- 146

- 7

The rule for length contraction is: ## x = \gamma x ^\prime ## and ## x ^\prime = \frac {1} {\gamma} x ##

But the lorentz transformations for distance are ## x = \gamma x ^\prime + \gamma vt ^\prime ## and ## x ^\prime = \gamma x - \gamma vt ##

Why in the lorentz transformations above, unlike in the rule for length contraction, aren't we multiplying ## x ## by ## \frac {1} {\gamma} ## in order to get ## x ^\prime ## ? We are always multiplying by ## \gamma ## whether or not we want to go from x to ## x ^\prime ## or vice versa.

It's true that the rule for length contraction involves length, and the lorentz transfortions for distance involve finding the distance from an observer to an event. But isn't distance the same thing as length? When we say that a fire cracker happened at ## x ^\prime = 10 ## in Bob's frame, aren't we saying that the length between the fire cracker and Bob is 10 as measured by Bob?

Also, here's an example using the lorentz transformations for distance where the answers don't seem to make sense: take an event that happened at ## (x ^\prime = 10, t ^\prime = 0) ## Say for example, Bob in a train passes Alice at 0,0 in both frames, and an explosion happens for Bob at ## x ^\prime = 10, t ^\prime = 0 ## as he passes Alice. Using the lorentz transformations to find x in Alice's frame, with a relative velocity between the frames at v = .6, we'd get ## x = \gamma 10 + 0 ##, x =10.25. Now say we have another example where this time it is Alice who observes the explosion to happen at x = 10 and t= 0. If we try and find where the explosion happens for Bob we get ## x ^\prime = \gamma 10 - 0 ##, ## x ^\prime = 10.25 ##. Why are they both getting the same distance? Isn't distance frame dependent?