1. Introduction

2. Accelerated Frames

3. Gravitational Perturbations

3.3. The Gravitational Perturbation Of Clocks

It is not possible to use a single clock to compare measurements between systems of variable gravitational potential. To compare physical systems with differing gravitational potentials we use two separate clocks, one as standard and another as a system variable. Let a laboratory clock T_L that is fixed in space serve as standard and let a second movable clock T_S be introduced as a system variable. The system clock T_S may be subjected to different gravitational potentials and then compared with the laboratory clock T_L to determine how the clock rate changes. Clocks placed in locations of higher gravitational potential speed up with respect to a stationary clock, while clocks in uniform relative motion slow down. Thus, the state of the system T_S is determined by its velocity and its gravitational potential, both of which are observed and measured in continuous increments. Measurements of the time coordinate with respect to T_L are performed without making physical contact, a normal procedure for time measurements in both relativity theory and classical theory, but in disagreement with quantum theory.

The same property of light that allows astronomers to measure the distance of objects at the edge of the universe (spatial linearity) and compare the period of oscillation of red shifts (temporal linearity) can now be used to extrapolate to the very small dimensions of an atomic clock to analyze space-time at the microscopic level. The linearity of the radiation allows the coordinate differences of oscillating electrons in atomic clocks to be easily measured with respect to a flat space-time. Consider a well-known test for relativity theory using cesium beam atomic clocks based on the Cs₁₃₃ isotope. The clocks are flown around the world first in an eastward and then in a westward direction, and later compared to a laboratory clock fixed on the earth's surface [6]. The experiment demonstrates a slowing of clocks Δt_k that is greatest in the eastward direction of flight due to the earth's rotation and a speeding up of clocks Δt_g at higher altitudes due to an increased gravitational potential. They are relativistic corrections to the time of the system clock T_S in the airplane which cannot be interpreted by a non-relativistic model of clocks because they are located in an accelerated system of coordinates. When Δt_k and Δt_g are summed at the end of the flight they will equal the time on the atomic clock T_L in the laboratory.

T_L=T_S−Δt_g+Δt_k

(1)

Because the clock rate is locally determined both on the airplane and at the laboratory the frequency of the Cs atoms, 9,192,631,770 Hz, does not change during the approximately one week duration, ΔT≈ 6x10⁵ sec, of the flights. However, during that time period T_S speeds up relative to T_L an amount equal to approximately 2.7 x 10^-7 seconds due to the combined influence of clock velocity and gravitational potential. The corrections are due to the relative velocity and altitude of the airplane, so they occur simultaneously.

The linearity of time allows the classical interval ΔT, or time of flight of the airplane, to be subdivided into a series of identically constituted clock cycles of period τ that sum linearly.

(2)

where τ_m is the first clock cycle and τ_n is the last clock cycle. The clock periods τ correspond to single ticks of the clock, complete cycles of a transitioning electron between excited and ground states. Each of the cycles includes an infinitesimal correction with respect to the laboratory clock and their sum gives the total correction factor in 1). The correction for each clock cycle is calculated by dividing the total correction by the approximate number of oscillations during the measurement period ΔT. The linear correction factor that must be applied to the clock period τ due to gravitational field is,

≤ 2.7 X10^-23 sec

This corresponds to an uncertainty in the electron’s path length for each period of oscillation of ct ≤ 8.1x10^-15 m. It is a distance that is much smaller than the indeterminacy of an electron relative to the nucleus due to the wavelength, λ=3.26x10^-2 m. The measurement of time is not governed by the uncertainty principle because clock ticks are quantum non-demolition measurements; that is, the uncertainty of the ticks does not increase from its measured value as the system evolves [7]. In other words, measurements disturb the system in a predictable way. It is a determination of the maximum thickness of the electron shell as opposed to a determination of the electron’s position on the shell. That level of spatial precision is necessary to enable periodic electron transitions to reproduce the ticks of a clock with the observed precision. If the surfaces of the atom and its excited states were not completely uniform electron transitions would not be of uniform duration and clock ticks would be irregular.

3.4. Fractional Shifts In Wavelength

A measurement to “test the interplay of quantum mechanics and general relativity on the millimeter scale” was made between optical atomic clocks separated by a difference in height of 0.33 m [8]. After 40,000 seconds of data the authors found that the difference in gravitational potential of the clocks had caused a “fractional frequency shift” of 4.1× 10^-17 cycles/second. This corresponds to a fractional shift in wavelength Δλ.

The fractional shift in wavelength provides an estimate of the “perceived” thickness of an electron shell that is over two magnitudes greater precision than the measurement described in the previous paragraph.

The most recent and also most accurate clock experiments use a single crystal of 100,000 strontium atoms and ultraviolet light to differentiate between the gravitational potential of the crystal’s upper and lower surfaces, a distance of one millimeter [2]. The fractional frequency instability given for that experiment, 7.6X10^-21, leads to the calculation of a fractional shift in wavelength Δλ, indicating the “tolerance” of electron shells during measurements of time.

= 5.3 X 10^-27 m

Therefore each cycle of the atomic clock’s “pendulum”, an oscillating electron, is carried out between surfaces of indefinite thickness Δx ≤ 5.3 X 10^-27 m. The authors conclude, “These results suggest that there are no fundamental limitations to inter-clock comparisons reaching frequency uncertainties at the 10^-21 level, offering new opportunities for tests of fundamental physics.” Apparently the measurement of gravitational potential using atomic clocks is only limited by our ability to differentiate between clocks spatially; that is, by the granularity of matter itself. Thus the interrelationship of gravity and clock time at the quantum level involves all of the fundamental properties of matter in a single experiment.

It is possible to measure clock rate change due to a gravitational potential of one millimeter. Moreover experimentalists have hypothesized that no minimum distance exists, and that clock rate changes are continuous with respect to changes in gravitational potential. To express the idea formally we describe clock function in terms of the Hamiltonian, H=T+V, where T refers to the electron’s ground state energy and V to the energy required to raise it to a higher energy level. Clock experiments show that if we increase the potential of the atomic clock by placing it in a higher gravitational potential it will cause a corresponding increase in the Hamiltonian δ that may be expressed as follows,

H=T+(V+δ)

(3)

where δ may be made arbitrarily small because there are no “fundamental limitations” to changes in clock rate in response to continuous changes of the gravitational field. In non-relativistic quantum mechanics the Hamiltonian operator corresponds to discrete values of a system’s energy spectrum, the eigenvalues, and continuous change does not occur.

In order to explain clock rate change in response to increases in gravitational potential of arbitrarily small amount, fundamental change is necessary in the way we perceive energy exchange at the quantum level. The increases in potential cause increased clock energy, as manifested not by changes in electron transition energy, an invariant, but by an increased number of ticks of the system clock relative to the laboratory clock. The additional ticks increase in number relative to the length of time of the experiment and they are due to an exchange of energy between gravitational and quantum systems. The exchanges of energy occur continuously, thereby contradicting nonrelativistic theory which proposes that exchanges of gravitational field energy are mediated discretely by gravitons. In order to account for continuous exchanges of energy relativistic equations of motion are required, a topic that will be taken up in the next section.

4. Lagrangian Quantum Mechanics

4.1. Differential Equations of Motion

The oscillating electron of an atomic clock transitions back and forth between discrete electron shells, but it moves continuously on its way from one shell to the other, and as it is moving it is being accelerated by the gravitational field. Due to the extremely short distance the electron travels, the tiny duration of its path, and infinitesimal changes produced by the gravitational potential; the curvature of space-time may be neglected. Therefore within the infinitesimally small space-time region of the atom the equivalence principle is not just an approximation it is exactly true and the motion of the electron is subject to the laws of special relativity.

We are now prepared to answer the question posed in the introduction: How is it possible for the electron in an atomic clock to transition within the atom in a regular way while processing inputs from two physically distinct external sources, velocity and gravitational potential? The electron is subject to constraints imposed in two distinct space-times. Its discrete properties are determined by the four-dimensional volume λτ between electron shells; but another infinitesimal part, due to the equivalence principle, is transmitted continuously by gravitational field acceleration. To derive local equations of motion for the electron that include both we use Minkowski space and invoke the concept of invariant spacetime intervals [9]. “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” Neither space nor time is assigned independent reality within the atom, rather a union of the two is preserved in the form of a spacetime interval.

The rate of an atomic clock does not speed up or slow down due to changes in the distance between electron shells. The distance between electron shells and the locally determined number of ticks per second of an atomic clock are invariant properties of matter because atomic structure is an invariant. Thus the electron in an atomic clock is subject to the invariant requirements of atomic structure due to the discrete nature of electron shells, yet at the same time it moves continuously in space-time during transitions between electron shells. For a complete determination of the equations of motion of the electron both discrete and continuous coordinates will be necessary.

From equations 1), 2), and 3) and the experimentally confirmed linearity of light we conclude that the electron of the atomic clock T_S oscillates at frequency ν_Cs with relativistic correction to each cycle due to angular velocity relative to the center of the earth and correction to each cycle due to an acceleration of coordinates between ticks of the clock. An electron in free space accelerates under the influence of a gravitational field due to its mass by undergoing a change in velocity; however, an atomic electron cannot accelerate in the same way as a free electron because the spacing between electron shells is an invariant. To describe the influence of gravitational fields on the electron of an atom we use Minkowski space, hold the spatial coordinates constant, and let time vary. The use of Minkowski space has the advantage that the spacetime distance of an electron transition is invariant, so that all local frames of reference will agree on the total distance in spacetime between the ticks of a clock. The invariant properties of Minkowski space make it indispensable to a description of atomic structure. It allows the microscopic equations of motion of a transitioning electron to be assigned a differential equation for the variation of time with respect to space,

(4)

where τ(x) is the invariant clock period of the atom in cycles per second, represents a continuously applied correction due to kinematical time dilation, and represents a continuously applied speeding up of time due to increases in gravitational potential. In ordinary space-time the motion of a particle is described by its change in position with respect to time. However, the motion of an electron in Minkowski space is given by changes in spacetime interval. Therefore perturbation experiments with atomic clocks in a gravitational field suggest that time is not determined absolutely as in Newtonian mechanics, rather its most fundamental expression is to be found in the relativistic function of clocks as determined by the equivalence principle, where the time between clock ticks is determined by the spacetime interval of an electron transition

4.2. Energy Absorption In A Conservative System

The differential equation of motion 4) describes the influence of a conservative force, gravity, on an atomic clock’s transitioning electron. Hamilton’s principle states that the differential equation of motion for a physical system can be reformulated as an equivalent integral equation, the time integral of a Lagrangian. The Lagrangian gives a complete accounting of the system’s energy with the velocity , the coordinates , and the time as inputs. The true evolution of the electron in Minkowski space is a path of minimum action between two fixed energy states and and two specified times t₁ and t₂.

(5)

The action is formulated in generalized coordinates so it is independent of any particular system of coordinates and it is invariant for all coordinate transformations thereby providing a description that is consistent with the formalism of general relativity theory.

4.3. Energy Absorption in A Non-Conservative System [10]

Due to the conservation of energy, absorption must occur before emission is possible. Quantization by a continuous absorption of energy is not amenable to a description by a purely field model because the fields are not bounded. Instead we use a particle model and describe the electron during continuous excitation between the states |1> and |2>. The Lagrangian for the electron is equal to the kinetic energy T minus the potential energy V, or L=T-V. The transition of an electron from the ground state to an excited state is characterized in generalized coordinates with six dimensions, three to describe its position on the electron shells R₁ and R₂, and three to describe its trajectory.

(6)

The electron initiates its motion at a point on the equipotential surface R₁ of the ground state at time t₁, is excited along a path r, and upon arriving at R₂ after a period of time, (t₂ – t₁) = τ, it assumes the experimentally determined orbital angular momentum, (T-V) = 2πE. In contrast to the Bohr semiclassical model of the atom this derivation of energy absorption assigns the correct value of zero for the ground state orbital angular momentum. The action, S[r(t)], is a functional that describes the absorption process in four dimensions. It has as its argument an infinite number of functions, the possible electron trajectories r(t). It differs from 5) by introducing physically defined end-points for the path as determined by the electron shells. The energy states |1> and |2> determine the energy and corresponding period of a field cycle. When we evaluate the action integral for a quantum mechanical change of state we obtain the familiar Bohr relation describing a photon,

(7)

which is equivalent to the more familiar time-averaged relation E=hν. By treating time symmetrically with the space coordinates in four dimensions a relativistic theory describing energy absorption is obtained that accurately describes the behavior of a quantum oscillator. Electron transitions evolve continuously between the observed discrete ticks of a clock. Thus atomic structure requires two four-dimensional space-times, one discrete and one continuous, to allow implementation of the equivalence principle upon a transitioning electron and permit unification of the electromagnetic and gravitational fields in the simplest possible way, by ordinary coordinate superposition.

4.4. Energy Emission

In the previous section we described in 6) the non-conservative process of energy absorption from the environment and electron excitation. In this section we will describe its natural continuation, electron decay and the emission of a photon. The properties of energy absorption and emission for a bounded electromagnetic system, the atom, are of particular interest in our discussion for we wish to compare them with the time evolution of the absorption and emission of energy in a bounded gravitational system, the galaxy. It is not possible to make a direct comparison of gravitational and electromagnetic systems. Not only are the fields of vastly different strengths, but gravitational potentials do not share differences in polarity that are characteristic of electric charge. Despite glaring differences in their gross outward physical appearance, there are similarities that exist on a geometrical level. The kinetic flow of mass whether linear or rotational creates a transversely directed gravitational field that is analogous to the transverse magnetic field caused by current flow. The induced gravitomagnetic field, or force, is directed perpendicular to the mass flow and has been used to explain properties of relativistic jets emanating from the cores of quasars and other active galactic nuclei [11]. It is postulated that due to mass-energy equivalence transversely directed forces occur in response to mass and energy flows of all types and that they may be compared to electromagnetic flows.

In order to compare the gravitational and electromagnetic energy flows we require an interpretation of quantum mechanics that treats energy not as an observable at single points in time as in non-relativistic theory, but as a continuous time evolution [12]. Dirac described the problem simply and elegantly. “There is an alternative formulation for classical dynamics, provided by the Lagrangian. This requires one to work in terms of coordinates and velocities instead of coordinates and momenta. The two formulations are, of course closely related, but there are reasons for believing that the Lagrangian one is the more fundamental. In the first place the Lagrangian method allows one to collect together all the equations and express them as the stationary property of a certain action function. (This action function is just the time-integral of the Lagrangian.) There is no corresponding action principle in terms of the coordinates and momenta of the Hamiltonian theory. Secondly the Lagrangian method can easily be expressed relativistically, on account of the action function being a relativistic invariant; while the Hamiltonian method is essentially non-relativistic in form, since it marks out a particular time variable as the canonical conjugate of the Hamiltonian function. For these reasons it would seem desirable to take up the question of what corresponds in the quantum theory to the Lagrangian method of the classical theory.”

In Dirac’s first study, “The Lagrangian and the Action Principle”, he theorizes, “We ought to consider the classical Lagrangian not as a function of the coordinates and velocities but rather as a function of the coordinates at time t and the coordinates at time t+dt”. Rather than pick out a particular point in time as occurs with the Hamiltonian function of nonrelativistic theory, Dirac is seeking compatibility with relativity theory by calculating change in action over a space-time interval between two points in time. The results of the study proved to be accurate for they gave Feynman the idea to pursue a path integral formulation of quantum mechanics.

In the next section Dirac develops an idea complementary to the particle model with “An Application to Field Dynamics” that we wish to expand upon. He begins, “We may treat the problem of a vibrating medium in the classical theory by Lagrangian methods which form a natural generalization of those for particles. We choose as our coordinates suitable field quantities or potentials.” To obtain an equation that describes the energy emission by an atomic oscillator using field dynamics we introduce a Lagrangian density given by the fields and its first derivatives L(ϕ_i ,ϕ_{i ,μ}). This allows for a complete accounting of the energy interactions that occur during an electron transition, where ϕ_i is the current density described radially and ϕ_i,μ is the electromagnetic field strength described transversely. The transformation from a classical to a quantum viewpoint is realized by localizing the fields over the “region of space-time” between electron shells. Continuing with Dirac’s analysis, “We introduce at each point of space-time a Lagrangian density, which must be a function of the coordinates and their first derivatives with respect to x,y,z, and t, corresponding to the Lagrangian in particle theory being a function of coordinates and velocities. The integral of the Lagrangian density over any (four-dimensional) region of space-time must then be stationary for all small variations of the coordinates inside the region, provided the coordinates on the boundary remain invariant.”

Dirac’s field interpretation is a natural continuation of our description of absorption energy 6) and the next step in the time evolution of radiation processes. Emission initiates from the excited state R₂ = (x₂,y₂,z₂) at time t₂ and it finalizes at the ground state R₁ = (x₁,y₁,z₁) at time t₁. Each of the electron shells R₂ and R₁ determines a locus of points where the fields vanish and therefore they represent invariant field boundaries. We continue with Dirac’s arguments by requiring the integral of the Lagrangian density over the region of space-time between the excited and ground states to be “stationary for all small variations of the coordinates inside the region.” Changes in action are evaluated by integrating the Lagrangian density four-dimensionally thereby yielding a relativistic formulation of emission that is invariant, the same for all observers.

(8)

The end points of the electron’s path are located on equipotentials, space-like surfaces, and the four-dimensional localization of the fields necessarily corresponds to the result from 7), and the reduced Planck’s constant ћ. The action S[ϕ_i(t)] is a functional, a function of the values of coordinates on the discrete boundaries of the space-time surfaces R₂ and R₁ which are in turn functions of the continuous space-time variables of the fields within the surface. The boundaries of the fields are uniquely fixed four-dimensionally by the volume ∫d³x and the time interval t₂-t₁. The field model given by 8) describes one cycle of fields, or photon emission, as opposed to the particle model 6) which describes the excitation of an electron and one cycle of the fields. Two electron transitions, excitation and decay, equal one clock cycle and two field cycles.

Comparing 6) and 8) we see that emission and absorption are not symmetric processes since it is impossible to obtain the field distribution leading to absorption from that which resulted due to emission. Absorbed fields are unbounded whereas the emissions are bounded in space and time by the wavelength and period. Combined they represent a fully relativistic model that explains the quantization of energy as a four-dimensional localization of field in space-time, the photon. The electron shells define spatial boundaries in a way that is similar to the way the event horizon defines spatial boundaries for a black hole.

5. Gravitational Field Energy

5.4. Gravitational Emission Equation

In equation 6) we described the four-dimensional localization of electromagnetic field energy due to linearly superposed field potentials and resulting photon creation. Space-like field boundaries in the form of electron shells and discrete times of transition localize fields consisting of both radial and transverse fields. The localization of gravitational field energy by means of a superposition of potentials is also characterized by the formation of a space-like field boundary, the horizon, with the passage of time Δt=0. In addition to the symmetry that exists in the above mentioned mass-energy conversion processes other symmetries are apparent when comparing field properties. The relativistic superposition of electromagnetic potentials is described by a Lagrangian density composed of radial and transversely oriented force fields with distinct physical origins. Similarly the superposition of gravitational potentials leads to a radially oriented baryonic force field observed in the galactic bulge and a transversely oriented force field that is manifested by the tangential acceleration of matter contained in the disc and observed as rotational velocity. Symmetry arguments based on a comparison of gravitational and electromagnetic potentials support a theory that includes superposed forces of independent physical origin; a localized, radially oriented 1/r² potential due to baryonic matter and a diffuse, transversely oriented 1/r potential which creates a tangential acceleration. Therefore a theory is favored that can explain galactic structure by means of a continuous field law in a way that is more closely aligned to the general relativistic concept of space-time than to a model based on dark matter or a modification of Newtonian laws.

We can show a formal relationship between the energy flow of photon creation, as expressed by the action integral in 8) and black hole formation, if it is assumed that the black hole represents a field source. The field is described by a Lagrangian density L(Φ_i,Φ_i,μ), where Φ_i represents radial fields due to localized baryonic matter and Φ_i,μ represents transverse fields generated independently by a different form of matter, localized within the same black hole. It differs from a MOND type theory of acceleration which postulates a modified Newtonian law with 1/r² acceleration within the galactic bulge that gradually transforms at large distances to a 1/r acceleration [16]. The concept of a “two component” source for black holes gives a different interpretation of the Tully-Fisher relation, which states that multiplying the luminosity of a galaxy by the period of rotation yields a constant [17]. The units of the constant, energy times time, are of action and may be compared due to symmetry arguments to the quantization of electromagnetic fields in equation 7); that is, to the equation Eτ=ћ. It is hypothesized therefore that because the laws of energy apply universally in nature there must be an analogous equation,

Eτ=H

describing the transformation of gravitational field energy that is equivalent to the quantization of electromagnetic field, where H has the units of action and is the gravitational equivalent of the reduced Planck's constant. It means that in a spiral galaxy we expect the two sources of field Φ_iand Φ_i,μ to be present within the black hole with the same intensity, just as the electric and magnetic fields of a photon have the same intensity. It also correctly predicts that observations of increased galactic emission energy, or energy flow, are an indication of a higher rotational velocity and a lower period of rotation, as observed.

It is a short step from postulating that the Tully-Fisher correlation represents solutions of an emission equation to obtaining the equation. Symmetry arguments based on equation 8) suggest that the action integral of a galaxy’s energy flow is a function of the values of the space-like coordinates on the event horizon R which are in turn a function of the continuous space-time variables of the matter within the surface, where integration is performed from zero to R and for one complete revolution of period τ.

(10)

The action integral S[Φ(t)] is a functional describing the transformation of matter and energy localized within the event horizon of a black hole into the emission energy of a galaxy with rotational period τ. Because the properties of energy, when described four-dimensionally with time, are independent of the material system which supports them we can compare the the gravitational emission equation 10) with the electromagnetic emission equation 8). The existence of an emission equation governing galaxy energy flow suggests that galaxy structure is the manifestation of a field emanating from a supermassive black hole, and it consists of a radial component Φ_i due to baryonic matter causing gravitational attraction and a transverse component Φ_i,μ from a distinct source causing the tangential component of radial acceleration. The proportion of baryonic mass in a black hole contributing to radial acceleration as compared to the mass contributing to rotational acceleration determines the observed bulge to disc ratio of galaxy structure.

The photon is the most elementary form of baryonic matter Φ_i and it travels at speed c. It is hypothesized by analogy that the neutrino is the most elementary source of rotational acceleration Φ_i,μ due to its spin and that it also travels at speed c. Just as photon fields are highly complex and essential for understanding our local environment this model suggests that neutrino fields are complex and essential for understanding the cosmological environment. Continuing with our analogy, we assign the spin of the free neutrino left or right-handed helicity and an orientation in space from 0 to 2π in the same way that the photon is oriented by polarization. A high density of superposed neutrinos, localized within the black hole of a galaxy with their spins aligned, exert rotational acceleration on matter in the disc in accordance with 10) by means of a 1/r field that extends to infinity. Due to action/reaction symmetry free neutrinos surrounded by matter experience a rotational force in the opposite direction of the localized effect, where the energy of a neutrino determines its resistance to rotation due to inertia in compliance with the conservation laws. The orientation in space of a free neutrino determines whether it is able to interact with an electron, muon, or tau lepton; where the time of passage and density of surrounding matter determines the rotation of a neutrino, hence its final orientation. Neutrinos with fields of intermediate orientation are “sterile” since they do not interact with leptons. The field model differs from the standard model because it describes neutrino oscillation not as a result of a quantum mechanical superposition of neutrino eigenstates, but as the classical superposition of neutrinos with distinct field orientations acting in harmonious conjunction with the matter surrounding it. The classical neutrino model hypothesizes that flavor oscillation is dependent upon the density of the matter it traverses. This provides a means for testing the standard model which describes it as a property of flavor eigenstates.

The hypothesis from 10) that galaxy structure is determined by the localized matter and energy of a black hole is supported by experimental observations noting that every galaxy has a black hole at its center and the radial acceleration of stars in the disk is directly proportional to galaxy luminosity [18]. Because the influence of a galaxy’s localized field energy extends throughout space to infinity it governs the dynamics of galaxy clusters as well. The forces are mapped by astronomical observations on a flat background space in a reference system whose origin resides at the center of the black hole. The electromagnetic and gravitational emission equations 8) and 10) are action functionals which describe the transformation of localized energy to energy manifested externally by fields with radial and transverse components. The fact that they are derived relativistically demonstrates a formal symmetry that is absent from non-relativistic theories that use ad hoc methods to postulate the existence of unobserved forms of “dark” matter.

6. Discussion

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