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:(1)where ; , , , , and is a positive constant. In a subsequent section, we'll deduce that must be the square of the Hubble constant. denotes the invariant interval associated with this metric where assuming 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 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 forward (i.e. replace where 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 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 in Appendix S1. From these calculations it is clear that this space-time geometry is indeed flat. Taking the Rindler transformation equations, , , we find our metric equation becomeswhere speed of light in the local Minkowski reference frame [25]. If one operates under the assumptions of special relativity, would in fact equal , 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 and right wedge for [26]. For the rest of our analysis, however, we no longer assume that special relativity is valid throughout globally flat space-time (again, ) 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 and , respectively. Additionally, as , 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 in Appendix S1, we find for the equations of motion of a particle within a particular inertial system (, where our ‘proper velocity’ in component form is ):(2)(3)(4)(5)And our norm for the ‘four-velocity’ is given by(6)whereand for massive particles. Notice that our ‘four-velocity’ in this theory is dimensionless for spatial components and has units of [time]/[distance] for our time component since (and therefore ) has units of [distance]. Multiplying each term in our radial equation of motion by and plugging in (6),(7)But to remain at a constant radial distance away from our inertial center: , . Therefore, only massless particles can have circular orbits.

Possible geodesic paths obey the relation from (6), and solving for and , we find thatExamining our equation of motion (4), we see that a particle remains at a constant value of for non-zero angular velocity in if and only if and , , . Consequently, the angular velocity of a particle traveling in the equatorial plane () of this inertial reference frame is bound by the range:(8)Then, for a photon traveling in a circular orbit in the equatorial plane, we find(9)Later, we'll see that a massless particle can have circular orbits only for (orbits with 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:And solving for the radial acceleration, we find that(10)In this limit, the inertial motion of our point-particle is described by a spatial acceleration in 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 ) is dominated by angular velocities with approximately circular radial motion (). Our equations of motion reduce toPlugging and into our expressions for and :Now, if we assume and integrate:where for light taking a circular orbit in the equatorial plane and thus for . Plugging in for , we have(11)where is a constant. In the large limit:(12)(13)(14)As long as the particle is not located at either of the poles (, ), we see a sinusoidal spatial angular acceleration that decreases with and moves the object toward . One can then picture spiral galaxy formation resulting from objects orbiting an inertial center with large angular velocity in .

If we refer back to our expression for , we find for the orbital velocity () of a particle in this limit:(15)And for , 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 , 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 and therefore . 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 by analyzing the inherent redshift that occurs in these inertial frames (i.e. we'll take to be the Hubble constant). Since , this value will also give us an upper limit for the slope of our orbital velocity curves at large . 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 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 in Appendix S1 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:(16)(17)(18)Applying Noether's theorem (),(19)(20)(21)Plugging into (6) and solving for , we find that(22)where(23)requiringNotice, for massive particles moving radially in the equatorial plane (, , , and ), this constraint reduces to: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 would equal the energy of the particle divided by its rest mass (i.e. in special relativity, would be equal to where is the energy of the particle). Using (19), (20), (21), and (22), we find(24)(25)(26)where(27)(28)(29)For light traveling radially in the equatorial plane, , and (24) reduces to(30)giving us(31)where is a constant signifying the radial position of the photon at . 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 region of our inertial system to which we regard as our inertial center point. Integrating from to ,Solving for ,(32)For a photon traveling radially inward, the sign of the root is negative, and it reaches inConsequently, not even light can reach 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 . However, to evaluate all of these velocity expressions for , we return to symmetry equations (19) and (20):Therefore,(33)Plugging into (23),which one can plug back into (22) to find consistency with our expressions for above. Yet we see from (27), (28), and (29) that(34)(35)(36)which implies for that all of these terms vanish. Then our spatial velocity terms for becomewhere we have used (33) in these limit expressions. So we see that our velocity terms are not necessarily divergent for . However, we'll address the issue of motion for small 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 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 (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 as . First, we determine the limiting value of :Then for massive particles () assuming or ,(37)which is just the equation for a massless particle traveling radially inward. When or , we must return to conservation equations (20) and (21). We find andPlugging in (19) and solving when , we again find (37). We now understand that eventually all massive particles move toward . 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 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 back to the rate at which physical clocks are observed to tick.

However, we progress onward and look at the large limits for both and . Beginning with the former, we findPlugging into and examining for massive particles,(38)And for or ,(39)Solving for when or using (25) and ,At the poles, particles have no angular velocity in nor angular momentum in (). Lastly for our large limits, we have :Plugging into ,For massive particles and assuming or ,(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 and angular momentum in , slowing down in radial velocity as they approach the center point about which they orbit.

Looking back at our expression for , we ask ourselves the question: for what values of is most positive? For positive , we have particles moving radially outward, and maximizing this expression with respect to 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 since and in this limit:But the largest positive value of occurs if we minimize the denominator of the first term in brackets with respect to . Clearly, this term needs to be re-evaluated when or . Returning to conservation equations (19) and (20), we solve for the radial motion of a particle through the poles by plugging into (6) ( or and ):For large proper radial motion, we assume (as is our analogue of the rest mass condition from Einstein's special relativity). Then, our expression for radial motion through the poles reduces to(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 values for which 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 , and , . For the two, we obtain from (19)(42)In the former situation (, ), we substitute (42) and (21) into (6) and arrive at(43)Whereas for the latter case (, ), we plug (42) and (20) into (6) to findwhich 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 , 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 of a photon with momentum () emitted/received by an observer traveling with proper velocity in component form given by is (see Chapter 6.3 of [24])where 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 , we havewhere and 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. in this localized region), it appears necessary to assume that, in our theory of inertial centers, the wavelength of a photon with wave-vector emitted/received by an observer with ‘proper velocity’ is given bywhere we emphasize to the reader that in our theory the component form of the ‘four-velocity’ for our observer is affinely parametrized by (i.e. ), in direct contrast to special and general relativity for which the four-velocity of an observer would be affinely parametrized by proper time . Proceeding with our radial treatment, the wave-vector for this photon is of the form,And the wavelength observed by a radially traveling individual is given byUsing the Killing vector field in (16), we obtain the conservation law ():But for a photon, . So,where the positive root corresponds to light traveling away from and negative to light traveling inward. Solving for the motion of the observer in this particular inertial reference frame,And for an observer nearly at rest with respect to the inertial center about which he/she orbits,Taking time to move forward, we find thatBut from our earlier analysis, we found that a radially traveling photon abides by the equation, . Thus,(44)Then for a light signal sent between two observers at rest in this inertial frame, the redshift factor is given by the expression:(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 equal to(46)where . We notice that(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 ,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 and ). 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 ( 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 ( 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.

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Table 1. Redshift from objects within the Local Group.

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Table 2. Redshift from galaxies within h of 0 h in RA within Local Group.

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Until now, we have assumed that our coordinate time can take values between without explicitly examining the motion of particles in the region. Reducing our analysis to solely radial motion away from the poles, we analyze geodesic paths in Minkowski coordinates () first for simplicity. However, we must be very clear that under our assumptions 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 . Then our equations of motion reduce toleading to the straight lines that we expect in Minkowski coordinates:(48)where is a constant bounded by . We leave the physical interpretation of the Minkowski constant in this theory of inertial centers for the next section. However, using our transformation equations, we find in radial Rindler coordinates(49)One immediately notices that for massive particles (), both limiting cases of result in the particle heading inward toward the 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 center point of the particular inertial frame in question (see Figure 1).

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Figure 1. Geodesic paths in Minkowski coordinates.

https://doi.org/10.1371/journal.pone.0078114.g001

Reduction to special relativity

Taking the differential of both Rindler transformation equations:wherePlugging in these expressions above, we find thatIf we localize our view of space-time such that all differential terms(50)we will then haveFurther, we require for this local patch of space-time that(51)and our transformation equations reduce to(52)(53)with differential expressions(54)(55)

For the observer remaining a radial distance 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 and in this small limit. If one takes 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 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 away from the inertial center point about which he/she orbits at , will find:(56)If we treat 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 , 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, for stationary observers located at any particular radial distance 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. ), we have where is given by (56), effectively ensuring that our coordinate time progresses at the same rate as the proper time 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 asAnd 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:where , , and for and , , and 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. 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 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).

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Figure 2. Local approximation of the inertial frame of reference.

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Furthermore, it is clear that in order to express 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 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(57)where 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 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, where ). 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 . 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 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 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 (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. 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:where() describe our local gravitational system and () 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 objectwhere , , and are constants, is taken to be small, and we only examine the region of the inertial system. Thus, 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 flat space-time Minkowski coordinates given by () in the local stationary limit, we employ the Schwarzschild metric noting that our affine parameter is approximately given by . As well, 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 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 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 (, ), we havewhere is the wavelength measured by our observer. Applying conservation laws for and using the time-translationally invariant Killing vector field for the Schwarzschild metric, :And taking into account the motion of the observer and photon ( and )Plugging into our expression for the observed wavelength of the photon,(58)where and refer to the photon/observer traveling radially outward/inward () in the local system (in ). If we assume the observer to be nearly at rest in the local frame (), then and expression (58) reduces towhere in the following we approximate in the small 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 . Therefore, we find a slight modification to the Schwarzschild redshift factor:(59)where refers to the radial position of the absorber/emitter in the inertial reference frame (i.e. relative to the inertial center) and to the radial position relative to the center of our massive object 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 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 iswhere our values in this example refer to local radial distances away from the center of the Sun and 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 relative to changes in , we havewhereSince Pioneer 10 was on course to travel away from the center of the Milky Way in the general direction of Aldebaran [4], 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, where 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. ). 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 approximatelyto 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 s. The consistency of this value with that of the Hubble constant [20] lends support to the argument that the time-scale 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 . 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]:where is the derivative operator compatible with the metric (i.e. ), , is the mass associated with our field, and 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 havewhere and refers to the Minkowski metric components. Making the substitution , 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 bywhere now 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 and not by proper time . Unfortunately, there does not appear to be a natural operator substitution for . 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 to find(60)where and 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 in Appendix S1, we must “slice” our manifold into space-like hypersurfaces each indexed by (). For our radial Rindler chart, the future-directed unit normal to each is given by(61)where the absolute value is necessary to keep future-directed for all values of , 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 ( and ), and thus the inner product of our Klein-Gordon extension is given by(62)where the bar symbol indicates complex conjugation (i.e. is the complex conjugate of ), is the union of these two radial Rindler wedge space-like hypersurfaces, is the unit normal to our space-like hypersurface , is the induced Riemannian metric on (; denotes the matrix associated with these Riemannian metric components), and 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 (the absolute value is not a smooth function) as well as the undefined behavior of our unit normal for , 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 over the range: , where 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 (positive and negative) to cover our inertial reference frame and naively ignore the issues with 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. ). We find that our inner product is given by(63)Our remaining task reduces to solving for solutions () to our extension of the Klein-Gordon equation (60). From Appendix F in Appendix S1 which utilizes [45], [46], [47], [48], and [49], we find(64)where and . is the spherical harmonic of degree and order . We maintain convention and use to denote the order of . However, this is a quantum number very different from the mass of our scalar field. The mass term is contained solely in our expression for . is the Macdonald function (modified Bessel function) of imaginary order . is the Legendre function of degree and imaginary order .

Notice, we allow to have domain: where can take both positive and negative values. Physically, this interpretation requires the existence of the field in both the negative and positive regions of the inertial system which brings us back to the discussion earlier in this section of our concern with . From [50], the limiting behavior of expressed aswhere and is the gamma function along withshows that oscillates for small when and exponentially decays for large . In addition, from Figure 3, we see that our radial ‘wave function’ spreads out away from for larger ‘momentum’ values of , allowing for oscillatory behavior at larger values of and thus an increased likelihood of observing quanta farther away from the inertial center of the reference frame in question.

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Figure 3. Plots of for , and .

https://doi.org/10.1371/journal.pone.0078114.g003

Our Heisenberg field operator can be expanded in the following manner (see Chapters 3.1 and 3.2 of [43]):(65)where the annihilation and creation operators in terms of our inner product areand 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):

A very important point for the reader to take away from our analysis in this section is that our field operator as defined in (65) exists in both the and portions of space-time. In other words, we take space-time to be comprised of both the and 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 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 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, ). Then from our classical analysis of geodesic paths, our observer experiences a radial acceleration according to (10) ofwhere coincides with the proper time 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 bywhere the () 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(66)Thus,where the 's refer to the Minkowski constants for each particular reference frame (56). Plugging in above,(67)where .

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 (; 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:(68)Then our Hamiltonian (; see Chapter 8 of [52]) for this system is given by(69)where is the mass of our observer and is the kinetic energy associated with his/her motion as observed in the external frame. If our observer is on the order of m [53] away from his/her local inertial center and is found in the external frame to be m/s [54], we find s. 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. 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?