The Lorentz transformation (LT) is the cornerstone of Einstein’s Special Relativity Theory (SR). What the great majority of physicists have not understood is that it is fatally flawed.  One can easily see this from a critical examination of the light speed postulate on which it is based, as will be shown in the following discussion.  It is also true that the LT is in violation of the Law of Causality, the ancient principle that is embodied in all laws of physics. 

The Law of Causality is a critical principle that has been a key stimulus for progress in the physical sciences.  It states that no physical change occurs without the application of some external force.  In other words, nothing happens by chance in physics.  Newton’s First Law of Kinetics (Law of Inertia) is a prime example of how this principle works in actual practice.  It is claimed that a body in motion will continue indefinitely with the same constant speed and in the same direction in space until it is acted upon by some unbalanced external force. 

This line of argument can be extended to the properties of an object such as the rate of a clock.  According to the Law of Causality, the rate of the clock will not change spontaneously as long as it remains in such a constant state of motion.  For the sake of concreteness, we will refer to this in the following as an inertial clock. The Lorentz transformation (LT) of Einstein’s SR [1] makes use of two inertial clocks in different rest frames in formulating its equations. In particular, they are used to measure the elapsed times Δt and Δt’ in eq. [1] given below:

eq. (1) exponents: Δt’ = (1-v2c-2)-0.5 (Δt –vc-2Δx) = γ (v)  (Δt –vc-2Δx).

In this equation c =299792458 ms-1, v is the relative speed of the two rest frames under consideration, and Δx is the distance separating one of the two observers from the object in question. Consistent with Newton’s First Law, one expects that an inertial clock cannot change its rate spontaneously, that is, without the application of some unbalanced external force. [2], [3] the ratio of the rates of any two inertial clocks must therefore be constant.  This means that the elapsed times Δt and Δt’ measured for a given event must always occur in the same ratio (Q) as their rates, i.e. Δt’=Δt/Q.  This equation is referred to as Newtonian Simultaneity because it is evident therefrom that if the events occur simultaneously for one of the clocks (Δt’=0), they must also be simultaneous based on the other (Δt =0).  The attribution to Newton is appropriate because of his longstanding belief that events occurring anywhere in the universe must always occur at the same time.  The Δt’=Δt/Q  equation is also referred to as the Clock-rate Corollary to Newton’s First Law.[2],[3]  The space-time mixing in eq. [1] shows that the LT is unable to satisfy the Newtonian Simultaneity requirement since it precludes the possibility that Δt’ will always be proportional to Δt. This constitutes proof that the LT violates the Law of Causality.

Moreover, there is another problem with the LT, namely its reliance on Einstein’s light-speed postulate (LSP).[4],[5] It states that the speed of the light pulse has the same constant value of c for all observers independent of their state of motion and that of the light source. Consider the following case, however, in which a light source leaves the laboratory with speed v at the same time that it emits a light pulse in the same direction. One can see that this is an untenable assumption by calculating the respective distances separating the light pulse from each rest frame after a certain time T has passed.  The value of this distance is seen to be CT in each case. But this is impossible, since the source and stationary observer are no longer at the same position in space.  In arriving at this conclusion, it clearly does not matter how great, whether it is just a few milliseconds or many thousands of years.  In summary, Einstein’s light speed postulate is completely unrealistic.

There is also another effective way to use the above (“distance reframing”) procedure.  Consider again what happens when some time T has elapsed since the light source began to move with speed v.  After this time has passed the source is found at a distance vT from the stationary observer, while the light pulse is again located at a distance cT from the source.  The corresponding distance separating the light pulse from the stationary observer is obtained by simply adding these two partial distances together, in which case the answer is clearly vT+cT.  We don’t need Newton or Galileo to deduce this value, nor the ancient Greek and Roman philosophers.  It involves the same “theory” as we use to measure the length of a room with a meter stick.  We measure out the various portions of the room in meters and just add the results. 

Since the motion of the light pulse and source occur at the same time T, it is possible by definition to calculate the speed of the light pulse relative to the stationary observer, namely as the ratio of the distance travelled to the amount of elapsed time, i.e. as (vT+cT)/T=v+c.  This result is exactly what one obtains when one applies the classical (Galilean) velocity transformation (GVT).  It therefore stands in clear contradiction to another of Einstein’s conclusions from SR [1], namely that the GVT is not applicable for light or other fast moving objects.  The GVT is known in standard mathematical language as the vector addition of velocities. 

Moreover, it can be stated without fear of contradiction that, just as for vector addition, it applies to motion in all three (not four!) spatial directions. It was used by Bradley in the 17th century to deduce a key aspect of astronomical measurements, namely the aberration of starlight from infinity.  Einstein concluded on the basis of his light speed postulate that the angle of aberration is tan^-1 (γv/c), whereas the correct value that Bradley obtained by vector addition is tan^-1 (v/c).  The maximum speed observable speed in free space is not c as SR would have one believe, but rather 2c when two light pulses approach each other head-on.  Each pulse travels a distance of cT in time T, so their total closing distance is 2cT.  There is no reason to doubt this,

Einstein was aware of the fact [6], first pointed out by Poincaré,[7] that the LT predicts that events do not always occur simultaneously for two observers in relative motion to one another (remote non-simultaneity or RNS).  His famous example [8] of two lightning strikes on opposite sides of a train as it passes by the station platform was intended to bolster belief in RNS.  Examination of his argument, however, shows that is based on his LSP which has been shown above to be unreliable.  When the GVT is used to analyze the problem instead, however, the strikes are found to occur simultaneously. [5] 

The impetus for treating the speed of light differently than for other objects can be traced to the Fresnel-Fizeau light-damping experiment in the early 19th century. [9] It leads to the conclusion that the speed of light in a medium with a refractive index close to n=1 is independent of the medium’s speed v in the laboratory; c (v) =c.  It was recognized that this behavior is inconsistent with the GVT.  After the Michelson-Morley experiment [10] carried out in 1887 appeared to be in agreement with the above relation, Voigt [11] suggested that a suitable transformation could be obtained by simply introducing a free parameter into the GVT equations.  His result was inconsistent with Galileo’s Relativity Principle (RP), however.  Larmor [12] and Lorentz [13] were able to modify Voigt’s transformation so as to remove this objection and Einstein [1] used the resulting transformation, the LT, in developing his version of relativity theory, i.e. SR.

The point which has not been appreciated is that this success in no way removes the necessity of using the GVT for other purposes.[14] A distinction can be been made between experiments of two different types: One (Type A) in which two observers in relative motion obtain different values for the speed of light emitted from a given source, and another (Type B) in which a single observer measures the light speed under two different conditions, such as occurs in the Fresnel-Fizeau light-damping experiment.  For the latter purpose, one must use the relativistic velocity transformation (RVT), which is easily derived from both the Voigt transformation and the LT.  The ranges of applicability for the GVT and RVT are seen to be mutually exclusive. 

The Newton-Voigt transformation (NVT) [15] is consistent with the Δt’=Δt/Q relation (Newtonian Simultaneity) and, unlike both the LT and the Voigt transformation, is therefore consistent with the Law of Causality.  The corresponding (different from the LSP) light speed postulate assumes that the speed of light in free space is always equal to c relative to its source, independent of the states of motion of both the observer and the light source; the NVT also satisfies this requirement. It is also consistent with the results of the Michelson-Morley experiment [10] and it also satisfies the condition required by the Galilean RP. 

In order to apply the NVT in a given case, it is necessary to know not only the relative speed v of the two observers involved in the transformation but also the value of the ratio Q of the rates of their respective clocks.  The latter value must be obtained experimentally.  The results of the Ives-Stilwell experiment [16], [17] and the various studies of the lifetimes of muons and pions [18] [19] [20] [21] [22] were in agreement with Einstein’s time dilation prediction from SR.[1]  It was found that the value of Q depends on the speed v of the light source relative to the laboratory, namely as γ(v)=(1-v²/c²)^-0.5.The Hay-et al. ultra-centrifuge experiment with x-ray radiation [23] [24] [25] showed, however, that time dilation is not symmetric.  The LT prediction of a red shift being observed in all cases was contradicted in this study (Einstein Symmetry Principle), although this was not recognized by the authors. 

The atomic clock experiments on board circumnavigating airplanes that were carried out by Hafele and Keating [26] [27] a decade later ruled out the possibility that Einstein’s 1907 Equivalence Principle [28] satisfactorily explains what occurs in general.  It was found that the clocks on the eastward flying airplane ran slower than those flying westward.  The explanation is that the speed v of each clock that determines the clock rate is taken to be relative to the earth’s center of mass (ECM).  This fact shows that Einstein’s Symmetry Principle is not viable and instead that time dilation is an asymmetric phenomenon,

The parameter Q in the Newton Simultaneity formula is thus seen to be the ratio of the corresponding γ(v) factors.  An inverse proportionality therefore exists between a given elapsed time measured with each clock and the associated γ(v) factors.  This relationship is referred to as the Uniform Time Dilation Law (UTDL).[29] [30] [31] To apply it in a given case, it is necessary to specify a rest frame, referred to as the objective rest frame or ORS, [32] relative to which the speeds of the clocks (v and v’) are to be referenced in each case.  It is the laboratory in the Hay et al. x-ray study, the ECM in the Hafele-Keating experiment with circumnavigating atomic clocks, or more generally, as the rest frame from which an object has been accelerated.  With these definitions, it is possible to define Q as the ratio of γ(v’) to γ(v).  The latter (Q) is most effectively seen as a conversion factor between the rates of the clocks.

It is also possible to prove that the same conversion factor applies to distances.  If an object of length L is accelerated, length expansion must accompany time dilation in order that the speed of light in free space has the same value c in both rest frames, This is the opposite relationship expected based on the FitzGerald length contraction (FLC) prediction of SR. [1] Moreover, again unlike the case for the FLC prediction, the amount of the expansion must be independent of the orientation of the object.

The experiments of Bucherer [33] in 1908 with electrons accelerated to speed v in an electromagnetic field found that the inertial mass of the electrons increased in proportion to γ(v).  On this basis it can be concluded that inertial mass also scales with factor Q.  The conversion factors of all other physical properties can therefore be deduced to have conversion factors which are integral multiples of Q.  For example, speed is the ratio of distance to speed, so the conversion factor for speed is Q/Q=Q⁰=1, that is, it is independent of the state of motion of the observer.  This is of course consistent with the light speed postulate stated above, namely that the speed of light in free space relative to its source is always equal to c.  The conversion factor for frequency is Q^-1 based on the fact that it is defined to be the reciprocal of the period of clocks.  Accordingly, energy scales as Q since it is defined as the product of inertial mass and the square of speed.

The scaling procedure outlined above is consistent with the Principle of Rational Measurement (PRM) [34] which states that the measurement process is always perfectly objective.  It is the basis of the Uniform Scaling method as a whole. [35]  A key aspect of Uniform Scaling is that the reverse conversion factor Q’ is always the reciprocal of the original (Q’=1/Q),  It is clearly distinguished from Einstein’s Symmetry Principle which states that two clocks can both be running slower than each other at the same time. 

A consequence of the perfect objectivity of the Uniform Scaling method is that it allows one to deduce the value of Q for any two rest frames (2 and 3) from the respective Q values of another rest frame (1): Q(2,3)=Q(1, 3)/Q(1,2)=Q(2,1)Q(1,3).  It should also be noted that the rest frames do not have to be inertial in order to apply the Uniform Scaling method.  The Hafele-Keating airplane experiment [26], [27] shows that the UTDL is valid for atomic clocks that are constantly accelerating.  The values of the speeds are those measured instantaneously at the current time.

There is an analogous scaling procedure for differences in gravitational potential in this case the quantity Ai=GM/c2ri plays the same role as γ (v) for kinetic scaling.  The corresponding conversion factor S is equal to Ao/Ap.  The two factors Q and S are independent of one another.  This is again seen from the Hafele-Keating study in which the effect of gravity on the clock rates is simply added to the corresponding kinetic effect.  This is a key observation since physicists have traditionally believed that the two effects are intertwined.  A typical unfounded assertion is that the effects of gravity cannot be “painted” onto SR.

A corollary to Newton’s First Law of Kinetics has been presented which asserts that the rates of inertial clocks cannot change spontaneously, i.e. without the application of some unbalanced external force.  It is based on the Law of Causality.  As a consequence, the elapsed times Δt and Δt’ for two events measured by two such clocks must always occur in the same ratio: Δt’= Δt/Q. where Q is the ratio of the two rates. This relationship is referred to as Newtonian Simultaneity in view of Newton’s belief that two events occurring anywhere in the universe must occur at the same time for any pair of observers.  Accordingly if one elapsed time Δt’=0, the corresponding elapsed time registered on the other clocks must also be equal to zero (Δt=0).   In view of this relationship, it follows that the Lorentz transformation (LT) is inconsistent with the Law of Causality because of the space-time mixing in its eq. (1).

Another problem with the LT is its use Einstein’s light-speed postulate (LSP).  It states that the speed of light is the same for all observers independent of their state of motion or that of the light source from which it has been emitted.  As a consequence, one must believe that after elapsed time T the distance travelled by the light from both a stationary observer and the light source moving at speed v and same direction as the light moving away from him is the same, namely cT.   That is clearly impossible since the observer is no longer located at the same position in space as the light source after this time has elapsed.

A satisfactory replacement for the LT is obtained by incorporating Newtonian Simultaneity in its equations and replacing Einstein’s LSP with a different postulate which states that the speed of light relative to the source is always equal to c.  The resulting space-time transformation, referred to as the Newton-Voigt transformation (NVT), satisfies Galileo’s Relativity Principle (RP) and also guarantees that the speed of light is the same under two separate conditions such as occur in the Fresnel-Fizeau light damping experiment (v=0 and v≠0 of the medium).  The NVT is consistent with the relativistic velocity transformation (NVT) which von Laue used successfully to derive the empirical formula for light damping. [9]

Nonetheless, the Galilean velocity transformation (GVT) must be applied in order to correctly deduce that the speed of light is different for two observers who are in relative motion to one another with speed v. namely v+c.  As a result, it is seen that the use of the two transformations is mutually exclusive; the GVT must he used when the speeds of light relative to different observers are measured, whereas the RVT must be used when only a single observer measures the speed of light under two different conditions such as occur in the light-damping experiment.

The proportionality relationship of Newtonian Simultaneity is just one example of the Uniform Scaling method.  Conversion factors between the values of physical properties in different rest frames are always integral multiples of the parameter Q in the latter relationship for each physical property.  There is an inverse proportionality between an elapsed time obtained on the basis of one clock and the value of γ (v) for the latter’s speed relative to some well-defined Objective Rest Frame (ORS), such as the ECM in the Hafele-Keating circumnavigating atomic clock study.  This relationship is referred to as the Universal Time-dilation Law because it is found to be quite generally applicable.  It is used in the application of the GPS navigation method in order to guarantee that the rates of clocks on satellites are the same as their counterparts on the earth’s surface or elsewhere. 

There is also a corresponding relationship involving objects located at different gravitational potentials.  In this case a parameter S is determined as the ratio of two factors defined as GM/c²r, where r is distance of the object from the active gravitational mass such as the sun and M is the associated gravitational mass.  The two factors S and Q are completely independent of one another, as demonstrated by the results of the Hafele-Keating experiment.

1. Einstein, Zur Elektrodynamik bewegter Körper. Ann. Physik 1905; 322:891-921.
2. J Buenker, Clock-rate Corollary to Newton’s Law of Inertia, East Africa Scholars J. Eng. Comput. Sci. 2021; 4:138-142.
3. J. Buenker, Proof That the Lorentz Transformation Is Incompatible with the Law of Causality, East Africa Scholars J. Eng. Comput. Sci. 2022; 5:53-54.
4. J. Buenker, Proof That Einstein’s Light Speed Postulate Is Untenable, East Africa Scholars J. Eng. Comput. Sci. 2022; 5:51-52.
5. J. Buenker, Dissecting Einstein’s Lightning-Strike Example: Proof That Einstein’s Light Speed Postulate Is Untenable, East African Scholars. J. Eng. Compute. Sci. 2022; 5:51-52.
6. Pais, ‘Subtle is the Lord…’ The Science and Life of Albert Einstein, (Oxford University Press, Oxford, 1982:126-128.
7. Poincaré, Rev. Métaphys. Morale 6, 1 (1898).
8. Einstein, Relativity: The Special and the General Theory, Translated by R. W. Lawson (Crown Publishers, New York, 1961:25-27.
9. Pais, ‘Subtle is the Lord…’ The Science and Life of Albert Einstein, (Oxford University Press, Oxford, 1982:144-145.
10. A. Michelson and E. W. Morley, Am. J. Sci. 1887;34:333-334. L. Essen, Nature 1955; 175:793-794
11. Voigt, Goett. Nachr. 1887:41.
12. Larmor, Aether and Matter, Cambridge University Press, London, 1900:173-177.
13. A. Lorentz, Proc. K. Ak. Amsterdam 6, Collected Papers, 1904; 5:172.
14. J. Buenker, Stellar aberration and light-speed constancy, J. Sci. Discov. 2019; 3:1-15.
15. J. Buenker, Voigt's conjecture of space-time mixing: Contradiction between non-simultaneity and the proportionality of time dilation, BAOJ Physics 2:27, 1-9 (2017).
16. I. Ives and G. R. Stilwell. An experimental study of the rate of a moving clock. J. Opt. Soc. Am. 1938; 28:215-226.
17. I. Ives and G. R. Stilwell. An experimental study of the rate of a moving clock. J. Opt. Soc. Am. 1941; 31:369-374.
18. Rossi, K. J. Greisen, J. C. Stearns, D. Froman and P. Koontz, Phys. Rev. 1942; 61:675.
19. S. Ayres, D. O. Caldwell, A. J. Greenberg, R. W. Kenney, R. J. Kurz and B. F. Stearns. Comparison of π+ and π- lifetimes, Phys. Rev. 1967; 157:1288.  A. J. Greenberg, thesis, Berkeley, 1969;23:1473
20. H. Durbin, H. H. Loar and W.W.Harens, Phys. Rev. 1952; 88:179.
21. M. Lederman, E. T. Booth, H. Byfiel and J. Kessler. Phys. Rev. 1951; 83:685.
22. C. Burrowes, D. O. Caldwell, D. H. Frisch, D. A. Hill, D. M. Ritson ans R. A. Schluter, Phys. Rev. Letters. 1959; 2:117.
23. J. Hay, J. P. Schiffer, T. E. Cranshaw and P. A. Egelstaff. Measurement of the red shift in an accelerated system using the Mössbauer effect in Fe57. Phys. Rev. Letters. 1960; 4:165-166.
24. Kuendig W, Measurement of the transverse Doppler effect in an accelerated system. Phys. Rev. 1963; 129:2371-2375.
25. C. Champeney, G.R. Isaak and A. M. Khan, Measurement of relativistic time dilatation using the Mössbauer effect, Nature 1963; 198:1186-1187.
26. C. Hafele and R. E. Keating, Around-the-world atomic clocks: predicted relativistic time gains, Science. 1972; 177:166-168.
27. C. Hafele and R. E. Keating, Science. 1972; 177:168-170.
28. Einstein, Jahrb. Radioakt. u. Elektronik 1907; 4:411.
29. J. Buenker, Relativity Contradictions Unveiled: Kinematics, Gravity and Light Refraction (Apeiron, Montreal, 2014)50.
30. J. Buenker, The universal time dilation law: objective variant of the Lorentz transformation, BAOJ Physics, 2017; 3; 23:1-6.
31. J. Buenker, The new relativity theory: Absolute simultaneity, modified light speed constancy postulate, uniform scaling of physical properties, International Journal of Applied Mathematics and Statistical Sciences. 2021; 10:151-162.
32. J. Buenker, Time dilation and the concept of an objective rest system, Apeiron. 2010; 17:99-125.
33. H. Bucherer, Phys. Zeit. 1908; 9:755.
34. J. Buenker, Relativity theory and the principle of the rationality of measurement, Int. J. App. Math. & Stat. Sci. 2021; 10:1-12.
35. J. Buenker,Relativity Contradictions Unveiled: Kinematics, Gravity and Light Refraction (Apeiron, Montreal, 2014)71-77.  
36. J. Buenker, R, J, Buenker, Extension of Schiff’s gravitational scaling method to compute the precession of the perihelion of Mercury, Apeiron 2008;15:509-532.