Two myths about special relativity
In a recent article, Two myths about special relativity[1], Ralph Baierlein provides a lucid discussion on two subtleties of special relativity. You may try to answer these questions by yourself and have a little fun:
- Does the Lorentz transformation reduce to the Galileo transformation for
v/c << 1? - What does really state Einstein’s second postulate on the velocity of light? That it is independent from the observer?
As it happens (and despite many authors answering both questions in the affirmative) the answer to both questions is ‘no’. In the first case, just take a look at Lorentz’s time transformation:
t1′ - t0′ = (1 - v^2/c^2)^{-1/2} ((t1 - t0) - (x1 - x0) v/c^2)
and note that, while you can safely approximate the square root as 1 when v/c is small, the term proportional to the spatial distance cannot be so mindlessly dropped: you can always find points for which it’s first order. Dropping it would be like saying that a rotation about the X axis reduces to a rotation about, say, the Z axis for small angles.
As for the second question, Einstein’s original postulates were: a) the principle of relativity, and b) that the velocity of a light ray respect of a given observer is independent of the emitter’s velocity. From these two principles (quite reasonable as they stand, specially if you think of light as some sort of wave), Einstein deduced (as you can do yourself with a little thought) that the velocity of light is the same for all observers (which is a less intuitive result): you don’t actually need to take that as a postulate.
Admittedly, nothing earth-shakening here, but its find to review the basis (and clear up lurking misconceptions) every now and then.
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[1] Unfortunately, you’ll need a subscription, a library with one, or a kind soul [2], to access the article.
[2] I wouldn’t say i have a mean soul.
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