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User:Brews ohare/Definitions of the metre

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See also: Speed of light, Metre, and Second

A change in the meaning of the term speed of light as used in the SI system of units occurred in 1983. In 1983 the 17th Conférence Générale des Poids et Mesures defined the metre to be the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second.[1] The reasons for using this definition are stated in Resolution 1.[2] In other words, the speed of light is c = 299,792,458 m/s exactly.

The modern situation can be summarized as follows:[3]

“One fallout of this new definition was that the speed of light was no longer a measured quantity; it became a defined quantity. The reason is that, by definition, a meter is the distance light travels in a designated length of time, so however we label that distance - one meter, five meters, whatever - the speed of light is automatically determined. And measuring length in terms of time is a prime example of how defining one unit in terms of another removes a constant of nature by turning c into a conversion factor whose value is fixed and arbitrary.”

— James Jespersen, Jane Fitz-Randolph, From sundials to atomic clocks: understanding time and frequency, p. 280

This article compares the earlier definition of the metre with the modern version, and explains in somewhat more detail than Speed of light what is behind the decision to make the change, and some differences in perspective it entails.

Relation of speed and time standards to the metre

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The definition of the metre has become the length traveled by light in a specified fraction of a second. Clearly matters concerning the speed of light are immediately translated to issues about the metre. For example, if the speed of light is inaccurately realized, the metre also will be inaccurately realized. If the speed of light varies with the cosmological evolution of the universe, the standard metre also will evolve. Thus, a study of the standard for the metre immediately involves one in any queries that may arise concerning the speed of light.

Likewise, the metre now depends upon the standard of time, and improvements in the accuracy of measurement of time intervals will improve the accuracy of the metre. On the other hand, the standard of time must be carefully realized to avoid inaccuracies in the metre. For example, gravitational dilation in the times measured by atomic clocks must be accounted for (see time dilation in GPS systems). Likewise, if hypothetical evolution in the value of the fine structure constant actually does occur, it will cause the frequency of an atomic clock to change.

Separate length and time standards

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In ordinary life, time and length are considered separate matters, and speed is derived from these two measurements. Thus, we lay out a mile length and measure how long it takes to run it. Then our speed is the mile length divided by our time to traverse it.

A consequence of this approach is that the speed is not known exactly: the mile is measured only with a certain accuracy, say 1 mile ± an inch, and the time, say 4 minutes, is measured only with a certain accuracy, say ± 1 sec. Then the speed could be as fast as (1 mile + 1 inch)/(4 minutes − 1 second), or it could be as slow as (1 mile − 1 inch)/(4 minutes + 1 second). In particular, if we were to measure the speed of light, it would be stated with an error bar as, say, c = 299,710±22 km/s.

Time-of-flight definition of length

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A standard of length can be avoided by using a standard speed. It could be the speed of sound in a specified medium, for example. Then a distance is determined as a "time-of-flight", that is, how long it takes the sound wave to travel the length. Such a system requires only a time unit, the second, and all lengths are determined in terms of times-of-flight in seconds.

Although speed is defined in this system, and so has no uncertainty, time still is measured with an experimental uncertainty. Thus, assuming a exact value for the speed of sound in our medium, a length still would be uncertain because the time-of-flight cannot be found exactly, but has an error ±Δt. An additional uncertainty is whether the standard speed has been realized. Thus, if the medium used in the measurement differs somewhat from the standard, a correction to the standard speed is needed, and this correction is not exactly known, so the speed is estimated, say, as the standard speed ±Δv . Thus, the length could be as long as (v+Δv)(t+Δt), or as short as (v−Δv)(t−Δt)

Comparison of the two methods

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In choosing between these two systems, it is not their "correctness" that is at stake. What is involved is practical matters, most notably, how large an uncertainty is associated with each method. A secondary matter is how easily the method can be used, as a very complex laboratory procedure using specialized equipment complicates the use of the standard, adds to its expense, and limits its availability.

In the case of a measured length and a measured time, a comparison of lengths involves the measurement error in the lengths, a comparison of times involves the measurement error in the times, and a comparison of speeds involves both errors in length and in time.

In the case of a defined speed, a comparison of times has exactly the same error as in using the other method. A comparison of lengths has the error associated with time comparisons, and the error associated with realization of the standard speed. In the case of a standard speed based upon the propagation of sound, an issue is how readily and accurately one can ascertain that the standard medium has been realized. Likewise, in a comparison of some particular speed with the standard speed one must consider how accurately the standard speed has been realized.

It is when considering this matter that the speed of light becomes so highly recommended, because experiment has shown the speed of light to be readily realized in a great variety of circumstances, available to all observers without undue concern and preparation. Nonetheless, in a laboratory or other setting, one cannot realize the standard speed of light using any medium whatsoever, because the speed of light depends upon the properties of the medium. Therefore, in a terrestrial medium, one measures the properties of the medium and uses theoretical formulas to correct for the influence of the medium. For example, a measurement comparison using light in a partial vacuum involves measurement of the partial pressures of contaminants. The influence of residual gases, for example, is corrected for by using theoretical formulas describing the effect of residual gases upon the speed of light. Alternatively, the functional form for this dependence is fitted to the observed results as a function of the amount of residual gas and extrapolated to zero contamination.

Measuring the speed of light

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An interferometric determination of length. Top: constructive interference; bottom: destructive interference.

The article Speed of light describes a variety of historical attempts to determine the speed of light. In this article only more modern attempts are described to provide some background for understanding the decision to switch from separate standards of length and time to a time-of-flight standard based upon a standard for the second alone.

Modern length measurement is based upon the interferometer, schematically depicted in the figure. A light beam, preferably of one frequency or wavelength, is split in two and sent along a reference path (left) and a comparison path (right). At the ends of the the two paths are mirrors that reflect the light to rejoin in an interference pattern, seen at the bottom of each panel. If the comparison path is the same number of wavelengths as the reference, as in the top panel, the two beams reinforce each other, reconstituting the original beam in constructive interference. However, if the two paths differ by half a wavelength, they cancel each other in destructive interference, and no light is seen. By moving the mirror on the right, the interference pattern can be observed to vary as the number of wavelengths in the comparison path is changed. In that way we can "count fringes" as the mirror on the right is moved, and determine how many wavelengths longer the comparison path is than the reference path.

A time interval separates the emission of the light and reassembly of the split beams. To observe a sharp pattern the light must maintain its phase relation for this time interval. The time interval, say Δt, at which the phase undergoes a random change is called the coherence time, and the corresponding coherence length is cΔt. A single-frequency laser is necessary to obtain a coherence length as long as a metre.[4]

In 1960, the metre was defined as 1 650 763.73 wavelengths in vacuum of the krypton-86 atomic transition between the levels 2p10 and 5d5 as measured using an interferometer with a traveling microscope to detect the interference fringes. The discharge was held at the triple point of nitrogen (≃ 63K). [5] The estimated accuracy of realizations of this standard, which uses a discharge lamp containing solid krypton at a temperature of 64 K, is about 4 parts per billion, or an error of about two metres in the Earth-Moon separation.[3][6]

Due largely to increased experience with lasers in such measurements, the stability and frequency control of lasers rapidly improved. Modern HeNe/CH4 and CO2/OsO4 systems show reproducibilities near 1 × 10−13 and stabilities better than 1 × 10−14.[7] Because a wavelength of light λ is related to its frequency f through its speed of propagation, a wavelength is λ = c/f, providing a way to establish the wavelengths being counted by fringes. The frequency of a laser is determined by the energy difference between atomic levels involved in a transition, and is not a sharply defined value, but depends upon the lifetime that an atom can spend in each level. The CIPM has published a list of atomic transitions and their expected accuracy based upon specified preparation of the emitting atoms.[6] The frequencies are established by comparison with the time-standard 133Cs atomic clock. As examples, several transitions in 127I2 are listed with relative uncertainties of 2.5 × 10−11 to 4.5 × 10−10. The lowest listed uncertainty is about 6 × 10−13 for the 1S0–3P1 transition in laser stabilized 40Ca atoms.[6]

The CGPM changed the definition of the meter in its 17th General Conference of 1983, listing a number of deficiencies of the krypton 86 lamp as a source that led to poor accuracy in reproducibility of the metre that would be improved by a switch to lasers.[8] The krypton lamp suffered with regard to power, coherence length and reproducibility of frequencies. In its later 2002 Revision of the practical realization of the definition of the metre CGPM suggested that besides a time-of-flight implementation, the metre could be realized in terms of the wavelength in vacuum of a plane electromagnetic wave determined as λ = c/f with c = 299 792 458 m/s, and tabulated a list of radiations with their frequencies and estimated relative accuracies. Thus, the change of definition did not change the notion that the metre is a certain number of wavelengths, but allowed flexibility in deciding what wavelengths would be used. The new definition made comparison of sources easier than counting fringes, however, and more accurate, because frequencies could be compared with a relative accuracy of 10−14 or so, depending upon frequency.[9]

A difficulty in measuring wavelengths is the departure from plane waves caused by inhomogeneity of the optical properties of the paths, or due to diffraction. That deviation introduces uncertainty in the wavelength. On the other hand, the frequency of the radiation is not affected by such matters, making it a more accurately observable property of an electromagnetic wave.[10]

What do we know about the speed of light?

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The most important practical matter in using a time-of-flight method is just how reproducibly and how accurately the standard speed can be realized by an observer. A great deal of effort has gone into determining the factors affecting the speed of light. Among the issues considered are these:

A major conclusion is that the answer to all these questions is "yes" with one exception. If the light travels in free space, then the answer to all these questions is "Not as far as we can tell with today's measurement abilities."

Two other questions are:

The answer to this question is that as far as we can tell, no such effects are found.

  • 7. Is the speed of light altered by gravity?

The answer to this question is "yes, light speed is reduced in the presence of a gravitational field."[11] Hence, a correction for gravity must be used to refer a ‘time-of-flight’ measurement to the standard speed of light.

Consequently, the speed of light can be counted upon to have the same value for all observers, provided they observe the light in free space and correct for gravity. As a practical matter, free space is not readily obtained (in fact, it is unobtainable although it can be approached). The common practice, then, is to make measurements in a medium, such as air or "partial vacuum", and to correct these measurements based upon theoretical formulas that express the effect of the non-ideal nature of the medium. Of course, these corrections are subject to errors, if for no other reason than we must measure the medium's nonidealities, and so they are known only approximately. It must be noted, however, that these experimental difficulties in achieving the standard speed of light have no bearing upon its defined value, which being defined is exactly 299,792,458 m/s.

How can the speed of light be monitored in a system where it has a defined value?

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In the present SI units system where the speed of light has a defined value of c = 299,792,458 m/s exactly, what does monitoring the ‘speed of light’ mean? Obviously, it doesn't mean watching the meetings of the BIPM to see whether the defined value is changed (although, of course, that is a matter under observation). Instead, what is meant is keeping a watch over the actual speed of propagation of light. That can be done by observing the time-of-flight of light over a fixed distance, for example. Naturally, this is an experimental measurement, and subject to three errors: first, the accuracy with which the time can be measured, second the accuracy with which the length can be maintained at a fixed value, and third the accuracy with which the medium of propagation can be held constant. An uncertainty is whether the length of the path changes with time (for example, as the universe expands, there is evidence that distances change; see Metric expansion of space) and whether the time provided by the standard of time changes (see Fine structure constant). Much more mundane factors can change the fixed length, for example, a bar changes length with temperature, flexure (changes in bending because of changed gravitational field or movement of supports), and so forth. These effects are small, but the anticipated variation in the speed of light is really small, so very extreme care must be taken in looking at all the possible sources of uncertainty.

Is the speed of light actually a constant?

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Main article: Variable speed of light

The possibility exists that over the lifetime of the universe the speed of light may change, for example, if it depends in some way upon the size of the universe or upon its composition, both of which have varied since the big bang and continue to change. A number of areas of ongoing research have predicted such changes, for example, quantum gravity, but the predicted changes are too small for modern methods to detect.

References

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  1. ^ "Base unit definitions: Meter". NIST. Retrieved 2009-08-22.
  2. ^ "Resolution 1". Conférence Générale des Poids et Mesures. BIPM. 1983. Retrieved 2009-08-23.
  3. ^ a b James Jespersen, Jane Fitz-Randolph (1999). From sundials to atomic clocks: understanding time and frequency (Reprint of NIST Monograph 155 (1999), 2nd revised ed.). Dover Publications. p. 280. ISBN 0486409139.
  4. ^ John F. Ready (1997). "Figure 2-16: Fringe visibility in interferometric experiments". Industrial applications of lasers (2nd ed.). Academic Press. p. 56. ISBN 0125839618.
  5. ^ S. V. Gupta (2010). Units of measurement: past, present and future : international system of units. Springer. p. 57. ISBN 3642007376.
  6. ^ a b c The relative uncertainty in this transition frequency is estimated as 4 × 10−9. See TJ Quinn (1999). "Practical realization of the definition of the metre (1997)" (PDF). Metrologia. 36: 211–244. §2.1 p. 214.
  7. ^ John L. Hall and Jun Ye (2003). "Optical Frequency Standards and Measurement". IEEE Transactions on instrumentation and measurement. 52 (April). IEEE: 227. {{cite journal}}: More than one of |number= and |issue= specified (help)
  8. ^ Barry N. Taylor and Ambler Thompson, ed. (2008). "The international system of units; NIST Special Publication 303" (PDF) (2008 ed.). NIST: 70ff & 77ff. {{cite journal}}: |chapter= ignored (help); Cite journal requires |journal= (help)
  9. ^ Gordon WF Drake (2006). "Table 30.1: Limits on time variations of frequencies of different transitions". Springer handbook of atomic, molecular, and optical physics, Volume 1 (2nd ed.). Springer. p. 459. ISBN 038720802X.
  10. ^ Wolfgang Demtröder (2008). "§9.7 Absolute optical frequency measurement and optical frequency standards". Laser Spectroscopy: Vol. 2: Experimental Techniques (4th ed.). Springer. p. 523. ISBN 3540749527.
  11. ^ Ta-Pei Cheng (2005). Relativity, gravitation, and cosmology: a basic introduction. Oxford University Press. p. 49. ISBN 0198529570.

General background

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