Intiutions for General Relativity

General Relativity is a theory of gravity that describes how gravity affects the curvature of spacetime. It builds on the principles of Special Relativity, following the natural consequences of the constancy of the speed of light further and leading to the emergence of gravity and curved spacetime.

Space-time coordinates

Instead of thinking only in spatial coordinates, we can think in space-time coordinates. One axis is movement through time, and the other is movement through space. This coordinate system is called Minkowski space. Before continuing, it is critical to remember that in this coordinate system we think of an object moving at any speed as having a larger component along the space axis. So forget the direction of motion for now.

Let’s first consider the spatial component for our example. From Brock’s perspective, he is at rest—his space-time vector has zero space component. Ash is moving at speed $v$ relative to Brock, so his space-time vector has a component along the space axis proportional to $v$.

   * Brock
   *------* Ash
+---------------→ Space

Now we consider the time component—this is where the magic happens. Let’s “tick” time forward by one instant. From the last section, we know that Ash will experience slightly less time than Brock.

Time
▲     Brock
│  *  (1,11)
│  |
│  |        Ash
│  |      * (8,8)
│  |     /
│  |    /
│  |   /
│  |  /
│  | /
│  |/
│  * (1,1)
+---------------→ Space

Working out the math, we find that the length of both space-time vectors is exactly $c$.