## Abstract

The problem of measuring the velocity of light by signals sent in one direction, instead of the usual method, by signals sent out and back, is studied. This requires two clocks, and the necessary steps to set the two clocks are discussed. It is found desirable to distinguish between “velocity” in its elementary and traditional sense, and the “rod-clock-quotient” directly obtained by measurement with rods and clocks experiencing the Fitzgerald-Larmor-Lorentz contractions. The precise formula derived for the one-way measurement of the velocity of light involves two rod-clock-quotients. The use of one-way light signal measurements in the Lorentz transformations and the Special Theory of Relativity is discussed.

© 1948 Optical Society of America

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1. The frequent assertion that “the Michelson-Morley experiment abolished the ether” is a piece of faulty logic. When Maxwell predicted a positive result from the experiment he did so on the basis of two assumptions; the first, that the light waves were transmitted through a medium, the second, which was not realized until pointed out by Fitzgerald, that the measuring instruments would not be affected by motion. The nul result of the experiment proved some assumption made in predicting a positive result to be wrong. The experimental demonstration of the variation of measuring instruments with motion, in exactly the way to produce a nul result, shows that it was the second assumption alone that was wrong; leaving the evidence for a transmitting medium, as derived from aberrational and rotational phenomena, as strong, if not stronger, than ever.
2. While more than one clock may be involved in a measurement, all clocks in a stationary medium can be given the same rate and setting by light signals, using for setting one-half the transit time out and back. Hence all clocks, being identical in indication, may be included under the above “a” clock. The question whether it is practically possible to determine that a clock or rod is stationary in the medium is no bar to using such rods and clocks in developing a theoretical argument.
3. The demonstration of the contraction of clock rates is described in “An Experimental study of the rate of a moving atomic clock,” H. E. Ives and G. H. Stilwell, J. Opt. Soc. Am. 28, 215 (1938), and J. Opt. Soc. Am. 31, 369 (1941). For the proof of the contraction of lengths the Kennedy-Thorndyke experiment (Phys. Rev. 42, 400 (1932)) serves. This assumed the contractions of length and showed that the nul result of the experiment demanded that clock rates should vary: With the just quoted positive establishment of the clock rate variation, the Kennedy-Thorndyke experiment proves the assumed length contraction.
[Crossref]
4. For the derivation of the F.L.L. contractions in terms of motion through the light transmitting medium from Maxwell’s radiation pressure and the conservation laws, see “Derivation of the Lorentz transformations,” H. E. Ives, Phil. Mag. 36, 392 (1945).
5. This is one of the commonest ways of measuring velocity. It is used by the mariner with his chronometer, by the automobilist in traversing the “measured mile,” and by the train traveller who counts telegraph poles passed in the interval given by the watch in his hand. By it the measured velocity of light is infinity.
6. W. H. McCrea, Relativity Physics (Methuen, London, and Company, Ltd.1935), p. 8.
7. In Einstein’s original presentation he declared the value “c” for the velocity of light to be “firmly established by experiment.” This was not a legitimate citation, for he was discussing the one-way transmission of signals, while the only existing experimental values were those from out-and-back measurement.
8. P. Drude, Lehrbuch der Optik (Hirzel, Leipzig, 1900) p. 426.
9. J. Larmor, “Aether and Matter,” (Cambridge University Press, 1900) p. 178.
10. Larmor uses the term “velocity” in exactly the sense adhered to in the present paper.
11. If we transform our origin of coordinates to the light transmitting medium instead of the refracting material we get the relationV+W=[cμ(1-W2c2)/(1+Wcμ)]+W=(cμ+W)/(1+Wcμ).This formula is commonly credited to the Special Theory of Relativity. That the considerably earlier formula of Larmor (17), derived from analysis of the effect of moving charges on the velocity of electromagnetic waves through the surrounding ether, is identical in content, appears to have been generally overlooked.

#### 1945 (1)

For the derivation of the F.L.L. contractions in terms of motion through the light transmitting medium from Maxwell’s radiation pressure and the conservation laws, see “Derivation of the Lorentz transformations,” H. E. Ives, Phil. Mag. 36, 392 (1945).

#### Drude, P.

P. Drude, Lehrbuch der Optik (Hirzel, Leipzig, 1900) p. 426.

#### Ives, H. E.

For the derivation of the F.L.L. contractions in terms of motion through the light transmitting medium from Maxwell’s radiation pressure and the conservation laws, see “Derivation of the Lorentz transformations,” H. E. Ives, Phil. Mag. 36, 392 (1945).

#### Larmor, J.

J. Larmor, “Aether and Matter,” (Cambridge University Press, 1900) p. 178.

#### McCrea, W. H.

W. H. McCrea, Relativity Physics (Methuen, London, and Company, Ltd.1935), p. 8.

#### Phil. Mag. (1)

For the derivation of the F.L.L. contractions in terms of motion through the light transmitting medium from Maxwell’s radiation pressure and the conservation laws, see “Derivation of the Lorentz transformations,” H. E. Ives, Phil. Mag. 36, 392 (1945).

#### Other (9)

This is one of the commonest ways of measuring velocity. It is used by the mariner with his chronometer, by the automobilist in traversing the “measured mile,” and by the train traveller who counts telegraph poles passed in the interval given by the watch in his hand. By it the measured velocity of light is infinity.

W. H. McCrea, Relativity Physics (Methuen, London, and Company, Ltd.1935), p. 8.

In Einstein’s original presentation he declared the value “c” for the velocity of light to be “firmly established by experiment.” This was not a legitimate citation, for he was discussing the one-way transmission of signals, while the only existing experimental values were those from out-and-back measurement.

P. Drude, Lehrbuch der Optik (Hirzel, Leipzig, 1900) p. 426.

J. Larmor, “Aether and Matter,” (Cambridge University Press, 1900) p. 178.

Larmor uses the term “velocity” in exactly the sense adhered to in the present paper.

If we transform our origin of coordinates to the light transmitting medium instead of the refracting material we get the relationV+W=[cμ(1-W2c2)/(1+Wcμ)]+W=(cμ+W)/(1+Wcμ).This formula is commonly credited to the Special Theory of Relativity. That the considerably earlier formula of Larmor (17), derived from analysis of the effect of moving charges on the velocity of electromagnetic waves through the surrounding ether, is identical in content, appears to have been generally overlooked.

The frequent assertion that “the Michelson-Morley experiment abolished the ether” is a piece of faulty logic. When Maxwell predicted a positive result from the experiment he did so on the basis of two assumptions; the first, that the light waves were transmitted through a medium, the second, which was not realized until pointed out by Fitzgerald, that the measuring instruments would not be affected by motion. The nul result of the experiment proved some assumption made in predicting a positive result to be wrong. The experimental demonstration of the variation of measuring instruments with motion, in exactly the way to produce a nul result, shows that it was the second assumption alone that was wrong; leaving the evidence for a transmitting medium, as derived from aberrational and rotational phenomena, as strong, if not stronger, than ever.

While more than one clock may be involved in a measurement, all clocks in a stationary medium can be given the same rate and setting by light signals, using for setting one-half the transit time out and back. Hence all clocks, being identical in indication, may be included under the above “a” clock. The question whether it is practically possible to determine that a clock or rod is stationary in the medium is no bar to using such rods and clocks in developing a theoretical argument.

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Fig. 1

### Equations (24)

$distance traveled in the medium time taken to travel the distance ,$
$l = l 0 [ 1 - ( v 2 / c 2 ) ] 1 2 ,$
$ν = ν 0 [ 1 - ( v 2 / c 2 ) ] 1 2 .$
$t 1 = D ′ [ 1 - ( W 2 / c 2 ) ] 1 2 c - W ,$
$t 2 = D ′ [ 1 - ( W 2 / c 2 ) ] 1 2 c + W ,$
$t 1 + t 2 = 2 c D ′ [ 1 - ( W 2 / c 2 ) ] c 2 - W 2 .$
$t 1 + t 2 = t ′ [ 1 - ( W 2 / c 2 ) ] 1 2$
$c = 2 ( D ′ / t ′ ) .$
$Q 1 = c .$
$Y = D τ = D ′ τ ′ [ 1 - W 2 c 2 ] 1 2 [ 1 - ( W + Y ) 2 c 2 ] 1 2 .$
$Y = q [ 1 - ( W 2 / c 2 ) ] [ 1 + ( q 2 / c 2 ) ] 1 2 + ( q W / c 2 ) ,$
$τ ′ = τ [ 1 - ( W 2 / c 2 ) ] 1 2 - W D c 2 [ 1 - ( W 2 / c 2 ) ] 1 2 [ 1 + ( q 2 / c 2 ) ] 1 2 ,$
$Δ = τ [ 1 - ( W 2 / c 2 ) ] 1 2 - τ [ 1 - ( W 2 / c 2 ) ] 1 2 - W D c 2 [ 1 - ( W 2 / c 2 ) ] 1 2 ( 1 + q 2 c 2 ) 1 2 = ( τ - W 2 c 2 ) 1 2 [ ( 1 + q 2 c 2 ) 1 2 - 1 ] + W D c 2 [ 0 - ( W 2 / c 2 ) ] 1 2 ( 1 + q 2 c 2 ) 1 2 .$
$τ = [ D ′ [ ( 1 + q 2 c 2 ) 1 2 + W q c 2 ] ] / [ q ( 1 - W 2 c 2 ) 1 2 ] ,$
$Δ = D ′ q [ ( ( 1 + q 2 c 2 ) 1 2 - 1 ) + W q c 2 ] .$
$t ′ = t [ 1 - ( W 2 / c 2 ) ] 1 2 - Δ$
$t = ( t ′ + Δ ) / ( 1 - W 2 c 2 ) 1 2 .$
$V = Q 2 ( 1 - W 2 c 2 ) 1 + Q 2 W c 2 + Q 2 q [ ( 1 + q 2 c 2 ) 1 2 - 1 ] .$
$Q 2 = ( c ) / [ 1 - c q ( ( 1 + q 2 c 2 ) 1 2 - 1 ) ] ,$
$c = [ 2 Q 2 ( Q 2 - q ) ] / [ 2 Q 2 - q ] .$
$c = Q 2 .$
$V = c μ ( 1 - W 2 c 2 ) / ( 1 + W c μ )$
$Q 2 = c μ / [ 1 - c μ q [ ( 1 + q 2 c 2 ) 1 2 - 1 ] ]$
$V+W=[cμ(1-W2c2)/(1+Wcμ)]+W=(cμ+W)/(1+Wcμ).$