Hey! I'm trying to understand time dilation in terms of minkowski diagrams. Below I've added a diagram showing the two coordinate systems where the primed one moves relative to the unprimed one with a speed v. My reasoning in this diagram is that when the observer in the unprimed system compares the clock in the primed system with his own that corresponds to comparing the space-time coordinates with eachother. He therefore assigns (t,x) while the primed observed assigns (t',0). Or in other words, as he sees t on his clock he simultaneously sees t' on the primed observers clock. But from my diagram it clearly seams like t' > t which is not consitent with the result that t = (gamma)t' > t; i.e the unprimed observer should see the moving clock run slow not vice versa. What is wrong with my reasoning here?
In your spacetime diagram, " t' > t " is not correct because you must measure intervals (spacetime separations) with Minkowskian spacetime geometry---not Euclidean geometry. In Minkowski spacetime, " t' < t ". [Note that in a PHY 101 position-vs-time diagram (which has an underlying [and underappreciated] Galilean spacetime geometry) we have " t' = t ".]
But should it not be possible to analyse the problem in euclidian geometry aswell? I mean in two dimentional space time the lorentz transformations has a corresponding picture: http://en.wikipedia.org/wiki/File:Minkowski_diagram_-_asymmetric.svg
Although the spacetime diagram is drawn on a plane (like a familiar Euclidean geometry diagram is), the problem is how to measure "lengths" along worldlines (physically representing durations of elapsed time). One way is to draw hyperbolas (for Minkowski spacetime) instead of circles (for Euclidean geometry). These represent the locus of points [events] that are "equidistant" from a given point [event] in the corresponding geometry. Another way is to trace out light-rays from light-clocks [as done in my avatar]. What is interesting about that diagram in my avatar is that the spacetime-volume enclosed between each tick [the volume contained inside the intersection of the future light cone of one tick and the past light cone of the next tick] is Lorentz invariant.
So could I then conclude that any kind of analysis in such a diagram which really involves lengths or differences in coordinates is invalid? Is for example a similar analysis on length contraction also invalid? This in contrast to analysis only which involves separate space-time events like this one: https://www.physicsforums.com/showthread.php?t=407012 Is that kind of analysis valid because it does not involve distance? Btw: Could you recommend any material where I could read up on the properties of Minkowski space you're takling about? :)
Note that diagrams in the thread you linked to has scales drawn along each line... and that those scales are not consistent with Euclidean geometry in the plane. In other words, along lines through the origin, the points at "1 tick mark from the origin" do not trace out a circle... but they do trace out a hyperbola. Can you see your diagram in the diagrams in that posted-thread? t'=0.8 and t=1.0... thus t' < t.
You need to project t' onto the t axis in order to determine how the (x,t) observer views the (x',t') observer's clock. This projection should be taken along the line of simultaneity in the (x',t') observer's frame: draw a line through the event that's parallel to the x' axis. Read off where this line intersects the t-axis.
Nah? The (x,t) observer must surely compare the two clocks simulatneously occording to him and not to the other observer? Yes, this starts to make sence. Thank you! How do one arrive at the conclusion to draw hyperbolas relating the points rather than circles? I do know that there are several things in minkowski space that look 'hyperbolic', but could you recommend any good texts or derivations?
Check out Taylor & Wheeler: http://www.eftaylor.com/download.html#special_relativity [see part two]. Dray: http://www.physics.oregonstate.edu/coursewikis/GSR/ Moore: http://www.amazon.com/Ideas-That-Shaped-Physics-Frame-Independent/dp/0072397144 or http://www.amazon.com/Travelers-Guide-Spacetime-Thomas-Moore/dp/0070430276 Ellis & Williams: http://www.amazon.com/Flat-Curved-Space-Times-George-Ellis/dp/0198511698 I've got something brewing... but it's not quite ready yet.
Here are three different frames showing relationship to the hyperbolic calibration curves. I'm showing the event labled "NOW" for each observer such that each observer is located on the same hyperbolic curve (same proper time). Notice that the simlultaneous X1 space for each observer is a line tangent to the hyperbolic curve. So, you can see how the projections from frame to frame give the time dilations.
I think it's easier to see the time dilations using a symetric space-time diagram. Have two observers moving in opposite directions with the same speed relative to a rest frame as shown in the diagram below. Then you can see the derivation of the hyperbolic calibration curves.