SPEED OF LIGHT IN VACUUM

The question of the value of the speed of light in vacuum, measured again and again with the aim of refining it, continues to attract close attention. Evidence of this is provided, for example, by the abundance of critical surveys that appeared during 1951–1952[^1–^4], as well as reviews[^5–^11] addressed chiefly to a broad circle of scientific workers. The reason for this heightened interest should be seen not only in the fact that the speed of light is one of the fundamental physical constants, but also in the radical revision, carried out in recent years, of the accepted value of this quantity. It should be noted at once that this revision is connected not with the discovery of some error in earlier work, but with a sharp increase in the accuracy of individual measurements. The rapid development of experimental technique, especially the technique of microwave measurements, made it possible by the end of the 1940s to carry out a number of measurements of the speed of light with an accuracy exceeding by 1–2 orders of magnitude

accuracy of the preceding measurements. At the same time, both the measuring technique and the frequency ranges for which the measurements were carried out proved to be incomparably more varied than before. Along with the usual measurements of the propagation velocity of waves of the visible range in air, radiolocation measurements, measurements of resonant frequencies in the fields of cylindrical waveguides, and other methods were brought in for determining the speed of light. Two years ago we already had occasion to acquaint the readers of our journal with the methodology and the first results of such measurements¹². However, since then a number of new works have appeared that deserve attention. Let us briefly recall the essence of the matter.

Until the end of the nineteen-forties, the generally accepted value of the speed of light in vacuum was considered to be

\[ c = 299\,776 \pm 4\ \text{km/sec}, \]

obtained by Birge¹³ as the result of a careful analysis of a very large number of measurements carried out before 1941. A distinctive feature of these data was that the errors of the individual measurements proved to be very large, and the comparatively modest probable error indicated by Birge was the result of statistical processing of an enormous body of material. The principal result of the most recent measurements of the speed of light unexpectedly proved to be not a decrease in the probable error, but a substantial change in the value itself (on average by 16 km/sec), far beyond the limits of the error indicated by Birge (see ¹², as well as Table I). Later data, published over the past two years, have confirmed and refined this result.

The greatest interest among these data is undoubtedly presented by the results obtained by Hansen and Bol², ¹⁰, ¹⁴, Aslakson¹⁵, and Froome²³. The method used by Hansen and Bol¹⁰, ¹¹ is, in general, analogous to Essen’s method described in ¹²: the resonant frequency was measured in a hollow cylindrical resonator from which the air had been evacuated. Some difference in the details of carrying out the experiment and in the determination of the instrumental constants is of interest insofar as comparison of Essen’s results with those of Hansen and Bol makes it possible to judge the absolute error of the method. In this connection we note that Essen’s estimate of the limiting absolute error as ±1 km/sec*) proved to be excessively optimistic. The value obtained by Hansen and Bol is as follows:

\[ c = 299\,789.3 \pm 0.4\ \text{km/sec}. \]

The details of Hansen and Bol’s experiments have not yet been published; however, their measurements apparently are distinguished by exceptional care. Thus, Birge and Watts², critically considering the data of various authors, assign them a statistical weight 14 times greater than that of Essen’s data and 156 times greater than the value obtained by Dorsey from a critical analysis of all experiments performed before 1941**).

*) The values of possible errors in the measurements of Essen and Gordon-Smith (±9 km/sec) and Essen (±3 km/sec) given in ¹² were subsequently reduced by Essen³ to ±3 and ±1 km/sec, respectively, as a result of taking into account certain previously unconsidered factors. This error is considered by Essen not as probable, but as the maximum possible.

**) The value obtained by Dorsey differs somewhat from the value obtained by Birge—see Table III.

Measurements by Aslakson1 are a direct continuation of his earlier work2 on radio geodesy (see 3). However, repetition of the measurements with more advanced apparatus and a revision of the data on the refractive index of air for microwaves (for dry air at 760 mm Hg and \(0^\circ\)C, \(n = 1.0002876\) was assumed) led him to the value

\[ c = 299\,794.2 \pm 1.4\ \text{km/sec}, \]

which differs appreciably from that obtained earlier (see Table II).

In Froome’s experiments4 the phase velocity of microwaves in air was measured. The value of the velocity was obtained as a result of simultaneous determination of the frequency and the wavelength of the radiation. The frequency was found by comparison with the higher harmonics of a standard quartz oscillator. To measure the wavelength, an arrangement analogous to a Michelson interferometer was used. The radiation of a stabilized klystron (24,000 Mc/s) was directed by a dividing device into two arms of the interferometer. One of the arms (the short one) remained unchanged. The second consisted of a waveguide leading the radiation into a very large room, in which (at a distance of up to 21.5 m) there was a metallic mirror reflecting the radiation back into the interferometer. Displacement of the mirror along the beam changed the path difference of the interfering waves and, consequently, the intensity of the radiation incident on the detector. Mirror positions corresponding to a minimum of the detector readings could be fixed with an accuracy of up to \(\pm 3\ \mu\). Since the total displacement of the mirror could reach 162 cm (about 260 wavelengths), it was thereby possible to determine the wavelength with an accuracy of up to \(\pm 3 \cdot 10^{-6}\) of the measured quantity. The most substantial errors arose from diffraction phenomena in the interferometer (they were eliminated by varying the mean distance to the mirror) and from the influence of echoes produced by fixed objects located in the same room as the mirror. The results obtained were recalculated to vacuum, on the basis of the available data on the refractive index of air and water vapor for the frequency interval used.

The final value obtained by Froome is as follows:

\[ c = 299\,792.6 \pm 0.7\ \text{km/sec}, \]

where the indicated error limits take into account both random errors and the authors’ estimate of the possible systematic error.

A method similar in idea, but quite different in execution, was used by Rank and his collaborators5. As is known, the rotational energy levels of a diatomic or polyatomic linear molecule are determined by the relation \(\nu = BJ(J+1) - D J^2(J+1)^3 + \ldots\), where \(\nu\) is the frequency corresponding to the transition from the ground level to the given one, \(J\) is the rotational quantum number, \(B = \dfrac{h}{8\pi^2 I}\), and \(I\) is the moment of inertia for the given vibrational state. The methods of microwave absorption spectroscopy make it possible to determine directly the frequency \(\nu\) corresponding to the \(0 \to 1\) transition for a number of molecules (in this case \(\nu = 2B - 4D\)). At the same time, the values \(B\) and \(D\) can be determined independently from an analysis of the rotational structure of absorption bands in the infrared region of the spectrum. In the latter case, however, it is not the frequencies, but the wavelengths that are measured directly, and the values of \(B\) and \(D\) are obtained expressed in reciprocal centimeters. Thus, the ratio of the value of \(B\) obtained by microwave methods to the value of the same quantity obtained spectroscopically is equal to the speed of light in vacuum.

Rank and his collaborators carried out optical measurements of the rotational structure for two HCN bands. HCN vapor filled a tube of length,

8 meters; under a pressure of 35 to 70 mm Hg, and the light beam, before entering the spectrograph, crossed this tube several times. The spectrograph had a grating of 15,000 lines per inch, 6.5 inches long. The focal length was 10 meters. The first-order spectrum was subjected to further analysis by means of a Fabry–Perot interferometer of thickness 21.35 mm.

As a result the authors obtained the following values:

\[ B = 1.47830 \pm 0.00025\ \mathrm{cm}^{-1},\qquad D = 3.1\cdot 10^{-6}\ \mathrm{cm}^{-1}. \]

Since the data available in the literature from microwave measurements of these quantities did not have sufficient accuracy, such measurements were carried out anew by Towne and co-workers \(^{25}\), and the following value was obtained:

\[ B = 44315.9 \pm 0.25\ \mathrm{Mc}. \]

From comparison of the indicated values of \(B\), for the speed of light in vacuum one obtains:

\[ c = 299\,776 \pm 7\ \mathrm{km/sec}. \]

The authors indicate that there are possibilities for further refinement of the result.

Of considerably less interest are the measurements of Houston \(^{18}\) and Mac Kinley \(^{19}\). Both authors determined the speed of propagation of light in air visually, using a somewhat modified Fizeau method.

Mac Kinley, to modulate the light beam, used, instead of a Kerr cell, the effect of rotation of the plane of polarization of light in a quartz plate under the action of an electric field applied to the plate (modulation frequency 8 Mc). The result he obtained is as follows:

\[ c = 299\,780 \pm 70\ \mathrm{km/sec}. \]

Houston’s method is more original. A beam of light was passed through a plate of piezoquartz in which a standing ultrasonic wave was excited (frequency 115 Mc), and modulation was effected by utilizing the dependence of the intensity of the diffracted beam of the first order (deflection angle \(35'\)) on the phase of the ultrasonic vibrations. The value he obtained was:

\[ c = 299\,782 \pm 9\ \mathrm{km/sec}. \]

As can be seen, the last three values are closer to Berg’s value than to the results of other more recent measurements, and the error of the measurements is considerably larger. These works (especially those of Roonk and Houston) should be regarded, however, not so much as attempts to refine the value of the speed of light, but rather from the point of view of the development of methods for measuring this quantity.

Finally, one cannot fail to mention two works devoted to measuring the speed of motion of \(\gamma\)-quanta in air \(^{20,21}\). Since the methodology of these measurements has already been described in our journal, we shall not dwell on it. Let us note only that in both cases the accuracy of the measurements is very small—only about 1%, i.e. close to the accuracy of the earliest measurements carried out by other methods (apart from microwave ones; see Table 1). However, these works are of interest, since they extend the range of frequencies covered by direct measurements. In fact, at the present time data are already available relating to the quasistatic field (measurements of the ratio of electromagnetic units to electrostatic ones), to short radio waves (\(\nu \sim 5\text{–}75\) Mc), to the microwave range (\(\nu \sim 220\text{–}10\,000\) Mc), to visible light (\(\nu \sim 10^8\) Mc), and to \(\gamma\)-radiation (\(\nu \sim 10^{18}\text{–}1\) Mc).

Table I

Results of the principal measurements of the speed of light in vacuum, carried out before 1952*)

1. By the ratio of electrostatic units to electromagnetic units. Quasi-stationary field.

*) Some of the results of the measurements were later corrected, taking into account new values of other constants used by the authors in determining \(c\).

Continuation of Table I

2. Propagation velocity in free space

*) According to modern measurements \(300\,870 \pm 100\) km/sec^23.

Continuation of Table I

Continuation of Table I

3. By the velocity of propagation in a waveguide

4. By the product of frequency and wavelength

Thus, the measurements cover (with small gaps) the frequency interval from zero to \(10^{22}\) cycles. As was to be expected, there is no basis for doubting the absence of dispersion of the speed of light in a vacuum throughout this enormous frequency interval. The results of the principal measurements of the speed of light performed before 1952 are summarized in Table I.

We also point to the recently published detailed description of Bergstrand’s measurements\({}^{26}\), the results of which were published earlier (see p. 12). Table II gives the results of the most careful measurements carried out after 1941. Finally, Table III contains the most probable values of the speed of light obtained by various authors as a result of a critical analysis of the experimental data. When considering this table, two circumstances at once catch the eye: twice (between 1929 and 1934 and between 1948 and 1951) the accepted most probable value underwent changes far beyond the limits of the probable error. Exactly the same picture is revealed when considering Table II—the differences between the data of different authors

Table II

Results of the most careful recent measurements of the speed of light
in a vacuum

Table III

Results of critical analysis of experimental data on the speed of light
in a vacuum

significantly greater than the probable errors indicated by them. This is perhaps one of the most striking examples of the difficulty and imperfection of methods for estimating the possible error of measurements, showing with what caution one should approach the results of such estimates. Returning to Table III, let us note that the authors of all the most recent summaries¹—⁴ agree on the value

\[ c = 299\,790\ \text{km/sec}. \]

The exception is the value proposed by Aslakson.¹⁵

The point is that Aslakson considered all data obtained before 1941 to be faulty, except for the value obtained by him himself in 1951 and not included in other summaries.

Since Froome’s data, likewise not included in the published summaries, agree considerably better with the value indicated by Aslakson, it seems probable that the latter is closer to reality.

In conclusion, let us recall that the measurements of Bergstrand, Aslakson, and Froome relate to the propagation of electromagnetic waves in air, and the figures obtained by them are burdened by errors in determining the refractive indices of air. The results of Essen, Hansen, and Bol, however, are free from these errors, for the measurements were carried out directly in vacuum.

G. Rozenberg

CITED LITERATURE

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