When is a straight-line the slowest way to arrive anywhere?

* I have been gone for a few months nursing a healing brain. This post may make your brain ache a bit.

Q: When does moving in straight line from point A to point B take the longest duration in time to get from A to B?

A: Always. Huh?? This thinking really hurts, so hang in there.

Time, no matter whether it is measured in months, weeks, or even nanoseconds, is indeed the longest distance between any two points.

Newtonian physics, before Einstein, saw space and time as separate and distinct. There is distance between two places in space and elapsed time between two events in time. Not surprising, the shortest path in space between two points in three-dimensional space is a straight line and every 12-year-old will tell you so. Accordingly, you can make the distance you actually travel between two points in space as long as you wish by taking a wildly winding path. Got it – no problems yet.

However, Special Relativity says this is not right and the answer hinges on Einstein’s Special Relativity and the fact that the three dimensions of space combine with time to make “spacetime” (which is an interval between two events in our four-dimensional universe). Modern physics combines the three basic directions in which a person is free to move from place to place, with the unfolding sequence of moments in his life, into a unified universe of four basic directions (or “dimensions”). Recognizing that space and time are not separate leads to the seemingly contradictory finding that the longest TIME between to points is a STRAIGHT LINEyea, ouch, it hurts my brain too. So how do we get there?

Most of us live in a world where questions involving the speed of light never enter into our daily lives. In this slower than light speed world moving is much like a running back covering the distance between the 10-yard line and the 40-yard line by running straight down the field never swerving left or right. Here the straight line is the shortest distance between two points and we assume it represents the fastest elapsed time to get from the 10 to the 40 yard line. In doing so we are accepting the notion that time is a universal feature of the universe, but it is not. Time is relative – i.e. clocks moving at different velocities will record different elapsed times. A clock on a jet traveling around the world will reflect a shorter elapsed time compared to a clock that is stationary on earth. The key here is velocity.

Moving in a straight line – an unaccelerated trajectory moving at constant velocity, results in the longest duration possible. Doing the opposite – zipping all over the place accelerating as fast as you can, but taking care to reach your destination at an appointed time, you will experience a shorter duration. Restated, longest elapsed time between two points is a straight line.

Notes:

  1. See, From Eternity to Here, by Sean Carroll, © 2010. A great read on time from an award-winning theoretical physicist.
  2. At the end of The Glass Menagerie by Tennessee Williams, Tom claims to have traveled much farther by standing still on the fire escape than if he had gone to the moon, “for time is the longest distance between two places.” – an incredible quote because:  (a) it is exactly in line with the understanding of modern physics, and (b) it is from a playwright who likely knew little of physics.