### Time Dilation and GPS Satellites

#### by James Buckland

The term *mathematics* is often accompanied by the derogatory epithet *pure*; indicating that the subject has little to no real-world application, and is therefore of no value to society. It is true that many mathematical fields are without application; I would merely add *yet.* Many times throughout history has an isolated and insular field seen invaluable, if delayed, employment. A good example is George Boole‘s work (circa 1850) on a field of algebraic logic now known as *boolean algebra*, which saw massive application over a hundred years later in the form of *digital logic gates*, the AND, OR, and NOTs that are so employed by modern computer circuits, transistors, and diodes.

This can be a very valuable perspective to maintain; one of the features of the massive parallel engine that is the modern academic world is that it is so encapsulated; workers sometimes do not see the fruits of their labor within their lifetime; they are considered lucky to receive recognition. The most famous recipient of this recognition was Albert Einstein, whose 1905 work on special relativity introduced a highly counterintuitive concept known as time dilation; as expressed in his *Principle of Invariant Light Speed, *

light is always propagated in empty space with a definite

~~velocity~~[speed]cwhich is independent of the state of motion of the emitting body.

Relativity exists in two forms: general and special, the former operating on macroscopic scales; the latter, microscopic. By 1916, physicists were able to predict two effects of *time dilation:*

- one, that a reference frame with sufficient velocity will experience a slower rate of time relative to an inertial reference frame, and;
- two, that a reference frame with sufficiently low gravitational potential will experience a similarly slowed rate of time relative to an inertial reference frame.

At the time, this work was, of course, internationally recognized and acclaimed; Einstein became a celebrity even to the layperson; however, this fame was due to partially to the mass-energy equivalence and partially to his work on the atomic bomb and its political implications. Time dilation, needless to say, did not breach the public sphere.

Around forty years later, in 1956, Friedwardt Winterberg proposed a real-world text of relativity which involved the placement of highly accurate atomic clocks on very fast, very high-altitude satellites; this would have the effect of readily testing both effects of time dilation. This idea was kicked around the defense-contractor circuit for about ten years, and eventually resulted in TRANSIT, in 1964, an early satellite navigation system.

(An aside: GPS and other satellite navigation systems operate on a sort of three-dimensional triangulation; this involves sending up signals to all nearby satellites, which ping back the time and location of the signal’s reception; this defines a sphere (see: trilateration) around the GPS unit, which can calculate its own location using a set of surprisingly simple equations.)

However, without compensation for the time-dilation effects experienced as such high velocity (and such high relative gravitational potential), GPS errors would accumulate by around 10km a day. Thus, all GPS systems are programmed with time-dilation compensation algorithms built-in, using the Lorentz transformation, which really isn’t so bad after all.

Now imagine momentarily an alternate historical timeline, in which we sent up GPS satellites, uninspired by Winterberg’s predictions; merely to measure location. After all, the technology involved in building the satellites doesn’t involve relativity at all; merely strong engineering principles, rocket science, propulsion, and other comparatively intuitive sciences. Imagine that we used the same triangulation methods and discovered a 38 microsecond drift per day! There would have been absolutely no way of explaining it, until astrophysics arrived at the explanation on its own.

To reiterate: often, pure mathematics (and, by extension, Physics, the most virginal of the sciences) will involve a field with no obvious applications. I take it on a mixture of gambler’s fallacy and faith that, because there are so many examples of previously unapplied mathematics finding vital applications decades later, all mathematics *must* have some application; it had merely not been discovered yet.

[…] rulers are mathematical structures (rulers, really) which almost perfectly demonstrate my earlier article on the oft-delayed values of pure mathematics. A perfect Gombol ruler will be a set of markings […]