I watched this lecture on Lorentz transformation (https://www.youtube.com/watch?v=EhXWiAJBmzc). I'd say the tutor employed a simplistic and elegent approach to derive the transformation. But I also got these questions: How did Einstein know the transformation of an event in two frames was a Lorentz transformation which already prescribed time dilation and length contraction? How did he know it would not involve higher order relationships or other non-linear relationship? Another question: is it possible to derive length contraction and time dilation using a single reference frame and classical kinetics? Hope someone could enlighten me on this rudimentary questions.
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1$\begingroup$ From en.wikipedia.org/wiki/Lorentz_transformation#History Many physicists—including Woldemar Voigt, George FitzGerald, Joseph Larmor, and Hendrik Lorentz himself—had been discussing the physics implied by these equations since 1887. $\endgroup$– PM 2RingCommented Dec 3, 2019 at 6:33
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$\begingroup$ hermes.ffn.ub.es/luisnavarro/nuevo_maletin/… $\endgroup$– UmaxoCommented Dec 3, 2019 at 7:04
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2$\begingroup$ Have you read Einstein’s original 1905 paper on the subject? $\endgroup$– Bob DCommented Dec 3, 2019 at 7:33
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$\begingroup$ See "Chasing the Light Einstein's Most Famous Thought Experiment" by John D.Norton. "pitt.edu/~jdnorton/papers/Chasing.pdf" $\endgroup$– Cinaed SimsonCommented Dec 3, 2019 at 8:19
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$\begingroup$ He may not have "known", he figured it out based on a couple simple principles and lot of algebra. Also, Lorentz had figured out these transforms before Einstein. $\endgroup$– user196418Commented Dec 5, 2019 at 0:26
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2 Answers
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Thanks everyone for your comments. Special relativity is a theory with assumptions and verified by result of experiments.
I found the following derivatation uses least assumption and is easiest to understand:
Since space is assumed to be homogeneous, the transformation must be linear. The most general linear relationship is obtained with four constant coefficients, A, B, γ, and b:
$$x'=\gamma x + b t \;$$ $$t'=A x + B t. \,$$
https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations
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$\begingroup$ You may also want to have a look at this wonderful website where Einstein's original papers are collected: einsteinpapers.press.princeton.edu . see e.g. einsteinpapers.press.princeton.edu/vol2-trans/154 $\endgroup$– QuilloCommented Mar 31, 2023 at 8:23
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One can find two different approaches from Einstein, the first in his book, "Zur Elektrodynamik Bewegter Körper von Albert Einstein, 1905", and the second in " Ueber the spezielle und allgemeine Relativitaetstheorie" . In the first he starts with the definition of time for a point at a constant distance from the origin of the moving system, "t1=0.5(t0+t2)". If you read this approach, you'd find it mathematically correct up to the point on page 13, where he multiplies the result by "a= ((c^2-v^2)^0.5)/c " . It is not clear, why can't "a" have any other value. The second approach is mathematically wrong. the most simple approach I found myself is to solve 2 equations x'=Ax+Bt (!) t'=ct+Dx (2) by applying the conditions for relative velocity and equal appearance of equal long stabs. THIS results in x'=Ax -Avt (1) t'=At-(A-1/A)x/v (2) if we put x=ct and x'=ct', you get A=gamma and you'd have LT. By A=1 one gets the Galileo transform
