Notes on Relativity
The Principle of Relativity (Galileo, 1632)
In Galileo’s language: Absolute motion cannot be detected.
In modern language: The laws of nature must be the same in any two reference frames if one is moving at a constant velocity with respect to the other.
Newtonian Mechanics (Newton, 1687)
The three laws of motion: F = ma, etc.
Maxwell’s theory of Electromagnetism (Maxwell, 1865-73)
The full theory consists of the four Maxwell equations in either integral or differential form. Today we would consider the Lorentz force equation, which came later, as an essential part of the theory as well. A consequence of the theory, sometimes used in discussions of relativity as an abbreviation for it: electromagnetic fields propagate as waves with speed c = (ε0μ0)–1/2.
Coordinate transformations
A coordinate transformation is the set of equations for converting the space and time coordinates of an event as measured in one reference frame to the coordinates measured in another frame. In special relativity we restrict ourselves to frames moving with constant velocities with respect to each other. (Frame S' has a constant velocity as seen from frame S, and vice versa.)
The Galilean transformation is implied by the writings of Galileo, although he did not write the equations. It is in accordance with “common sense.” Velocities add, etc.
The Lorentz transformation was derived by several people independently, including Larmor, Poincaré, Lorentz, and finally, Einstein.
Note that as (v/c)2 —> 0, the Lorentz transformation —> the Galilean transformation.
Consistency
Newton’s theory of mechanics is consistent with the Principle of Relativity if the Galilean transformation is used to convert from one frame to another, but not if the Lorentz transformation is used.
Maxwell’s theory of electromagnetism is consistent with the Principle of Relativity if the Lorentz transformation is used to convert from one frame to another, but not if the Galilean transformation is used.
Which is right?
Einstein, in 1905, declared that the Principle of Relativity and the Maxwell equations must be right. Therefore the Lorentz transformation must be the correct one to use, and Newtonian mechanics must be replaced by a new mechanics which keeps the same form under a Lorentz transformation and also yields essentially the same results as the old theory in realms where it was known to work well, i.e., at speeds small compared with the speed of light. He invented the new mechanics. You can read a translation of his original paper online.
All experiments to date are consistent with Einstein’s assumption and with the modified theory of mechanics he proposed. Many, many experiments contradict Newtonian mechanics and the Galilean transformation. The first such was performed by Michelson and Morley in 1887.
Consequences
The Lorentz transformation is equivalent to length contraction and time dilation (which had been proposed by FitzGerald and by Lorentz earlier). They can be derived from the so-called Einstein postulates: the Principle of Relativity and the validity of Maxwell’s equations. Another consequence, published a few months later in a three-page paper, is E = mc2. You can read a translation of this paper online also.
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