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What am I looking at?
This is a spacetime diagram. It's like a position-time graph with the axes
inverted; time is the vertical axis and position is the horizontal axis. A
vertical line represents something at rest (i.e. maintaining the same
location as time progresses). The diagram shows how time differs for
observers in relative motion. There is a black frame of reference and a
red frame of reference. Each frame has a clock, which is at rest at the
origin within its respective reference frame.
What are the circles?
These represent clocks. There is a single black clock and a single red
clock shown at successive moments in time. The numbers inside represents
the clocks' time readings.
What are the diagonal green dashed lines?
These represent light signals. Since nothing can move faster than speed of
light, each clock's worldline cannot tilt from the vertical more than the
dashed lines.
What are "lines of simultaneity?"
Everything on the same black dashed line happens at the same time for
someone at rest in the black frame - those things all happen at the
instant that the black clock on the line has its given value. If red is in
motion relation to black, its notion of simultaneity differs, and the red
dashed line do not coincide with the black ones.
How does this diagram show time dilation?
We can see that the greater the speed of a clock, the more vertical
spacing between circle, which means more time elaspes between ticks.
How does this show the equivalence of different inertial frames?
When the lines of simultaneity are displayed, you can see the symmetry in
time dilation. If black and red are in relative motion, each judges the
other's clock to be running more slowly.
How does this diagram show the Doppler effect?
If you display the light signals send from one clock to the other, you can
see frequency at which they are received is lower than the frequency they
are emitted. Note that this effect is symmetric between the two frames and that
the factor by with the frequency is reduced is different than the factor
by which each clock judges the other to be slowed.