### Special relativity, part 1

Warning: I'm so depressingly out of practice at mathematical things that I could well have stupid mistakes in the following. Sorry! If you spot anything wrong, please mail and correct me...

So, I'm slowly working through Taylor and Wheeler's Spacetime Physics, which is a reasonable pile of fun. I reached the halfway point during my gardening leave, and have basically been on hold while I've been learning the new job. However, I thought it worth starting to write up some notes on the things I've been thinking about while learning special relativity.

• Invariance of interval is a great starting point for the book. I'd previously read explanations based on Lorentz transformation, and this is a much more intuitive explanation.
• Change of reference frame The fundamental point is that a change of reference frame that changes velocity is no longer a shear, but a hyperbolic rotation. Talk of hyperbolic functions was in the first edition of the book (which I "trialed" before I bought the paper copy), but sadly taken out of the second edition.
• Lorentz contraction is weird If we go past each other at speed, you seem shorter, with time going slower to me, and you see me as shorter with time going slower? Yep. This is not entirely alien - you get the same thing in a non-relativistic world where, if we run past each other while making a noise, we both hear the same Doppler effect coming from the other person. It does seem a bit odd, doesn't it, though?
• The garage paradox I finally understand the question about "Imagine you drive a long car really fast into a short garage. The car fits from the reference frame of the garage, why can't you park it?" If you want to stop the front and rear of the car at the same point in time from the reference frame of the garage, you'll be stopping them at different times in the reference from of the car - namely stopping the front rather earlier than stopping the back!
• How come we get contraction if the Lorentz transform increases coordinates? The Lorentz transformation has x' = (x + vt) / (1-v^2)^0.5. The term on the bottom is less than 1, so we're stretching out x coordinates. But we're supposed to have a contraction. What's going on? While the transform changes x coordinates, it also changes t coordinates. The transformed points that have the same t' coordinates have x' coordinates that are closer together. I should draw a diagram.

I've got some more complicated bits to cover (even before I get onto the second half of the book), but that'll do for now.

Posted 2015-12-19.