docs: curved time

Author: philiplinden

docs/relativity/general/00-index.md

@@ -1,4 +1,4 @@
-# Gravity, Curved Space, & Curved Time
+# General Relativity
 
 Now let's bring our understanding of relative to the general case where there
 could be a gravity well nearby (aka _general_ relativity).
@@ -10,12 +10,58 @@ further and leading to the emergence of gravity and the idea that space and time
 are not independent axes, but rather our reference frames are composed of a
 continuous _spacetime_.
 
-## Curved spacetime
+## Curved time creates gravity
+
+### Reframing our understanding of gravity
+
+In Newtonian physics, gravity is a force that pulls objects together. On Earth,
+this would look like a uniform force acting on all objects pulling them towards
+the ground. But our exploration of Einstein's postulate on
+[inertial reference frames](relativity/00-index.md#inertial-reference-frames)
+showed us that we can "flip" our reference frame and the physics are the same
+By this definition it is impossible to tell if you are at rest or in motion. So
+taking that to the extreme, maybe you are at rest and the Earth is accelerating
+upwards towards you!
+
+Now consider a photon. It travels in a straight line in space. In the reference
+frame where Earth is accelerating upwards, after a moment the photon is still
+traveling in a straight line but the ground is closer to it. More time passes
+and eventually the ground intersects the photon's path. From the perspective of
+the Earth, it looks like the photon is bending towards the ground!
+
+Video: [Gravity bends light even though it has no mass](https://www.youtube.com/watch?v=05jFhuRs-w0&t)
+
+Now let's get a laser and trace the path of its photons in the Earth's reference
+frame. We observe that the photons are bent towards the ground, like before.
+
+This time let's measure the path of a photon at the top of the beam and at the
+bottom. To us, it appears that the photon at the top takes a longer path to
+reach the ground compared to the bottom photon. Our special relativity
+intuitions remind us that the speed of light is constant, so time must be slower
+for the top photon compared to the bottom photon. This is called [gravitational
+time dilation](https://en.wikipedia.org/wiki/Gravitational_time_dilation).
+
+I find this explanation profoundly unsatisfying. We can do better.
 
-```admonish warning
-This section is a work in progress.
-```
+### Geometric explanation
+
+A more intuitive way to think about it is to think about it geometrically.
+
+I don't have the time or skills to make a beautiful animation for this, but the
+following video does an excellent job of explaining it:
+
+Video: [Gravity comes from time curving space](https://youtu.be/OpOER8Eec2A)
+
+![alt text](./assets/image.png)
+
+## Curved spacetime
 
 A key insight that will help us later is to understand that this definition
 holds true _even if we morph the reference frame_. So on the grid, an object
 would still move from point A to point B with respect to the reference frame.
+This is what we mean when we say that _curved_---all objects move along straight
+lines unless acted upon by a force, but the grid itself is getting morphed by
+gravity.
+
+Video: [Curved spacetime changes trajectories of light and
+matter](https://www.youtube.com/watch?v=S78h8zQwQe0)