constant}$),

$\displaystyle
E=\sqrt{\Lambda}r\cosh(\sqrt{\Lambda}t)\frac{dt}{d\sigma}+\sinh(\sqrt{\Lambda}t)\frac{dr}{d\sigma}$
(19)
$\displaystyle\Omega=\sqrt{\Lambda}r^{2}\cos{\theta}\frac{dt}{d\sigma}+r^{2}\sin{\theta}\sinh(\sqrt{\Lambda}t)\cosh(\sqrt{\Lambda}t)\frac{d\theta}{d\sigma}$
(20) $\displaystyle
L=r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}\frac{d\phi}{d\sigma}$ (21)

Plugging into (6) and solving for $dt/d\sigma$, we find that

$\frac{dt}{d\sigma}=\frac{1}{\sqrt{\Lambda}}\bigg{[}\frac{E}{r}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{r^{2}}\cos{\theta}\pm\sqrt{}|_{dt/d\sigma}\bigg{]}$
(22)

where

$\displaystyle\sqrt{}|_{dt/d\sigma}=\bigg{\\{}(\frac{E}{r}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{r^{2}}\cos{\theta})^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}\bigg{(}\frac{E}{r}\bigg{)}^{2}+k\bigg{(}\frac{\sinh(\sqrt{\Lambda}t)}{r}\bigg{)}^{2}\bigg{)}$
$\displaystyle+\bigg{(}\frac{\Omega}{r^{2}}\bigg{)}^{2}+\bigg{(}\frac{L\tanh(\sqrt{\Lambda}t)}{r^{2}}\bigg{)}^{2}\bigg{]}\bigg{\\}}^{1/2}$
(23)

requiring

$(\frac{E}{r}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{r^{2}}\cos{\theta})^{2}\geq\sin^{2}{\theta}\bigg{[}\bigg{(}\frac{E}{r}\bigg{)}^{2}+k\bigg{(}\frac{\sinh(\sqrt{\Lambda}t)}{r}\bigg{)}^{2}\bigg{]}+\bigg{(}\frac{\Omega}{r^{2}}\bigg{)}^{2}+\bigg{(}\frac{L\tanh(\sqrt{\Lambda}t)}{r^{2}}\bigg{)}^{2}$

Notice, for massive particles moving radially in the equatorial plane ($k=1$,
$\theta=\pi/2$, $\Omega=0$, and $L=0$), this constraint reduces to:

$E^{2}\geq 1$

which is just our analogue of the statement in special relativity that the
energy of an object must be greater than or equal to its rest mass [27] since
in special relativity one would assume this constant $E$ would equal the
energy of the particle divided by its rest mass (i.e. in special relativity,
$E$ would be equal to $\tilde{E}/mc^{2}$ where $\tilde{E}$ is the energy of
the particle). Using (19), (20), (21), and (22), we find

$\displaystyle\frac{dr}{dt}=\frac{\sqrt{\Lambda}r}{\sinh(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{Er}\cos{\theta}\pm\sqrt{}|_{dr/dt}}-\cosh(\sqrt{\Lambda}t)\bigg{]}$
(24)
$\displaystyle\frac{d\theta}{dt}=\frac{\sqrt{\Lambda}}{\sin{\theta}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\frac{Er}{\Omega}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\cos{\theta}\pm\sqrt{}|_{d\theta/dt}}-\cos{\theta}\bigg{]}$
(25)
$\displaystyle\frac{d\phi}{dt}=\frac{\sqrt{\Lambda}L}{\sin^{2}{\theta}\cosh^{2}(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{Er\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\Omega\cos{\theta}\pm\sqrt{}|_{d\phi/dt}}\bigg{]}$
(26)

where

$\displaystyle\sqrt{}|_{dr/dt}=\bigg{\\{}(\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\frac{\Omega}{Er}\cos{\theta})^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}1+k\bigg{(}\frac{\sinh(\sqrt{\Lambda}t)}{E}\bigg{)}^{2}\bigg{)}+\bigg{(}\frac{\Omega}{Er}\bigg{)}^{2}$
$\displaystyle+\bigg{(}\frac{L\tanh(\sqrt{\Lambda}t)}{Er}\bigg{)}^{2}\bigg{]}\bigg{\\}}^{1/2}$
(27)
$\displaystyle\sqrt{}|_{d\theta/dt}=\bigg{\\{}(\frac{Er}{\Omega}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\cos{\theta})^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}\bigg{(}\frac{Er}{\Omega}\bigg{)}^{2}+k\bigg{(}\frac{r\sinh(\sqrt{\Lambda}t)}{\Omega}\bigg{)}^{2}\bigg{)}$
$\displaystyle+1+\bigg{(}\frac{L\tanh(\sqrt{\Lambda}t)}{\Omega}\bigg{)}^{2}\bigg{]}\bigg{\\}}^{1/2}$
(28)
$\displaystyle\sqrt{}|_{d\phi/dt}=\bigg{\\{}(Er\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)+\Omega\cos{\theta})^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}(Er)^{2}+k(r\sinh(\sqrt{\Lambda}t))^{2}\bigg{)}$
$\displaystyle+{\Omega}^{2}+(L\tanh(\sqrt{\Lambda}t))^{2}\bigg{]}\bigg{\\}}^{1/2}$
(29)

For light traveling radially in the equatorial plane, $\Omega$, $L=0$ and (24)
reduces to

$\frac{dr}{dt}\bigg{|}_{k=0,\Omega=0,L=0}=\pm\sqrt{\Lambda}r$ (30)

giving us

$r(t)|_{k=0,\Omega=0,L=0}=r_{0}\textrm{exp}({\pm\sqrt{\Lambda}t})$ (31)

where $r_{0}$ is a constant signifying the radial position of the photon at
$t=0$. One could have arrived at this expression for the general case of light
traveling radially even outside of the equatorial plane simply from (6) for
null geodesics. Let us now pose the question of whether or not it is possible
for light to travel from the $r>0$ region of our inertial system to $r=0$
which we regard as our inertial center point. Integrating $dr/dt$ from $0$ to
$\Delta t$,

$\Delta r=\pm\sqrt{\Lambda}r_{0}\int^{\Delta
t}_{0}{\textrm{exp}(\pm\sqrt{\Lambda}t)dt}=r_{0}[\textrm{exp}(\pm\sqrt{\Lambda}\Delta
t)-1]$

Solving for $\Delta t$,

$\Delta t=\ln{\bigg{(}\frac{\Delta r+r_{0}}{r_{0}}\bigg{)}^{\pm
1/\sqrt{\Lambda}}}$ (32)

For a photon traveling radially inward, the sign of the root is negative, and
it reaches $r=0$ in

$\lim_{r_{{\rm final}}\rightarrow 0}\Delta
t=\ln{\bigg{(}\frac{(0-r_{0})+r_{0}}{r_{0}}\bigg{)}^{-1/\sqrt{\Lambda}}}=\ln{\bigg{(}\frac{r_{0}}{-r_{0}+r_{0}}\bigg{)}^{1/\sqrt{\Lambda}}}\rightarrow\infty$

Consequently, not even light can reach $r=0$ in a finite amount of time. But
what about the inertial behavior of massive particles in these systems? At
first glance, (24) and (25) appear to be divergent for $t=0$. However, to
evaluate all of these velocity expressions for $t=0$, we return to symmetry
equations (19) and (20):

$E=\sqrt{\Lambda}r\frac{dt}{d\sigma}\bigg{|}_{t=0}\indent{\rm
and}\indent\Omega=\sqrt{\Lambda}r^{2}\cos{\theta}\frac{dt}{d\sigma}\bigg{|}_{t=0}$

Therefore,

$\frac{E}{r}\cos{\theta}=\frac{\Omega}{r^{2}}\bigg{|}_{t=0}$ (33)

Plugging into (23),

$\displaystyle\sqrt{}|_{dt/d\sigma}|_{t=0}=\bigg{\\{}\bigg{(}\frac{E}{r}\sin^{2}{\theta}+\frac{E}{r}\cos^{2}{\theta}\bigg{)}^{2}-\bigg{[}\sin^{2}{\theta}\bigg{(}\frac{E}{r}\bigg{)}^{2}+\bigg{(}\frac{E}{r}\cos^{2}{\theta}\bigg{)}^{2}\bigg{]}\bigg{\\}}^{1/2}$
$\displaystyle=\sqrt{\bigg{(}\frac{E}{r}\bigg{)}^{2}-\bigg{(}\frac{E}{r}\bigg{)}^{2}}=0$

which one can plug back into (22) to find consistency with our expressions for
$dt/d\sigma|_{t=0}$ above. Yet we see from (27), (28), and (29) that

$\displaystyle\sqrt{}|_{dr/dt}=\frac{r}{E}\cdot\sqrt{}|_{dt/d\sigma}$ (34)
$\displaystyle\sqrt{}|_{d\theta/dt}=\frac{r^{2}}{\Omega}\cdot\sqrt{}|_{dt/d\sigma}$
(35) $\displaystyle\sqrt{}|_{d\phi/dt}=r^{2}\cdot\sqrt{}|_{dt/d\sigma}$ (36)

which implies for $t=0$ that all of these terms vanish. Then our spatial
velocity terms for $t=0$ become

$\displaystyle\frac{dr}{dt}\bigg{|}_{t=0}=\frac{\sqrt{\Lambda}r}{\sinh(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\sin^{2}{\theta}+\cos^{2}{\theta}}-1\bigg{]}=\frac{0}{0}$
$\displaystyle\frac{d\theta}{dt}\bigg{|}_{t=0}=\frac{\sqrt{\Lambda}}{\sin{\theta}\sinh(\sqrt{\Lambda}t)}\bigg{[}\frac{\cos{\theta}}{\sin^{2}{\theta}+\cos^{2}{\theta}}-\cos{\theta}\bigg{]}=\frac{0}{0}$
$\displaystyle\frac{d\phi}{dt}\bigg{|}_{t=0}=\frac{\sqrt{\Lambda}L}{\sin^{2}{\theta}}\bigg{[}\frac{1}{Er\sin^{2}{\theta}+Er\cos^{2}{\theta}}\bigg{]}=\frac{\sqrt{\Lambda}L}{Er\sin^{2}{\theta}}$

where we have used (33) in these limit expressions. So we see that our
velocity terms are not necessarily divergent for $t=0$. However, we’ll address
the issue of motion for small $\sqrt{\Lambda}t$ later when we relate
Einstein’s special relativity to our theory of inertial centers. One must also
keep in mind that expressions (24), (25), and (26) represent a set of complex
differential equations that we unfortunately will not be able to solve in this
paper. The purpose of the following portion of this section is in fact to
evaluate the large $t$ behavior of all spatial velocities where it is not
explicitly apparent how to evaluate this limit if one were to work in
Minkowski coordinates while keeping in mind the notion that he/she must relate
back to the radial Rindler chart for inertial time as $d\chi^{2}\neq
c^{2}d\tau^{2}$ (we’ll elaborate further on the term “inertial time” in our
next section). It does appear easier to proceed in this manner of working in
Minkowski and relating back to radial Rindler for solely radial motion as we
shall do later in this section.

Yet, we return to our velocity expressions from Noether’s theorem in order to
examine the general expression for $dr/dt$ as $t\rightarrow\infty$. First, we
determine the limiting value of $\sqrt{}|_{dr/dt}$:

$\lim_{t\rightarrow\infty}{\sqrt{}|_{dr/dt}}=\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)\sqrt{1-\frac{k}{E^{2}\sin^{2}{\theta}}}$

Then for massive particles ($k=1$) assuming $\theta\neq 0$ or $\pi$,

$\lim_{t\rightarrow\infty}{\frac{dr}{dt}}=-\sqrt{\Lambda}r$ (37)

which is just the equation for a massless particle traveling radially inward.
When $\theta=0$ or $\pi$, we must return to conservation equations (20) and
(21). We find $L=0$ and

$\frac{dt}{d\sigma}\bigg{|}_{\theta=0,\pi}=\frac{\Omega}{\sqrt{\Lambda}r^{2}\cos{\theta}}$

Plugging in (19) and solving when $t\rightarrow\infty$, we again find (37). We
now understand that eventually all massive particles move toward $r=0$. Yet as
the object approaches the center, its speed decreases as well and will only
stop moving inward when it reaches this inertial center point in an infinite
amount of time. Thus, with this large $t$ behavior, we apparently inherit the
ability to model the observed anomalous effects of ‘dark flow’ [3]. In our
next section, we will provide an interpretation for the physical significance
of our coordinate time in our theory of inertial centers, relating $t$ back to
the rate at which physical clocks are observed to tick.

However, we progress onward and look at the large $t$ limits for both
$d\theta/dt$ and $d\phi/dt$. Beginning with the former, we find

$\lim_{t\rightarrow\infty}{\sqrt{}|_{d\theta/dt}}=\frac{Er}{\Omega}\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)\sqrt{1-\frac{k}{E^{2}\sin^{2}{\theta}}}$

Plugging into $d\theta/dt$ and examining for massive particles,

$\lim_{t\rightarrow\infty}{\frac{d\theta}{dt}}=\frac{-\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)}\cot{\theta}$
(38)

And for $\theta\neq 0$ or $\pi$,

$\lim_{t\rightarrow\infty}{\frac{d\theta}{dt}}\bigg{|}_{\theta\neq 0,\pi}=0$
(39)

Solving for $d\theta/dt$ when $\theta=0$ or $\pi$ using (25) and $L=0$,

$\frac{d\theta}{dt}\bigg{|}_{\theta=0,\pi}=\frac{\sqrt{\Lambda}}{\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)}\tan{\theta}=0$

At the poles, particles have no angular velocity in $\theta$ nor angular
momentum in $\phi$ ($d\theta/d\sigma,L=0$). Lastly for our large $t$ limits,
we have $d\phi/dt$:

$\lim_{t\rightarrow\infty}{\sqrt{}|_{d\phi/dt}}=Er\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)\sqrt{1-\frac{k}{E^{2}\sin^{2}{\theta}}}$

Plugging into $d\phi/dt$,

$\lim_{t\rightarrow\infty}\frac{d\phi}{dt}=\frac{\sqrt{\Lambda}L}{\sin^{2}{\theta}\cosh^{2}(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{Er\sin^{2}{\theta}\cosh(\sqrt{\Lambda}t)(1\pm\sqrt{1-\frac{k}{E^{2}\sin^{2}{\theta}}})}\bigg{]}$

For massive particles and assuming $\theta\neq 0$ or $\pi$,

$\lim_{t\rightarrow\infty}\frac{d\phi}{dt}=0$ (40)

We have a clearer picture of the inertial trajectories of massive particles
over time in the context of our redefined inertial reference frames. As time
progresses, massive objects will eventually move radially inward losing
angular velocity in $\theta$ and angular momentum in $\phi$, slowing down in
radial velocity as they approach the center point about which they orbit.

Looking back at our expression for $dr/dt$, we ask ourselves the question: for
what values of $\theta$ is $dr/dt$ most positive? For positive $dr/dt$, we
have particles moving radially outward, and maximizing this expression with
respect to $\theta$ provides us with the easiest possible path to be ejected
away from our inertial center. Examining particles with large radial ‘proper
velocities’ relative to their own angular ‘proper velocities’ which from (19),
(20), (21) implies $Er\gg\Omega,L$ since $dr/d\sigma\gg
r\cosh(\sqrt{\Lambda}t)\cdot d\theta/d\sigma$ and $dr/d\sigma\gg
r\cosh(\sqrt{\Lambda}t)\cdot d\phi/d\sigma$ in this limit:

$\displaystyle\frac{dr}{dt}\bigg{|}_{Er\gg\Omega,L}=\frac{\sqrt{\Lambda}r}{\sinh(\sqrt{\Lambda}t)\cosh(\sqrt{\Lambda}t)}\bigg{[}\frac{1}{\sin^{2}{\theta}\pm\sqrt{\sin^{4}{\theta}-\frac{\sin^{2}{\theta}}{\cosh^{2}(\sqrt{\Lambda}t)}(1+k\frac{\sinh^{2}(\sqrt{\Lambda}t)}{E^{2}})}}-\cosh^{2}(\sqrt{\Lambda}t)\bigg{]}$

But the largest positive value of $dr/dt|_{Er\gg\Omega,L}$ occurs if we
minimize the denominator of the first term in brackets with respect to
$\theta$. Clearly, this term needs to be re-evaluated when $\theta=0$ or
$\pi$. Returning to conservation equations (19) and (20), we solve for the
radial motion of a particle through the poles by plugging into (6) ($\theta=0$
or $\pi$ and $d\theta/d\sigma=0$):

$\displaystyle\frac{dr}{dt}\bigg{|}_{\theta=0,\pi}=\frac{\sqrt{\Lambda}r}{1+\frac{k}{E^{2}}\sinh^{2}(\sqrt{\Lambda}t)}\cdot\bigg{\\{}-\frac{k}{E^{2}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\pm\bigg{[}\bigg{(}\frac{k}{E^{2}}\bigg{)}^{2}\cosh^{2}(\sqrt{\Lambda}t)\sinh^{2}(\sqrt{\Lambda}t)$
$\displaystyle+\bigg{(}1+\frac{k}{E^{2}}\sinh^{2}(\sqrt{\Lambda}t)\bigg{)}\bigg{(}1-\frac{k}{E^{2}}\cosh^{2}(\sqrt{\Lambda}t)\bigg{)}\bigg{]}^{1/2}\bigg{\\}}$

For large proper radial motion, we assume $E^{2}\gg
k\cosh^{2}(\sqrt{\Lambda}t)$ (as $E^{2}\geq 1$ is our analogue of the rest
mass condition from Einstein’s special relativity). Then, our expression for
radial motion through the poles reduces to

$\frac{dr}{dt}\bigg{|}_{\theta=0,\pi}\approx\pm\sqrt{\Lambda}r$ (41)

We see that massive particles can travel at speeds near that of photons
through the poles, and therefore it appears that the easiest way for particles
to be ejected radially outward away from an inertial center would be through
the poles of the inertial system. If we imagine a supernova occurring near the
center point of an inertial system, we find that a simple potential scenario
for the occurrence of relativistic jets [28] in this reference frame would be
the expulsion of stellar remnants through the poles. Consequently, if we use
this logic to provide an alternative for relativistic jet production, we must
then require that each of our inertial frames have a particular orientation
governed by the location of these poles and embodied mathematically by the
spatial positions for which particular metric components vanish. In other
words, when describing a particular inertial frame, these are the $\theta$
values for which $\sin{\theta}=0$ previously referred to as “coordinate
singularities” (e.g. see Chapter 5.1 of [29]) but taken here as a physical
attribute of the inertial system reflecting the idea that the radial Rindler
chart is the “natural” coordinate system for an inertial reference frame in
flat space-time. Thus we must ask ourselves the following question. How is
this orientation established in the theory of inertial centers? As we shall
mention later in our paper, this is an open question which we will have to
address in future work.

Back to our circular orbit analysis, we solve for the radius at which light
can have circular paths in a particular inertial system for possibly both
$d\theta/dt=0$, $d\phi/dt=\pm\sqrt{\Lambda}/\cosh(\sqrt{\Lambda}t)$ and
$d\theta/dt=\pm\sqrt{\Lambda}/\cosh(\sqrt{\Lambda}t)$, $d\phi/dt=0$. For the
two, we obtain from (19)

$\frac{dt}{d\sigma}\bigg{|}_{k=0,{\rm
circular}}=\frac{E}{\sqrt{\Lambda}r\cosh(\sqrt{\Lambda}t)}$ (42)

In the former situation ($\theta=\pi/2$, $\Omega=0$), we substitute (42) and
(21) into (6) and arrive at

$r_{\theta=\pi/2,{\rm circular}}=\frac{L}{E}$ (43)

Whereas for the latter case ($\phi=\phi_{0}$, $L=0$), we plug (42) and (20)
into (6) to find

$r=\frac{\Omega\cosh(\sqrt{\Lambda}t)}{E[\cos^{2}{\theta}\pm\sin{\theta}\sinh(\sqrt{\Lambda}t)]}$

which is not constant. Consequently, in our inertial systems, light can travel
in circular orbits only in the equatorial plane with angular velocity given by
(9) at a radius given by (43). The type of lensing expected from a black hole
or ‘dark matter’[30] is evidently reproduced in a similar manner by light
traveling with angular velocity about an inertial center point. Although in
the analysis above, we studied circular orbits where light remains at a
constant $r$, the logic applies similarly for the case where the photon has
both radial and angular velocity components.

We come to the redshift factor for light traveling radially. Before we begin
with this analysis, we must refer back to our procedure for determining the
observed wavelength of a photon when operating under the assumptions of
special and general relativity. In general relativity, the observed frequency
$f$ of a photon with momentum $p^{a}$ ($p^{a}=\hbar k^{a}$) emitted/received
by an observer traveling with proper velocity in component form given by
$u^{\mu}=dx^{\mu}/d\tau$ is (see Chapter 6.3 of [24])

$-2\pi f=k_{a}u^{a}\bigg{|}_{P}$

where $P$ is the location in space-time at which the event in question occurs
(i.e. emission/absorption). Dividing through by the Minkowski constant for the
speed of light $c$, we have

$-\frac{2\pi}{\lambda}=\sum_{\mu}k_{\mu}\cdot\frac{1}{c}\frac{dx^{\mu}}{d\tau}$

where $c=\lambda f$ and $\lambda$ is the wavelength of the photon
emitted/received by our observer. Since we require that our theory in flat
space-time reduce to special relativity within a localized region of our
respective inertial system (i.e. $d\chi^{2}\rightarrow c^{2}d\tau^{2}$ in this
localized region), it appears necessary to assume that, in our theory of
inertial centers, the wavelength of a photon with wave-vector $k^{a}$
emitted/received by an observer with ‘proper velocity’ $U^{a}$ is given by

$-\frac{2\pi}{\lambda}=k_{a}U^{a}\bigg{|}_{P}$

where we emphasize to the reader that in our theory the component form of the
‘four-velocity’ for our observer is affinely parametrized by $\chi$ (i.e.
$U^{\mu}=dx^{\mu}/d\chi$), in direct contrast to special and general
relativity for which the four-velocity of an observer would be affinely
parametrized by proper time $\tau$. Proceeding with our radial treatment, the
wave-vector for this photon is of the form,

$k^{\mu}\rightarrow\langle k^{t},k^{r},0,0\rangle$

And the wavelength observed by a radially traveling individual is given by

$-\frac{2\pi}{\lambda}=k^{a}U_{a}=-\Lambda
r^{2}k^{t}\frac{dt}{d\chi}+k^{r}\frac{dr}{d\chi}$

Using the Killing vector field in (16), we obtain the conservation law
($-\rho_{0}=k^{a}\rho_{a}$):

$\rho_{0}=\sqrt{\Lambda}r\cosh(\sqrt{\Lambda}t)k^{t}+\sinh(\sqrt{\Lambda}t)k^{r}$

But for a photon, $0=k^{a}k_{a}\Longrightarrow k^{r}=\pm\sqrt{\Lambda}rk^{t}$.
So,

$k^{t}=\frac{\rho_{0}}{\sqrt{\Lambda}r[\cosh(\sqrt{\Lambda}t)\pm\sinh(\sqrt{\Lambda}t)]}$

where the positive root corresponds to light traveling away from $r=0$ and
negative to light traveling inward. Solving for the motion of the observer in
this particular inertial reference frame,

$-1=-\Lambda
r^{2}\bigg{(}\frac{dt}{d\chi}\bigg{)}^{2}+\bigg{(}\frac{dr}{d\chi}\bigg{)}^{2}$

And for an observer nearly at rest with respect to the inertial center about
which he/she orbits,

$\frac{dt}{d\chi}=\pm\frac{1}{\sqrt{\Lambda}r}$

Taking time to move forward, we find that

$\frac{2\pi}{\lambda}=\frac{\rho_{0}}{\cosh(\sqrt{\Lambda}t)\pm\sinh(\sqrt{\Lambda}t)}$

But from our earlier analysis, we found that a radially traveling photon
abides by the equation,
$r(t)=r_{0}\textrm{exp}(\pm\sqrt{\Lambda}t)=r_{0}[\cosh(\sqrt{\Lambda}t)\pm\sinh(\sqrt{\Lambda}t)]$.
Thus,

$\lambda\propto r$ (44)

Then for a light signal sent between two observers at rest in this inertial
frame, the redshift factor $z$ is given by the expression:

$z=\frac{\lambda_{\rm absorber}-\lambda_{\rm emitter}}{\lambda_{\rm
emitter}}=\frac{r(t_{{\rm absorber}})}{r(t_{{\rm emitter}})}-1$ (45)

Consequently, we see large shifts from emitters much closer to the center of
the system (assuming the absorber position remains the same).

Suppose, within the framework of this theory, we examine light propagating at
the scale of the inertial reference frame associated with our observable
universe. Then analogous to the manner in which the expression for the Hubble
parameter [13] is derived in the Friedmann-Lemaître-Robertson-Walker (FLRW)
metric[31][32][33], we set the Hubble constant $H_{0}$ equal to

$H_{0}=\bigg{|}\frac{\dot{r}}{r}\bigg{|}=\sqrt{\Lambda}$ (46)

where $\dot{r}=dr/dt$. We notice that

$\frac{\ddot{r}}{r}=\Lambda>0$ (47)

producing a positive value for the acceleration of cosmological redshift and
therefore replicating the observed effects assumed to be the result of ‘dark
energy’ [12]. Thus, for any particular inertial reference frame, we should see
a shift in wavelength similar to the Hubble constant for the radial motion of
photons. We’ll use this conclusion later when we take the Pioneer anomaly as
support for our theory in the context of the inertial system associated with
the Milky Way. However, we should be concerned with our expression for
$\dot{r}/r$,

$\frac{\dot{r}}{r}=\pm\sqrt{\Lambda}$

as this theory then requires that we also have blueshifted objects if the
absorber is in fact closer than the emitter to the center of the inertial
frame within which the light signal in question propagates (i.e. negative
values for $\dot{r}/r$ and $z$). Nevertheless, if we apply our analysis to
objects at the scale of the Local Group [34][35] as in Table 1, we would
require an alternative interpretation for the observed significant blueshifts.
Whereas in current models, this blueshift would be interpreted as the Doppler
effect and thus for example as Andromeda (Messier 31) moving with velocity
toward the Milky Way [36], in our theory of inertial centers one could
interpret a portion of this blueshift (we say portion as the motion of our
observers within an inertial system also affects wavelength) as the
possibility that Andromeda is farther away from the inertial center associated
with the Local Group than we are. In support of these observations, we refer
to Table 2 where there appears to be an orientation associated with our
redshift values. For similar values of right ascension ($\pm 2$ h), we see a
steady change in wavelength shift from blue ($-$) to red ($+$) as one proceeds
from large positive values of declination to large negative values of
declination. In our theory, we would still need to consider differences in
radial distance associated with these objects and not just spatial
orientation. However, given that our distance modulus values are very much
similar for most of these entries ($\approx 24$ mag), it seems that this
interpretation for an orientation to the Local Group should be taken into
consideration. On the other hand, even if there does appear to be an
orientation associated with the Local Group, we must question why we have not
seen significant blueshifts at much larger scales. We will come back to these
ideas later in our work.

Until now, we have assumed that our coordinate time can take values between
$-\infty<t<\infty$ without explicitly examining the motion of particles in the
$t<0$ region. Reducing our analysis to solely radial motion away from the
poles, we analyze geodesic paths in Minkowski coordinates ($T,R$) first for
simplicity. However, we must be very clear that under our assumptions $T$ does
not represent inertial time as previously stated and in our theory of inertial
centers corresponds to an “unnatural” time coordinate for flat space-time
combining both physically observable clock time and spatial distance as
$cT=r\sinh(\sqrt{\Lambda}t)$. Then our equations of motion reduce to

$0=\frac{d^{2}T}{d{\sigma}^{2}}\indent\indent{\rm and}\indent\indent
0=\frac{d^{2}R}{d{\sigma}^{2}}$

leading to the straight lines that we expect in Minkowski coordinates:

$R=v\cdot(cT)+R_{0}$ (48)

where $v$ is a constant bounded by $|v|\leq 1$. We leave the physical
interpretation of the Minkowski constant $c$ in this theory of inertial
centers for the next section. However, using our transformation equations, we
find in radial Rindler coordinates

$r(t)=\frac{R_{0}}{\cosh(\sqrt{\Lambda}t)-v\sinh(\sqrt{\Lambda}t)}$ (49)

One immediately notices that for massive particles ($|v|<1$), both limiting
cases of $t\rightarrow\pm\infty$ result in the particle heading inward toward
the $r=0$ center point of the inertial system. This produces a scenario for
inertial motion of massive objects beginning at a center point in the far
past, coming to a maximum radial distance away at a later time, and then
heading back inward to eventually return to the same center point. In other
words, classically, all particles must also originate from the $r=0$ center
point of the particular inertial frame in question (see Figure 1).

### Reduction to special relativity

Taking the differential of both Rindler transformation equations:

$cdT=dr\sinh(\sqrt{\Lambda}t)+\sqrt{\Lambda}r\cosh(\sqrt{\Lambda}t)dt$
$dR=dr\cosh(\sqrt{\Lambda}t)+\sqrt{\Lambda}r\sinh(\sqrt{\Lambda}t)dt$

where

$\sinh(\sqrt{\Lambda}t)=\sqrt{\Lambda}t+\frac{(\sqrt{\Lambda}t)^{3}}{3!}+\frac{(\sqrt{\Lambda}t)^{5}}{5!}+\ldots$
$\cosh(\sqrt{\Lambda}t)=1+\frac{(\sqrt{\Lambda}t)^{2}}{2!}+\frac{(\sqrt{\Lambda}t)^{4}}{4!}+\ldots$

Plugging in these expressions above, we find that

$cdT=dr\bigg{(}\sqrt{\Lambda}t+\frac{(\sqrt{\Lambda}t)^{3}}{3!}+\ldots\bigg{)}+\sqrt{\Lambda}rdt\bigg{(}1+\frac{(\sqrt{\Lambda}t)^{2}}{2!}+\ldots\bigg{)}$
$dR=dr\bigg{(}1+\frac{(\sqrt{\Lambda}t)^{2}}{2!}+\ldots\bigg{)}+\sqrt{\Lambda}rdt\bigg{(}\sqrt{\Lambda}t+\frac{(\sqrt{\Lambda}t)^{3}}{3!}+\ldots\bigg{)}$

If we localize our view of space-time such that all differential terms

$\mathcal{O}(\Lambda)=0$ (50)

we will then have

$cdT\approx\sqrt{\Lambda}\bigg{(}tdr+rdt\bigg{)}$ $dR\approx dr$

Further, we require for this local patch of space-time that

$tdr\ll rdt$ (51)

and our transformation equations reduce to

$\displaystyle cT\approx\sqrt{\Lambda}rt$ (52) $\displaystyle R\approx r$ (53)

with differential expressions

$\displaystyle cdT\approx\sqrt{\Lambda}rdt$ (54) $\displaystyle dR\approx dr$
(55)

For the observer remaining a radial distance $r=r_{0}$ away from the center of
his/her reference frame, the radial Rindler chart will be accurately
approximated by Minkowski coordinates under conditions (50) and (51) as $R=r$
and $T\propto t$ in this small $\sqrt{\Lambda}t$ limit. If one takes
$\sqrt{\Lambda}$ to be a fundamental property of each inertial system in
question, it must be that the measured Minkowski value for the speed of light
constant $c$ is a byproduct of the reference frame we wish to locally
approximate. In other words, in the Minkowski approximation for the radial
Rindler chart, an observer, located a radial distance $r=r_{0}$ away from the
inertial center point about which he/she orbits at $t=0$, will find:

$c=\sqrt{\Lambda}r_{0}$ (56)

If we treat $t=0$ as the point at which we determine the initial conditions
for the particle that we are observing (i.e. boundary conditions for position
and velocity), then our object will appear to move along straight line
geodesics for small values of $\sqrt{\Lambda}t$, but as we continue to observe
for longer periods of time, the properties of the radial Rindler chart which
we are approximating become more and more relevant.

In order for us to relate our theory of inertial centers to special
relativity, we must require that coordinate time in the radial Rindler chart
progress at the same rate as the proper time of an observer stationary
relative to the center of the inertial system within which we are analyzing
events (i.e. inertial time). In other words, $dt/d\tau=1$ for stationary
observers located at any particular radial distance $r=r_{0}$ away from an
inertial center. However, keep in mind that stationary observers do not follow
along geodesic paths from equation (7). Then, for observers which we can
consider as nearly stationary relative to the center of a particular inertial
system (i.e. $r={\rm constant}$), we have $d\chi^{2}\rightarrow
c^{2}d\tau^{2}$ where $c$ is given by (56), effectively ensuring that our
coordinate time $t$ progresses at the same rate as the proper time $\tau$ of a
stationary observer. Consequently, we find under (50) and (51) in addition to
our stationary observer assumption that our line element can be treated
approximately as

$-c^{2}d\tau^{2}=-c^{2}dT^{2}+dR^{2}+R^{2}d\Omega^{2}$

And therefore in this “stationary” limit (relative to the inertial center),
when not operating about the poles, we come upon time- and spatial-
translational invariance within our local region where the origin of our
coordinate system is located at the inertial center of this reference frame.
Because it appears that we have now recovered time- and spatial-translational
invariance in this limit, we can naively assume that we have the ability to
translate our coordinate system in any way we prefer (e.g. moving the center
of our reference frame away from the inertial center). In other words, we can
approximate when our motion is not near the poles of our global inertial
system with the metric:

$-c^{2}d\tau^{2}=-c^{2}dT^{2}+dX^{2}+dY^{2}+dZ^{2}$

where $X=R\sin{\theta}\cos{\phi}+X_{0}$, $Y=R\sin{\theta}\sin{\phi}+Y_{0}$,
and $Z=R\cos{\theta}+Z_{0}$ for $R>0$ and $X_{0}$, $Y_{0}$, and $Z_{0}$ are
constants (see Figure 2). Thus, this local stationary approximation reduces
our theory to special relativity (see Chapter 4.2 of [24]). Additionally, we
see from these transformation equations that the Minkowski chart is not able
to cover all of space-time in our theory of inertial centers (i.e. $r<0$
values are neglected by the Minkowski chart). We will come back to this idea
later in our work. Physically, our localization conditions require that the
time-scale constant $\sqrt{\Lambda}$ for our inertial systems be small enough
such that we as observers here on Earth would observe only the stationary
limit in our “everyday lives”. Of course, this statement also assumes that we
are nearly stationary relative to the inertial center about which we orbit
taken in our next section to be the center of the Milky Way. Yet, given our
redshift analysis, it appears that the Hubble constant provides the necessary
scale [20] for this requirement from (46).

Furthermore, it is clear that in order to express $d\chi^{2}$ in terms of the
proper time of the observer whose motion we wish to analyze in the relevant
inertial frame (i.e. the particular system in which the observer can be
treated as a point-particle orbiting an inertial center point) and still have
our invariant interval reduce to $c^{2}d\tau^{2}$ in our stationary limit
where the observer’s distance to the inertial center point about which he/she
orbits is very nearly constant, we must have

$d\chi^{2}=\Lambda r^{2}d\tau^{2}$ (57)

where $r$ represents the physical distance to the center point of the inertial
system in question. In addition, according to our theory of inertial centers,
the value that we use in special and general relativity for the constant $c$
in our massive geodesic equations relies on the particular inertial reference
frame in which we can regard the object whose local behavior we wish to
examine as a point-particle orbiting an inertial center point (in special and
general relativity, $g_{ab}u^{a}u^{b}=-c^{2}$ where $u^{\nu}=dx^{\nu}/d\tau$).
Therefore, the local Minkowski constant that we measure for the speed of light
is dependent upon our position in our most local inertial reference frame
(i.e. the frame in which we can be treated as a point-particle orbiting an
inertial center). As well, for two different stationary observers orbiting
about the same inertial center point, the clock of the observer located closer
to their shared inertial center will appear to run faster when examined from
the perspective of the more distant observer. Meaning, not only do observable
clock rates differ due to the relative velocity of individuals as in special
relativity, but they also differ due to the difference in distance of each
individual from the inertial center about which each orbits. Thus, initially
synchronized clocks that are stationary relative to the shared inertial center
about which both orbit do not remain synchronized if they are located at
different distances from this inertial center.

### Application to a local gravitational system

Before we present the approximations of this section, it seems necessary to
provide remarks as to how gravitation fits into the theory of inertial
centers. The formulation of our theory of inertial centers detailed in
previous sections deals with the structure of flat space-time ignoring
possible issues with curvature. So what we are really asking is the following.
How does an object move in flat space-time when absolutely no external forces,
fields, etc. are present to affect said object? Nevertheless, we still assume
within our model that all objects cause curvature in space-time due to their
intrinsic energy-momentum content, but this curvature we take to be a local
effect within the far larger inertial system that we are attempting to
redefine in this work. However, as presented in the previous section
“Reduction to special relativity”, we claim that locally the structure of flat
space-time within our redefined inertial reference frames reduces to the flat
space-time of Einstein’s theory of special relativity as long as the observer
remains at very nearly the same distance away from the inertial center about
which he/she orbits. As discussed above, we term this the “local stationary
approximation”, and in this approximation the observer sees space-time locally
within a region located at the same radial distance as the observer away from
the inertial center about which he/she orbits as approximately special
relativistic, where the speed of light in this confined region of space-time
is given by (56) and our affine parameter reduces to $\chi=c\cdot\tau$. If one
then considers the influence of an object on the structure of space-time in
this local region where special relativity approximately holds, we assume in
the theory of inertial centers that this object will bend space-time locally
according to Einstein’s general relativity. Meaning, gravity remains a
consequence of local curvature in space-time in the theory of inertial
centers. However, when we take a perspective far away from our massive object
so that the curvature this object induces in space-time looks approximately
insignificant for the purpose of examining motion at these larger scales, we
claim that we can treat the object very nearly as a particle in a flat space-
time inertial reference frame as formulated above, where the inertial motion
of the object is dictated by the geodesics of our model. But again, if we
focus our attention on the local region around the massive object while
disregarding the existence of the larger inertial system, we will still
observe the effects resulting from the curvature the object induces in space-
time and thus the gravitational effects it has on other objects around it
(i.e. general relativity holds locally).

In the following, we give an approximate method under our stationary
localization conditions as described in the previous section for determining a
potential implication of our theory of inertial centers with regard to the
observables of a local gravitational system. We take the view that the
Schwarzschild metric [37] applies in the small $\sqrt{\Lambda}t$ limit within
confined regions of our inertial reference frame for observers nearly
stationary with respect to the inertial center about which they orbit as one
would expect from the well-established accuracy of general relativity[38].
Below, our “mixing” of the Schwarzschild metric with the radial Rindler chart
is an approximate way of expressing the fact that locally in the inertial
reference frame of our theory the observer can treat the speed of light as
nearly constant if one were to remove the massive object and work in flat
space-time (i.e. set $M=0$ in the Schwarschild metric) as well as the idea
that general relativity holds locally. But the observer must always keep in
mind that the inertial frame of Einstein’s special relativity is actually not
an inertial frame of the theory of inertial centers, and thus this local
system is located in the more globally relevant inertial system where the
speed of light is not constant. Then if we no longer take $M=0$ (i.e. return
the massive object to the local system), we should still expect gravitation as
formulated by Einstein when we examine locally and disregard the larger
inertial system from our model. In other words, the Schwarzschild metric still
applies locally in the theory of inertial centers when examining motion about
an uncharged non-rotating spherically symmetric massive object. However, if we
move our observer farther and farther away from the gravitational source, the
local limit will no longer apply since we have to take into consideration the
structure and properties of the larger inertial system as well as the fact
that in our theory objects move inertially along geodesics different from
those of Einstein’s theories of special and general relativity, even though
locally these different geodesics appear to be very similar (i.e.
$d\chi^{2}\rightarrow c^{2}d\tau^{2}$ in the local stationary limit).

Additionally, we need some approximate way to take into account the fact that
the speed of light is not constant throughout the inertial system while still
keeping in mind that locally the observer may experience gravitational effects
from a massive object nearby. We admit that the methods in this section are
rough at best, but it is our hope that in future work we will be able to model
far more accurately this transition from the local approximation of general
relativity to the more global application of the theory of inertial centers.

Our metric equation takes the form of the Schwarzschild solution:

$-d\chi^{2}=-Bc^{2}dT^{2}+\frac{1}{B}dR^{2}+R^{2}d\Omega_{l}^{2}$

where

$B(R)=1-\frac{2MG}{c^{2}R}\indent\indent{\rm and}\indent\indent
d{\Omega_{l}}^{2}=d{\theta_{l}}^{2}+d{\phi_{l}}^{2}\sin^{2}{\theta_{l}}$

($T,R,\theta_{l},\phi_{l}$) describe our local gravitational system and
($t,r,\theta,\phi$) refer to the global inertial reference frame within which
the local system is located. In other words, we assume that the observer takes
the coordinate transformations away from the inertial center to cover local
space-time in the same manner as outlined in our previous section. Meaning,
ignoring the existence of the massive object

$\displaystyle cT=r\sinh(\sqrt{\Lambda}t)$ $\displaystyle
X=r\cosh(\sqrt{\Lambda}t)\sin{\theta}\cos{\phi}+X_{0}$ $\displaystyle
Y=r\cosh(\sqrt{\Lambda}t)\sin{\theta}\sin{\phi}+Y_{0}$ $\displaystyle
Z=r\cosh(\sqrt{\Lambda}t)\cos{\theta}+Z_{0}$

where $X_{0}$, $Y_{0}$, and $Z_{0}$ are constants, $\sqrt{\Lambda}t$ is taken
to be small, and we only examine the $r>0$ region of the inertial system.
Thus, $T\approx t$ where we employ equation (56) for the local stationary
limit. Then taking into account the existence of this massive object in the
local region with $M=0$ flat space-time Minkowski coordinates given by
($T,X,Y,Z$) in the local stationary limit, we employ the Schwarzschild metric
noting that our affine parameter is approximately given by $\chi=c\cdot\tau$.
As well, $c$ refers to the speed of light in the local system at the point in
the global inertial frame where the observer and photon meet, and $M$ is the
mass of the object. Then we will proceed through a standard treatment of the
gravitational redshift for the Schwarzschild metric (see Chapter 6.3 of [24]).
However, we keep the Minkowski constant $c$ in all of our expressions as we
intend to investigate the implications of the variable nature of the speed of
light in flat space-time from our theory of inertial centers. For an observer
and photon both traveling radially in this local system
($U^{\mu}=dx^{\mu}/d\chi\rightarrow\langle U^{T},U^{R},0,0\rangle$,
$k^{\mu}\rightarrow\langle k^{T},k^{R},0,0\rangle$), we have

$-\frac{2\pi}{\lambda}=-Bc^{2}U^{T}k^{T}+\frac{1}{B}U^{R}k^{R}$

where $\lambda$ is the wavelength measured by our observer. Applying
conservation laws for $U^{a}$ and $k^{a}$ using the time-translationally
invariant Killing vector field for the Schwarzschild metric,
$\xi^{a}=(\partial/\partial T)^{a}$:

$U^{T}=\frac{E}{Bc^{2}}\indent\indent{\rm and}\indent\indent
k^{T}=\frac{\rho_{0}}{Bc^{2}}$

And taking into account the motion of the observer and photon ($0=k^{a}k_{a}$
and $-1=U^{a}U_{a}$)

$U^{R}=\pm\sqrt{\bigg{(}\frac{E}{c}\bigg{)}^{2}-B}\indent\indent{\rm
and}\indent\indent k^{R}=\pm\frac{\rho_{0}}{c}$

Plugging into our expression for the observed wavelength of the photon,

$\frac{2\pi}{\lambda}=\frac{E\rho_{0}}{B}\bigg{[}\frac{1}{c^{2}}-\bigg{(}\pm\bigg{)}\bigg{|}_{{\rm
photon}}\cdot\bigg{(}\pm\frac{1}{Ec}\sqrt{\bigg{(}\frac{E}{c}\bigg{)}^{2}-B}\bigg{)}\bigg{|}_{{\rm
observer}}\bigg{]}$ (58)

where $(\pm)|_{{\rm photon}}$ and $(\pm)|_{{\rm observer}}$ refer to the
photon/observer traveling radially outward/inward ($+/-$) in the local system
(in $R$). If we assume the observer to be nearly at rest in the local frame
($U^{T}\gg U^{R}$), then $B=(E/c)^{2}$ and expression (58) reduces to

$\lambda\propto c\sqrt{B}$

where in the following we approximate in the small $\sqrt{\Lambda}t$ limit
with our equation for the local speed of light in the inertial reference frame
(56). We employ this “trick” as the Schwarzschild metric is just an
approximation in our model valid under confined regions of the particular
inertial system within which the gravitational source is located. However, one
should be able to experimentally detect with an apparatus of the necessary
sensitivity that these photons progress along the geodesics of our theory of
inertial centers (and not straight lines) bent locally due to the curvature in
space-time caused by our massive object $M$. Therefore, we find a slight
modification to the Schwarzschild redshift factor:

$z=\frac{\lambda_{\rm absorber}-\lambda_{\rm emitter}}{\lambda_{\rm
emitter}}=\frac{r_{{\rm absorber}}}{r_{{\rm
emitter}}}\sqrt{\frac{1-\frac{2MG}{R_{{\rm absorber}}}\cdot\frac{1}{\Lambda
r^{2}_{{\rm absorber}}}}{1-\frac{2MG}{R_{{\rm emitter}}}\cdot\frac{1}{\Lambda
r^{2}_{{\rm emitter}}}}}-1$ (59)

where $r_{{\rm absorber/emitter}}$ refers to the radial position of the
absorber/emitter in the inertial reference frame (i.e. relative to the
inertial center) and $R_{{\rm absorber/emitter}}$ to the radial position
relative to the center of our massive object $M$ in the local gravitational
system. Consequently, we should see a modified redshift factor consisting of
the Schwarzschild expression (Chapter 6.3 of [24]) scaled by the solution
found in our flat space-time vacuum analysis.

Let us then apply this analysis to the case of a space probe traveling out of
our solar system where the $r_{\rm absorber}/r_{\rm emitter}$ factor should
have a larger impact on our observations. In our crude example, we treat both
the probe and the absorber as essentially stationary. Referring to expression
(59) for observers at rest, the absorber wavelength in terms of the emitter is

$\lambda_{{\rm absorber}}|_{{\rm Modified}}=\lambda_{{\rm
emitter}}\cdot\frac{r_{{\rm absorber}}}{r_{{\rm
emitter}}}\sqrt{\frac{1-\frac{2MG}{R_{{\rm absorber}}}\cdot\frac{1}{\Lambda
r^{2}_{{\rm absorber}}}}{1-\frac{2MG}{R_{{\rm emitter}}}\cdot\frac{1}{\Lambda
r^{2}_{{\rm emitter}}}}}$

where our $R$ values in this example refer to local radial distances away from
the center of the Sun and $r$ to distances away from the center of the Milky
Way. For the ratio between the Schwarzschild wavelength and the modified value
above assuming the term under the square root remains approximately the same
for small changes in $r$ relative to changes in $R$, we have

$\frac{\lambda_{{\rm absorber}}|_{{\rm Schw}}}{\lambda_{{\rm absorber}}|_{{\rm
Modified}}}\approx\frac{r_{{\rm emitter}}}{r_{{\rm absorber}}}$

where

$\lambda_{\rm absorber}|_{\rm Schw}=\lambda_{\rm
emitter}\cdot\sqrt{\frac{1-\frac{2MG}{R_{\rm
absorber}}\cdot\frac{1}{c^{2}_{\rm absorber}}}{1-\frac{2MG}{R_{\rm
emitter}}\cdot\frac{1}{c^{2}_{\rm emitter}}}}$

Since Pioneer 10 was on course to travel away from the center of the Milky Way
in the general direction of Aldebaran[21], we can approximate the path of our
photon as nearly a radial one in our galactic inertial reference frame.
Therefore, if we naively ignore the two-way nature of the Doppler residuals,
$r_{\rm absorber}\approx r_{\rm emitter}\cdot e^{-\sqrt{\Lambda}\Delta t}$
where $\Delta t$ is the time it takes the massless particle to travel from the
emitter to the absorber, assuming time measured by the emitter progresses at
nearly the same rate as that measured by the absorber in this short distance
calculation (i.e. $\tau_{a}\approx\tau_{e}=t$). Notice, these photons
travelled inward for Pioneer 10, so the root is negative. Plugging into our
expression above, the fractional difference in wavelength predicted here on
Earth is approximately

$\frac{\lambda_{{\rm absorber}}|_{{\rm Schw}}}{\lambda_{{\rm absorber}}|_{{\rm
Modified}}}=\frac{1}{e^{-\sqrt{\Lambda}t}}\approx 1+\sqrt{\Lambda}t$

to first order where we assume that our modified expression coincides with our
experimental values. Then the observed “time acceleration” reported in[39] and
[40] provides an estimate for the time-scale of our galaxy of
$\sqrt{\Lambda}|_{{\rm MW}}=2.92\times 10^{-18}$ s-1. The consistency of this
value with that of the Hubble constant[20] lends support to the argument that
the time-scale $\sqrt{\Lambda}$ is universal for all inertial reference frames
as we had implicitly assumed from our proposed form of the affine parameter
presented in our introduction. However, further experiment is necessary in
order to verify this claim.

Clearly, the two-way nature of the Doppler residuals of the Pioneer
experiments as well as the difference in clock rates for varying positions
within an inertial system in our theory will complicate our analysis further.
However, the purpose of this section is to illuminate to the reader the idea
that we may have evidence from experiments within our own solar system that
support the relevance of this theory of inertial centers and suggest that
possibly all inertial reference frames as defined within this theory abide by
the same fundamental time-scale constant $\sqrt{\Lambda}$. Nevertheless,
others have argued as in [41] that the Pioneer anomaly is a consequence of the
mechanics of the spacecrafts themselves instead of evidence of “new physics”.
Therefore, to gain more support for the theory of inertial centers, we must
address in future work not only the two-way nature of the Doppler residuals as
both Pioneer 10 and Pioneer 11 appear to report blueshifted wavelengths even
when they traveled in opposite directions with respect to the galactic
inertial frame of reference but also the possibility that our theory can
succinctly explain the other astrometric Solar System anomalies outlined in
[23] and [40].

### Quantization of a real scalar field

We begin our extension into quantum field theory from the covariant form of
the Klein-Gordon equation [42]:

$\nabla_{a}\nabla^{a}\phi-\mu^{2}\phi=0$

where $\nabla_{a}$ is the derivative operator compatible with the metric
$g_{ab}$ (i.e. $\nabla_{a}g_{bc}=0$), $\mu=mc/\hbar$, $m$ is the mass
associated with our field, and $\hbar$ is the reduced Planck constant. First,
we explore how one can intuitively arrive at this equation of motion given our
classical assumptions. In special relativity, we have

$-m^{2}c^{2}=p^{a}p_{a}=\sum_{\nu,\beta}\eta_{\nu\beta}p^{\nu}p^{\beta}$

where $p^{\nu}=m\cdot dx^{\nu}/d\tau$ and $\eta_{\nu\beta}$ refers to the
Minkowski metric components. Making the substitution
$p^{a}\rightarrow-i\hbar\nabla^{a}$, we come upon the Klein-Gordon equation
above for a scalar field.

However, in our theory of inertial centers, the equation of motion in terms of
‘momentum’ is given by

$-m^{2}=p^{a}p_{a}$

where now $p^{\nu}=m\cdot dx^{\nu}/d\chi$ and so we have a major difference in
our ‘momentum’ terms. In contrast with our experience in relativity, the
‘four-velocity’ for massive particles in our theory is parametrized by $\chi$
and not by proper time $\tau$. Unfortunately, there does not appear to be a
natural operator substitution for $dx^{\nu}/d\chi$. Yet, if we use expression
(57), we have a potential extension of the Klein-Gordon equation when
analyzing motion at a particular scale. It appears that one should substitute
$c\rightarrow\sqrt{\Lambda}r$ to find

$\Box\phi-\tilde{\mu}^{2}r^{2}\phi=0$ (60)

where $\tilde{\mu}=m\sqrt{\Lambda}/\hbar$ and $\Box=\nabla_{a}\nabla^{a}$ is
the Laplace-Beltrami operator. Notice that in our equation of motion we have
explicit reference to the particular inertial reference frame in which we are
analyzing the behavior of the field as opposed to the Klein-Gordon equation
which has no explicit reference to any inertial system. This seems to be
consistent with the idea that the proper time is not the invariant quantity
associated with our theory of inertial centers, and therefore our choice of
proper time reflects the choice of scale in which we must work to analyze the
progression of our field within this inertial system. One can also apply this
substitution in an analogous manner to other equations of motion/Lagrangians,
yet in the following we will only address the simple case of a free real
massive scalar field.

Then, as outlined in Chapter 4.2 of [43] and briefly reviewed in Appendix D,
we must “slice” our manifold $M$ into space-like hypersurfaces each indexed by
$t$ ($\Sigma_{t}$). For our radial Rindler chart, the future-directed unit
normal to each $\Sigma_{t}$ is given by

$n^{a}=\frac{1}{\sqrt{\Lambda}|r|}\bigg{(}\frac{\partial}{\partial
t}\bigg{)}^{a}$ (61)

where the absolute value is necessary to keep $n^{a}$ future-directed for all
values of $r:0<r^{2}<\infty$, allowing for positive and negative values. We
will interpret the physical significance of this relaxation on the domain
restrictions for our radial coordinate later in our analysis. We see that our
hypersurface can be decomposed into the union of two surfaces for each of the
Rindler wedges ($r<0$ and $r>0$), and thus the inner product of our Klein-
Gordon extension is given by

$\displaystyle(\phi_{1},\phi_{2})=-i\Omega([\bar{\phi}_{1},\bar{\pi}_{1}],[\phi_{2},\pi_{2}])$
$\displaystyle=-i\int_{\Sigma_{{\rm I}}\cup\Sigma_{{\rm
II}}}d^{3}x\sqrt{|h|}[\phi_{2}n^{a}\nabla_{a}\bar{\phi}_{1}-\bar{\phi}_{1}n^{a}\nabla_{a}\phi_{2}]$
(62)

where the bar symbol indicates complex conjugation (i.e. $\bar{\phi}_{i}$ is
the complex conjugate of $\phi_{i}$), $\Sigma_{0}=\Sigma_{{\rm
I}}\cup\Sigma_{{\rm II}}$ is the union of these two radial Rindler wedge
space-like hypersurfaces, $n^{a}$ is the unit normal to our space-like
hypersurface $\Sigma_{0}$, $h_{ab}$ is the induced Riemannian metric on
$\Sigma_{0}$ ($h=\mathrm{det}(h_{\nu\beta})$; $(h_{\nu\beta})$ denotes the
matrix associated with these Riemannian metric components), and $\Omega$
refers to the symplectic structure for our extension of the Klein-Gordon
equation.

We should be rather concerned considering the discontinuous nature of the
time-orientation of $n^{a}$ (the absolute value is not a smooth function) as
well as the undefined behavior of our unit normal for $r=0$, the location of
our inertial center. However, given the solutions we find below, it seems to
be an important question whether or not we are forced to treat each Rindler
wedge separately as its own globally hyperbolic space-time or the combination
of these wedges as the entire space-time over which we must analyze solutions
to our extension of the Klein-Gordon equation. The difference between these
two formulations will be that in the former we must define separate creation
and annihilation operators for each wedge as in the analysis of [44]. Whereas
in the latter, we have one set of creation and annihilation operators for all
values of $r$ over the range: $0<r^{2}<\infty$, where $r$ can take both
positive and negative values. It also seems likely that a greater
understanding of our inertial centers and their physical significance (i.e.
how are these inertial centers established?) will provide far more insight
into the proper way to treat this situation. In this paper, however, we assume
the latter approach requiring that we use all values of $r$ (positive and
negative) to cover our inertial reference frame and naively ignore the issues
with $r=0$ mentioned above. This approach seems to be far more consistent with
the idea implicit in our theory of inertial centers that the radial Rindler
chart covers the entire flat space-time manifold for the inertial system in
question, except of course for the location of each of our inertial centers
(i.e. $r=0$). We find that our inner product is given by

$\displaystyle(\phi_{1},\phi_{2})=-i\int_{-\infty}^{\infty}dr\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\bigg{[}r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin{\theta}\bigg{(}\frac{\phi_{2}}{\sqrt{\Lambda}|r|}\partial_{t}\bar{\phi}_{1}-\frac{\bar{\phi}_{1}}{\sqrt{\Lambda}|r|}\partial_{t}\phi_{2}\bigg{)}\bigg{]}\bigg{|}_{t=0}$
$\displaystyle=-\frac{i}{\sqrt{\Lambda}}\cosh^{2}(\sqrt{\Lambda}t)\bigg{[}\int_{0}^{\infty}rdr\int_{0}^{\pi}\sin{\theta}d\theta\int_{0}^{2\pi}d\phi\bigg{(}\phi_{2}\partial_{t}\bar{\phi}_{1}-\bar{\phi}_{1}\partial_{t}\phi_{2}\bigg{)}$
$\displaystyle-\int_{-\infty}^{0}rdr\int_{0}^{\pi}\sin{\theta}d\theta\int_{0}^{2\pi}d\phi\bigg{(}\phi_{2}\partial_{t}\bar{\phi}_{1}-\bar{\phi}_{1}\partial_{t}\phi_{2}\bigg{)}\bigg{]}\bigg{|}_{t=0}$
(63)

Our remaining task reduces to solving for solutions ($\phi_{i}$) to our
extension of the Klein-Gordon equation (60). From Appendix F which utilizes
[45], [46], [47], [48], and [49], we find

$\phi_{\alpha,l,m}=\sqrt{\frac{\tilde{\mu}\alpha}{2\pi\cosh(\pi\alpha)}}\cdot\sqrt{1-\eta^{2}}\cdot
P_{l}^{-2i\alpha}(\eta)\cdot\frac{K_{i\alpha}(\frac{\rho^{2}}{2})}{\rho}\cdot
Y_{l}^{m}(\theta,\phi)$ (64)

where $\eta=\tanh(\sqrt{\Lambda}t)$ and $\rho=\sqrt{\tilde{\mu}}r$.
$Y_{l}^{m}$ is the spherical harmonic of degree $l$ and order $m$. We maintain
convention and use $m$ to denote the order of $Y_{l}^{m}$. However, this $m$
is a quantum number very different from the mass of our scalar field. The mass
term is contained solely in our expression for $\tilde{\mu}$. $K_{i\alpha}$ is
the Macdonald function (modified Bessel function) of imaginary order $\alpha$.
$P_{l}^{-2i\alpha}$ is the Legendre function of degree $l$ and imaginary order
$-2\alpha$.

Notice, we allow $K_{i\alpha}(\frac{\rho^{2}}{2})/\rho$ to have domain:
$0<\rho^{2}<\infty$ where $\rho$ can take both positive and negative values.
Physically, this interpretation requires the existence of the field in both
the negative and positive $r$ regions of the inertial system which brings us
back to the discussion earlier in this section of our concern with $n^{a}$.
From [50], the limiting behavior of $K_{i\alpha}$ expressed as

$\lim_{y\rightarrow
0^{+}}K_{i\alpha}(y)=-\bigg{(}\frac{\pi}{\alpha\sinh(\alpha\pi)}\bigg{)}^{1/2}\bigg{[}\sin(\alpha\ln(y/2)-\phi_{\alpha,0})+\mathcal{O}(y^{2})\bigg{]}$

where $\phi_{\alpha,0}=\arg\\{\Gamma(1+i\alpha)\\}$ and $\Gamma(z)$ is the
gamma function along with

$\lim_{y\rightarrow\infty}K_{i\alpha}(y)=\bigg{(}\frac{\pi}{2y}\bigg{)}^{1/2}e^{-y}\bigg{[}1+\mathcal{O}\bigg{(}\frac{1}{y}\bigg{)}\bigg{]}$

shows that $K_{i\alpha}(\frac{\rho^{2}}{2})/\rho$ oscillates for small
$|\rho|$ when $\alpha\neq 0$ and exponentially decays for large $|\rho|$. In
addition, from Figure 3, we see that our radial ‘wave function’ spreads out
away from $\rho=0$ for larger ‘momentum’ values of $\alpha$, allowing for
oscillatory behavior at larger values of $|\rho|$ and thus an increased
likelihood of observing quanta farther away from the inertial center of the
reference frame in question.

Our Heisenberg field operator can be expanded in the following manner (see
Chapters 3.1 and 3.2 of [43]):

$\displaystyle\hat{\Phi}(t,r,\theta,\phi)=\int_{0}^{\infty}d\alpha\sum_{l=0}^{\infty}\sum_{m=-l}^{l}[\phi_{\alpha,l,m}\hat{a}(\bar{\phi}_{\alpha,l,m})+\bar{\phi}_{\alpha,l,m}\hat{a}^{\dagger}(\phi_{\alpha,l,m})]$
$\displaystyle=\int_{0}^{\infty}d\alpha\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\sqrt{\frac{\tilde{\mu}\alpha}{2\pi\cosh(\pi\alpha)}}\cdot\frac{K_{i\alpha}(\frac{\rho^{2}}{2})}{\rho}\cdot\sqrt{1-\eta^{2}}\cdot\bigg{[}\hat{a}(\bar{\phi}_{\alpha,l,m})P^{-2i\alpha}_{l}(\eta)Y_{l}^{m}(\theta,\phi)$
$\displaystyle+\hat{a}^{\dagger}(\phi_{\alpha,l,m})P^{2i\alpha}_{l}(\eta)\bar{Y}_{l}^{m}(\theta,\phi)\bigg{]}$
(65)

where the annihilation and creation operators in terms of our inner product
are

$\displaystyle\hat{a}(\bar{\phi}_{\alpha,l,m})=(\phi_{\alpha,l,m},\hat{\Phi})$
$\displaystyle\hat{a}^{\dagger}(\phi_{\alpha,l,m})=-(\bar{\phi}_{\alpha,l,m},\hat{\Phi})$

and $\\{\phi_{\alpha,l,m}\\}$ comprise an orthonormal basis of the “positive
frequency” solutions to the extended version of the Klein-Gordon equation for
the theory of inertial centers. For a real scalar field, these annihilation
and creation operators satisfy the commutation relations (bosonic statistics):

$[\hat{a}(\bar{\phi}_{\alpha,l,m}),\hat{a}^{\dagger}(\phi_{\alpha^{\prime},l^{\prime},m^{\prime}})]=(\phi_{\alpha,l,m},\phi_{\alpha^{\prime},l^{\prime},m^{\prime}})=\delta(\alpha-\alpha^{\prime})\delta_{ll^{\prime}}\delta_{mm^{\prime}}$

A very important point for the reader to take away from our analysis in this
section is that our field operator as defined in (Quantization of a real
scalar field) exists in both the $r>0$ and $r<0$ portions of space-time. In
other words, we take space-time to be comprised of both the $r>0$ and $r<0$
regions of the inertial system, and thus the Minkowski chart is not able to
cover all of space-time in our theory. It then appears that a potential
explanation for the matter/antimatter asymmetry in our observable universe
within the framework of our theory of inertial centers would be that there
exists a parallel region of each inertial system embodied mathematically above
by the existence of our field operator in the hypothetical $r<0$ region of
space-time. Logically, if we exist in our region of space-time with an
imbalance toward matter, one would then assume that in this parallel region
there exists an imbalance in favor of antimatter as the total charge of the
field throughout all of space-time should be conserved. We are, of course,
operating under the assumption that the solutions to our equation of motion
extend in a similar manner as in special relativity when one allows for
complex fields of non-zero spin (e.g. solutions to a Dirac equation [51]
extension are also solutions to our Klein-Gordon extension) since we should
not worry about antiparticles with a real scalar field. Therefore, we must
extend our work on the theory of inertial centers to incorporate spin in order
to see the full significance of this possible explanation for the
matter/antimatter asymmetry in our observable universe.

To conclude our discussion, we assume throughout the rest of this section that
$\sqrt{\Lambda}$ is a universal constant for all inertial systems, taken to be
the Hubble constant as proposed in our introduction, and imagine that there
exists an observer very near to an inertial center point such that his/her
motion in this particular reference frame is approximately stationary (i.e.
spatial ‘four-velocities’ are very much outweighed by ‘velocity’ in time,
$dt/d\chi$). Then from our classical analysis of geodesic paths, our observer
experiences a radial acceleration according to (10) of

$\frac{d^{2}r}{dt^{2}}=-\Lambda r$

where $t$ coincides with the proper time $\tau$ for our nearly stationary
observer in this system. However, say we wish to understand our observer’s
motion not in terms of his/her proper time in this particular inertial frame
but instead in terms of his/her proper time in an external inertial frame of
reference where these two different systems do not share a common inertial
center point. We know that our invariant interval is given by

$-d\chi^{2}=-\Lambda r_{l}^{2}d\tau_{l}^{2}=-\Lambda r_{e}^{2}d\tau_{e}^{2}$

where the $e$ ($l$) subscript refers to quantities in the external (local)
inertial reference frame. Assuming our observer is nearly stationary in both
inertial systems (i.e. coordinate times for each system coincide with proper
times within each reference frame respectively), his/her clock in the local
frame progresses by

$\frac{dt_{l}}{dt_{e}}=\frac{c_{e}}{c_{l}}$ (66)

Thus,

$\frac{d}{dt_{l}}=\frac{c_{l}}{c_{e}}\cdot\frac{d}{dt_{e}}$

where the $c$’s refer to the Minkowski constants for each particular reference
frame (56). Plugging in above,

$\frac{d^{2}r_{l}}{dt_{e}^{2}}=-\Lambda_{{\rm eff}}\cdot r_{l}$ (67)

where $\sqrt{\Lambda_{{\rm eff}}}=\sqrt{\Lambda}\cdot c_{e}/c_{l}$.

According to Newtonian mechanics which is a good approximation here since we
assume our observer is nearly stationary in the local inertial system, one
would attribute this radial acceleration to a ‘force’ (even though we know
that there really is no force here), and associated with this ‘force’ is a
potential ($\vec{F}=-\vec{\nabla}V$; see Chapters 1 and 2 of [52]). So for the
acceleration above, one would assume while working in Newtonian mechanics that
there exists a potential causing this movement of the form:

$V=\frac{1}{2}m\Lambda_{{\rm eff}}\cdot r_{l}^{2}$ (68)

Then our Hamiltonian ($H=T+V$; see Chapter 8 of [52]) for this system is given
by

$H=\frac{p^{2}}{2m}+\frac{1}{2}m\Lambda_{{\rm eff}}\cdot r_{l}^{2}$ (69)

where $m$ is the mass of our observer and $T=p^{2}/2m$ is the kinetic energy
associated with his/her motion as observed in the external frame. If our
observer is on the order of $10^{-15}$ m[53] away from his/her local inertial
center and $c_{e}$ is found in the external frame to be $\approx 3.0\times
10^{8}$ m/s[54], we find $\sqrt{\Lambda_{{\rm eff}}}\sim 10^{23}$ s-1. We
remark for the reader less acquainted with nuclear theory that the Hamiltonian
above is referred to as the isotropic harmonic oscillator and was used as a
starting point for nuclear shell models due to its ability to reproduce the
“magic numbers” associated with stable configurations of nucleons within the
nucleus (see Chapter 4 of [55] and Chapter 3.7 of [56]). In addition, the
energy scale associated with the Hamiltonian above (i.e.
$\hbar\cdot\sqrt{\Lambda_{{\rm eff}}}\sim 10^{8}$ eV) is of a similar order as
the scale inputted into these isotropic harmonic oscillator models for the
magnitude of the nuclear ‘force’ [57]. Thus, our ability to replicate the same
features as those of the simplest nuclear shell model compels us to ask the
following question with regard to the theory of inertial centers: Is there an
inertial center point at the center of the nucleus of every atom?

## Limitations of the study, open questions, and future work

There is a plethora of data for us to critically investigate the validity of
this theory of inertial centers. Nevertheless, we have chosen to leave these
detailed investigations for future work as the purpose of this paper is to lay
out the theoretical foundations to illicit these types of rigorous comparisons
with experiment for all aspects of our model. As we have mentioned briefly at
certain points within our discussion, there are many open questions that must
be addressed. The most pressing of these appears to be how to explain the
cosmic microwave background (CMB) within our theory of inertial centers. One
may be tempted to immediately point to the Fulling-Davies-Unruh effect[58] as
the source of this cosmic radiation since the Unruh effect predicts that an
“accelerating” observer in Minkowski vacuum, who can be described by orbits of
constant spatial coordinate in the classic Rindler chart, detects black-body
radiation that appears to be nearly homogeneous and isotropic with predicted
anisotropies due to the orientation of this observer throughout his/her
“accelerated” path [59]. However, we must keep in mind that the scale
associated with the temperature of Unruh radiation [58][44]

$T=\frac{\hbar a}{2\pi k_{B}c}$

requires $a\sim 10^{20}$ m/s2 to produce a temperature on the order of the
CMB, $T\approx 2.7$ K [60], where $k_{B}\approx 1.38\times 10^{-23}$ J K-1 is
Boltzmann’s constant, $\hbar\approx 1.05\times 10^{-34}$ J s[54], and $a$ is
the proper acceleration of the observer. If we approximate the original
analysis of [58] by working in 1+1 space-time (i.e. 1 time and 1 spatial
dimension), the acceleration would be proportional to the inverse of $r=r_{0}$
for observers moving along orbits of constant $r$ [44]. This then requires
$r_{0}=c^{2}/a\sim 10^{-3}$ m for the CMB temperature scale, which clearly
makes no sense since we would be millimeters away from the center of our
observable universe. Nevertheless, the analysis used to derive the Unruh
effect implicitly operates under the assumption of the validity of special
relativity in flat space-time and therefore takes $d\chi^{2}=c^{2}d\tau^{2}$.
Yet, as we have emphasized repeatedly above, in our theory of inertial
centers, the invariant interval associated with the metric is given in terms
of proper time by $d\chi^{2}=\Lambda r^{2}d\tau^{2}$. Therefore, we must
extend these ideas to apply to our model where we are observers existing
within multiple inertial systems (universe $\rightarrow\ldots\rightarrow$
Local Group $\rightarrow$ Milky Way). In addition, for our situation, this
radiation would not be interpreted physically as due to the “acceleration” of
the observer as in the case of [58], but instead one would have to think of
this effect as simply the result of the restriction of the Minkowski vacuum to
each of the radial Rindler wedges (see Chapters 4.5 and 5.1 of [43]). We are
still encouraged that this course of action may result in a plausible
interpretation as experimental evidence of large-scale temperature anomalies
appears to suggest a significant orientation to the CMB [61].

At this point in our discussion, we offer a brief review of the literature
concerning both the Pioneer anomaly as well as the other known astrometric
anomalies within our own solar system. First, however, we mention other
theories which contrast with our own study but are relevant for the discussion
below. The authors of [62] and [63] investigate the potential effects of an
expanding universe which could be induced on objects within our solar system.
Furthermore, [64] attempts to model the consequences of an extra radial
acceleration on the orbital motion of a planet within our solar system. As
well, [65] provides an alternative model for gravitation resulting in an
additional “Rindler-like” term at large distances which the author claims can
potentially model the plateauing nature of observed orbital velocity curves.
We must stress that the model proposed in [65] is in fact very different from
the model that we have proposed above as our theory of inertial centers does
not attempt to reformulate gravity. As we emphasized earlier, our model is an
attempt to reformulate the motion of objects when no net external forces are
acting upon said objects in empty flat space-time. Nevertheless, [66], [67],
[68], [69], and [70] use these ideas of an additional “Rindler-like” term in
gravitation to examine the possible observable effects of the aforementioned
extension to general relativity. For a background reference concerning
phenomenology in the context of general relativity, we refer the reader to
[71] as preparation for our presentation of the known anomalies exhibited
within our own solar system.

Besides the Pioneer anomaly, there are experimental claims of possible
anomalies alluding to inconsistencies with our current model for the Solar
System. These include:

1\. An anomalous secular increase in the eccentricity of the orbit of the Moon

2\. The “flyby” anomaly

3\. An anomalous correction to the precession of the orbit of Saturn

4\. A secular variation of the gravitational parameter $GM_{\odot}$ where
$M_{\odot}$ is the mass of the Sun

5\. A secular variation of the astronomical unit (AU)

The anomalous secular increase in the eccentricity of the orbit of the Moon
was originally found in the experimental analysis of the Lunar Laser Ranging
(LLR) data in [72] and expanded upon in [73], [74], [75], and [76]. The
“flyby” anomaly refers to an anomalous shift in the Doppler residuals received
from spacecrafts when comparing signals before and after these spacecrafts
undergo gravitational assists about planets within the Solar System
[40][77][78]. The anomalous perihelion precession of Saturn appears to be a
more controversial claim as the work of [79] and [80] seems to suggest the
validity of this observation with further investigation in [81] and [82].
However, work such as [83], [84], and [85] seems to show that this reported
anomaly is an experimental artifact. Finally, the last two anomalies of a
secular variation in the product of the mass of our Sun and the gravitational
constant $G$ as well as the astronomical unit are more difficult claims to
understand in the context of our model as there are many complex mechanisms
which could affect our measurements of these quantities (e.g. rate of mass
accretion of the Sun from infalling objects versus depletion through expelled
radiation resulting from nuclear fusion) in addition to the fact that our
measurement of the AU is implicitly linked to our measurement of
$GM_{\odot}$[86]. Nevertheless, [86] and [87] are useful references for these
anomalies. Additionally, [23] provides a detailed summary of the majority of
the anomalies listed above.

Returning to the Pioneer anomaly, the reader may have concerns with our
earlier analysis as recent simulations such as [41] suggest that this anomaly
should be taken as a thermal effect from the spacecraft itself instead of
evidence linked to “new physics”. For a selection of work concerning the
possible thermal explanation of the Pioneer anomaly, see [41], [88], [89],
[90], [91], [92], and [93]. Nevertheless, this analysis still does not address
the asymmetric nature of the “flyby” anomaly [40][77] as well as the other
significant astrometric Solar System anomalies summarized in [23]. By
“asymmetric nature”, we are referring to the fact that the magnitude of the
“flyby” anomaly appears to depend upon the direction of approach of the space
probe toward Earth as well as the angle of deflection away after “flyby”.
Furthermore, as mentioned in [94], the “onset” of the Pioneer anomaly after
Pioneer 11’s encounter with Saturn is still of concern when explaining these
observables as the result of systemic thermal effects. While [41] briefly
addresses this “onset” in their conclusion, future analysis of the early data
points for Pioneer 11 near its gravitational assist about Saturn appears to be
of the utmost importance, especially considering before its encounter with
Saturn this spacecraft moved nearly tangentially to the direction of
Sagittarius A*, whereas after it traveled nearly toward the Milky Way center.
Thus, in the context of our own model, this “onset” has the potential to be
interpreted as the consequence of the spacecraft’s change in direction
relative to the inertial center associated with the center of the Milky Way,
similar to ideas we will have to explore for the asymmetric nature of the
Earth “flyby” anomalies (for potential connections between the Pioneer and
“flyby” anomalies, see [40]). Therefore, we choose not to rule out the
possibility that the Pioneer anomaly may be support for our theory of inertial
centers as this effect as modeled in our earlier analysis in fact must be
observed in order for our theory to have physical relevance. As mentioned
earlier, we will have to address in far more rigorous detail in future work
the dual nature of the Pioneer residuals in order to possibly explain the
blueshifts from both Pioneer 10 and Pioneer 11 data.

In addition, others such as [65], [66], and [67] have used a “Rindler-like
force” emanating from the center of a gravitational source to supplement
general relativistic gravity as a model that can potentially explain orbital
velocity curves as well as the Pioneer anomaly [68][69]. For a review of how
this and other gravitational supplements would impact current expectations for
the orbits of other major bodies in the Solar System, see [95], [79], [96],
[97], [98], [99], [100], [101], [102], [103], [67], [104], [105], [106],
[107], [108], and [109]. However, these supplements all require spherical
symmetry about the center of the gravitational source in question and are very
different from our reformulation of flat space-time where in our theory we do
not assume that there exists a gravitational source at the center of galaxies,
groups, clusters, etc. Recall that we are concerned with reformulating
inertial motion and inertial reference frames in flat space-time (i.e. our
description of the way in which objects move in flat space-time when subjected
to no net external forces). Additionally, we maintain that locally within
confined regions of the inertial system of our theory of inertial centers
Einstein’s version of gravitation seen as the consequence of space-time
curvature induced by the energy-momentum of a massive object in his theory of
general relativity still applies in the same manner. In other words, in our
theory of inertial centers, this observed deviation from assumed special
relativistic flat space-time geodesics arises from our redefinition of the
inertial system itself instead of some modification to gravitation.
Consequently, when attempting to explain these astrometric Solar System
anomalies in the context of our theory, we focus on the difference in
geodesics in the galactic inertial reference frame when compared to assumed
special relativistic geodesics for flat space-time and assume that all of the
objects in our Solar System including the Sun orbit about the inertial center
point associated with the center of the Milky Way (again, we assume that there
is no gravitational source at the center of our galaxy). Meaning, the Pioneer
anomaly is not taken to be a phenomenon due to gravity in the theory of
inertial centers. Instead the Pioneer anomaly and possibly the other
astrometric Solar System anomalies which we have listed above are taken to be
the result of our redefinition of inertial systems as well as the change in
our expectations for what constitutes inertial motion. Consequently, the
relative acceleration between massive objects in our solar system is nearly
unchanged from what one would expect from general relativity as all objects
within our solar system orbit about the center of the Milky Way along
relatively similar paths. Therefore, we are not modifying our expectations for
the interactions between objects within the Solar System. We are modifying our
expectations for the paths of all objects in the Solar System through the
Milky Way. While internally within our solar system the planets remain nearly
unchanged in their paths as they move slowly in the “Newtonian limit” (i.e.
their speeds are much less than that of light), light propagating between
these massive objects in our theory won’t behave as one would expect from
general relativity as at these speeds one must take into account the
properties of the larger inertial system associated with our galaxy.

One must bear in mind that these anomalies are linked to the propagation of
electromagnetic radiation throughout our solar system as our experimental
apparatuses use light for precision measurements. While the work of [62]
attributes the Pioneer anomaly to the local effects of light signal
propagation in an expanding universe as expressed by a “post-Friedmannian”
metric decomposition, these claims would not be able to explain the asymmetric
nature of the wavelength shift residuals in the “flyby” anomaly as the FLRW
metric requires homogeneous and isotropic expansion of space in all directions
[31]. However, there is no expansion in our theory of inertial centers and our
inertial reference frames do have an orientation. Therefore, we must take into
consideration, when comparing with our own model in future work, two important
ideas: in this theory of inertial centers, the speed of light is not constant
in flat space-time and objects follow inertial paths described by geodesics
about inertial centers in the radial Rindler chart, where we assume that the
inertial center associated with the Milky Way is in the direction of
Sagittarius A*. Thus, in our model, the observables associated with the
astrometric Solar System anomalies listed above do not necessarily reflect the
existence of an additional acceleration in the Solar System since our theory’s
radial acceleration would be imposed on all objects within the Solar System
including the Sun and in the same direction toward the center of the Milky Way
(10) with seemingly negligible difference in magnitude depending upon the
position of the massive object in question (i.e. changes in position within
our solar system are negligible relative to the distance of our solar system
from the center of the Milky Way when considering the motion of massive
satellites, planets, etc.). In other words, in sharp contrast with the
analysis in papers such as [65], [109], [97] and [95], we assume that there is
no additional acceleration associated with the Sun’s gravitational pull on
other objects within the Solar System, and thus the relative acceleration of a
satellite, planet, etc. with respect to the center of the Sun remains nearly
unaffected in our model when we compare with general relativity. Instead, it
appears that in the theory of inertial centers these anomalies should more
likely be interpreted as a consequence of the non-constant nature of the speed
of light within our galactic inertial system as well as of the expected shifts
in wavelength when light propagates between differing distances from an
inertial center point. Future experiments within the vicinity of our solar
system to test the validity of the theory of inertial centers could include
sending a spacecraft to the outer edges of our solar system along a closed
orbit about the Sun or using identical spacecrafts along open orbits in
different directions with respect to the galactic center (e.g. one travels
tangentially to the direction of the center of the Milky Way while another
moves directly toward/away from the center; for a hyperbolic orbit proposal,
see [110]). To test the positional dependence aspects for electromagnetic
radiation in this theory, these hypothetical missions should measure the
potential variations in wavelength shift and time delay for light signals sent
and received at different positions along these orbits with respect to the
center of the Milky Way. As well, future theoretical work will require us to
explicitly detail observational effects on our astrometric measurements of the
planetary ephemerides that are unique to the theory of inertial centers. One
could then potentially find these predicted deviations from current models
when comparing with the experimental work of [83] and [85].

Using the measured value for the speed of light on Earth ($c_{\rm
Earth}\approx 3.0\times 10^{8}$ m/s) and the value for the time-scale given
from the “time acceleration” in [40], we find that our distance to the center
of the Milky Way is approximately $r_{0}|_{{\rm MW}}\approx 1.03\times
10^{23}$ km. We see that the value obtained for our galactic radial distance
is far larger than the predicted value from models requiring a supermassive
black hole at the center of the Milky Way (intimidatingly, nearly six orders
of magnitude [111]). It is imperative then that we reconcile this calculated
value with observational data. Not only will this maintain consistency with
experiment but it will also provide accurate distance scales within our
galaxy. This will allow us to further understand the large observed wavelength
shifts near Sagittarius A* within the framework of our theory of inertial
centers and potentially explain the paradox of youth [112] through concrete
analysis of star formation near the Milky Way center.

Addressing our classical inertial motion analysis, one can immediately tell
from the theoretical approach in our discussion that this paper is limited by
the lack of necessary quantitative comparison with orbital velocity curves,
redshift surveys, and lensing observations. Future work will require modeling
using computer simulations of our equations of motion not only to produce
orbital velocity curves that will facilitate comparison with data but to also
give us a far more thorough understanding of classical inertial motion outside
of the limiting behavior examined in this paper. To implement, it appears that
we should use a finite difference method with the component form of our
geodesic equation parametrized in terms of the proper time of the object in
question within a particular inertial system as expanded upon at the end of
Appendix B. Furthermore, we will have to apply this same finite difference
method to our normalization condition for the ‘four-velocity’ but parametrized
in terms of the proper time in this inertial frame. We also have to attend to
a pressing issue with regard to the “Hubble behavior” associated with
wavelength shifts within our inertial system. As outlined earlier, this theory
requires that we observe both significant redshifts and blueshifts, yet on
scales larger than the Local Group, blueshifted emitters are reportedly
scarce. Thus, if our theory is to be considered seriously, we must provide an
explanation for why there is such an imbalance towards reported redshifted
emitters at the largest observable scales. Nevertheless, one apparent
resolution lies in the possible alternative “blueshift interpretation” of
spectroscopic profiles as mentioned and subsequently applied in [113], [114],
and [115] with possible support for the re-examination of spectroscopic
profiles in the blueshifted emission lines found in other work such as [116].

Proceeding to our quantum concerns, our seemingly shocking proposal that at
the center of the nucleus of every atom there could potentially exist an
inertial center point raises many more questions for our theory of inertial
centers. Of course, this type of claim requires thorough and rigorous
justification in both future theoretical work and even more importantly in
comparison with experiment. For example, a simple comparison with experiment
would be to determine how accurate of a fit our “n-particle amplitudes”
(reviewed in Appendix D) with individual solutions for quantum numbers
$(\alpha,l,m)$ given by (64) are with current experimental knowledge of the
nucleus. Nevertheless, we have chosen to mention these ideas in this paper in
order to highlight to the reader how much of a potential impact this
redefinition of inertial motion and inertial reference frames could possibly
have on our understanding of structure formation for all scales from the
largest to the smallest. As for questions: for one, can we reconcile these
claims with our current knowledge of the electronic and nuclear structure of
the atom when we factor in charge, spin, and electromagnetism? Additionally,
how much of our current model for the nucleus is affected by these ideas? It
also becomes ever more important to answer the following: What establishes one
of these inertial centers as well as the orientation of one of our inertial
systems?

## Conclusions

All of our assumptions within this work in one way or another are built upon
the idea that objects do not move in a straight line at a constant speed when
no external forces are acting upon them in empty flat space-time. In other
words, we assume that Newton’s first law does not give the correct
characterization of inertial motion. Therefore, we essentially “start from
scratch” and concentrate on how to incorporate all of the following observed
features into a revised understanding of inertial motion: accelerated
redshifts and the Hubble relation, plateauing orbital velocity curves at large
distances from a central point about which objects move, consistent velocity
“flow” on the largest of scales directed toward a central point, and an
orientation associated with each of these central points. We take an inertial
frame of reference to be the system within which objects follow these revised
inertial trajectories and begin our reformulation with the knowledge that our
theory of globally flat space-time must reduce to special relativity within
confined regions of our newly defined inertial systems. Consequently, it
appears natural to approach this reformulation from the notion that we should
have a metric theory of flat space-time, and within this metric theory objects
still follow along geodesic trajectories when no external forces are acting
upon them as in special and general relativity. However, in order to
distinguish our metric theory of flat space-time from special relativity, we
must require that our affine parameter not be proper time globally throughout
these reference frames. In addition, we find that we are able to reproduce the
previously listed features with the radial Rindler chart as the coordinate
parametrization of our flat space-time manifold, thereby assuming the physical
significance of special central points which we deem “inertial center points”
situated throughout all of space-time. As one would expect from their given
name, these inertial center points describe the centers of each of our
inertial systems, and our inertial trajectories are then assumed to be the
orbits of objects about these inertial centers. Meaning, inertial motion must
be thought of relative to both the center point and the orientation (i.e.
location of the poles) of each of these inertial reference frames.
Consequently, it is assumed that the observed motion of objects about central
points on the largest of scales (e.g. stars orbiting the center of a galaxy,
galaxies orbiting the center of a group/cluster, etc.) is not due to
gravitational effects but is instead a manifestation of inertial motion within
our theory of flat space-time, which we term our “Theory of Inertial Centers”.

This redefinition of inertial motion then allows us to no longer assume the
existence of ‘dark energy’, ‘dark matter’, and ‘dark flow’. Furthermore, as we
have the ability to model the Hubble relation within our theory, we do not
require the occurrence of a ‘Big-Bang’ event, and therefore we also do not
require ‘inflation’ nor an expanding universe (i.e. we do not operate under
the assumptions of $\Lambda$CDM). The cornerstone of our theory is embodied in
the statement that within our inertial systems, time and space are
fundamentally intertwined such that time- and spatial-translational invariance
are not inherent symmetries of flat space-time. Meaning, our invariant
interval associated with the metric incorporates both time and spatial
distance. Therefore, observable clock rates depend upon not only the relative
velocity of observers within these inertial systems but also on the difference
in distance of each observer from an inertial center, expressed mathematically
by relation (57). Given this relation, we find that our theory of globally
flat space-time in fact reduces to special relativity for observers which we
can consider as nearly stationary with respect to the inertial center point
about which they orbit (i.e. the local stationary limit). As well, our ideas
then require that the local speed of light which we measure within a confined
region of these newly defined inertial systems is linearly dependent upon our
distance away from the inertial center about which we orbit (56). Thus, the
speed of light throughout each of these redefined inertial systems in flat
space-time is not constant.

With these theoretical foundations presented, we proceeded by examining the
local consequences of our theory for a gravitational system located within one
of these inertial systems as an observer should be able to measure with a
detector of the necessary sensitivity the deviation of an object’s
(specifically light’s) inertial path in flat space-time away from special
relativistic geodesics and into the geodesics of our theory as outlined in the
local stationary limit. Thus, within the framework of the theory of inertial
centers, we interpret the Pioneer anomaly as an observable consequence of our
revised ideas on inertial motion. However, as mentioned later in our paper,
there are many open questions that must be answered with regard to the
propagation of light signals within our solar system in the context of our
theory. Specifically, can our revision of inertial motion and inertial
reference frames explain the other known astrometric Solar System anomalies
(i.e. “flyby” anomaly, the anomalous increase in the eccentricity of the Moon,
and the variation in the AU)? And, can we explain the blueshifted nature of
both Pioneer 10 and Pioneer 11 Doppler data once we factor in the two-way
nature of these residuals as well as the change in clock rates for observers
located at different distances from the center of the Milky Way in our model?

Furthermore, after quantizing for a real massive scalar field, we came upon a
potential explanation for the asymmetry between matter and antimatter in our
observable universe within the context of our theory of inertial centers. If
we allow for the possibility that our field exists in both radial Rindler
wedges (i.e. $r>0$ and $r<0$), it appears that a logical explanation for the
observable imbalance toward matter would be that our antimatter counterparts
are located in the “other” radial Rindler wedge for each of our inertial
systems, as the charge of each field in these systems should be conserved
(e.g. abundance of electrons in one wedge should imply an abundance of
positrons in the “other” wedge). Nevertheless, this logic relies on the
consistency of our extension for a real scalar field to complex fields with
spin. Thus, in future work, we will have to address the validity of this
interpretation when we extend our analysis (e.g. Dirac spinors). In addition,
we concluded our discussion by examining the nearly stationary limit for
particles close to an inertial center point. Using expression (10), we chose
to work naively under Newton’s assumptions and take this acceleration on our
observer to be the result of a Newtonian force derived from a conservative
potential. Then, the stationary Hamiltonian associated with this Newtonian
approximation would take the form of the isotropic harmonic oscillator. Taking
the perspective of an observer exterior to the inertial system in question
(i.e. the external observer orbits a different inertial center), we found the
observed oscillator energy scale using relation (57) while operating under the
assumption that the time-scale for each inertial system is a universal
constant and therefore the same for each. A simple potential explanation for
the ability of the isotropic harmonic oscillator to explain the “magic
numbers” associated with stable arrangements of nucleons within the nucleus of
an atom then arose in the context of our model. Since both the form of our
stationary Hamiltonian as well as the determined energy scale match that of
the starting point for our nuclear shell models, it appears that we must
seriously consider the possibility that there exists an inertial center point
at the center of the nucleus of every atom when working under the assumptions
of our theory of inertial centers as, in our stationary limit, the
acceleration of each particle within the inertial system mimics what one would
find if he/she naively assumed a Newtonian Hamiltonian of the form of the
isotropic harmonic oscillator. In other words, within the context of our
theory, the ability of the isotropic harmonic oscillator to model the simplest
nuclear configurations would be interpreted as a consequence of the physical
existence of an inertial center located at the center of the nucleus of every
atom, where these simple configurations of nucleons arise from the stationary
limit for objects very near to an inertial center. Although these claims are
radical in nature, we are still compelled to question whether or not the
nuclear ‘force’ is even really a force within the framework of our model.
Future theoretical and experimental work will be required in order to fully
understand the nature of these ideas.

## Acknowledgments

I would like to thank several anonymous reviewers as well as both editors for
useful comments and critiques which very much helped to improve the clarity of
this work.

##  Appendix A Affine connection terms, Ricci and Riemann tensors

Following [24] for our general expressions below, we work in the metric

$ds^{2}=-\Lambda
r^{2}dt^{2}+dr^{2}+r^{2}\cosh^{2}(\sqrt{\Lambda}t)[d\theta^{2}+d\phi^{2}\sin^{2}{\theta}]$

Our affine connection tensor for this choice of coordinates is given by the
expression:

$\Gamma^{c}_{ab}=\frac{1}{2}g^{cd}[\partial_{a}g_{db}+\partial_{b}g_{da}-\partial_{d}g_{ab}]$

where in component form $\partial_{\mu}=\partial/\partial x^{\mu}$ is our
ordinary partial derivative and in radial Rindler coordinates our metric
components gathered in matrix form are given by

$\displaystyle(g_{\mu\nu})=\begin{pmatrix}-\Lambda r^{2}&0&0&0\\\ 0&1&0&0\\\
0&0&r^{2}\cosh^{2}(\sqrt{\Lambda}t)&0\\\
0&0&0&r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}\end{pmatrix}$ (70)
$\displaystyle(g^{\mu\nu})=\begin{pmatrix}-\frac{1}{\Lambda r^{2}}&0&0&0\\\
0&1&0&0\\\ 0&0&\frac{1}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)}&0\\\
0&0&0&\frac{1}{r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}}\end{pmatrix}$
(71)

with $(g^{\mu\nu})$ corresponding to the inverse of the matrix associated with
$(g_{\mu\nu})$ ($g_{tt}=-\Lambda
r^{2},g_{rr}=1,g_{\theta\theta}=r^{2}\cosh^{2}(\sqrt{\Lambda}t),g_{\phi\phi}=r^{2}\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}$).
For the reader who may be unfamiliar with abstract index notation, we look for
each affine connection term associated with the tensor above by examining this
expression in component form:

$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}\sum_{\rho}g^{\lambda\rho}[\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}]$

where we use greek indices (e.g. $\lambda,\mu,\nu$) for components and latin
indices (e.g. $a,b,c$) for the tensor itself. Just as an example, we look for
the component $\Gamma^{t}_{tr}$:

$\Gamma^{t}_{tr}=\frac{1}{2}\sum_{\beta}g^{t\beta}[\partial_{t}g_{\beta
r}+\partial_{r}g_{\beta t}-\partial_{\beta}g_{tr}]$

where we sum over like indices for $\beta=t$, $r$, $\theta$, $\phi$. Then,

$\displaystyle\Gamma^{t}_{tr}=\frac{1}{2}\bigg{\\{}g^{tt}[\partial_{t}g_{tr}+\partial_{r}g_{tt}-\partial_{t}g_{tr}]+g^{tr}[\partial_{t}g_{rr}+\partial_{r}g_{rt}-\partial_{r}g_{tr}]+g^{t\theta}[\partial_{t}g_{\theta
r}+\partial_{r}g_{\theta t}-\partial_{\theta}g_{tr}]$
$\displaystyle+g^{t\phi}[\partial_{t}g_{\phi r}+\partial_{r}g_{\phi
t}-\partial_{\phi}g_{tr}]\bigg{\\}}$

However, from (71), we know that $g^{tr}=g^{t\theta}=g^{t\phi}=0$. Therefore,
our expression reduces to

$\Gamma^{t}_{tr}=\frac{1}{2}g^{tt}[\partial_{t}g_{tr}+\partial_{r}g_{tt}-\partial_{t}g_{tr}]$

Yet from (70), we see that $g_{tr}=0$. This leaves us with

$\Gamma^{t}_{tr}=\frac{1}{2}g^{tt}\partial_{r}g_{tt}=\frac{1}{2}\bigg{(}-\frac{1}{\Lambda
r^{2}}\bigg{)}\cdot\partial_{r}(-\Lambda r^{2})=\frac{1}{r}$

For the reader who wishes to derive the rest of these affine connection
components, we notice from (70) that $g_{\mu\nu}=0$ for $\mu\neq\nu$ (our
metric is diagonal in radial Rindler coordinates). Then the expression for our
affine connection terms reduces to

$\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda\lambda}[\partial_{\mu}g_{\lambda\nu}+\partial_{\nu}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\nu}]$

Yet, because $\Gamma^{c}_{ab}$ is symmetric in $a\Leftrightarrow b$ (i.e. if
we swap $a$ and $b$ indices, our tensor remains the same as one can see above
since $g_{ab}$ is also symmetric under the same exchange by definition), our
possibilities for the affine connection terms are limited to three cases:
$\lambda=\nu\neq\mu$; $\lambda=\nu=\mu$; $\mu=\nu\neq\lambda$. For
$\lambda=\nu\neq\mu$,

$\Gamma^{\lambda}_{\mu\lambda}=\frac{1}{2}g^{\lambda\lambda}[\partial_{\mu}g_{\lambda\lambda}+\partial_{\lambda}g_{\lambda\mu}-\partial_{\lambda}g_{\mu\lambda}]=\frac{1}{2}g^{\lambda\lambda}\partial_{\mu}g_{\lambda\lambda}\indent(\lambda\neq\mu)$

where we used the diagonal property of our metric parametrization in the last
equality. Applying similar logic to our other two cases, we obtain

$\Gamma^{\lambda}_{\lambda\lambda}=\frac{1}{2}g^{\lambda\lambda}\partial_{\lambda}g_{\lambda\lambda}\indent{\rm
and}\indent\Gamma^{\lambda}_{\mu\mu}=-\frac{1}{2}g^{\lambda\lambda}\partial_{\lambda}g_{\mu\mu}\indent(\lambda\neq\mu)$

Using these identities, one finds that our non-zero affine connection terms
are:

$\displaystyle\Gamma^{t}_{tr}=\frac{1}{r}\indent\Gamma^{t}_{\theta\theta}=\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\indent\Gamma^{t}_{\phi\phi}=\frac{1}{\sqrt{\Lambda}}\cosh(\sqrt{\Lambda}t)\sinh(\sqrt{\Lambda}t)\sin^{2}{\theta}$
$\displaystyle\Gamma^{r}_{tt}=\Lambda
r\indent\Gamma^{r}_{\theta\theta}=-r\cosh^{2}(\sqrt{\Lambda}t)\indent\Gamma^{r}_{\phi\phi}=-r\cosh^{2}(\sqrt{\Lambda}t)\sin^{2}{\theta}$
$\displaystyle\Gamma^{\theta}_{\theta t}=\Gamma^{\phi}_{\phi
t}=\sqrt{\Lambda}\tanh(\sqrt{\Lambda}t)\indent\Gamma^{\theta}_{\theta
r}=\Gamma^{\phi}_{\phi
r}=\frac{1}{r}\indent\Gamma^{\theta}_{\phi\phi}=-\sin{\theta}\cos{\theta}\indent\Gamma^{\phi}_{\phi\theta}=\cot{\theta}$

We define the curvature tensor by the action of the linear map
$(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})$ on a dual vector field
$\omega_{c}$ (for more information on vector fields, see Chapter 2 of [24]):

$\nabla_{a}\nabla_{b}\omega_{c}-\nabla_{b}\nabla_{a}\omega_{c}={R_{abc}}^{d}\omega_{d}$

where $\nabla_{a}$ is the derivative operator compatible with our metric (or
covariant derivative; $\nabla_{a}g_{bc}=0$) and we refer to ${R_{abc}}^{d}$ as
the Riemann curvature tensor. Our Riemann curvature tensor can be expressed in
terms of the affine connection associated with a particular choice of
coordinate chart:

${R_{abc}}^{d}=\partial_{b}\Gamma^{d}_{ac}-\partial_{a}\Gamma^{d}_{bc}+\Gamma^{e}_{ca}\Gamma^{d}_{be}-\Gamma^{e}_{cb}\Gamma^{d}_{ae}$

And in component form,

${R_{\mu\nu\rho}}^{\sigma}=\partial_{\nu}\Gamma^{\sigma}_{\mu\rho}-\partial_{\mu}\Gamma^{\sigma}_{\nu\rho}+\sum_{\lambda}\bigg{[}\Gamma^{\lambda}_{\rho\mu}\Gamma^{\sigma}_{\nu\lambda}-\Gamma^{\lambda}_{\rho\nu}\Gamma^{\sigma}_{\mu\lambda}\bigg{]}$

Using our affine connection terms, we discover ${R_{\mu\nu\rho}}^{\sigma}=0$
$\forall\mu,\nu,\rho,\sigma$ as we should expect since the radial Rindler
chart is just a coordinate transformation away from the Minkowski chart (the
geometric properties of the manifold are independent of coordinate
parametrization). Therefore,
$R_{\mu\beta}=\sum_{\lambda}{R_{\mu\lambda\beta}}^{\lambda}=0$ and
$\sum_{\lambda,\mu,\nu,\beta}R^{\lambda\mu\nu\beta}R_{\lambda\mu\nu\beta}=0$.
The metric satisfies the Einstein field equations in vacuum without a
cosmological constant [6] and represents flat space-time.

##  Appendix B Equations of motion

$0=U^{a}\nabla_{a}U^{b}$ (72)

where $\nabla_{a}$ is the derivative operator compatible with our metric (or
covariant derivative; $\nabla_{a}g_{bc}=0$). For the reader who may be
unfamiliar with concepts in differential geometry, the action of this
derivative operator on an arbitrary vector field (a vector field is an
assignment of a vector at each point on the manifold) can be expressed in
terms of our more familiar partial derivatives through the affine connection
tensor associated with a particular coordinate system. When our derivative
operator acts on an arbitrary vector field $v^{a}$, we have

$\nabla_{a}v^{b}=\partial_{a}v^{b}+\Gamma^{b}_{ac}v^{c}$

and for this same derivative operator acting upon a dual vector field,

$\nabla_{a}v_{b}=\partial_{a}v_{b}-\Gamma^{c}_{ab}v_{c}$

Without going into further detail with regard to vector spaces, the reader may
feel more informed to know that we can relate a vector with its dual space
counterpart through the metric:

$v_{a}=g_{ab}v^{b}$

In addition, a vector $v^{a}$ given at each point on a curve $C$ is said to be
parallelly transported as one moves along this curve if

$t^{a}\nabla_{a}v^{b}=0$

where $t^{a}$ refers to the tangent vector to the curve. We then define a
geodesic to be a curve whose tangent denoted $U^{a}$ satisfies (72) (for more
on parallel transport, see Chapter 3.3 of [24]) and assume that our particles
travel along these curves when subjected to no net external forces.
Additionally, a parametrization of a curve which yields (72) is called an
affine parametrization, and thus by definition a geodesic is required to be
affinely parametrized.

For a geodesic along which one of our particles moves denoted
$x^{\mu}(\sigma)$ in our particular coordinate system and parametrized in
terms of the affine parameter $\sigma$, our tangent vector to this curve in
component form is given by $U^{\mu}=dx^{\mu}/d\sigma$ (where $\sigma=\chi$ for
massive particles) and is said to be the ‘proper velocity’ or ‘four-velocity’
of this particle. We also define

$\chi=\int(-g_{ab}T^{a}T^{b})^{1/2}dt$

where $T^{a}$ is the tangent vector to any particular time-like (i.e.
$g_{ab}T^{a}T^{b}<0$) curve and $t$ is an arbitrary parametrization of this
curve. Thus, along a time-like geodesic affinely parameterized by $\chi$, we
have

$g_{ab}U^{a}U^{b}=-1$

Applying all of the above concepts to expand our equation of motion (72) in a
particular coordinate system,

$0=U^{a}\bigg{[}\partial_{a}U^{b}+\Gamma^{b}_{ac}U^{c}\bigg{]}$

Using our expression for the ‘proper velocity’ in component form, we come upon
the geodesic equation of motion for particles in terms of our affine
connection terms:

$\displaystyle
0=\sum_{\alpha}\frac{dx^{\alpha}}{d\sigma}\cdot\frac{\partial}{\partial
x^{\alpha}}(\frac{dx^{\nu}}{d\sigma})+\sum_{\mu,\rho}\Gamma^{\nu}_{\mu\rho}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\rho}}{d\sigma}$
$\displaystyle=\frac{d^{2}x^{\nu}}{d\sigma^{2}}+\sum_{\mu,\rho}\Gamma^{\nu}_{\mu\rho}\frac{dx^{\mu}}{d\sigma}\frac{dx^{\rho}}{d\sigma}$

Therefore, for our radial Rindler metric, we can plug in the affine connection