I just have a quick question about time dilation/proper time because my physics book makes it a little confusing. Let's say we have an observer on Earth, and then an observer on a space ship. The space ship leaves Earth, flies to the Moon, and then returns to Earth. Who is the person measuring the proper time and why? I know that a clock "runs slower" when it is in motion because it is in frame S' which is the rest frame of the clock, but doesn't the observer on Earth also have a clock that is in it's rest frame?
Each observer has their own proper time measured by the clock in their rest frame. However, one man's proper time is not another man's proper time. Time dilation means that each observer will see the other observer's clock running slower (compared to their own proper time measuring clock). But everything is perfectly symmetric from either observer's point of view as long as the relative motion is uniform. You measure your clock ticking at the "normal rate" (your proper time) and you see the other person's clock ticking at a slower rate. Similarly the other person measures their clock ticking at the "normal rate" (their proper time) and they see your clock ticking at a slower rate.
This is all well and nice, but it gets interesting when the two compare their clocks after one of them does a round trip. This means that one of them necessarily had to accelerate and decelerate and was not in uniform motion (technically, was not on a geodesic). Now you have an opportunity to actually compare those two clocks and you'll always find that the person in uniform motion (in this case, the observer at rest on Earth) was the one whose clock has ticked the most, and hence aged the most.
The best way to understand this is to realize that the length of paths in spacetime is measured by the total proper time along that path (measured by that path-traveller's clock in their rest frame). One can show that the paths of uniform motion (geodesics) have that length maximized, so any path that deviates from a geodesic (because of accelerations), will necessarily measure a shorter total proper time after a round trip.
EDIT AFTER FIRST COMMENT: Time dilation isn't the appropriate effect to consider in this particular problem -- length contraction is. In Nick's frame, a length contracted ship passes by at speed $v$. In Molly's frame, a point-object (heh) Nick passes by an uncontracted ship at speed $v$. Clearly, this should happen quicker in Nick's frame because of the length contraction. Thinking in terms of time dilation simply doesn't help here. Think from the point of view of each observer and it will be quickly obvious which effect to use.
time moves slower for the moving observer especially if they are travelling at super high relativistic speeds, and time is moving faster for the stationary observer relative to the moving one. that is the correct understanding. one interesting thing. if you live in a 4th floor flat, the gravity for you is minutely less than those on the ground floor being higher up-as earths gravity is strongest at the centre of the earth. so time for the 4th floor tenant would travel minutely quicker, again because they are in their own time-space reference frame. the results because of the minute difference in gravity would be so small. When we walk up a flight of stairs, time is at war with itself. Being farther from the pull of Earth's gravity causes our clock to tick faster, but moving counteracts this effecT, as moving causes your time to travel slower compared to the stationary observer. one reason for this is that despite all these personalised reference frame, the law of the universe is that the speed of light (c) is constant, it is the same for all observers-the moving and the stationary. even gravity does not effect light speed. it can bend light (gravitational bending)but not slow it down, because photons are massless. SR is just really complicated much more than GR. They are the most important two theories in physics and in my view in the whole of science, with the second being the nature of quantum level of reality with phenomena such as the uncertainty principle, and the dual nature of particles. GR AND SR are how reality work on the macro level-the level of stars and planets and even galaxies, while the quantum theory is how stuff works at the smallest level, where the laws of physics break down, where it is like reality is in flux, that it has not been made yet. another phenomena that occurs at high velocity is length contraction, a type of lorentz transformation. I understand that even less.that is I know that it is happens, I know the equations for it, but I don't understand WHY? Recently I have come to understand the WHY of SR-that is, for a long time the reason why relativistic velocity causes time dilation stumped me, whereas GR time dilation from gravity is for me very plainly logical. Now i understand that as we travel through space we also interact with the time dimension so it stands to reason that our experience of time will therefore change. the speed of light is a fundamental constant of the universe which I believe has to do with the fundamental structure of our four dimensional reality. it may be necessary to merely memorise these concepts,and a day will come when the WHY will just click. Good luck.