Understanding Special Relativity on a Geometric and Intuitive Level

[MetaMusic]

So here's the deal guys: I have a bachelor's in physics and have gotten an A in an undergraduate special relativity course but I do not feel that I fully understand the subject.

I can do the problems in special relativity which require the various formulas involved in the subject and I even understand the traveling twin paradox where one twin ages less due to his acceleration.

Still I don't have an intuitive grasp of the subject. I've come to figure out that to really understand the theory you need to look at the geometry and not just the equations. So far I understand this:

If time is a dimension similar to space in the same way that an interval is similar to a length (similar, but an interval can only get larger and not smaller with time since it is abstracted from direction), then a time axis and a spatial axis can be rotated around one of the other spatial dimensions and when we do this we find that the time axes from the two reference frames differ and the slope between them is seen by us as relative velocity.

Consider two objects standing still: Since the time axes of these two reference frames are angled, one frame's time becomes space seen from the second frame. So the other particles is traveling down his time axis but to us he's traveling down a space axis (to an extent)!

So far so good. But then my mind fails me when I try to conceptualize the notion that simultaneity breaks between reference frames. Obviously if you have two frames of reference with a 30 degree angle between them you can draw a horizontal line in one frame but it won't be horizontal in the other, which means events on the left side of the line will be seen before the other side is seen in the other reference frame since the line isn't horizontal in that frame. But I still don't feel like I understand everything as a whole. Maybe my picture leaves out length contraction. When two 2D Minkowski reference frames have an angle between them the time axes each see the other one as short but the same happens for the space axes. So that would explain length contraction as well as time dilation. I still think I'm missing something though. I can't fully picture it all at once. As though I don't have enough working memory or something.
;(

 

[UltrafastPED]

Try Taylor & Wheeler's "Spacetime Physics"; E.F. Taylor has provided free access to some chapters:
http://www.eftaylor.com/download.html#special_relativity

 

[A.T.]

Maybe my picture leaves out length contraction.

You should not try to visualize the individual relativistic effects separately, but rather the entrie Lorentz transformation as a whole. The below animation after 1:00 shows a visual comparison of Galilean and Lorentz transformations, and how the relativistic effects arise from the later:

https://www.youtube.com/watch?v=C2VMO7pcWhg