…three conventional dimensions of space, was still able to piece together this key physical insight. Increasing your motion through space decreased your motion through time, and increasing your motion through time decreased your motion through space. All measurements of space and time are only meaningful relative to the observer in question, and depend on the relative motion of the observer to the observed.
The thought that all objects move at the speed of light in four-dimensional space-time correlates to the principle of least action. The straightest path is always the path that minimizes the “action.” Objects seek a to adopt a rest state such that the total potential energy is minimized. This is another way of saying that an object will seek out the lowest energy state. A ball rolls down a hill until it finds itself stuck or at the bottom of a valley.
When a system is in motion it will attempt to integrate this momentum over the distance traveled. In this way, the principle of least action seeks to find the most efficient way of minimizing the conversion of potential energy to momentum or momentum to potential energy.
As an object “chooses” a trajectory with the shortest path, it attempts to choose a path that minimizes the amount of action along its trajectory through space-time. This requires an object to pick the path that maximizes the passage of time.
This can be thought of as maximizing the amount of clicks on a wristwatch that moves along a given trajectory. A trajectory which maximizes for time is one that minimizes its action along that trajectory. We often think of a straight line as the shortest distance path but, in Minkowski space a “straight line” is the path with the longest duration in the time dimension.
The below quotes are me paraphrasing Sean Carroll from his Q & A video on gravity (see link here).
“In general relativity, inertial trajectories are geodesics. Geodesics are the extremal length paths through a space-time diagram.”
“You extremize along all the possible paths to maximize for time. I say ‘extremize’ because in relativity the physical paths will actually maximize the amount of time. This is to say, the amount of clicks on an actual wristwatch that moves along that trajectory.”
This realization about geodesics has some profound consequences.