Good questions! You are falling into the trap that, actually, the majority of people coming to us with special relativity questions have. You've learned about length contraction and time dilation (which are very easy to glean off the internet), but you haven't learned about loss of simultaneity (which is more subtle, but just as important). Without additionally accounting for loss of simultaneity, you will run into all kinds of paradoxes and contradictions.

Besides linking to the hundreds of previous questions along these lines, all I can say is, find any good book (absolutely not anything popular and internet-based, these are all oversimplified, and they generally make less sense the smarter you are). I promise they will explain literally everything about all the questions you have asked so far; you are going down a very well-trodden road.

Examples of good books include Spacetime Physics, Special Relativity: For the Enthusiastic Beginner, and _ An Illustrated Guide to Relativity_. Actually, just about any book is good, as long as it's a real book.

@knzhou thank you for the advice. I'll let you know how it goes!

@knzhou so I took your advice started reading an illustrated guide to relativity...and I'm pretty confused. I'm on page 96...(in case you have the book)...where we have a space time diagram from the point of view of a tree, and a car moving past the tree. I understand that slanted lines parallel to the car's world-line in the tree diagram are in the same place from the car's point of view, since they correspond to points that are all the same distance away from the car at any point in time. Additionally, I understand that slanted lines, slanted in such a way that two photons shot from the...

...origin when the car passed the tree are the same distance away from the car at any point in time, correspond to points that are all happening at the same time for the car. So, events that are moving to the right as time passes for the tree, are staying in the same location for the car, and events that are happening later in time for the tree that are in the direction the car is moving, are already happening for the car.

@knzhou (and I'm putting this in a separate comment because this is the heart of my question) I understand that when we switch into the car's diagram, lines at the same time for the car must become horizontal, and lines at the same place must become vertical. But why is it that when we switch into the car's diagram, we need to do so in such a way that the areas stay the same? Thanks!

@JoshuaRonis What do you mean, "areas stay the same"? The area of what?

@knzhou In the tree's frame,the area enclosed by the parallelogram made by two parallel lines slanted lines, each signifying events at the same place for the car, and two other parallel slanted lines, each signifying events at the same time for the car, stays the same when we make those lines vertical and horizontal respectively (when we switch into the car's space time diagram).

Just sit tight and keep reading! There are multiple ways to set up special relativity, i.e. multiple postulates that you can start with to get to the same result. "Same area" is one of them, "constancy of speed of light" is another. You have to ultimately take something to start with, because if you don't make any assumptions, then you can't make any deductions from those assumptions. Ultimately all of these things will be neatly unified in the geometric picture of special relativity, but that will only make sense after reading more.

@knzhou Thanks, I will! I gave up with the illustrated guide though :( ....I got too confused...I'm going through Morin's now, I really like his lively style. I didn't really like SpaceTIme Physics, because his approach was wayyy too convoluted (almost like he needs props to explain things...I don't know why the author brings in so many external references. Do you have any other recommendations? Thanks!

Yeah, a lot of young people seem to like Morin. Another one I liked was the later chapters of Kleppner and Kolenkow (1st edition). There are lots of options, and after finishing one good one, all the other ones will become pretty easy to read.

@knzhou holy shish kebabs I just had a breakthrough in understanding. Okay, so Alice is shooting past Bob in her space-ship at a velocity of $v$. She's going to travel a distance of $L_p$ according to Bob, and a distance of $\frac{L_p}{\gamma}$ according to her. The time it will take for her to travel that distance is $\frac{L_p}{v}$ according to Bob, and $\frac{L_p}{v\gamma}$ according to her. Alright, so far so good...however, what was confusing me was how it was possible that from her point of view, if $\frac{L_p}{v\gamma}$ seconds passed according to her, $\frac{L_p}{v\gamma^2}$ seconds...

...must've passed for Bob, while at the same time it was possible for Bob to conclude that if $\frac{L_p}{v}$ seconds happened for him, $\frac{L_p}{v\gamma}$ seconds passed for her? How can it be that Bob thinks more time passed for him than for her, and Alice thinks more time passed for her than for him? AND NOW I FINALLY UNDERSTOOD! When Alice was first passing past Bob, events happening for her NOW a distance of $\frac{L_P}{\gamma}$ away (from her point of view) and a distance of $L_P$ away (from Bob's point of view) were still in Bob's future! Bob still had to wait $\frac{L_Pv}{c^2}$...

...for those events to happen! Since she was travelling towards a location where, in her NOW was still in Bob's future, its entirely possible that she concludes more time passes for her than passes for Bob, while Bob concludes that more time passed for him, because as she moved away from him, she was literally moving into locations that were in Bob's future but her now! Its like when Alice flies past Bob, she's not only moving away from him in time, she's also moving towards his future! What the heck?!! Relativity is so cool!

Yup, that's the right way to see it! That's the first big "jump" people always have to make when learning relativity. Now you, too, can answer a thousand special relativity questions on this site.

Just ask another normal question, that's how this site is meant to work!

@knzhou awesome! Thank you for helping me thus far! I want to just ask these last questions to make sure I'm getting everything correct...*(I'd start a chat, since I know we're not supposed to have such long comments, but I don't know how, and its also just not a normal question...I just want to see if I have things right):* Alice flies past Bob, as described in the situation in the comments above, at a velocity of $V$, and travels a distance (as measured by Bob) of $L_p$ away from him. $\frac{L_p}{v}$ - The time which Bob measures passes. $\frac{L_p}{v\gamma}$ - The time which...

...Alice measures passes for her. $\frac{L_p}{v\gamma^2}$ - The time which Alice measures passes at the location $L_P$ away from Bob. $\frac{L_pV}{c^2}$ - the amount by which Alice's now is in Bob's future (or he is in her past, same thing) at a distance of $L_P$ (according to Bob) in the direction in which Alice is travelling as she passes him, and correspondingly the amount by which Alice is in Bob's past (or he is in her future, same thing) as she gets to that distance away from him, at his position. Am I missing anything, or is that all? Thanks again @knzhou !!!!!!!!!

Sounds fine! Test your understanding with more of the practice problems from Morin, which are elaborations of this idea, and which come with solutions.

Will do! Thanks again, I'll probably be bothering you again with questions soon, as we're starting relativity in class soon! Adios and (idk where u are, but its night here) goodnight!

@knzhou alright, that wasn't so long, but I slept on it, and now this whole Alice being in Bob's future thing is really blowing my mind.

Lets say Bob is $L_p$ away from Alice, as measured by him, and she's travelling towards him at a really, really fast velocity $V$.

Therefor, the Bob she sees at a distance of $\frac{L_p}{\gamma}$ away (due to length contraction) is a Bob $\frac{L_pV}{c^2}$ time units (whatever they are) in the future from the Bob that's currently there, seeing her a distance of $L_p$ away from him.

Lets say that $\frac{L_PV}{C^2}$ time intervals into the future, a bee stands on his shoulder. Bob is allergic to bees. So, Alice sends a light pulse to Bob that indicates to him to brush the bee off, so that the bee doesn't sting him.

Hmm...actually, nevermind. I was thinking maybe there was some way for Jill to see Jack react to her sending the message, and then not send it to see what would happen...but now I can't come up with it.

.....okay, scratch all that...back to the basic Alice and Bob scenario, because I confused myself again... :(

Say that Bob has a "left clock" and a "right clock". They are a distance of $L_P$ away from one another, according to him, and are synchronized with one another, according to him. Bob is stationed at his right clock.

At a time $t=0$, Alice passes by his left clock, and she's travelling towards him at a really, really fast velocity $V$.

At the instant that Alice passes Jack's left clock, she sees Bob's right clock $\frac{L_PV}{C^2}$ seconds ahead from his left clock. What this means is that for Alice, at the instant she passes the left clock, the Bob she is looking at is $\frac{L_PV}{C^2}$ seconds older than the Bob that is seeing her pass that left clock actually is.

There goes my first question...what does the Bob which Alice is seeing as she passes that left clock see? Presumably, he sees her some distance ahead from the left clock...right? How far ahead?

Well, anyways, lets continue. When Alice reaches Bob, he will have measured a time of $\frac{L_P}{V}$ to have passed. Alice will have measured a time of $\frac{L_P}{V\gamma}$ to have passed for her, while she will have seen Bob age by $\frac{L_P}{V\gamma^2}$, which is less than Bob felt himself age by, but is possible because the Bob she saw at when she first passed the left clock was $\frac{L_PV}{C^2}$ seconds into the future of the Bob that was looking at her as she passed the left clock.

Sure enough, $\frac{L_P}{V\gamma^2} + $\frac{L_PV}{C^2} = $\frac{L_P}{V}$$ They agree on the time on the right clock once Alice gets there.

So I understand all that...pretty much, I understand everything from Alice's point of view, but I'm confused about Bob's point of view...when Alice first passes the left clock, Alice sees $\frac{L_PV}{C^2}$ seconds into the Bob that sees her future. However, from the point of view of the Bob that sees her, he is seeing $\frac{L_PV}{C^2}$ into Alice's future...oh, wait, that makes sense! The Alice that sees the Bob that sees her passing the left clock hasn't yet passed the left clock!

Hmm...sorry, the above isn't very coherent, and I haven't posed any questions well...I'm still organizing my thoughts, but just typing this out is helping me a lot. I'll try thinking a bit more and seeing if I can pose concrete questions.

Question 1: Thte Bob which Alice sees as she passes the left clock is $\frac{L_PV}{C^2}$ seconds ahead from the Bob that sees Alice as she passes the left clock. Additionally, the Alice which Bob sees as she passes the left clock is......how far ahead from the Alice which sees THAT Bob?

I hope I'm explaining myself well there - at first, I thought the answer would just be $\frac{L_PV}{C^2}$ seconds ahead...but it can't be, since the Alice that sees THAT Bob (the one that sees her pass the left clock at $t=0$ on the left clock) must be some distance behind the left clock...