docs: special relativity

docs/relativity/01-special.md

@@ -92,7 +92,7 @@ distance and time are _not absolute_, they are _relative_. The time or distance
 can change depending on the observer's reference frame, but **distance traveled
 by a photon divided by elapsed time is always $c$.**
 
-## Time is relative
+## Truth is in the eye of the ~~beholder~~ observer
 
 The time experienced by an observer in its inertial frame is called its _proper_
 _time_. Something interesting happens when we compare the proper time of a body
@@ -106,6 +106,9 @@ Time dilation is the lengthening of the time interval between two events for an
 observer in an inertial frame that is moving with respect to the rest frame of
 the events (in which the events occur at the same location).
 
+The proper time interval $\Delta \tau$ between two events is the time interval
+measured by an observer for whom both events occur at the same location.
+
 ([source](https://openstax.org/books/university-physics-volume-3/pages/5-3-time-dilation))
 ```
 
@@ -115,4 +118,54 @@ photon makes one full round trip. Since the speed of light is constant in all
 inertial reference frames, an observer will **always** observe the photon moving
 at the speed of light, $c$, regardless of the motion of its source.
 
-[ insert visualization of a photon clock ]
+```admonish example
+An astronaut observes a photon in the light clock on a spaceship moving at a
+constant velocity, $v$, relative to the Earth. An astronomer on Earth also
+observes the same photon in the light clock. Both observers measure the time it
+takes for the photon to make one full round trip.
+
+- The astronaut observes the photon moving at exactly the speed of light, $c$.
+- The astronomer observes the photon moving at exactly the speed of light, $c$.
+- The astronaut and astronomer disagree on the time it takes for the photon to
+  make one full round trip in the light clock.
+
+**Problem**: Whose round-trip duration measurement is correct?
+
+**Solution**: The astronaut and astronomer are both correct! It's not a trick
+question, this time---we can solve it with 8th-grade math and disciplined logic.
+
+In the astronaut's inertial frame:
+- The photon moves at speed $c$.
+- The light clock is at rest so the photon travels $2D$ in the time it takes to
+  complete one round trip.
+- The time it takes for the photon to make one full round trip is
+  $\Delta t = \frac{2D}{c}$.
+
+In the astronomer's inertial frame:
+- The photon moves at speed $c$.
+- The light clock is moving so the photon travels a longer distance in the time
+  it takes to make one full round trip.
+  - The photon travels sideways a distance $v \Delta t$ in addition to the
+    distance $2D$ it travels up and down between the mirrors.
+  - The total distance the photon travels can be found by pythagorean theorem.
+    One side of a right triangle is $2D$ and the other side is $v \Delta t$. The
+    hypotenuse is the total distance the photon travels over one tick, $s$.
+    $$
+    s = \sqrt{(2D)^2 + (v \Delta t)^2}
+    $$
+- The time it takes for the photon to make one full round trip is
+  $\Delta t = \frac{2 \sqrt{D^2 + (v \Delta t)^2}}{c}$.
+```
+
+We see here that both of Einstein's postulates are satisfied, and we have not
+violated any laws of physics. Yet two observers measured the photon taking
+different durations to make the same trip in space. How can this be?
+
+```admonish help
+Remember that physics and science do not describe the _truth_, they describe
+math and logical axioms that are consistent with the observed data. As
+counterintuitive as it may be,
+[special relativity overwhelmingly agrees with scientific observations](https://en.wikipedia.org/wiki/Tests_of_special_relativity).
+```
+
+### Exploring time dilation