The derivation of the Lorentz transformation (LT) presented by Einstein in 1905 is based on two postulates, Galileo’s relativity principle and the constancy of the speed of light c in free space. There is an additional assumption that is essential for this result, however, that is of a mathematical nature. Lorentz pointed out in 1899 that the condition of constancy of light speed is only sufficient to define the relationship between two space-time vectors to within a constant (normalization) factor ε. Poincaré published an argument in 1905 that first made use of this extra assumption, namely that ε could only be a function of the relative speed u of the two observers. This (hidden) postulate admits of only one solution (ε=1). It has a number of attractive features from a purely theoretical point of view, but it also leads inevitably to the non-simultaneity condition of Einstein’s special theory of relativity through the LT equation below (given in differential form):
dt = εγ (dt’ + udx’/c2) = γ (dt’ + udx’/c2), (1)
where γ = (1-u2/c2)-0.5.
The above assumption regarding the functionality of the normalization constant ε is too restrictive, however, to be justified on a suitably general basis. The purpose of any such transformation is to describe the relationship between the respective measured space-time values obtained by two observers in relative motion for another moving object. Therefore, there is no a priori reason for assuming that ε is not also a function of the object’s speed, rather than just of the observers’ relative speed u. Recognition of this point allows us to employ a different condition for determining ε, one that insists on strict adherence to the principle of simultaneity of events, that is, by setting dt=Qdt’ in eq. (1):
dt = εγ (dt’ + udx’/c2) = Qdt’, (2)
which leads to the solution:
ε (u, u·v’) = Qη (u·v’)/γ (u), (3)
with η (u·v’) = (1 + u·v’/c2) ^-1 = (1 + udx’/c2 dt)-1 (4)
Using this value for the normalization constant of the LT leads then to the alternative space-time transformation (ALT) given below:
dx = Qη (dx’ + u dt’) (5a)
dy = Qη dy’/γ (5b)
dz = Qη dz’/γ, (5c)
dt = Qdt’. (5d)
The ALT also satisfies the constancy of light speed requirement of special relativity and leads to exactly the same velocity transformation as the LT. It therefore is consistent with all known measurements that are otherwise claimed as proof of the viability of Einstein’s original theory. But the ALT also leads to the conclusion that events occur simultaneously for all observers, regardless of their state of motion. Two light pulses that travel an equal distance dr’ on an airplane also travel an equal distance dr for someone on the ground, independent of the direction of the pulses, for example, unlike the prediction of the LT, as can be seen by application of eqs. (5a-c).
The ALT demonstrates that it is possible to satisfy the famous two postulates in Einstein’s 1905 paper without sacrificing the principle of simultaneity of events. It also frees one from the necessity of arguing that two clocks can both be running slower than each other at the same time or that two rods can both be smaller than each other (Einstein’s symmetry principle). Instead, the ALT allows one to return to the ancient principle of rationality (and objectivity) of measurement (PRM), that is, that all observers must agree on the ratio of any two physical quantities of the same type. The PRM is the essential basis for introducing a rationalized set of units such as the mks or cgs systems. Experiments with clocks on airplanes, rockets, centrifuges and satellites (GPS technology) indicate strongly that measurement is not symmetric but instead rational, and especially in the case of GPS, that events do occur simultaneously for all observers after taking account of differences in the rates of clocks used to make the respective measurements. The latter conclusion is perfectly consistent with the relativity principle, but the ALT also emphasizes that the units in which the various laws of physics are expressed vary systematically from one system to another depending on their state of motion and position in a gravitational field.
All these points are discussed in detail in R. J. Buenker, Simultaneity and the constancy of the speed of light: Normalization of space-time vectors in the Lorentz transformation, Apeiron 16, 96-146 (2009)
dt = εγ (dt’ + udx’/c2) = γ (dt’ + udx’/c2), (1)
where γ = (1-u2/c2)-0.5.
The above assumption regarding the functionality of the normalization constant ε is too restrictive, however, to be justified on a suitably general basis. The purpose of any such transformation is to describe the relationship between the respective measured space-time values obtained by two observers in relative motion for another moving object. Therefore, there is no a priori reason for assuming that ε is not also a function of the object’s speed, rather than just of the observers’ relative speed u. Recognition of this point allows us to employ a different condition for determining ε, one that insists on strict adherence to the principle of simultaneity of events, that is, by setting dt=Qdt’ in eq. (1):
dt = εγ (dt’ + udx’/c2) = Qdt’, (2)
which leads to the solution:
ε (u, u·v’) = Qη (u·v’)/γ (u), (3)
with η (u·v’) = (1 + u·v’/c2) ^-1 = (1 + udx’/c2 dt)-1 (4)
Using this value for the normalization constant of the LT leads then to the alternative space-time transformation (ALT) given below:
dx = Qη (dx’ + u dt’) (5a)
dy = Qη dy’/γ (5b)
dz = Qη dz’/γ, (5c)
dt = Qdt’. (5d)
The ALT also satisfies the constancy of light speed requirement of special relativity and leads to exactly the same velocity transformation as the LT. It therefore is consistent with all known measurements that are otherwise claimed as proof of the viability of Einstein’s original theory. But the ALT also leads to the conclusion that events occur simultaneously for all observers, regardless of their state of motion. Two light pulses that travel an equal distance dr’ on an airplane also travel an equal distance dr for someone on the ground, independent of the direction of the pulses, for example, unlike the prediction of the LT, as can be seen by application of eqs. (5a-c).
The ALT demonstrates that it is possible to satisfy the famous two postulates in Einstein’s 1905 paper without sacrificing the principle of simultaneity of events. It also frees one from the necessity of arguing that two clocks can both be running slower than each other at the same time or that two rods can both be smaller than each other (Einstein’s symmetry principle). Instead, the ALT allows one to return to the ancient principle of rationality (and objectivity) of measurement (PRM), that is, that all observers must agree on the ratio of any two physical quantities of the same type. The PRM is the essential basis for introducing a rationalized set of units such as the mks or cgs systems. Experiments with clocks on airplanes, rockets, centrifuges and satellites (GPS technology) indicate strongly that measurement is not symmetric but instead rational, and especially in the case of GPS, that events do occur simultaneously for all observers after taking account of differences in the rates of clocks used to make the respective measurements. The latter conclusion is perfectly consistent with the relativity principle, but the ALT also emphasizes that the units in which the various laws of physics are expressed vary systematically from one system to another depending on their state of motion and position in a gravitational field.All these points are discussed in detail in R. J. Buenker, Simultaneity and the constancy of the speed of light: Normalization of space-time vectors in the Lorentz transformation, Apeiron 16, 96-146 (2009)
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