## Abstract

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Full Article | PDF Article**Journal of the Optical Society of America**- Vol. 29,
- Issue 5,
- pp. 183-187
- (1939)
- •doi: 10.1364/JOSA.29.000183

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- Distances and times in this paper are measured, unless otherwise specified, by rods and clocks which are unaffected by motion (Lorentz’ “stationary” rods and clocks). The light signals are transmitted through the luminiferous ether, with respect to which velocities are measured by rods and clocks of the above type.

- We here assume that the ellipsoid which is equivalent to the interferometer does not change the direction of its axes. Actually the ellipse is tilted so that its major axis remains always normal to the direction of the field. This adds to tan−1Rdθ/dR a small angle, approximately=12dθ, the effect of which in the formula would be only to add small times of high order.

- This is considered sufficient check on the general validity of the formula to permit omission of the very complicated case of light signals going around two sides of a triangle to return to the diagonal mirror, instead of along the direction of travel of the interferometer, as in this case.

- In this connection reference may be made to a paper by H. B. Phillips, “Note on Einstein’s Theory of Gravitation” (J. Math. Phys. 1, 177–190 (1922)) where the development is also made without the use of the tensor calculus or (explicitly) of invariance. Phillips assumes the equivalence hypothesis, but interprets its implications differently than is here done. His assumption is that “fixed” distances and times in the gravitational field will measure the same as if measured from the equivalent moving particle. The assumption in the present paper is that rods and clocks in a stationary gravitational field behave like rods and clocks on the equivalent moving particle. While superficially the same in appearance, the two assumptions are effectively different.In order to make measurements of distances and times, rods and clocks must be used, and must be transported on the measuring platforms (J. Opt. Soc. Am. 27, 263 (1937)). Rods and clocks transported on the moving (freely falling) platform, do not yield the same measurements as do rods and clocks of identical behavior transported on the stationary platform in the gravitational field. Phillips’ formula for ds is in terms of distances and times measured on the gravitating body, while ours is in the invariant system (Note 1). Our formula corresponding to his would be of the same form as in a field-free space (i.e., no γ’s) and would embody the assumption that the measured velocity of light is constant (invariance of interferometer behavior). Phillips’ formula for ds thus calls for variation of interferometer indications in a gravitational field.

[CrossRef] - In this connection the following statement by T. Levi-Civita (Nuovo Cimento 13, 45 (1936)) is of interest: “During a conversation which I had with Einstein a few years ago I asked him … if it is possible to give any concrete interpretation … to the elementary chronotopic interval ds2 …. He replied to me that he did not know of any physical meaning ….”

[CrossRef] - For an alternative treatment, using Hamilton’s theorem of ordinary dynamics, see H. A. Lorentz, Problems of Modern Physics, Note 18, p. 278.

In this connection the following statement by T. Levi-Civita (Nuovo Cimento 13, 45 (1936)) is of interest: “During a conversation which I had with Einstein a few years ago I asked him … if it is possible to give any concrete interpretation … to the elementary chronotopic interval ds2 …. He replied to me that he did not know of any physical meaning ….”

[CrossRef]

In this connection reference may be made to a paper by H. B. Phillips, “Note on Einstein’s Theory of Gravitation” (J. Math. Phys. 1, 177–190 (1922)) where the development is also made without the use of the tensor calculus or (explicitly) of invariance. Phillips assumes the equivalence hypothesis, but interprets its implications differently than is here done. His assumption is that “fixed” distances and times in the gravitational field will measure the same as if measured from the equivalent moving particle. The assumption in the present paper is that rods and clocks in a stationary gravitational field behave like rods and clocks on the equivalent moving particle. While superficially the same in appearance, the two assumptions are effectively different.In order to make measurements of distances and times, rods and clocks must be used, and must be transported on the measuring platforms (J. Opt. Soc. Am. 27, 263 (1937)). Rods and clocks transported on the moving (freely falling) platform, do not yield the same measurements as do rods and clocks of identical behavior transported on the stationary platform in the gravitational field. Phillips’ formula for ds is in terms of distances and times measured on the gravitating body, while ours is in the invariant system (Note 1). Our formula corresponding to his would be of the same form as in a field-free space (i.e., no γ’s) and would embody the assumption that the measured velocity of light is constant (invariance of interferometer behavior). Phillips’ formula for ds thus calls for variation of interferometer indications in a gravitational field.

[CrossRef]

[CrossRef]

For an alternative treatment, using Hamilton’s theorem of ordinary dynamics, see H. A. Lorentz, Problems of Modern Physics, Note 18, p. 278.

[CrossRef]

[CrossRef]

[CrossRef]

For an alternative treatment, using Hamilton’s theorem of ordinary dynamics, see H. A. Lorentz, Problems of Modern Physics, Note 18, p. 278.

Distances and times in this paper are measured, unless otherwise specified, by rods and clocks which are unaffected by motion (Lorentz’ “stationary” rods and clocks). The light signals are transmitted through the luminiferous ether, with respect to which velocities are measured by rods and clocks of the above type.

We here assume that the ellipsoid which is equivalent to the interferometer does not change the direction of its axes. Actually the ellipse is tilted so that its major axis remains always normal to the direction of the field. This adds to tan−1Rdθ/dR a small angle, approximately=12dθ, the effect of which in the formula would be only to add small times of high order.

This is considered sufficient check on the general validity of the formula to permit omission of the very complicated case of light signals going around two sides of a triangle to return to the diagonal mirror, instead of along the direction of travel of the interferometer, as in this case.

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$$(1-{V}^{2}/{c}^{2}{)}^{{\scriptstyle \frac{1}{2}}}=(1-2GM/R{c}^{2}{)}^{{\scriptstyle \frac{1}{2}}},$$

$${L}^{\prime}=L(1-2GM/R{c}^{2}{)}^{{\scriptstyle \frac{1}{2}}},$$

$${\tau}^{\prime}=\tau /(1-2GM/R{c}^{2}{)}^{{\scriptstyle \frac{1}{2}}},$$

$$\begin{array}{l}{L}^{\prime}=L{\gamma}^{{\scriptstyle \frac{1}{2}}},\\ {\tau}^{\prime}=\tau /{\gamma}^{{\scriptstyle \frac{1}{2}}}.\end{array}$$

$$2L/{c}_{T}=2L{\gamma}^{{\scriptstyle \frac{1}{2}}}/{c}_{R}=\tau /{\gamma}^{{\scriptstyle \frac{1}{2}}}=2L/c{\gamma}^{{\scriptstyle \frac{1}{2}}},$$

$$\begin{array}{l}{C}_{T}=c{\gamma}^{{\scriptstyle \frac{1}{2}}},\\ {c}_{R}=c\gamma .\end{array}$$

$$\begin{array}{ll}\hspace{0.17em}\hfill & t=b/c\gamma \hfill \\ \text{and}\hfill & t=a/c{\gamma}^{{\scriptstyle \frac{1}{2}}}\hfill \\ \text{or}\hfill & a=tc{\gamma}^{{\scriptstyle \frac{1}{2}}},\hfill \\ \hspace{0.17em}\hfill & b=tc\gamma .\hfill \end{array}$$

$${r}^{2}={a}^{2}{b}^{2}/({a}^{2}\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\theta +{b}^{2}\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}\theta ).$$

$${{c}_{\theta}}^{2}=\frac{{r}^{2}}{{t}^{2}}=\frac{{a}^{2}{b}^{2}}{{t}^{2}({a}^{2}\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}\theta +{b}^{2}\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}\theta )}=\frac{{c}^{2}{\gamma}^{2}}{{\text{cos}}^{2}\hspace{0.17em}\theta +\gamma \hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}\theta}.$$

$$\begin{array}{l}t=\frac{L{\gamma}^{{\scriptstyle \frac{1}{2}}}(1-{v}^{2}/{c}^{2}{\gamma}^{2}{)}^{{\scriptstyle \frac{1}{2}}}}{c\gamma +v}+\frac{L{\gamma}^{{\scriptstyle \frac{1}{2}}}(1-{v}^{2}/{c}^{2}{\gamma}^{2}{)}^{{\scriptstyle \frac{1}{2}}}}{c\gamma -v}\\ =\frac{2L}{c{\gamma}^{{\scriptstyle \frac{1}{2}}}(1-{v}^{2}/{c}^{2}{\gamma}^{2}{)}^{{\scriptstyle \frac{1}{2}}}}.\end{array}$$

$${{c}_{\theta}}^{2}=\frac{{c}^{2}{\gamma}^{2}({L}^{2}+{v}^{2}{t}^{2}/4)}{{v}^{2}{t}^{2}/4+\gamma {L}^{2}}.$$

$$4({L}^{2}+{v}^{2}{t}^{2}/4)=\frac{{c}^{2}{\gamma}^{2}{t}^{2}({L}^{2}+{v}^{2}{t}^{2}/4)}{{v}^{2}{t}^{2}/4+\gamma {L}^{2}}$$

$$t=\frac{2L}{c{\gamma}^{{\scriptstyle \frac{1}{2}}}(1-{v}^{2}/{c}^{2}{\gamma}^{2}{)}^{{\scriptstyle \frac{1}{2}}}},$$

$$t=\frac{2L}{c{\gamma}^{{\scriptstyle \frac{1}{2}}}(1-{v}^{2}/{c}^{2}\gamma {)}^{{\scriptstyle \frac{1}{2}}}},$$

$$\begin{array}{l}{r}^{2}={L}^{2}\xb7{L}^{2}\gamma /\left(\frac{{L}^{2}d{R}^{2}}{d{R}^{2}+{R}^{2}d{\theta}^{2}}+\frac{\gamma {L}^{2}{R}^{2}d{\theta}^{2}}{d{R}^{2}+{R}^{2}d{\theta}^{2}}\right)\\ ={L}^{2}\gamma (d{R}^{2}+{R}^{2}d{\theta}^{2})/(d{R}^{2}+\gamma {R}^{2}d{\theta}^{2}).\end{array}$$

$$\begin{array}{l}{{c}^{\prime}}^{2}/{r}^{2}={c}^{2}\gamma /{L}^{2}\\ \text{or}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{{c}^{\prime}}^{2}=\frac{{r}^{2}{c}^{2}\gamma}{{L}^{2}}=\frac{{c}^{2}{\gamma}^{2}(d{R}^{2}+{R}^{2}d{\theta}^{2})}{d{R}^{2}+\gamma {R}^{2}d{\theta}^{2}}.\end{array}$$

$${\left(1-\frac{\frac{d{P}^{2}}{d{T}^{2}}}{{{c}^{\prime}}^{2}}\right)}^{{\scriptstyle \frac{1}{2}}}:1={\left(1-\frac{d{R}^{2}/d{T}^{2}+{R}^{2}d{\theta}^{2}/d{T}^{2}}{\frac{{c}^{2}{\gamma}^{2}(d{R}^{2}+{R}^{2}d{\theta}^{2})}{d{R}^{2}+\gamma {R}^{2}d{\theta}^{2}}}\right)}^{{\scriptstyle \frac{1}{2}}}:1.$$

$$dT=\frac{r{\left(1-\frac{(dP/dT{)}^{2}}{{{c}^{\prime}}^{2}}\right)}^{{\scriptstyle \frac{1}{2}}}}{{c}^{\prime}+dP/dT}+\frac{r{\left(1-\frac{(dP/dT{)}^{2}}{{{c}^{\prime}}^{2}}\right)}^{{\scriptstyle \frac{1}{2}}}}{{c}^{\prime}-dP/dT},$$

$$dT=\frac{2L}{c{\gamma}^{{\scriptstyle \frac{1}{2}}}{\left(1-\frac{d{R}^{2}+\gamma {R}^{2}d{\theta}^{2}}{{c}^{2}{\gamma}^{2}d{T}^{2}}\right)}^{{\scriptstyle \frac{1}{2}}}}.$$

$$d{s}^{2}={c}^{2}\gamma d{T}^{2}-{R}^{2}d{\theta}^{2}-d{R}^{2}/\gamma .$$

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