User:Brews ohare/Speed of light (Example)

Example

An example illustrates how the new metre and the conversion-factor speed of light work in practice. If c is used to denote the real physical speed of light (the distance between points divided by the time it takes for a signal to transit them) and c₀ = 299 792 458 m/s, the SI-units post-1983 conversion factor (also referred to as the "speed of light" in the SI system of units), then the example shows that logically c ≠ c₀, even though numerically c and c₀ are nearly the same.

Take two points A & B. Suppose they are some fixed distance apart. (The actual distance between A & B can be measured, for example, using interferometry to determine the separation in units of wavelengths of some atomic transition). Suppose (hypothetically) measurement skills increase and the transit time of light between points A & B is measured to be a time t_AB that is a slightly shorter time than previously measured with older technique.

In that case the real speed of light as determined from the relation real speed = (actual distance between A & B)/ t_AB will be measured as larger, because points A & B have not changed position, and the time-of-transit t_AB has shortened.

However, the SI units conversion factor c₀ = 299 792 458 m/s is not affected by the new measurement technique. ^[1]^[2] It is an exact value, set by definition.^[3]

This example shows that the real speed of light (for example, expressed in units of the number of wavelengths traveled per second) is different in principle (logically different) from the SI conversion factor of c₀ = 299,792,458 m/s.

The distance between A & B in the SI units system is given by ℓ_AB = 299 792 458 m/s ·t_AB,^[4] which is a smaller number of metres than previously because the time is shorter (even though the real separation has not changed). Equivalently, in SI units the metre is longer. The shorter distance in metres from A to B combined with the shorter time-of-transit results in a speed in SI units that is always the conversion factor 299 792 458 m/s, regardless of any advance in technique.^[5]

In both cases the measured time has shortened. In nature, the separation between A & B is fixed, so the shorter time results in a larger measured value for the real speed of light. In SI units the distance in units of metres is shorter, and the light travels at the same rate of 299,792,458 m/s. The real speed of light can change (for example, with improvement in measurement technique), but the conversion factor is fixed by definition, lies outside measurement, and is not a property of nature;^[6] rather the conversion factor c₀ = 299,792,458 m/s is a property of the SI system of units.

Notes

^ Jespersen, J; Fitz-Randolph, J; Robb, J (1999). From Sundials to Atomic Clocks: Understanding time and frequency (Reprint of National Bureau of Standards 1977, 2nd ed.). Courier Dover. p. 280. ISBN 0486409139. One fallout of the new definition was that the speed of light was no longer a measured quantity … defining one unit [length] in terms of another [time] removes a constant of nature by turning c into a conversion factor whose value is fixed and arbitrary.
^ Sullivan, DB. "Speed of Light From Direct Frequency and Wavelength Measurements" (PDF). NIST. p. 191. Retrieved 2009-08-22. A consequence of this definition is that the speed of light is now a defined constant, not to be measured again.
^ SI Units Brochure from BIPM § 2.1.1.1, page 112 “It follows that the speed of light in vacuum is exactly 299 792 458 m/s, c₀ = 299 792 458 m/s.”
^ BIPM mise en pratique method (a) “length is obtained from the measured time t, using the relation ℓ = c_o·t and the value of the speed of light in vacuum c₀ = 299 792 458 m/s”
^ Rindler, W (2006). Relativity: Special, General, and Cosmological (2nd ed.). Oxford University Press. p. 41. ISBN 0198567316. Note that [...] improvements in experimental accuracy will modify the meter relative to atomic wavelengths, but not the value of the speed of light!
^ Edwin F. Taylor, John Archibald Wheeler (1992). Spacetime physics: introduction to special relativity (2nd ed.). Macmillan. ISBN 0716723271. Is 299, 792, 458 a fundamental constant of nature? Might as well ask if 5280 is a fundamental constant of nature.

[Jesp-1] Jespersen, J; Fitz-Randolph, J; Robb, J (1999). From Sundials to Atomic Clocks: Understanding time and frequency (Reprint of National Bureau of Standards 1977, 2nd ed.). Courier Dover. p. 280. ISBN 0486409139. One fallout of the new definition was that the speed of light was no longer a measured quantity … defining one unit [length] in terms of another [time] removes a constant of nature by turning c into a conversion factor whose value is fixed and arbitrary.

[NIST_pub-2] Sullivan, DB. "Speed of Light From Direct Frequency and Wavelength Measurements" (PDF). NIST. p. 191. Retrieved 2009-08-22. A consequence of this definition is that the speed of light is now a defined constant, not to be measured again.

[SI_Brochure-3] SI Units Brochure from BIPM § 2.1.1.1, page 112 “It follows that the speed of light in vacuum is exactly 299 792 458 m/s, c₀ = 299 792 458 m/s.”

[pratique-4] BIPM mise en pratique method (a) “length is obtained from the measured time t, using the relation ℓ = c_o·t and the value of the speed of light in vacuum c₀ = 299 792 458 m/s”

[W_Rindler-5] Rindler, W (2006). Relativity: Special, General, and Cosmological (2nd ed.). Oxford University Press. p. 41. ISBN 0198567316. Note that [...] improvements in experimental accuracy will modify the meter relative to atomic wavelengths, but not the value of the speed of light!

[JA_Wheeler-6] Edwin F. Taylor, John Archibald Wheeler (1992). Spacetime physics: introduction to special relativity (2nd ed.). Macmillan. ISBN 0716723271. Is 299, 792, 458 a fundamental constant of nature? Might as well ask if 5280 is a fundamental constant of nature.

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