Rongqing Dai
Abstract
Since 1913 the world has witnessed a very special and strange logical inference of disproving the validity of Galilean superposition by proving the validity of Galilean superposition. While it might sound like a typical symptom of failed answers in previous final exams that high school teachers would be keen to mention to new classes, it has actually happened at the highest level of academics around the world for more than a century without being much challenged up to now.
Keywords: Galilean Superposition, Sagnac Experiment, Speed of Light, Special Relativity, Critical Steps
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In classic mechanics, the Galilean superposition could be expressed as follows: when there are two objects A and B moving with respect to the coordinate origin O, if the velocity of A with respect to O is VA, the velocity of B with respect to O is VB, and the velocity of B with respect to A is VBA, then we have:
VB = VA + VBA (1)
When applying this formula to the speed of light, assuming the speed of the light source A with respect to coordinate origin O is VA and the speed of light with respect to the light source is c, then the speed of light with respect to the coordinate origin O would be:
CO = VA + c (2)
The special theory of relativity is constructed by denying the validity of formula (2), and accordingly the speed of light in vacuum c is assumed by special relativity to be the limit of all speeds in the universe. Thus, if formula (2) could be experimentally proved true, then we can draw two conclusions directly: 1) Special relativity is wrong; 2) The speed of light c in vacuum is not the speed limit.
Now without presuming the speed of a massive object VA to be faster than the speed of light, that is, assuming VA < c, as long as the c and VA are in the same direction, from (2) we will definitely get:
CO > c (3)
If VA ≈ c, then we have:
CO ≈ 2c (4)
If the speed of light with respect to its source A to be c, A moves at V1 with respect to object B1, and B1 has a velocity of V2 with respect to B2,… B n-1 has a velocity Vn with respect to Bn, then according to the above Galilean superposition rule, we know that the velocity Cn of light with respect to object Bn would be:
Cn = c + V1 + V2 +…… + Vn (5)
If we have Vi ≈ c for any number i from 1 to n, then we have:
Cn ≈ (n+1) c (6)
As n goes to infinity, Cn goes to infinity.
Has there ever been any experimental verification of Galilean Superposition for the speed of light?
The answer is yes. Formula (2) was experimentally verified by Sagnac as early as in 1913, but then the result was hilariously interpreted by the academia of physics as the proof of supporting the constancy of speed of light, i.e. the proof of denying the Galilean superposition [[1]]. More hilariously, as shown in the following image captured from Wikipedia [[2]], a critical step in all mathematical derivations to demonstrate that Sagnac experiment proves the constancy of speed of light would apply the Galilean superposition:
Image1. Critical steps of proving that Sagnac experiment denies the Galilean superposition. (Source: Wikipedia [2]).
The mathematical steps shown in the above image are typical Galilean superposition that is familiar to any good student in high school introductory physics class; but they are also indispensible critical steps for all mathematical works claiming to be the demonstration that Sagnac experiment proves the constancy of speed of light in vacuum, i.e. disproves the Galilean superposition, although different authors might present the relevant steps with different mathematical symbols or different expressions just like different text books might present the same Galilean superposition in different forms.
The reason for the above shown steps to be indispensible for the “proof” of Sagnac’s support to the constancy of speed of light is because we would have t1 = t2, or Δt = 0 in the result if we use the constancy of speed of light instead of the Galilean superposition in the derivation, which means there would be no more Sagnac effect in the result at all.
That is to say the academics of physics have been using Galilean superposition to directly “prove” a conclusion that would deny the validity of Galilean superposition all along since 1913, a kind of false operation normally only seen in failed test sheets of disqualified high school students.
[[1]] Dai, R. (2023). “When Philosophy is Disparaged (2023 ed)”. Retrieved from: https://www.academia.edu/104514666/When_Philosophy_is_Disparaged_2023_ed
[[2]] Wikipedia. “Sagnac effect”. Retrieved from: https://en.wikipedia.org/wiki/Sagnac_effect. Last edited on 4 January 2024, at 16:42 (UTC).
The World of Science and Science fiction of Ron Dai 
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