From Euclid to GPS

Jean-François COLONNA
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www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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[en français/in french]

Keywords: Euclidian Geometry, Géométrie euclidienne, GPS, Relativity, Relativité.

[See an abstract]

1-Euclidean Geometry:

Euclid lived in ancient Greece two or three centuries before our era. He is universally known as the author of The Elements, one of the oldest works in mathematics, which presented, starting from axioms, rigorously proven theorems.

These axioms [01] were considered by Euclid and many philosophers, including Socrates, as statements derived from experience, self-evident and whose meaning was beyond doubt. The axioms of what is called Euclidean geometry are:

A1-For any two distinct points, there exists a line containing both points and this line is unique [02].
A2-A line segment can be extended indefinitely.
A3-A circle can be drawn given a point as its center and a distance as its radius.
A4-All right angles are equal.
A5-For any line L and any point P not on L, there exists a line passing through P that does not intersect L and this line is unique [03].

From these axioms and previously proven theorems, new theorems can be demonstrated. One of the most famous is certainly the Pythagorean theorem:

which, moreover, holds a prominent place in The Elements.

For more than two millennia, numerous new theorems in geometry were proven, but despite this, an intense debate remained unresolved: Could the fifth axiom (A5) be derived as a theorem, or not?

In a plane, given a point P and a line D not passing through that point, there exists exactly one line passing through P and parallel to the first.

Throughout all those centuries, it was, in a way, both an axiom and a conjecture. But for physicists, whether one lived before...

or after Copernicus

One thing was certain: space was obviously Euclidean and absolute [04]. This was the view held by Newton, and at that time, light posed no particular problem...

2-Non-Euclidean Geometries:

By the end of the 19th century, thanks to the work of Gauss, Lobachevsky, Riemann and others, it became clear that it was possible to replace Axiom A5 with others, which then gave rise to new, non-contradictory geometries, thereby confirming that A5 could not be a theorem of Euclidean geometry.

In place of the axiom stating the uniqueness of the parallel line, two alternative versions of Axiom A5 could be used:

A51-Through a point P outside a line D, no line parallel to D can be drawn: this is known as spherical geometry.
A52-Through a point P outside a line D, an infinite number of lines parallel to D can be drawn: this is known as hyperbolic geometry.

In Euclidean geometry, the sum of the angles in a triangle is equal to 180 degrees, but this is no longer true in these new geometries: it is either greater or smaller, depending on whether one is in spherical or hyperbolic geometry. Moreover, all metric results, such as those involving lengths, including the Pythagorean theorem, are no longer valid.

But could these new geometries have real-world applications?

At the turn of the 19th and 20th centuries, even though William Thomson [05] famously declared in 1900: In the clear blue sky of Physics there remained on the horizon just two small clouds of incomprehension that obscured the beauty and clearness, suggesting that Physics was essentially "complete", two problems remained unresolved: On the one hand, the interpretation of the Michelson and Morley experiments [06] and on the other, the explanation of the inconsistency of calculations concerning blackbody radiation [07].

3-General Relativity:

At the beginning of the 20th century, physicists now had not just one geometry, the one they had been using for centuries, but several. In 1915, Albert Einstein published a revolutionary new paper [08] on General Relativity, in which, as in his 1905 Special Relativity, space and time could no longer be treated separately but instead formed quadridimensional spacetime. Moreover, this spacetime was curved, made possible by non-Euclidean geometries. The fundamental equation shows that, simultaneously, mass/energy curves spacetime and the curvature of spacetime determines the paths of mass/energy:

Very quickly, many physicists embraced the theory and in particular Georges Lemaître, who realized that, despite Albert Einstein's belief, the Universe could not be static. In 1929, observations by Edwin Hubble (in collaboration with Milton Humason [09]) and Vesto Melvin Slipher showed that other galaxies appeared to be moving away from us, the Milky Way, at speeds that increased with their distance [10]. Therefore, the Universe was not static but expanding, and if one plays the "film in reverse," it must show a contraction toward a singularity (of zero dimension and infinite density, within the framework of General Relativity, disregarding Quantum Mechanics...). This singularity was nicknamed the "Big Bang" by Fred Hoyle in the 1950s, intending to mock the theory in favor of his own steady-state model. However, it was later proven that his model was not consistent with observations and measurements.

General Relativity thus allows for the description of the Universe in its entirety:

but also of phenomena that are difficult to grasp due to how cataclysmic they are, such as black holes or gravitational waves [11]:

But in our daily lives, it seems quite useless and far removed from our everyday concerns...

4-GPS (Global Positioning System) [12]:

The GPS is a satellite-based positioning system designed and developed at the initiative of the Pentagon starting in the late 1960s. It allows the precise determination of the position of any point relative to the Earth. The principle is straightforward and relies on a trivial Euclidean geometry problem, which we will first describe in a two-dimensional plane to simplify the representation:

Let's consider a non-degenerate triangle ABC in a plane, which will serve as our reference frame (left-hand figure). Now, in the same plane, let's take a point P whose distances to the three vertices of the triangle are known (middle figure). Can we then determine the position of point P? The answer is obviously yes: point P lies at the intersection of the three circles centered at {A,B,C} with radii {AP,BP,CP} respectively (right-hand figure).

In three-dimensional space, we simply replace the triangle with a non-degenerate tetrahedron ABCD:

Point P lies at the intersection of four spheres centered at {A,B,C,D} with respective radii {AP,BP,CP,DP}.

Now, how can this be implemented in practice? Building a giant pyramid is obviously a ridiculous idea, particularly because its four vertices couldn't be visible from every point on the globe. The solution, therefore, is to use a number of satellites. Four satellites are clearly insufficient, since, just like the pyramid, they wouldn't be visible from all points on Earth. It is therefore "enough" to deploy a constellation of satellites. In the case of the GPS, around thirty satellites [13] are used, orbiting at an altitude of 10,900 nautical miles [14], distributed across six orbital planes inclined at 55 degrees to the equator. Their orbital period is twelve hours.

Thus, at any given moment, any observer (our point P from earlier) on the Earth's surface has more than four satellites (our previous "pyramid") in view. But this raises the question: how can the distances be measured? Even though the method actually used differs, it is based on the following principle: replace the evaluation of distance with a measurement of time. For example, if satellite A sends the message "the time is H1" and this message reaches point P at time H2, then the transmission duration is H2 - H1. Finally, knowing the speed of signal propagation [15], the distance is straightforward to compute. Of course, this simplifies the situation, ignoring the fact that the medium of propagation is neither a vacuum nor homogeneous.

The (perhaps apocryphal) story goes that this was the way the U.S. military initially planned to program the satellites. However, some physicists warned them that it would not work properly. Nevertheless, the satellites were launched, but the physicists secured the ability to reprogram the satellites from Earth and fortunately so, because this indeed became necessary.

But what problem(s) had the physicists pointed out? Quite simply, relativistic effects had been ignored:

1-Special Relativity: In the case of uniform relative motion, there is time dilation. If V and C denote the velocity [16] of a satellite relative to the Earth and the speed of light respectively and if Dt_T and Dt_S are the time intervals measured on Earth (P) and onboard the satellite (A), then the following relation holds:
```
                               Dt_S
                    Dt_T = -------------
                                  V²
                           √(1 - ----)
                                  C²
```
Hence, V being strictly less than C:
```
                    Dt_T > Dt_S
```
And so, the duration measured onboard a satellite (Dt_S) is shorter than on Earth (Dt_T), meaning that time passes more slowly in orbit than when stationary on the ground -effect 1-.

2-General Relativity: Here, the opposite effect is observed. Indeed, the gravitational potential of the Earth is obviously weaker at higher altitudes than on the surface. The equations of General Relativity show that the weaker this potential is, the faster time flows. Therefore, in orbit, time passes faster than on the ground -effect 2-.

Of course, on the one hand, there is no reason for these two temporal effects (-1-: slowdown and -2-: acceleration) to cancel each other out and on the other hand, they cannot be neglected, or there will be a significant loss of precision [17].

Thus, these theories, which one might think are only relevant for describing extraordinary phenomena like the Big Bang, black holes, or even gravitational waves, cannot be ignored in our daily lives. They are present in everyday life and in the pockets of many Earthlings!

Finally, it is worth noting that the GPS system is based on two of the three pillars of contemporary Physics, namely Special and General Relativity, as we have just seen. However, the third pillar, Quantum Mechanics, is not absent: indeed, on the satellites, in the five ground stations [18] and in the portable receivers, there are numerous integrated circuits that themselves contain a multitude of transistors, products of Quantum Mechanics. Furthermore, since the GPS principle relies on time measurements, it is essential to have the most precise clocks [19] in the satellites and ground stations, clocks that can only be achieved through the principles of Quantum Mechanics. Therefore, the GPS is one of the greatest syntheses of contemporary Physics!

5-The Importance of Research in Mathematics:

Pure Mathematics and Applied Mathematics are often distinguished and opposed. Pure Mathematics draws its inspiration from highly abstract questions, disconnected from Reality, while Applied Mathematics responds to concrete questions posed by Science and Industry. One might therefore consider that only Applied Mathematics has real value. But as this document shows, this is not the case. In the past, the debate over the parallel postulate might have been considered futile, but who could have imagined that, several centuries later, it would lead to such a concrete and "democratized" application?

And today, no one knows what the study of major current conjectures might lead to, or what revolutionary applications it might offer us.

[01] Also called postulates.
[02] After a "refinement" made by David Hilbert.
[03] If a line intersects two other lines such that the interior angles on the same side are less than two right angles, the two lines, if extended infinitely, will meet on the side where the angles are less than two right angles.
[04] Just like time.
[05] Lord Kelvin.
[06] The Michelson and Morley interferometric experiments, conducted between 1881 and 1887, were designed to demonstrate the existence of the luminiferous aether, a hypothetical medium that would carry light waves (just as air carries pressure waves, or sound). However, what they showed, since the Earth was supposed to move relative to the aether, was that the speed of light was independent of the direction in which the measurement was made, which contradicted the classical law of addition of velocities (Earth relative to the aether and light relative to Earth in these experiments). Understanding came in 1905 with Albert Einstein's paper on Special Relativity.
[07] A black body is an ideal object that absorbs all electromagnetic energy it receives without reflecting or transmitting it, but it still radiates due to the thermal agitation of its components. The "classical" calculations made at the end of the 19th century predicted an implausible amount of radiated energy, an infinite amount. Understanding came in 1900 with Max Planck's hypothesis of quanta.
[08] Let us recall that during his career, Albert Einstein published a number of articles that revolutionized Physics and our view of reality, including: Brownian motion (1905), Special Relativity (1905), General Relativity (1915), gravitational waves (1918), the photoelectric effect (1905, Nobel Prize 1921) and the EPR effect (Albert Einstein, Boris Podolsky and Nathan Rosen, 1935).
[09] A former muleteer who worked on the construction of the Mount Wilson Observatory!
[10] This can be seen by a spectroscopic analysis of their light, which shows a redshift in the frequencies (similar to the classical Doppler effect, but with very different causes), interpreted as a "flight"...
[11] It should be noted that Albert Einstein had predicted the possible existence of gravitational waves, adding that they would probably never be detected. In fact, it took a century before the two LIGO interferometers in the USA (in Washington state and Louisiana) "saw" the first ones on 09/14/2015. For me, this is the finest example of a discovery made thanks to Mathematics, as without them, these waves would not have been detected by chance, unlike the expansion of the Universe, which could have been observed simply by the redshift of the light from distant galaxies, even if Georges Lemaître had not yet highlighted it thanks to General Relativity...
[12] There are other systems with similar functionality: Beidou in China (1983), Galileo in Europe (1998), and Glonass in Russia (1976).
[13] The exact number of satellites may vary depending on the date, the launch of new units, or failures.
[14] The satellites are not in geostationary orbit (35.786 kilometers above the Earth's surface) possibly to reduce the energy consumption related to transmissions.
[15] The speed of propagation of an electromagnetic wave in a vacuum is equal to 299.792.458 meters per second.
[16] It can be considered uniform "locally"...
[17] Calculations show that neglecting Special and General Relativity causes a drift in measurements of more than 11 kilometers per day!
[18] The GPS uses five ground stations that continuously monitor the satellites in order to know, in particular, the tiny deviations from the "theoretical flight plans." They are located on the Hawaiian Islands, Ascension Island, Diego Garcia, Kwajalein and finally, Colorado Springs.
[19] It should be noted that the most advanced so-called atomic clocks have such precision that they are sensitive to the effects induced by General Relativity and thus, raising them by one meter vertically and thereby decreasing the gravitational potential they experience, alters their frequency of operation!