The speed of light in a vacuum, denoted by the symbol c, is a fundamental constant in physics, playing a crucial role in our understanding of space, time, and the universe itself. But why c? Why this specific letter to represent such a significant concept? While it’s universally recognized today, the journey to adopting c wasn’t straightforward. This article delves into the fascinating history behind the symbol c, exploring its origins and evolution from the 19th century to its current status as a cornerstone of scientific notation. Uncover the story of how a simple letter became synonymous with the ultimate speed limit of the cosmos.
Decoding ‘c’: A Concise Explanation
The symbol c for the speed of light is indeed derived from the Latin word celeritas, meaning “swiftness” or “speed”. This explanation, popularized by figures like Isaac Asimov, offers a direct and intuitive answer. However, the complete story is richer and more nuanced. While celeritas provides a convenient and perhaps later justification, historical evidence suggests a slightly different initial impetus.
In the 19th century, before c became standard, the uppercase letter V was commonly used, notably by James Clerk Maxwell in his groundbreaking work on electromagnetism. Einstein himself initially adopted V in his early papers on relativity. So, how did V transition to c?
The origins of c can be traced back to Wilhelm Weber and Rudolf Kohlrausch’s 1856 paper. They introduced a constant, denoted as c, in their electrodynamic force law. This “Weber’s constant” was experimentally determined and later found to be theoretically linked to the speed of light, specifically being approximately the speed of light multiplied by the square root of two.
Later, in 1894, Paul Drude refined the use of Weber’s constant, employing c to represent the speed of electromagnetic waves. While Drude continued to use V for the speed of light in optics, the use of c for electromagnetic wave speed gained traction. Influential physicists like Max Planck and Hendrik Lorentz gradually adopted c, contributing to its wider acceptance.
By 1907, when Einstein switched from V to c in his publications, c had become the established symbol for the speed of light in vacuum across electrodynamics, optics, thermodynamics, and relativity. While Weber’s original c likely stood for “constant” in his force law, the Latin association with celeritas provided a fitting and memorable interpretation as the symbol’s usage evolved to represent the speed of light. Therefore, the symbol c embodies both the “constant” nature of light speed and its inherent “celerity”.
The Journey of ‘c’: A Detailed Historical Perspective
The seemingly simple question of “Why c for the speed of light?” has a surprisingly complex and fascinating history. While Isaac Asimov’s explanation linking c to celeritas is widely known and appealing, a deeper historical dive reveals a more intricate evolution of this fundamental symbol.
Asimov, in his 1959 essay “C for Celeritas,” proposed that c was chosen as the symbol for the speed of light because celeritas is the Latin word for speed. This explanation, while concise and memorable, simplifies a more complex historical development. While celeritas may have been a reinforcing factor, the initial adoption of c appears to stem from a different source.
For much of the 19th century, the uppercase V was the prevalent symbol for the speed of light. James Clerk Maxwell, a pivotal figure in electromagnetism, consistently used V in his publications from 1865 onwards, including his seminal 1873 “Treatise on Electricity and Magnetism.” This established V as the standard notation, influencing subsequent generations of physicists. Even in the famous Michelson-Morley experiment report of 1887, which aimed to detect variations in the speed of light due to Earth’s motion, V was the symbol of choice. This notation extended across Europe, appearing in the works of Oliver Lodge, Joseph Larmor in England, Henri Poincaré in France, and Paul Drude and Hendrik Lorentz in Germany and the Netherlands respectively. Albert Einstein, in his groundbreaking 1905 papers on special relativity, initially followed this convention, also using V for the speed of light.
Einstein’s shift from V to c occurred rather abruptly in 1907. In his early relativity papers published in the Annalen Der Physik journal, he consistently used V. However, in a 1907 paper for the Jahrbuch der Radioaktivität und Elektronik, edited by Johannes Stark, Einstein adopted the lowercase c. This paper marked the debut of his iconic equation E = mc2 with the now-familiar c.
The question arises: why the sudden change? By 1907, Einstein was in correspondence with leading physicists like Max Planck and Hendrik Lorentz, both of whom were using c for the speed of light. Lorentz, a Nobel laureate, and Planck, a highly influential figure, wielded considerable authority in the physics community, particularly in German-speaking scientific circles. Adopting their notation would have been a pragmatic move for Einstein, especially as he sought academic recognition.
Furthermore, the lowercase c offered a practical advantage over V. In the equations of relativity, both the speed of light and the velocities of moving objects are frequently involved. Using both uppercase V and lowercase v could lead to confusion, particularly in handwritten notes. The more distinct symbol c for the speed of light and v for ordinary velocities improved clarity and reduced potential errors.
Looking further back into the late 1890s, we find Paul Drude and Max Planck among the early adopters of c. Drude, known for his meticulous work on physical constants, used c as early as 1894, citing a paper by Gustav Kirchhoff. Interestingly, Drude initially differentiated between V and c. In his “Theory of Optics” (1900), he used V to represent the experimentally measured speed of light in a vacuum, primarily in the optics section, while employing c for the theoretical speed of electromagnetic waves, particularly in the electromagnetism section. Although Maxwell’s theory predicted these values to be the same, relativity later solidified them as fundamentally identical.
Other notations were also in contention during this period. Waldemar Voigt used the Greek letter ω for the speed of light in 1887 when formulating transformations that closely resembled Lorentz transformations. This ω was occasionally used by others. Some physicists like Herglotz opted for a script B, while others, including Himstedt, Helmholtz, and Hertz, used A to represent the reciprocal of the speed of light in electrodynamic equations. However, in 1899, Planck’s adoption of c in a paper introducing Planck units, based on fundamental constants including the speed of light, lent further weight to the c notation. Drude and Planck, both editors of the prestigious Annalen Der Physik, played a crucial role in disseminating and popularizing the use of c.
Hendrik Lorentz also underwent a notational shift. Initially using A in 1887 and then Maxwell’s V, Lorentz eventually transitioned to c by 1903. Max Abraham also used c as early as 1902. The combined influence of Drude, Planck, and Lorentz solidified c‘s position as the dominant symbol in Germanic science by 1907, making Einstein’s adoption of it a natural progression.
In contrast to the German scientific community, French and English physicists often used lowercase v for the electromagnetic constant, a practice directly linked to Maxwell. Maxwell, in his treatise, presented experimental results for direct light speed measurements (denoted by V) and electromagnetic experiments (denoted by v). He outlined various electromagnetic experiments to determine v, which were actively pursued in England and France in the following decades. In this context, v consistently represented the measured electromagnetic quantity. However, the increasing complexity of relativistic equations, where v was already commonly used for the velocity of moving bodies, made using v for the speed of light increasingly impractical and confusing. A more distinct symbol was needed to differentiate the two.
Interestingly, Maxwell himself did use c in his 1873 treatise, albeit in a limited context. He discussed German electromagnetic theories, precursors to his own, developed by Gauss, Neumann, Weber, and Kirchhoff. These theories aimed to unify Coulomb’s and Ampère’s laws into a single force law. Weber’s 1846 formulation of this force law and the associated “Weber’s constant” became significant. The ratio between electrostatic and electromagnetic units of charge had the dimension of velocity, making its measurement crucial. Weber and Kohlrausch’s 1856 experiment provided the first accurate measurement of this ratio. To theoretically underpin this, they reformulated Weber’s force law using the symbol c for this measured constant. This “Weber’s constant” c appeared in numerous subsequent publications by German physicists throughout the 1860s and 70s.
Kirchhoff and Riemann, in separate papers, linked Weber’s constant to the propagation speed of electricity, finding it to be Weber’s constant divided by the square root of two, remarkably close to the measured speed of light. This, coupled with Faraday’s observations of magnetic field effects on light, fueled speculation that light might be an electromagnetic phenomenon, inspiring Maxwell’s groundbreaking work. Intriguingly, when Maxwell presented Weber’s force law in his treatise, he scaled down the c in the equation by a factor of the square root of two, effectively making it closer to the speed of light itself, although he defined it as the speed of electricity in wires.
Thus, c initially emerged as Weber’s constant, approximately the speed of light times the square root of two. Its evolution towards representing the speed of light involved two key steps: Maxwell’s scaling of c in Weber’s force law by the square root of two in 1873, and Drude’s explicit use of c in 1894 to denote the speed of electromagnetic waves, citing Kirchhoff’s work which still used c for Weber’s constant. Drude’s adjustment aligned c more directly with the speed of light as predicted by Maxwell’s theory, and he appears to be the first to explicitly define c as the velocity of electromagnetic waves, paving the way for Lorentz, Planck, and others to follow.
To fully understand Weber’s choice of c, we examine his 1856 paper, where he introduced the constant: “and the constant c represents that relative speed, that the electrical masses e and e must have and keep, if they are not to affect each other.” This suggests that Weber’s c likely stood for “constant” in his force law, rather than celeritas. Initially, it had no direct connection to the constancy of the speed of light.
Despite Weber’s likely intention, the association with celeritas may have played a role in the enduring popularity of c. Notably, c is also frequently used for the speed of sound and as the velocity constant in the wave equation, usages predating relativity.
Looking back further, Latin manuscripts from the 17th century, such as Galileo’s and Newton’s works, used celeritas for speed. While their style was primarily geometric and descriptive, limiting formulaic notation, an 18th-century example exists: Jacob Hermann’s 1716 “Phoronomia,” a Latin text on mechanics, used c for celeritas in his formulation of Newton’s second law.
Beyond relativity, the wave equation prominently features c as a speed constant. Jean d’Alembert’s 1747 work on vibrating strings introduced the wave equation without the velocity constant. Leonhard Euler generalized it in 1759, incorporating velocity denoted by a. However, Euler later used c as the velocity constant in his studies of drum vibrations and the 2-dimensional wave equation. Euler’s immense influence as a mathematician likely solidified c‘s association with velocity in the wave equation. The wave equation’s broad applicability, including to sound waves, led to c also being used for the speed of sound, as seen in Lord Rayleigh’s “Theory of Sound.”
Nineteenth-century physicists, educated in classical Latin texts, would have been familiar with celeritas and its association with c. Lorentz, for instance, used c in 1899 to represent Earth’s speed through the ether. Even Einstein, outside of relativity, used c for ordinary speeds in a letter regarding a flying machine patent.
In conclusion, while c‘s origin can be traced to Weber’s constant, where it likely signified “constant,” its continued use and eventual dominance as the symbol for the speed of light may have been reinforced by the convenient and fitting association with celeritas, the Latin word for speed and a pre-existing convention for velocity in scientific contexts. The precise intentions of Drude, Lorentz, Planck, and Einstein regarding their notation remain uncertain. Ultimately, the symbol c for the speed of light carries a dual legacy, representing both its constant nature and its fundamental role as a measure of celerity in the universe.
References
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