Relativity FAQ

Why is c the symbol for the speed of light?

"As for c, that is the speed of light in vacuum, and if you ask why c, the answer is that it is the initial letter of celeritas, the Latin word meaning speed."
Isaac Asimov in "C for Celeritas (1959)" [1]

A Short Answer

Weber apparently meant c to stand for "constant" in his force law, but there is evidence that physicists such as Lorentz and Einstein were accustomed to a common convention that c could be used as a variable for velocity. This usage can be traced back to the classic Latin texts in which c stood for "celeritas" meaning "speed". The uncommon English word "celerity" is still used when referring to the speed of wave propagation in fluids. The same Latin root is found in more familiar words such as acceleration and even celebrity, a word used when fame comes quickly.

Although the c symbol was adapted from Weber's constant, it was probably thought appropriate for it to represent the velocity of light later on because of this Latin interpretation. So history provides an ambiguous answer to the question "Why is c the symbol for the speed of light?" and it is reasonable to think of c as standing for either "constant" or "celeritas".

The Long Answer

In 1992 Scott I Chase wrote on sci.physics that "anyone who read hundreds of books by Isaac Asimov knows that the Latin word for 'speed' is 'celeritas', hence the symbol 'c' for the speed of light". Asimov had written an article entitled "C for Celeritas" in a sci-fi magazine in 1959 and had reprinted it in some of his later books [1]. Scott was the first editor of the Physics FAQ on Usenet and Asimov's explanation was later included in the relativity section as the "probable" answer to the question "Why is c the symbol for the speed of light?". Since then Asimov's answer has become a factoid repeated in many articles and books. But if you go back and read his essay you discover that Asimov merely stated his case in one sentence and made no further attempt to justify his theory for the origin of the "c" notation. So is his claim really born out by history, or was c originally introduced as a variable standing for something else? The special theory of relativity is based on the principle that the speed of light is constant so did c stand for "constant", or did it simply appear by accident in some text where all the other likely variables for speed had already been used up? These questions have been asked repeatedly on usenet and now after much searching through old papers and books the answers can be revealed.

A lower-case c has been consistently used to denote the speed of light in textbooks on relativity almost without exception since such books started to be written. For example, the notation was used in the earliest books on relativity by Lorentz (1909) [4], Carmichael (1913) [5], Silberstein (1914) [6], Cunningham (1915) [7], and Tolman (1917) [8]. That was not the case just a few years before. In his earliest papers on relativity from 1905-1907 Einstein began by using an upper-case V for the speed of light [9]. At that time he was also writing papers about the thermodynamics of radiation and in those he used up upper-case L [10]. All of these papers appeared in volumes of the German periodical Annalen Der Physik. Einstein's notation changed suddenly in 1907 in a paper for the Journal Jahrbuch der Radioaktivität und Elektronik [11]. There he used the lower case c and his most famous equation E = mc² came into being .

It is not difficult to find where the upper case V had come from. Maxwell used it extensively in his publications on electrodynamics from as early as 1865 [12]. It was the principle symbol for the speed of light in his 1873 treatise on electrodynamics [13]. By the 1890's Maxwell's book was in wide circulation around the world and there were translations available in French and German. It is no surprise then that the upper-case V is found in use in such papers as the 1887 report of Michelson and Morley on their attempt to find seasonal variations in the speed of light [14]. That was written in the United States but the same notation was also found across Europe, from papers by Oliver Lodge [15] and Joseph Lamor [16] in England to the lecture notes of Poincaré in France [17] and the textbooks of Paul Drude in Germany [18] and Lorentz in the Netherlands [19]. Einstein's education at the Polytechnik in Zurich had not covered Maxwell's theory of Electrodynamics in the detail he would have liked. But he had read a number of extra textbooks on the new Electrodynamics as self study so he would have been familiar with the standard notations. From 1905 he wrote his first papers on relativity and there is nothing extraordinary in his choice of the symbol V for the speed of light [9].

Why then, did he change it to c in 1907? At that time he still worked as a clerk in the Bern patent office, but for the previous two years he had been in regular corresponding with eminent physicists such as Max Laue, Max Planck, Wilhelm Wien and Johannes Stark. Stark was the editor of the Jahrbuch and had asked Einstein to write the article in which he was to first use the letter c. Einstein mentioned to Stark that it was hard for him to find the time to read published scientific articles in order to acquaint himself with all the work others have done in the field, but he had seen papers by Lorentz, Kohn, Monsegeil and Planck [20]. Lorentz and Planck in particular had been using c for the speed of light in their work. Lorentz had won the 1902 Nobel prize for physics and it is not surprising that physicists in Germany, had now taken up the same notation. It is also not surprising that Einstein who was looking for an academic position aligned himself to the same conventions at that time. Another reason for him to make the switch was that the letter c is simply more practical. The upper-case V would have been easily confused with the lower case v appearing in the equations of relativity for the velocity of moving bodies or frames of reference. Einstein must have found this confusion inconvenient especially in his hand written notes.

Looking back at papers of the late 1890's we find that Max Planck and Paul Drude in particular were using the symbol c at that time. The name of Drude is less well known to us today. He worked on relations between the physical constants and high precision measurements of their value. These were considered to be highly worthy pursuits of the time. Drude had been a student of Voigt who had used a Greek omega for the speed of light when he wrote down an almost complete form of the Lorentz transformations in 1887 [43]. Voigt's omega was later used by a few other physicists [44-45], but Drude did not use his teacher's notation. Drude first used the symbol c in 1894 and in doing so he referenced a paper by Kirchhoff [3]. As already mentioned, Paul Drude also used V. In fact he made a distinction of using V in the theory of optics for the directly measured speed of light in vacuum, whereas he used c for the electromagnetic constant that was the theoretical speed of electromagnetic waves. This is seen especially clearly in his book "Theory of Optics" of 1900 [21], which is divided into two parts with V used in the first and c in the second part. Although Maxwell's theory of light predicted that they had the same value, it was only with the theory of relativity that these two things were established as fundamentally the same constant. Other notations vied against Dride's and Maxwell's for acceptence. Herglotz [46] opted for an elaborate script B while Himstedt [47], Helmholtz [48] and Hertz [49] wrote the equations of electrodynamics with the letter A for the reciprocal of the speed of light. In 1899 Planck backed Drude by useing c when he wrote a paper introducing what we now call the Planck scale of units based on the constants of electrodynamics, quantum theory and gravity [22]. Drude and Planck were both editors of the prestigious Journal Annalen Der Physik so they would have had regular contact with most of the physicists of central Europe.

Lorentz was next to change notation. When he started writing about light speed in 1887 he used an upper case A [23] but then switched to Maxwell's upper case V [24]. He wrote a book in 1895 [25] that contained the equations for length contraction and was cited by Einstein in his 1907 paper. While Drude had started to use c, Lorentz was still using V in this book. He continued to use V until 1899 [26] but by 1903 when he wrote an encyclopedia article on electrodynamics [27] he too used c. Max Abraham was another early user of the symbol c in 1902 in a paper that was seen by Einstein [28]. From Drude's original influence, followed by Planck and Lorentz, by 1907 the c symbol had become the prevailing notation in Germanic science and it made perfect sense for Einstein to adopt it too.

In France and England the electromagnetic constant was symbolized by a lower case v rather than Drude's c. This was directly due to Maxwell who wrote up a table of experimental results for direct measurements of the speed of light on the one hand and electromagnetic experiments on the other. He used V for the former and v for the latter. Maxwell described a whole suite of possible experiments in electromagnetism to determine v. Those that had not already been done were performed one after the other in England and France over the three decades that followed [29]. In this context lower case v was always used for the quantity measured. But using v was doomed to pass away once authors had to write relativistic equations involving moving bodies because v was just too common a symbol for velocity. The equations were much clearer when something more distinct was used for the velocity of light to differentiate it from the velocity of moving bodies.

While Maxwell always used v in this way, he also had a minor use for the symbol c in his widely read treatise of 1873. Near the end he included a section about the German electromagnetic theory that had been an incomplete precursor to his own formulation [30]. This theory expounded by Gauss, Neumann, Weber and Kirchhoff attempted to combine the laws of Coulomb and Ampere into a single action-at-a-distance force law. The first versions appeared in Gauss's notes in 1835 [31], and the complete form was published by Weber in 1846 [32]. Many physicists of the time were heavily involved in the process of defining the units of electricity. Colomb's law of electrostatic force could be used to give one definition of the unit of charge while Ampere's force law for currents in wires gave another. The ratio between these units had the dimension of a velocity so it became of great practical importance to measure its value. In 1856 Weber and Kohlrausch published the first accurate measurement [2]. To give a theoretical backing they rewrote Weber's force law in terms of the measured constant and used the symbol c. This c appeared in numerous subsequent papers by German physicists such as Kirchhoff, Clausius, Himstedt and Helmholtz who referred to it as "Weber's Constant". That continued until the 1870's when Helmholtz discredited Weber's force law on the grounds of energy conservation and Maxwell's more complete theory of propagating waves prevailed.

Two papers using Weber's force law are of particular note. One by Kirchhoff [33] and one by Riemann [34] related Weber's Constant to the velocity at which electricity propagated. They found this speed to be Weber's Constant divided by the square root of two and it was very close to the measured speed of light. It was already known from experiments by Faraday that light was affected by magnetic fields so there was already much speculation that light could be an electrodynamic phenomena. This was the inspiration for Maxwell's work on electrodynamics so it is natural that he finally included a discussion of the force law in his treatise [30]. The odd thing is that when Maxwell wrote down the force law, he changed the variable c so that it was smaller than Weber's constant by a factor of the square root of two. So Maxwell was probably the first to use c for a value equal to the speed of light, although he defined it as the speed of electricity through wires instead.

So c was used as Weber's Constant having a value of the speed of light times the square root of two, and this can be related to the later use of c for the speed of light itself. Firstly, when Maxwell wrote Weber's force law in his treatise in 1873, he modified the scale of c in the equation so that it reduced by a factor of the square root of two. Secondly, when Drude first used c in 1894 for the speed of light [3], the paper by Kirchhoff that he cited [35] was using c for Weber's Constant, so Drude had made the same adjustment as Maxwell. It is impossible to say if Drude copied the notation from Maxwell, but he did go one step further in explicitly naming his c as the velocity of electrodynamic waves which by Maxwell's theory was also the speed of light. He seems to have been the first to do so, with Lorentz, Planck and others following suit a few years later.

So to understand why c became the symbol for the speed of light we now have to find out why Weber used it in his force law. In the paper of 1856 [2] Weber's constant was introduced with these words "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." So it appears that c originated as a letter standing for "constant" rather than "celeritas". However it had nothing to do with the constancy of the speed of light until much later.

Despite this, there could still be some substance to Asimov's claim that c is the initial letter of "celeritas". It is true, after all, that c is also often used for the speed of sound and it is commonly used as the velocity constant in the wave equation. Furthermore, this usage was around before relativity.

Starting with the Latin manuscripts of the 17th century, such as Galileo's "De Motu Antiquiora" or Newton's "Principia" we find that they often use the word "celeritas" for speed. However, their writing style was very geometric and descriptive. They did not tend to write down formulae where speed is given a symbol. But an example of the letter c being used for speed can be found from the eighteenth century. In 1716 Jacob Hermann published a Latin text called Phoronomia which means the science of motion [36]. In it he developed Newton's mechanics in a form more familiar to us now, except for the Latin symbols. His version of the basic Newtonian equation F = ma was dc = p dt, where c stands for "celeritas" meaning speed, and p stands for "potentia" meaning force.

Apart from relativity the most pervasive use of c to represent a speed today is in the wave equation. In 1747 Jean d'Alembert made a mathematical study of vibrating string and discovered the one dimensional wave equation, but he wrote it without the velocity constant. Euler generalised d'Alembert's equation to include the velocity and used the letter a to denote it [38]. The general solution is y = f(x - at) + f(x + at) representing two waves of fixed shape traveling in opposite directions at velocity a.

Euler was one of the most prolific mathematicians of all time. He wrote hundreds of manuscripts and most of them were in Latin. If anyone established a convention for using c for "celeritas" it has to have been Euler. In 1759 Euler studied the vibrations of a drum and moved on to the 2 dimensional wave equation. This he wrote in the form we are looking for with c now the velocity constant [39].

The wave equation became a subject of much discussion and was investigated by all the great mathematicians of the époque including Lagrange, Fourier, Laplace and Bernoulli. Through their works Euler's form of the wave equation with c for the speed of wave propagation was carved in stone for good. To a first approximation sound waves are also governed by the same wave equation in three dimensions so it is not surprising that the speed of sound also came to be denoted by the symbol c. This predates relativity and can be found for example in Lord Rayleigh's classic text "Theory of Sound" [40]. Physicists of the nineteenth century would have read the classic Latin texts on Physics and would have been aware that c could stand for "celeritas". As an example, Lorentz used c in 1899 for the speed of the Earth through the ether [41]. We even know that Einstein used it for speed outside relativity because in an letter to a friend about a patent for a flying machine he used c for the speed of air flowing at a mere 4.9 m/s [42].

In conclusion, although we can trace c back to Weber's force law where it most likely stood for "constant", it is possible that it's use persisted because c could stand for "celeritas" and had therefore become a conventional symbol for speed. We cannot tell for sure how Drude, Lorentz, Planck or Einstein thought about their notation so there can be no definitive answer for what it stood for then. The only logical answer is that when you use the symbol c it stands for whatever possibility you prefer.

References

[1] Isaac Asimov "C for Celeritas" in "The Magazine of Fantasy and Science Fiction", Nov-59 (1959), reprinted in "Of Time, Space, and Other Things", Discus (1975), and "Asimov On Physics", Doubleday, (1976)

[2] R Kohlrausch and W E Weber, "Ueber die Elektricitätsmenge, welche bei galvanischen Strömen durch den Querschnitt der Kette fliesst", Annalen der Physik, 99, p10 (1856)

[3] P Drude, "Zum Studium des elektrischen Resonators", Göttingen Nachrichten (1894), p189–223

[4] H A Lorentz, "The theory of Electrons and its applications to the phenomena of light and radiant heat." A course of lectures delivered in Columbia University, New York, in March and April 1906, Lieden (1909)

[5] R D Carmichael, "The Theory of Relativity", John Wiley & Sons (1913)

[6] L Silberstein, "The Theory of Relativity", Macmillan (1914)

[7] E Cunningham, "The Principle of Relativity", Cambridge University Press (1914)

[8] R C Tolman, "The Theory of the Relativity of Motion", Uiversity of California Press (1917)

[9] A Einstein, From "The Collected Papers, Vol 2, The Swiss Years: Writings, 1900-1909", English Translation, he wrote five papers using V, e.g. "On the Electrodynamics of Moving Bodies", Annalen Der Physik, 17, p891-921 (1905), "On the Inertia of Energy Required by the Relativity Principle", Annalen Der Physik, 23, p371-384 (1907)

[10] A Einstein, e.g. "On the Theory of Light Production and Light Absorption", Annalen Der Physik, 20, p199-206 (1906)

[11] A Einstein, "On the Relativity Principle and the Conclusions Drawn From It", Jahrbuch der Radioaktivität und Elektronik, 4, p411-462 (1907)

[12] J. Clerk Maxwell, “A dynamical theory of the electromagnetic field,” Philos. Trans. Roy. Soc. 155, p459–512 (1865). Abstract: Proceed-ings of the Royal Society of London 13, 531–536 (1864)

[13] J. Clerk Maxwell, "A Treatise on Electricity and Magnetism", Oxford Clarendon Press, (1873)

[14] A A Michelson and E W Morley, "On the Relative Motion of the Earth and the Luminiferous Ether." Amer. J. Sci. 34, p333-345 (1887), Philos. Mag. 24, p449-463 (1887)

[15] O Lodge, “Aberration Problems", Phil. Trans. Roy. Soc. 184, p729-804 (1893)

[16] J Larmor, "A Dynamical Theory of the Electric and Luminiferous Medium I", Phil. Trans. Roy. Soc., 185, p719-822, (1894)

[17] H Poincaré, "Cours de physique mathématique. Electricité et optique. La lumière et les théories électro-dynamiques" (1900)

[18] P Drude, "Physik des Äthers auf elektromagnetischer Grundlage", Verlag F. Enke, Stuttgart, (1894)

[19] H Lorentz, "Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern", Leiden, (1895)

[20] A Einstein, From "The Collected Papers, Vol 5, The Swiss Years: Correspondance, 1902-1914", English Translation, Doc 58.

[21] P Drude, "The theory of optics", translated from German by C. R. Mann and R. A. Millikan, New York, Longmans, Green, and Co., (1902)

[22] M Planck, "Uber irreversible Strahlungsvorgange", Verl. d. Kgl. Akad. d. Wiss., (1899)

[23] H A Lorentz, "De l'Influence du Mouvement de la Terre sur les Phenomenes Lumineux", Arch. Neerl. 21, 103 (1887)

[24] H A Lorentz, "On the Reflection of Light by Moving Bodies",Versl. Kon. Akad. Wetensch Amsterdam I, 74 (1892)

[25] H A Lorentz, "Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Körpern", Leiden (1895)

[26] H A Lorentz, "Théorie simplifiée des phenomènes electriques et optiques dans des corps en mouvement", Proc Roy Acad Amsterdam I 427 (1899)

[27] H A Lorentz, "Maxwells elektromagnetische Theorie" Encyclopädie der Mathematischen Wissenschaften. Leipzig, Teubner, (1903)

[28] M Abraham, "Prinzipien der Dynamik des Elektrons", Annalen der Physik, 10, p105-179 (1903)

[29] e.g J J Thomson and G F C Searle, "A Determination of 'v,' the Ratio of the Electromagnetic Unit of Electricity to the Electrostatic Unit", Proc Roy Soc Lond, 181, p583 (1890), M Hurmuzescu, "Nouvelle determination du rapport v entre les unites electrostatiques et electromagnetiques", Ann. de Chim. et de Phys., 7a serie T. X April 1897, p. 433. (1897)

[30] J. Clerk Maxwell, "A Treatise on Electricity and Magnetism", Oxford Clarendon Press,Vol II, chapter 23, section 849 (1873)

[31] K F Gauss, "Zur mathematischen Theorie der elektrodynamischen Wirkung" (1835), in "Werke", Göttingen, 1867, Vol. V, p. 602

[32] W Weber, “Elektrodynamische Maassbestimmingen uber ein allgemeines Grundgesetz der elektrischen Wirkung”, Abh. Leibnizens Ges., Leipzig, (1846)

[33] G Kirchhoff, "Ueber die Bewegung der Elektricität in Leitern" Ann. Phys. Chem. 102, 529–544 (1857)

[34] G F B Riemann, "Ein Beitrag zur Elektrodynamik", Annalen der Physik und Chemie , 131 (1867)

[35] G Kirchhoff, "Zur Theorie der Entladung einer Leydener Flasche", Pogg Ann 121, (1864)

[36] J Hermann, "Phoronomia", Amsterdam, Wetsten, (1716)

[37] J d'Alembert, ”Recherches sur les cordes vibrantes”, L’Académie Royal des Sciences (1747)

[38] L Euler, "De La Propagation Du Son" Memoires de l'acadamie des sciences de Berlin [15] (1759), 1766, p 185-209, in "Opera physica miscellanea epistolae. Volumen primum" p432

[39] L Euler, " Eclaircissemens Plus Detailles Sur La Generation et La Propagation Du Son Et Sur La Formation De L'Echo" Memoires de l'acadamie des sciences de Berlin [21] (1765), 1767, p335-363 in "Opera physica miscellanea epistolae. Volumen primum" p540

[40] J W Strutt, "Theory of Sound", Vol 1, p251, McMillan and co. (1877)

[41] H A Lorentz, "Stoke's Theory of Aberration in the Supposition of a Variable Density of the Aether", Proc. roy. Acad. Amsterdam, I, 443 (1899)

[42] A Einstein, "The Collected Papers, Vol 5, The Swiss Years: Correspondence, 1902-1914", English Translation, Doc 86 (1907)

[43] W Voigt, "Ueber das Doppler'sche Princip", Goett. Nachr., 2, p41 (1887)

[44] E Cohn, "Zur Elektrodynamik bewegter Systeme. II", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, der physikalisch-mathematischen Classe (1904)

[45] M Brillouin, "Le mouvement de la Terre et la vitesse de la lumière", comptes rendu, 140, p1674 (1905)

[46] G Herglotz, "Zur Elektronentheorie", Nachrichten von der Gesellschaft, 6, p357 (1903)

[47] F Himstedt, "Ueber die Schwingungen eines Magneten unter dem dämpfenden Einfluß einer Kupferkugel", Nachrichten von der Gesellschaft, 11, p308 (1875)

[48] H Helmholtz, Berlin: Verl. d. Kgl. Akad. d. Wiss., (1892)

[49] H Hertz, "Electric Waves", Macmillan (1893)