Devoted readers of this blog (assuming there are any) know that I’ve been studying relativity for some time — for why see https://luysii.wordpress.com/2011/12/31/some-new-years-resolutions/.
Probably some of you have looked at writings about relativity, and have seen equations containing terms like ( 1 – v^2/c^2)^1/2. You need a lot of math for general relativity (which is about gravity), but to my surprise not so much for special relativity.
Back in the early 50’s we were told not to study Calculus before reaching 18, as it was simply to hard for the young brain, and would harm it, the way lifting something too heavy could bring on a hernia. That all changed after Sputnik in ’58 (but too late for me).
I had similar temerity in approaching anything written by Einstein himself. But somehow I began looking at his book “Relativity” to clear up a few questions I had. The Routledge paperback edition (which I got in England) cost me all of 13 pounds. Routledge is a branch of a much larger publisher Taylor and Francis.
The book is extremely accessible. You need almost no math to read it. No linear algebra, no calculus, no topology, no manifolds, no differential geometry, just high school algebra.
You will see a great mind at work in terms you can understand.
Some background. Galileo had a theory of relativity, which basically said that there was no absolute position, and that motion was only meaningful relative to another object. Not much algebra was available to him, and later Galilean relativity came be taken to mean that the equations of physics should look the same to people in unaccelerated motion relative to each other.
Newton’s laws worked out quite well this way, but in the late 1800’s Maxwell’s equations for electromagnetism did not. This was recognized as a problem by physicists, so much so that some of them even wondered if the Maxwell equations were correct. In 1895 Lorentz figured out a way (purely by trying different equations out) to transform the Maxwell equations so they looked the same to two observers in relative motion to each other. It was a classic kludge (before there even were kludges).
The equation to transform the x coordinate of observer 1 to the x’ of observer 2 looks like this
x’ = ( x – v*t) / ( 1 – v^2/c^2)^1/2)
t = time, v = the constant velocity of the two observers relative to each other, c = velocity of light
Gruesome no ?
All Lorentz knew was that it made Maxwell’s equations transform properly from x to x’.
What you will see on pp. 117 – 123 of the book, is Einstein derive the Lorentz equation from
l. the constancy of the velocity of light to both observers regardless of whether they are moving relative to each other
2. the fact that as judged from observer1 the length of a rod at rest relative to observer2, is the same as the length of the same rod at rest relative to observer1 as judged from observer2. Tricky to state, but this just means that the rod is out there and has a length independent of who is measuring it.
To follow his derivation you need only high school algebra. That’s right — no linear algebra, no calculus, no topology, no manifolds, no differential geometry. Honest to God.
It’s a good idea to have figure 2 from p. 34 in front of you
The derivation isn’t particularly easy to follow, but the steps are quite clear, and you will have the experience of Einstein explaining relativity to you in terms you can understand. Like reading the Origin of Species, it’s fascinating to see a great mind at work.