"But there is another reason for mathematics's great reputation: it is mathematics that provides a level of security to the precise natural sciences that they would not be able to achieve without mathematics."
"Insofar as reality is concerned, mathematical laws are indeterminate; in so far as they are definite, they do not refer to reality."
—Albert Einstein
Einstein's theory of relativity revolutionized our understanding of the universe, fundamentally changing the way we perceive time and space. The theory is currently one of the twin foundations of contemporary physics, the other being quantum theory. However, its impact extends beyond the realm of theoretical physics into the practical realm of technology. In many practical applications, the Global Positioning System (GPS) embodies the practical effect of Einstein's theory.
Before examining the application of Einstein's theory of relativity to GPS, it is necessary to understand the key principles of his theory. The special theory of relativity, proposed in 1905, challenged the classical concepts of absolute space and time. Einstein proposed that the laws of physics are the same for all observers moving at a uniform speed (an inertial frame of reference) and that the speed of light in free space is constant regardless of the observer's speed.
The general theory of relativity, published a decade later in 1915, expanded these ideas to include gravity as a component of the curvature of space-time caused by mass and energy. The theory predicts phenomena such as gravitational time dilation, in which time passes more slowly in stronger gravitational fields, and the bending of light around massive objects known as gravitational lensing.
The key role of relativity in GPS
GPS is a network of satellites orbiting the Earth, constantly transmitting signals to provide precise location and time information to ground users.
A GPS receiver collects signals from at least four satellites, measures the time it takes for each signal to reach it, and calculates the distance to each satellite based on the speed of light. From these distance measurements, the receiver uses a mathematical technique called trilateration to pinpoint its exact location on Earth. This location data is then used to provide users with accurate information about their location, speed and even altitude.
GPS technology relies on precise timing synchronization, which is where Einstein's theory of relativity comes into play.
GPS was first conceived in the 1960s, and the United States launched the first GPS satellite, Navstar 1, in 1978. By the mid-1980s, with several satellites in orbit, there was a clear discrepancy between the predicted and observed positions provided. Through GPS system. These differences amount to tens of meters, making them unacceptably large for many practical applications, especially in the military and aerospace fields where precise positioning is critical.
That's when scientists and engineers working on GPS systems began investigating the causes of these inaccuracies. Two main relativistic effects stand out.
Special theory of relativity and satellite clocks
In his theory of special relativity, Einstein introduced the concept of time dilation, whereby time passes at different rates for observers moving at different speeds.
The satellites in the GPS constellation orbit the Earth at high speeds (approximately 14,000 kilometers per hour or 8,700 miles per hour) relative to an observer on the Earth's surface. Due to the satellite's higher speed, its onboard atomic clock keeps time slightly slower than the clock on the ground. If this relativistic effect is not taken into account, GPS accuracy will degrade rapidly, resulting in navigation errors of several kilometers.
General relativity and gravitational effects
Additionally, general relativity comes into play due to the Earth's gravitational field. Clocks with stronger gravitational fields experience time dilation—time passes more slowly. GPS satellites are farther from the center of the earth, where the gravitational field is weaker than on the surface.
Therefore, atomic clocks on satellites run slightly faster than clocks on the Earth's surface.
Accuracy of precise calculations
To ensure the accuracy of GPS systems, scientists and engineers must consider the effects of both special and general relativity.
In the late 1980s and early 1990s, scientists modified GPS systems to account for these relativistic effects. They adjusted the satellite's clocks to account for the time dilation effects of special relativity and the stronger gravitational fields of general relativity. With these adjustments, GPS accuracy improves significantly, providing precise navigation and positioning information to within a few meters.
Ground control stations regularly update satellite clocks to account for these relativistic effects.
Essentially, after the launch of the first GPS satellites, it took us several years to understand relativistic effects and incorporate them into the system to achieve the high levels of accuracy and reliability we enjoy today.
The application of relativity theory to GPS demonstrates the profound interplay between theoretical science and practical technology. Without a deep understanding of relativistic effects, GPS technology suffers from inaccuracies, making it much less useful for navigation and positioning