docs: intuitions and applications

docs/SUMMARY.md

@@ -4,17 +4,23 @@
 
 ---
 
+# Intuition
+
 - [Relativity](relativity/00-index.md)
   - [Special Relativity](relativity/special/00-index.md)
     - [The Basics](relativity/special/01-reference-frames.md)
     - [Time Dilation](relativity/special/02-time-dilation.md)
     - [Length Contraction](relativity/special/03-length-contraction.md)
-    - [Coordinate Transformations](relativity/special/04-transformations.md)
   - [General Relativity](relativity/general/00-index.md)
 - [Timekeeping](timekeeping/00-index.md)
 - [Network Topology](network-topology/00-index.md)
 - [Lunar PNT](lunar-pnt/00-index.md)
 
+# Application
+
+- [Relativistic Vector Math](relativity/vectors/00-index.md)
+  - [Lorentz Boosts](relativity/vectors/01-lorentz-boost-intro.md)
+
 ---
 
 - [Devlog](devlog.md)

docs/devlog.md

@@ -76,3 +76,14 @@ More relativity docs.
 
 More time making an animation to show time dilation with the light clock
 example.
+
+## 2025-01-05
+
+I've been working on the [fundamentals of relativity](relativity/00-index.md)
+for a while and I'm finally starting to dig in to the applications. I need to
+have a firm understanding of [relativistic vector
+math](relativity/vectors/00-index.md) so I can implement it in the app.
+
+I've split the docs into two sections: _Intuition_ and _Application_. The
+_Intuition_ section is for the fundamental concepts of relativity. The
+_Application_ section is for the hard math and implementation details.

docs/relativity/vectors/00-index.md

@@ -0,0 +1,20 @@
+# Applying Relativity to Vector Math
+
+Now that we've come to grips with the fundamental concepts and intuitions of
+relativity, we can start applying these concepts to vector math.
+
+```admonish note
+So far we've seen how the speed of light postulate leads to time dilation and
+length contraction, but we've only scratched the surface of the implications of
+special relativity.
+
+For example, _momentum_ and _energy_. Momentum, $p = mv$, and kinetic energy, $K
+= \frac{1}{2}mv^2$, are both conserved quantities in Newtonian mechanics. Pay
+close attention and you'll see that both of these quantities depend on the
+_velocity_ of the object---relativity will affect these quantities too!
+
+This rabbit hole goes _deep_, eventually leading to the classic $E=mc^2$
+equation and more, but we'll only cover the fundamentals and specific
+applications of relativity that relate to timekeeping and navigation in deep
+space.
+```

docs/relativity/special/04-transformations.md → docs/relativity/vectors/01-lorentz-boost-intro.md

@@ -77,13 +77,16 @@ significant compared to $c$ or we want to be very precise over long distances,
 we need a more accurate transformation that accounts for the effects of
 relativity.
 
-### Lorentz Transformations
+## Lorentz Boosts
 
 Good news, someone already figured this out! The Lorentz transformations are a
 set of equations that describe how to transform coordinates between two
 inertial reference frames that are in relative motion and are consistent with
 the speed of light postulate.
 
+Shorthand for this operation is to call it a _Lorentz boost_
+([source](https://en.wikipedia.org/wiki/Lorentz_transformation)).
+
 ```admonish example
 Recall our [previous example](02-time-dilation.md) with the astronaut and the
 astronomer. Let's say the astronomer's frame of reference is $S$ and the
@@ -121,12 +124,9 @@ $$
 x = vt + x' \sqrt{1 - \Big(\frac{v}{c}\Big)^2}, \quad \text{and} \quad x' =
 \frac{x - vt}{\sqrt{1 - \Big(\frac{v}{c}\Big)^2}}
 $$
-
-That was a lot of work, and it was only for the $x$ coordinate!
 ```
 
-Shorthand for this operation is to call it a _Lorentz boost_. The inverse
-Lorentz boost is the same thing but with the velocity reversed.
-([source](https://en.wikipedia.org/wiki/Lorentz_transformation)).
+## Vectorized Lorentz Boosts
 
-## Space-time interval
+[Wikipedia](https://en.wikipedia.org/wiki/Lorentz_transformation#Vector_transformations)
+has a great explanation of how to apply a Lorentz boost to a vector.