To my esteemed mentor and friend, Professor Dr. Ing. Eugen Wildermann.
- Introduction
The four fundamental forces of nature are the gravitational force, the electromagnetic force, the strong nuclear force, and the weak nuclear force. These forces are essential for understanding a wide range of physical phenomena and, together, constitute the basis of the Standard Model of physics of particles, which is currently the most complete theory for describing the forces and particles that make up the universe. In this essay, we will briefly review the evolution and development of the law of gravitation by Isaac Newton and some later refinements introduced by Albert Einstein. The objective is to describe the historical development of the laws of gravitation and highlight how both theories, despite their fundamental differences, continue to be pillars in the understanding and application of gravitation in various fields, especially in geodetic sciences. But before moving forward, let’s begin by clarifying in a basic way the terms “Gravitation” and “Gravity,” both fundamental concepts in physical geodesy. Gravitation is a more general term that encompasses the interaction (attraction) between any pair of masses in the universe. On the other hand, the term Gravity is the specific manifestation of the gravitational force on Earth and describes how objects are attracted to the center of the Earth due to its mass. Gravity is the result of two combined forces: the Gravitational force and the Centrifugal force generated by the Earth’s rotation.
- Background
The history of our understanding of gravitation dates back to antiquity, when various civilizations observed the sky to understand the movements of the planets among the stars. In this context, Aristotle (384-322 BC) proposed a geocentric model in which the Earth was considered the center of the universe. However, this model had limitations, as it could not adequately explain the apparent retrograde motion of some planets. Later, Claudius Ptolemy (100-170 AD) improved Aristotle’s geocentric model by introducing the system of epicycles and deferents, which sought to address the irregularities in the observed planetary movements. His approach was widely accepted for centuries.
In the 16th century, Nicolaus Copernicus (1473-1543) presented a revolutionary hypothesis in his work “De revolutionibus orbium coelestium.” Inspired by the ideas of Aristarchus of Samos (310-230 BC), Copernicus argued that the planets, including the Earth, revolved around the sun, thus establishing a heliocentric model that challenged existing beliefs.
Later, Tycho Brahe (1546-1601) made significant advances in astronomical observation by suggesting the precise measurement of the positions of the planets. In his observatory in Denmark, Brahe observed and meticulously recorded planetary movements for many years, accumulating data that would later be crucial for the studies carried out by Johannes Kepler (1571-1630), who used Brahe’s data and observations to formulate his three laws of planetary motion.
In parallel, Galileo Galilei (1564-1642) used telescopes to make astronomical observations that provided compelling evidence in support of Copernicus’ heliocentrism. However, this hypothesis was still questioned and considered heretical by the Church at the time. Furthermore, Galileo, known as the “father of modern science,” formulated the principle of inertia, which states that an object in motion will continue in motion at a constant speed and in a straight line, unless an external force acts upon it. A few decades later, Isaac Newton (1643-1727) consolidated these ideas by establishing that the movement of a body can only be changed through a force. Newton argued that if an object accelerates, it is because a force has been applied in the direction of the movement, and if its movement changes direction, a lateral force has been applied. Thus, Newton added the notion that it is necessary to apply a force to modify the speed or direction of the movement of an object, thus laying the foundations of classical mechanics.
- Isaac Newton’s Law of Universal Gravitation
The Law of Universal Gravitation, formulated by Isaac Newton in the 17th century, represents one of the most significant advances in the history of science. Newton published this law in 1687 in his masterpiece “Philosophiæ Naturalis Principia Mathematica” (Figure 1). This law established the basis for understanding gravitation as a fundamental force that affects all bodies in the universe. The Law of Universal Gravitation postulates that every particle of matter in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This relationship is expressed mathematically as (1):
This formula not only explained how objects attract each other but also provided a framework for understanding planetary orbits and the behavior of celestial bodies. By applying this law, Newton demonstrated that the same principle that causes an apple to fall to the ground is what keeps the Moon in orbit around the Earth and the planets in their trajectories around the Sun. This approach revolutionized cosmology by unifying the movements on Earth and in space under a single concept. Furthermore, the Law of Gravitation had a crucial impact on the development of mathematical physics. Through his analytical approach, Newton established methods that allowed later scientists to model and predict the movement of objects under the influence of gravitation. His work laid the foundations for calculus and the development of classical mechanics, fields that remain fundamental in modern science.
Shortly after the publication of “Philosophiæ Naturalis Principia Mathematica,” Newton faced significant criticism of his law of gravitation, including that of his contemporary Christiaan Huygens (1629-1695). Newton’s law implies that gravitation acts at a distance and instantaneously, a problematic concept from the perspective of modern physics, as no interaction is allowed to occur faster than the speed of light. This raises questions about the nature of the gravitation that Newton had proposed. Despite these criticisms, Newton’s Law of Gravitation remains a pillar in our understanding of gravitation and has provided a solid foundation for the advancement of physics.
- Albert Einstein’s Theory of General Relativity
In 1915, Albert Einstein (1879-1955) reformulated the understanding of gravitation through his Theory of General Relativity (GRT). This theory describes gravitation not as a conventional force, but as a curvature of space-time caused by the mass of objects. This new vision expanded the understanding of gravitational interactions and became consolidated as one of the cornerstones of modern physics. However, to better understand the GRT, it is necessary to consider some key background:
Background (19th Century – 1904)
1865: James Clerk Maxwell unifies electricity and magnetism, showing that light is an electromagnetic wave.
1887: The Michelson-Morley experiment fails to detect the “ether,” the hypothetical medium through which light was believed to propagate. This result suggests that the speed of light is constant for all observers.
1892-1904: Hendrik Lorentz and Henri Poincaré develop mathematical transformations to explain the results of the Michelson-Morley experiment, although they still maintain the idea of the ether.
1905: Einstein’s “Annus Mirabilis”
In 1905, Einstein published his Theory of Special Relativity (SRT) in the article “On the Electrodynamics of Moving Bodies” (Figure 2). The SRT emerged to resolve the inconsistencies between Newtonian mechanics and Maxwell’s equations of electromagnetism. Although the scientific community took years to accept the SRT, Max Planck (1858-1947) was one of the first to support it. The fundamental postulates of the SRT are:
- Principle of Relativity: The laws of physics are the same in all inertial (non-accelerated) frames of reference.
- Invariance of the speed of Light: The speed of light in a vacuum is constant for all observers, regardless of their relative motion.
The consequences of this new theory include:
Time Dilation: A moving clock runs slower than one at rest (2).
- Length Contraction: Objects in motion shorten in the direction of movement.
- Mass-Energy Equivalence: Mass can be converted into energy and vice versa, expressed in the famous equation (3):
- Relativity of Simultaneity: Events that are simultaneous for one observer may not be for another in relative motion.
1907–1915: Road to General Relativity
1907: Einstein has his “happiest thought”: the principle of equivalence, which postulates that a man in free fall does not feel gravitation. 1911: He predicts that light should deflect near the Sun, although his initial calculation turns out to be wrong. 1912: Einstein works with mathematician Marcel Grossmann to apply non-Euclidean geometry (curved space-time). 1915: In November, Einstein presents the field equations of General Relativity (GRT) at the Prussian Academy of Sciences. In December, he corrects his calculation on the deflection of light and explains the anomalous precession of Mercury, a problem not solved by Newton (Figure 3). The fundamental postulates of the GRT are:
- Fundamentals: The GRT generalizes special relativity, including gravitation as a property of the curvature of space-time.
- Principle of equivalence: There is no observable difference between a uniform gravitational field and uniform acceleration.
- Einstein’s field equations: These equations describe how matter and energy influence the curvature of space-time (4):
The consequences of the GRT include:
- Curvature of space-time: Massive objects deform space-time, creating what we perceive as gravitation.
- Gravitational lenses: Light curves when passing near massive objects.
- Black holes: Regions of space-time with such extreme curvature that nothing can escape them, not even light.
- Gravitational waves: Waves in space-time caused by violent events such as the collision of black holes.
- Gravitation in Modern Geodesy
Newton’s Law of Gravitation continues to be considered a valuable and effective tool in numerous practical and scientific applications, ranging from engineering to astronomy. Terrestrial gravity measurements are fundamental in geophysics and geodesy, as they allow studying the Earth’s internal shape and structure, defining reference systems, and monitoring temporal changes. These measurements are essential for:
- Geophysics: Exploration of resources, natural hazards, and terrestrial structure.
- Geodesy: Definition of the geoid, reference systems, and altimetry.
- Environment: Monitoring groundwater and climate changes.
Some instruments used in Geodesy and Astronomy are:
- Absolute gravimeters, for fundamental networks and calibration.
- Relative gravimeters for prospecting, temporal monitoring.
- Gradiometers for detailed oil exploration.
- Astrogeodetic zenith cameras measure the deflection of the vertical.
The Theory of Special Relativity (SRT) in Geodetic Sciences is mainly interconnected through high-precision timing, satellite positioning, and synchronization of reference systems. In general, the SRT is applied in geodesy as follows:
- Time Dilation and Clock Synchronization: The SRT predicts that moving clocks run slower due to time dilation (Lorentz factor). In Global Navigation Satellite Systems (GNSS, for example, GPS, Galileo), the atomic clocks of satellites (~3.9 km/s) run about 7 μs/day slower than terrestrial clocks. Without SRT corrections, GPS positioning errors would accumulate to about 11 km/day.
- Relativistic Doppler Effect and Signal Transmission: The Doppler shift of the Satellite Relation (SR) affects GNSS signals due to the movement of the satellite with respect to terrestrial receivers. Precise frequency stability requires adjustments to the Satellite Relation (SR).
- Reference Frame Synchronization (Sagnac Effect): The Sagnac effect (a consequence of the Satellite Relation) introduces delays in synchronization due to the Earth’s rotation when synchronizing clocks in different locations. Geodetic techniques such as VLBI must take this into account when comparing signals from distant radio telescopes.
- Determination of the Satellite Orbit and Relativistic Kinematics: the Satellite Relation (SR) contributes to: Velocity-dependent corrections in satellite dynamics, and Relativistic aberration (change of direction of signals due to the Earth’s rotation).
- Relativity in Terrestrial and Celestial Reference Systems: The International Terrestrial Reference System (ITRS) and the International Celestial Reference System (ICRS) must take the SRT into account when performing transformations between frames.
The Theory of General Relativity (GRT) and geodesy are deeply interconnected, as the GRT provides the fundamental theoretical framework for modern geodesy, especially in high-precision measurements and global positioning systems. These are the key relationships:
- Space-time curvature and terrestrial gravitational field: The GRT describes gravitation as the curvature of space-time caused by mass and energy. Geodesy models the terrestrial gravitational field, which is influenced by this curvature. Precise geodetic measurements (e.g., satellite orbits, GNSS signals) must consider relativistic effects.
- Geodesics in the GRT vs. Geodetic Reference Systems: In the GRT, geodesics are the shortest paths in curved space-time (e.g., free-fall trajectories of satellites). In geodesy, reference systems (such as the ITRS (International Terrestrial Reference System) must consider relativistic space-time distortions for greater precision. Global Navigation Satellite Systems (GNSS, e.g., GPS) rely on GRT corrections for accurate synchronization and positioning.
- Relativistic Geodetic Models: Modern geodesy uses relativistic reference systems (e.g., IAU resolutions and IERS conventions). The geoid (equipotential surface of the Earth’s gravitational field) is defined within the framework of the GRT. Space missions such as GRACE and GOCE measure the Earth’s gravitational field, which requires GRT-based data processing.
- Frame-dragging and Earth’s rotation: The GRT predicts the dragging of frames (Lense-Thirring effect) due to the Earth’s rotation, which subtly affects the orbits of satellites. Missions such as Gravity Probe B and LARES have tested this effect, perfecting geodetic models.
- Final considerations
Newton’s Law of Gravitation and Einstein’s Theories of Relativity are fundamental pillars in modern geodesy. These theories have enabled a precise understanding of the Earth’s gravitational field and have improved the accuracy of Earth’s navigation and observation systems. Although Newton’s theory and Einstein’s General Relativity address gravitation in different ways, they complement each other in many respects. Newton’s law is adequate for numerous practical applications, providing a solid basis for most gravitation calculations in everyday situations. However, the GRT becomes essential in contexts where gravitational fields are strong or when velocities close to the speed of light are considered, such as in the orbit of satellites or in the description of astrophysical phenomena. Scientific research continues in search of a unified theory that integrates the GRT with quantum mechanics. Such an advance could resolve some of the current contradictions in physics and open new avenues for a better understanding of the universe. I believe that the greatest greatness and legacy of these two giants of science transcends the limits of physics, astronomy, and geodesy. Thanks to their brilliant and revolutionary ideas, today we have calculus, physics, mechanics, quantum mechanics, and all the technology derived from these fields, which we often take for granted.
