The Principle of Equivalence: Periastron Precession, Light Deflection, Binary Star Decay, Graviton Temperature, Dark Matter, Dark Energy and Galaxy Rotation Curves

The nature of the principle of equivalence is explored. The path of gravitons is analyzed in an accelerating system equivalent to a gravitating system. The finite speed of the graviton results in a delay of the gravitational interaction with a particle mass. From the aberration found in the path of the graviton we derive the standard expression for the advancement of the periastron of the orbit of the mass around a star. In a similar way, by analysing the aberrations of the graviton and light paths in an accelerating reference frame, the expression for the deflection of light by a massive body is obtained identically to the standard result. We also examine the binary star system and calculate the decay in its orbital period. The decay is attributed to the redshift of the graviton frequency relative to the accelerating system. Here too, we obtain good agreement with experimental measurements. Also, hypothesizing that gravitons behave like photons, we determine the temperature of the gravitons in a binary star system and form the Bose-Einstein distribution. Finally, we show how the redshift of gravitons may be the source of dark matter, dark energy and flat line spiral galaxy rotation curves.

gravitons in an equivalent accelerating system using the principle of equivalence. The finite speed of the graviton causes a delay in the travel time in the interaction of the graviton with the planet. This travel time delay leads to an aberration of the graviton path relative to the planet. From this aberration, the standard expression is obtained for the periastron advance of the planet's orbit.
Similarly, the same analysis is applied to the deflection of a beam of light grazing a star. Due to the finite speeds of the graviton and photon, there are delays in their travel times. We show that the delays cause aberrations of the graviton path and the light path. Combining these aberrations yields an expression for the deflection of light passing through the gravitational field of a star, identical to the standard value.
We also look at the binary star system and focus on the decay of its orbital period. By accounting for the Doppler redshift of the graviton frequency in free fall in a gravitaional field, which produces a decrease in graviton energy, we obtain an expression for the decay of the period based on Newtonian orbital mechanics, which gives close results when compared with the report on the binary pulsar B1913+16. A course fit to data from eight PSRs produces good agreement throughout the varied types of binary systems. A comparison is made with the General Relativity analysis of the binary star systems studied. Likewise, assuming that gravitons behave like photons, we determine the temperature of the gravitons in a binary star system and describe the Bose-Einstein statistics. We show how the decrease of graviton energy during the expansion of the universe may be the source of dark energy.
Finally, we briefly describe how graviton redshift in spiral galaxies may appear as dark matter causing flat lining of rotation curves.
Except for the concepts that the graviton is the agent particle of gravitation, that a beam of light is composed of photons, that both particles travel at a fixed speed c in vacuum in any reference frame, and that of the equivalence of mass and energy, only Newtonian and Galilean principles are invoked in this study. This paper is based on an earlier publication by the same author [3]. We show that the earlier physical theories can go much further in the way of explaining certain phenomena which were thought to be only explainable by the General Relativity theory.

Planetary System Analyzed in an Accelerating Reference Frame
We seek to examine some essential properties of a gravitating system of a mass m orbiting a mass M, such as a planet orbiting a star. The equivalence of inertial and gravitational mass (Weak Equivalence Principle) has been proven experimentally to a high precision, ( ) ( ) ( ) 15 Titanium, Plutonium 1 9 stat 9 syst 10 . This physical principle allows us to transform into a frame S accelerating at the same rate as the gravitational source we are studying. The acceleration a of the reference frame S is shown in Figure 1, and is upward along the negative x i axis as viewed by an Journal of High Energy Physics, Gravitation and Cosmology inertial frame I. The infinitesimal travel time dt, as viewed from an inertial frame I, is given by where dr is the separation distance along the y axis between the graviton due to mass M interacting with particle mass m, and c is the speed of gravity, where c is also the speed of light in vacuum. The graviton g and particle mass m are aligned along the x axis direction and are in free fall in the direction of the positive x axis of frame S. The acceleration a of frame S is small so that all free fall velocities are much less than the speed of light. Therefore, we can ignore the effects of Special Relativity and use Galilean and Newtonian physics.
In frame S at time t 0 the initial velocity of free fall is v 0 . At time 1 0 d t t t = + the free fall velocity is v 1 and the velocity change dv is ( ) and the incremental distance dh of free fall of graviton g and mass m is Referring to Figure 1, consider the path of graviton g traveling to particle m. The angle q that the graviton path makes with the y axis of frame S, without aberration, is given by The speed of the graviton g is c and its square magnitude is given by where the component magnitudes in the x and y directions are given by, respectively, Since the reference frame S is moving at a non-zero relative velocity = when the observation is made at the time t 1 , there will be an aberration of the graviton path viewed from S, analogous to the aberration of light [5]. The total angle of aberration Δφ of the graviton g travel path for an observer in S, with respect to the mass m at the position (t 1 , dh, dr) when the speed of free fall along the x axis is dv in frame S, is given by

Aberration of a Planet Orbiting a Star
Assume that the acceleration is equated to Newtonian gravitation, 2 a GM r = . Then, we have for the incremental velocity and distance traveled in the time interval d d t r c = between t 0 and t 1 , where the graviton g travels to the particle m, and the angle that the photon makes is given by Assuming that the angle q is very small in this analysis, as it would be for typical orbits around stars, we can make the approximations and ( ) Then, from (8), the aberration Δφ of the graviton ray as seen at time t 1 in frame S at the particle m position is given by Assume that the mass m is a planet orbiting a star of mass M. The orbit will be an ellipse with periastron Rp and apastron Ra. The aberration measures the ef-fect of the delay in response of the mass m to mass M, which appears as a phase shift in the orbit. The total aberration δΨ for a full orbit is obtained by multiplying by dφ and integrating over φ from 0 to 2π and over r between positive distances Rp and Ra, expressed by where, since only the minimum and maximum radial distances are required, the integrals over r are integrated between from Rp to ∞ and from Ra to ∞. Equation (15) is also expressible in the form ( ) is the semilatus rectum, and where A is the semi-major axis and ε is the eccentricity of the orbit. For the integration over temporary variable u, the full integral is twice the value of the integral with u going from to r to ∞, one for the minimum and one for the maximum radial distances, hence the differential expression (2du). This is similar to what was done in (15). Equations (15) and (16) both give the standard expression for the advancement of the periastron of the planet's orbit [6].

Photons Grazing a Star Analyzed in an Accelerating Reference Frame
Now we seek to examine some essential properties of a gravitating system of a photon p passing by a mass M, such as a star. Again, as in the previous sections, we invoke the equivalence principle of inertial and gravitational mass and transform into a frame S accelerating at the same rate as the gravitational property of the star we are studying. The acceleration a of the reference frame S is shown in Figure 2, and is upward along the negative x i axis as viewed in an inertial frame I. As before, the infinitesimal travel time dt, as viewed from the inertial frame I, is given by where dr is the separation distance along the y axis between the graviton due to mass M interacting with the photon of mass m, and c is the speed of gravity, where c is also the speed of light in vacuum. The photon mass m p c = where p is the photon momentum. The graviton g and photon p are aligned along the x axis direction and are in free fall in the direction of the positive x axis of frame S. The acceleration a of frame S is small so that all free fall velocities are much less Journal of High Energy Physics, Gravitation and Cosmology than the speed of light. Therefore, we can ignore the effects of Special Relativity and use Galilean and Newtonian physics.
In frame S at time t 0 the initial velocity of free fall is v 0 . At time 1 0 d and the incremental distance dh of free fall of graviton g and mass m is Referring to Figure 2., consider the path of graviton g traveling to photon p. The angle q that the graviton path makes with the y axis of frame S, without aberration, is given by The speed of the graviton g is c and its square magnitude is given by where the component magnitudes in the x and y directions are given by, respec- (23) Looking at Figure 2, consider the path of photon p traveling to graviton g. The angle q that the photon path makes with the y axis of frame S, without aberration, is given by The speed of the photon p is c and its square magnitude is given by (27) Since the reference frame S is moving at a non-zero velocity when the observation is made at the time t 1 , there will be an aberration of the graviton and photon paths for an observer in S. The apparent angle Δφ that the photon path makes with the y axis relative to the star of mass M at the interaction point (t 1 , dh, dr), as observed in frame S, depends on the graviton angle, because the graviton gives the direction to the source. The tangent is given by the sum of the graviton and photon speeds in the x direction, plus the free fall speed relative to S, over the photon's speed in the y direction, expressed by, (28)

Deflection of a Photon Beam Grazing a Star
Referring to Figure 2, assume that the acceleration is equated to Newtonian gravitation, 2 a GM r = . Then, we have for the velocity and distance traveled in the time interval d d t r c = between t 0 and t 1 , where the graviton g travels to the photon p and the photon p travels to the graviton g, The angle that the graviton and the photon paths each make with the y-axis, using (19), is given by Assuming that the angle q is very small in this analysis, as it would be for typical photons grazing a star, we can make the approximations and ( ) Then, the total aberration Δφ of the photon interacting with the graviton as seen at time t 1 in frame S at the interaction position (t 1 , dh, dr), from (28), is given by Assume that the photon p is grazing a star of mass M and approaches the star from a great distance, making a closest approach of R from the center of the star and then continues away from the star to a great distance where observation of the total aberration angle (deflection angle) is determined. The total aberration δΨ for the photon's observed path is given by where we specified −dr in the first integral because the integration is in the direction of decreasing r, whilst the second integral specified +dr for an integration with increasing r. Equation (35) is the standard expression for the deflection of a beam of light from a distant source grazing a star [6].

Graviton Physics
Relative to an accelerating reference frame, for a small increment in velocity δv due to the motion of the frame, the change observed in the angular frequency ω due to the Doppler effect upon a graviton in free fall in the frame is given by where the change in the frequency δω is negative since it is red shifted because the velocity change δv is in the same direction as the motion of the graviton. The change in energy of the graviton is given by where ħ is Planck's constant over 2π. The change in momentum of the graviton is then given by The rate of momentum change is the force, which is given by where the mass m g of the graviton is given by , where λ is the graviton wavelength, we obtain ( ) where δξ is the energy decrease in the graviton due to a decrease in its frequency, where ( ) is the frequency of that energy decrease in the graviton. The graviton's free fall velocity change δv in the gravity field GM/r 2 is always in the same direction as the graviton's motion, implying that the frequency change 0 δω < and the graviton's energy always decreases.

Period Decay of a Binary Star System
In the astronomical observation of a binary pulsar, as in a neutron star binary system, a key equation to obtain is the expression for the evolution of the orbital period T as a function of time t. The starting point is Kepler's third law given by where G is Newton's gravitational constant, M 1 and M 2 are the masses which are assumed to remain constant, and A is the semi-major axis. Taking the derivative of (45) with respect to time, we get ( ) Another equation we utilize is for the orbital energy E, given by Take the time derivative of (47), yielding Substituting for dA/dt from (48) into (46) and simplifying, we get ( ) Equation (37) defines the energy change for a single graviton of frequency ω.
We submit that all gravitons experience the same frequency decrease due to Doppler redshift. Then, given that the total graviton energy Ξ is included in the total orbital energy E of (47), the change in the orbital energy due to the redshift of the graviton energy is expressed by where we applied the solution, where ε is the orbit eccentricity and the phase ( ) ( ) where ϕ is the true anomaly, accounts for the periastron advancement. dE/dt is the decrease in the graviton energy due to Doppler redshifting of the energy of the gravitons in free fall relative to the equivalent accelerating reference system. We do not yet possess an expression for the total graviton energy, but we can use physical principles to try to approximate it. We begin with a constant of the motion of an elliptic orbit, where the semi-major axis ( ) Examination of (54) reveals that it is in the form of kinetic energy 2 mv . Assuming the total energy of the gravitons is proportional to this, multiply the right hand side of (54) by where the first factor on the right hand side of (55) also accounts for the dual polarity of the gravitons. .
Equation (57) is equivalent in form to the derivation given by [7], which was derived from General Relativity with gravitational wave emission for energy decay. For comparison, we put that equation into the form  Except for the dependence on the eccentricity ε and proportionality factor, (57) and (58) are identical, due to the choices made for the form of the mass dependencies in (54).

Application to PSR B1913+16 and Other Binaries
We look at the report [8] on B1913+16, the Hulse-Taylor binary pulsar. This astronomical endeavor spanned 30 years of approximately yearly observations of the binary system.
The data for the system is as follows: ( ) is the solar mass, P b is the binary orbital period, Ṗ b is the orbital period change and ε is the orbital eccentricity.
For this paper we refer to the masses as primary M 1 and companion M 2 , the orbital period as T, the period change as dT/dt and the eccentricity as ε. The initial orbital period is 0 0.322997462727 day b T P = = . For the gravitational constant we use the value  Table 1, we list eight PSR's, [8]- [16]. Journal of High Energy Physics, Gravitation and Cosmology For all the PSR's in Table 1, the mean error and unbiased standard deviation of the mean error between the observed intrinsic dP b /dt values and this paper's predicted dT/dt values are Prediction Mean Error 0.1988 0.1178 . For a comparison with the standard GR gravitational wave emission theory, the mean error and unbiased standard deviation of the mean error are

Quantized Graviton Energy Distribution
From (55) with (45) we obtain the total graviton energy Ξ in terms of the semi-major axis A, Assume that the gravitons in a binary star system are confined to a conical frustum volume contained by the space between the two stars in the orbit. The volume V depends on the distance between the stars, r, and the diameters d 1 and d 2 of the masses M 1 and M 2 , respectively. The volume is given by, Then, with the graviton energy density V ρ = Ξ , the graviton field temperature T is obtained from the Stephan-Boltzmann law, By Wein's Displacement law, the frequency ω p where u(ω) peaks is given by For B1913+16, assuming the mass diameters are 22 km D d ≈ ≈ , by (62)-(65), the temperature of the graviton field is . In Table 2 we list the characteristics of the graviton Bose-Einstein statistics for the eight binary pulsars. Most binary NS diameters are assumed to be 22 km. PSR J0737-3039 is an eclipsing binary which enabled the determination of the companion diameter (18.1 km) [11]. Also, PSR J1012+5307 has a white dwarf companion whose diameter was also determined ( ) 0.094 sol R [15].

Gravitons as the Source of Dark Matter and Dark Energy
Consider the universe as a sphere of interior mass M with a thin spherical shell of mass m. The masses M and m are constants. The thin shell has a radius r(t) at time t. Only the mass interior to the shell has an effect on the shell. Define the graviton energy Ξ(t) within the shell at time t by ( ) ( ) GMm t r t Ξ = . (66) The distance r 0,k for any galaxy is identified by observing the leveling of the velocity rotation curve [18]. We show in a subsequent section how r 0 can be determined for any galaxy for which the Tully-Fisher relation is known [19]. This feature may not be observable for every galaxy but energy loss due to graviton redshift is assumed to affect every galaxy.
Then, for N galaxies, γ in (67) can be approximated by

Dark Energy
We define the cosmological graviton energy loss of ΔΞ de (t) at time t by the ex- where β is a constant with dimensionality [length] −2 and the second factor of the middle equality, is the cosmological graviton energy loss rate, which has the dimensionless form [ ] velocity c − .

Equation of the Expanding Universe
The total energy of the shell of mass m, having kinetic energy, gravitational potential energy, galaxies graviton redshift energy loss (67) and the cosmological graviton energy loss (71), is expressed by Journal of High Energy Physics, Gravitation and Cosmology is an acceleration. From (86) and (87) where the conversion factor converts from (m/km) 4 /(Msolkg/Solar mass) since in (86), M is in kilograms and V is in [m·s −1 ]. The definition of a 0 and its value are consistent with the MOND theory [22].
Though the effort to determine γ of (70) is beyond the scope of this present work, we may apply a bit of hypothetical experimentation by use of (69), ( ) Ω Ω ≈ = [23].

Conclusions
Applying the principle of equivalence has given us the shift in the orbital periastron and the stellar deflection of light. Both of these phenomena are well known, although we have derived these results in a non-traditional manner. The novel ingredient is in revealing the role that the graviton plays in all of this. The new result is in showing that, if they exist, the gravitons in a binary star system are continuously Doppler shifted, i.e., gravitationally redshifted, to lower energies that lead to a decay in the period of the orbit, having comparable magnitudes as observed for the binary pulsar systems that we cited. This challenges the hypothesis that the orbital decays in binary pulsar systems are due to the emission of gravitational radiation. We showed how the graviton field behaves as massless bosons in a volume contained by the two masses in orbit and that the peak frequencies of the gravitons are on the order of for the binary systems studied and the temperatures are on the order of 7 10 K T ≈ . We described how gravitational redshifts of gravitons can apply in the expansion of the universe, being the possible source for dark matter and dark energy. Gravitons fulfill the requirement of dark matter in that they interact with light and baryonic matter but, since they have yet to be observed, possibly do not emit light in the interaction process. And, graviton decay fulfills the behavior of dark energy, in which the gravitational energy loss results in an accelerated rate of expansion.
In the case of individual galaxies, where dark matter was invented to explain flat rotation curves, we show how the redshift of gravitons interacting with matter in the outer regions of a galaxy leads to a leveling of the orbital rotation velocity, in agreement with the baryonic Tully-Fisher relation.
The Extended Theories of Gravity [24], such as massless Scalar-Tensor Gravity and f(R) theories, endeavors to quantify the phenomena of dark matter, dark energy and the flattening of galaxy rotation curves as well as predicting additional polarizations of gravitational radiation beyond the quadrupole predicted by General Relativity. Perhaps a theory for the graviton that includes the ideas discussed in this paper can emerge from these extended theories.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.