What are the axioms of special relativity? Is there a book or paper that introduces the theory of special relativity in a rigorous manner, and proves that $E=mc^2$ after appropriate definitions?
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The Lorentz transformation needs to be differentiated and then solve for $dv$ Please see the link below.
YouTube video by a man called "Physics Reporter".
Deriving the mass energy equivalence formula
The axioms are simple: 1) intertial reference frames are equivalent-physical laws are the same in all of them. 2) the speed of light is the same in all these frames. From those you derive the Lorentz transformation. This is done in most texts that cover special relativity. Then you discover the energy-momentum four vector, which must have a modulus that is independent of reference frame. You get $E^2-p^2c^2=m_0^2c^4$ for its squared modulus. Evaluating in a convenient frame (where $p=0$), you get $E=m_0c^2$.
Einstein, A. (1905), "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?", Annalen der Physik 18 (13): 639–643
It's the first reference at Mass-energy equivalence. You might also try
Einstein, A. (1961), "Relativity: The Special and the General Theory", Three Rivers Press. 1961. ISBN: 0517025302
(Warning: There are apparently electronic versions of this text that lack illustrations. This will be a significant impediment to understanding whether the coordinate transforms are passive or active.)
I'm not sure if this should be asked on a math board.
@Ross Millikan answers "correctly"
The axioms are simple: 1) intertial reference frames are equivalent-physical laws are the same in all of them. 2) the speed of light is the same in all these frames. From those you derive the Lorentz transformation. This is done in most texts that cover special relativity.
But without me knowing the tedious details, already 95 years ago, Weyl wrote pedantically about the relation between physical axiomatizations and the Lorentz transformation. Apparently for uniquencess you must also set up axioms about some finiteness of the domains to be transformed to actually rule out fractional linear transformations, projective stuff, etc. etc.
Speaking of tedious rigour, as an answer to the question, someone wrote a PhD thesis about setting up special relativity in first order logic here: First-Order Logic Investigation of Relativity Theory with an Emphasis on Accelerated Observers.
Terence Tao has written in his blog about Einstein’s derivation of E=mc^2, explaining the physical intuitions and the mathematical derivation rigorous.
I've heard a lot of good things about Woodhouse's text, but I felt it was very sloppy especially being published in SUMS (Springer Undergraduate Mathematics Series). My favorite is The Special Theory of Relativity for Mathematics Students by Lorimer http://www.worldscientific.com/worldscibooks/10.1142/1125
I liked this because I could read this in about a week, and it was heavily based on basic linear algebra and didn't waste time getting caught up in nasty calculations like most textbooks. Certainly, this book is a great introduction but don't stop there (but certainly start here).