Distances contract as a result of motion?

[Desrib49]

Hello all,
I'll start out by saying that I am a fairly new space/cosmology enthusiast with no real background in the subject (high school physics is about it), so this question will likely seem quite elementary to most, if not all, of you.

Moving on. I recently bought Pedro Ferreira's https://www.amazon.com/dp/0753822563/?tag=pfamazon01-20 as a way to build a base level of knowledge. I was humming along just fine until I got to this concept that a stationary person will perceive that objects contract while they are in motion, and was wondering if somebody could shed some light on the subject.

To illuminate where I am struggling, here's the example Ferreira gives. If a train with a metal bar on board runs through a station, somebody on the platform can measure its length by clicking a stopwatch exactly when the front and back ends of the bar pass him (assume he knows the speed of the train and said speed is constant. No point making this more difficult than it needs to be). Likewise, somebody on the train can do the same: click a stopwatch on and off when the front and back of the bar pass the guy. No problems yet.

Then spacetime principles come into play. Ferreira states that the person on the train will perceive time as running more slowly on the platform than the person actually on the platform. Again, I understand this. Ferreira did a good job explaining how space and time are connected and how motion affects their relationship. However, Ferreira then states that the person on the train will perceive that more time has passed on the platform observer's stopwatch than the platform observer himself does. This is what I don't understand. These two assertions seem to be complete contradictions.

To me, it makes sense that since time "slows down" for people the faster they travel, the train observer should get a shorter time measurement than the platform observer, as stuff outside his reference frame is happening more rapidly than within it; for him, one second has passed when for somebody outside, two seconds have passed (and I am using hyperbole here to make it easy). Multiply that by the speed of the train and it seems that the platform observer should see the rod as expanding, not contracting.

Anyways, can anybody explain to me how I went wrong? Thanks in advance for your help

 

[Ich]

Likewise, somebody on the train can do the same: click a stopwatch on and off when the front and back of the bar pass the guy. No problems yet.

BIG problem there. This somebody may be at the front end or the rear end, but not at both ends at the same time. Information from the other end can't reach him faster than the speed of light, there's a time lag. Exactly in this lag lives "relativity of simultaneity". Depending on their state of motion, different events are being judged as simultaneous.

To me, it makes sense that since time "slows down" for people the faster they travel...

Makes sense, that's why you read it in the popular literature. But that's too easy. Think about the principle of relativity: you can't (and don't have to) say who's in motion and who not. So how could one clock run slower than the other? Which one?
The resolution is that time and space are a union. It's not enough to say when, you also have to say where.
Example: From the embankment, a clock fixed on the train shows increasingly less time than the embankment clock that is at the same position each time. One train clock, many embankment clocks needed.
This works backwards, too, with one stationary embankment clock and many train clocks running by. Embakment clock shows less time than the respective train clock.
This works because time in a different frame is different for each position. There's not one single "other time" that runs slow. Look up the Lorentz transformations.