1. DWT 23070885v98 0093382-000001
A Link Between the Lorentz Factor and Doppler-Related Anomalies?
Several Doppler-related anomalies identified in recent decades, including the few mm/s increased speeds
of Flyby spacecraft to the unexpected accelerations of Supernovae receding from Earth at near c, seem to
disappear when the Lorentz Factor in the Space-Time Doppler transforms used to measure speeds and
distances of these sources is replaced by “1.”
The Pioneer Anomaly: The Pioneer 10 and 11 spacecraft experience a fixed but anomalous acceleration
toward the Sun of 8.74+1.33x10-8
cm/s2
[1]. It has become generally accepted that these accelerations are
the result of thermal radiation forces [2]. However,a “pure numerical” coincidence between this
anomaly, which is based on round-trip propagation times of EM signals, and the Lorentz factor has been
noted [3].
The Flyby Anomaly: Doppler transforms for approaching and receding EM sources moving at v/c equal
(1+v/c)/[1-(v/c)2
]1/2
; i.e.,the ratio of linear and Lorentz factors. Thus, for EM sources approaching Earth
at speed v, this expression derives receding speeds for these sources that exceed their approaching speeds
by (2v/c)/[1-(v/c)2
]1/2
-2v/c; a violation of the conservation of energy. The mean increase in speeds
observed for these vehicles, ~6.393 mm/s, is close to the average of 6.554 mm/s imposed by the Lorentz
factor in the transform used to measure their speeds (Table 1).
Table 1.
Vehicle Speed at Infinity (km/s)[4] Observed Increases (mm/s)[4] Lorentz Increases (mm/s)
Galileo I 8.949 3.92 8.949xxxxx….
NEAR: 6.851 13.46 6.851xxxxx….
Rosetta 3.863 1.80 3.863xxxxx….
Mean Values: 6.554 6.393 6.554xxxxx….
Cosmic Expansion: Receding speeds of Supernovae suggest that the universe is expanding faster than
predicted by General Relativity [5]. For z = fr/fo-1, v in the red shift transform, (1-v/c)/[1-(v/c)2
]1/2
,
equals cz(z+2)/(z2
+2z+2). While the linear factor in this transform, cz/(z+2), dominates at all speeds, the
Lorentz factor increases receding speeds above ~0.1z, an increase that peaks at ~2.38z, and then returns to
near zero at z=~100; an increase close to the anomalous red-shifts reported for these EM sources. See
Figure. Further, the derivative of this red-shift transform, z[-z3
+4z+4]/[z3
+4z2
+4z+2]2
, peaks at
12.1308..% at z = ~0.343604...., versus ~12.93% at a “transition redshift” observed at z = 0.46+ 0.13 [6].
See Figure. While it is widely speculated that this effect results from an undetected force called Dark
Energy, replacing the Lorentz Factor with “1” appears to conform the universe’s expansion rate to one
consistent with that predicted by General Relativity.
The Cosmic Microwave Background: It has been generally assumed that the CMB would exhibit a
uniform temperature in all directions. It has been found, however, that this energy, derived from its
Doppler shifts by Earth-based observers,is not only variable but exhibits a dipole temperature anisotropy

2. DWT 23070885v98 0093382-000001
related to Earth’s velocity relative to the rest frame of the CMB with an amplitude of v/c of ~1.23x10-3
, or
v = 369 km/s [7]. Examination of data related to the CMB anisotropies should be undertaken to determine
whether the Lorentz factor in the Doppler transforms used to make these observations might be the source
of such anisotropies.
Further, anomalies uncovered based radar ranging observations such as the increasing Astronomical Unit
and an unexpected increase in the Eccentricity of the Moon’s orbit [8] may provide additional support for
the proposition that the Lorentz factor is the source of the above anomalies. For example, it has been
recently observed that round-trip propagation times of laser pulses from the Earth to the Moon and back
appear to have exceeded the accepted speed of light by ~200+10 m/s [9]; an increase consistent with
motion of an Earth-based receiver during signal propagation times. This increase is close to that imposed
by the Lorentz factor (in the linear rather than square root form) in the transform used to measure these
distances, (D/c)/[1-(v/c)2
]1/2
. Interestingly, such anomaly is much reduced--if not eliminated--when the
Lorentz factor in this transform is replaced by [1-(v/c)2
]. Thus, while differing from the spacecraft and
cosmic anomalies discussed above, this perceived increase in the speed of light may also be linked to the
Lorentz factor.

3. DWT 23070885v98 0093382-000001
References
[1] J.D.Anderson,P.A.Laing,E.L.Lau, A.S. Liu,M.M. Nieto,andS.G.Turyshev, Study of theanomalous
acceleration of Pioneer10 and 11, Phys.Rev.D 65 082004/1-50 (2002). [arXiv:gr-gc/0104064v5].
[2] SlavaG. Turyshev,ViktorT.Toth,Gary Kinsella,Siu-ChunLee,ShingM.Lok andJordan Ellis, Support
forthe thermalorigin of thePioneer anomaly,(2012). [arXiv:1204.2507v1].
[3] LiviuIvanescu, Curiousnumericalcoincidenceof thePioneerAnomaly, (2009). [arXiv:0.1641v1].
[4] J.D.Anderson,J.K.Campbell,J.E.Liukelund,J.Ellis,andJ.F.Jordan, Anomalous Orbital-Energy
ChangesObserved during SpacecraftFlybysof Earth,Phys.Rev.Lett.100.091102 (2008).
[5] S. Perlmutter,G.Aldering,etal:Measurementsof Ωand Λ from42 High-RedshiftSupernovae
(SupernovaCosmologyProject) Astrophys.J. 517, 565 (1999).
[6] Riess, et al, Type Ia Supernova Discoveries at z>1 From the Hubble Space Telescope: Evidence for
Past Deceleration and Constraints on Dark Energy Evolution (2004). [arXiv: 0402512v2].
[7] N.Aghanim,etal:Planck2013 results.XXVII.Doppler boosting of theCMB:Eppursi muove*(2013).
[arXiv:1303.5987v1].
[8] J.D.Anderson,M.M.Neito, AstrometricSolar-SystemAnomalies (2009). [arXiv:0907.2469v3 [gr-gc]].
[9] Daniel Y.Gezari, LunarLaser Ranging Testof theInvarianceof c (2009). [arXiv:0912.3934].