Abstract

The assumption of identity of properties of a body stationary in a gravitational field and of a body falling under an inverse square law of attraction, previously used to account for optical phenomena, is applied to dynamical phenomena by the extension of the same assumption to mass. Force equations corresponding to the gravitational mass

m0/(γ-r˙2/c2γ-r2θ˙2/c2)12

are derived, which when solved for a planetary orbit, give the observed advance of perihelion of Mercury.

© 1948 Optical Society of America

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References

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  1. H. E. Ives, “The behavior of an interferometer in a gravitational field,” J. Opt. Soc. Am. 29, 183 (1939).
    [Crossref]
  2. Ives, “Derivation of the Lorentz transformations,” Phil. Mag. 7, 392 (1945).
  3. A. S. Eddington, “Astronomical consequences of the electrical theory of matter,” Phil. Mag. 6, 321 (1917).
    [Crossref]
  4. Ives, “The physical significance of Birkhoff’s gravitational equations,” Phys. Rev. 72, 229 (1947).
    [Crossref]
  5. See W. H. McCrea, Relative Physics (Methuen and Company, Ltd., London, 1935), pp. 29–31.
  6. While this postulate of equivalence has been expressed in terms of velocity, from which the contraction factors are obtained, the primarily significant quantity is the corresponding acceleration, which is equivalent to the upward push experienced in a true gravitational field.
  7. It is obtained by combining Eqs. (38.8) and (39.62), pp. 85 and 86 of Eddington’s Mathematical Theory of Relativity (Cambridge, 1923).
  8. Combination of Eqs. (12b) and (13), Math. Phys.177 (1922).
  9. If one takes the equations quoted in the two preceding notes as the entire basis for developing force equations, there is no ground for favoring (22) and (23) over the simple and symmetrical equations (12) and (13), which, as noted, are essentially the equations of Birkhoff’s theory.

1947 (1)

Ives, “The physical significance of Birkhoff’s gravitational equations,” Phys. Rev. 72, 229 (1947).
[Crossref]

1945 (1)

Ives, “Derivation of the Lorentz transformations,” Phil. Mag. 7, 392 (1945).

1939 (1)

1922 (1)

Combination of Eqs. (12b) and (13), Math. Phys.177 (1922).

1917 (1)

A. S. Eddington, “Astronomical consequences of the electrical theory of matter,” Phil. Mag. 6, 321 (1917).
[Crossref]

Eddington,

It is obtained by combining Eqs. (38.8) and (39.62), pp. 85 and 86 of Eddington’s Mathematical Theory of Relativity (Cambridge, 1923).

Eddington, A. S.

A. S. Eddington, “Astronomical consequences of the electrical theory of matter,” Phil. Mag. 6, 321 (1917).
[Crossref]

Ives,

Ives, “The physical significance of Birkhoff’s gravitational equations,” Phys. Rev. 72, 229 (1947).
[Crossref]

Ives, “Derivation of the Lorentz transformations,” Phil. Mag. 7, 392 (1945).

Ives, H. E.

McCrea, W. H.

See W. H. McCrea, Relative Physics (Methuen and Company, Ltd., London, 1935), pp. 29–31.

J. Opt. Soc. Am. (1)

Math. Phys. (1)

Combination of Eqs. (12b) and (13), Math. Phys.177 (1922).

Phil. Mag. (2)

Ives, “Derivation of the Lorentz transformations,” Phil. Mag. 7, 392 (1945).

A. S. Eddington, “Astronomical consequences of the electrical theory of matter,” Phil. Mag. 6, 321 (1917).
[Crossref]

Phys. Rev. (1)

Ives, “The physical significance of Birkhoff’s gravitational equations,” Phys. Rev. 72, 229 (1947).
[Crossref]

Other (4)

See W. H. McCrea, Relative Physics (Methuen and Company, Ltd., London, 1935), pp. 29–31.

While this postulate of equivalence has been expressed in terms of velocity, from which the contraction factors are obtained, the primarily significant quantity is the corresponding acceleration, which is equivalent to the upward push experienced in a true gravitational field.

It is obtained by combining Eqs. (38.8) and (39.62), pp. 85 and 86 of Eddington’s Mathematical Theory of Relativity (Cambridge, 1923).

If one takes the equations quoted in the two preceding notes as the entire basis for developing force equations, there is no ground for favoring (22) and (23) over the simple and symmetrical equations (12) and (13), which, as noted, are essentially the equations of Birkhoff’s theory.

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Equations (38)

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m 0 / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2
d s = d t ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2
m g = m 0 / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2
m 0 { r ¨ - r θ ˙ 2 / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 + r ˙ d / d t [ 1 / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 ] } = F R
m 0 { 2 r ˙ θ ˙ + r θ ¨ / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 + r θ ˙ d / d t [ 1 / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 ] } = F θ .
m 0 r · r θ ˙ / ( γ - r ˙ 2 c 2 γ - r 2 θ ˙ 2 c 2 ) 1 2 = F θ r .
F θ r = m 0 h
γ = 1 - 2 G M / r c 2 .
m 0 r 2 θ ˙ / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 = m 0 h γ 1 2 .
m 0 { 2 r r ˙ θ ˙ + r 2 θ ¨ / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 + r 2 θ ˙ d / d t [ 1 / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 ] } = m 0 h d / d t γ 1 2 = F θ r
m 0 / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 .
F θ = G M m 0 / r 2 ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 · r r ˙ θ ˙ / c 2 γ .
F θ = - G M m 0 / r 2 ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 × ( 0 - r r ˙ θ ˙ / c 2 γ ) ,
F R = - G M m 0 / r 2 ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 × ( 1 + r 2 θ ˙ 2 / c 2 γ )
d d t ( r ˙ 2 + r 2 θ ˙ 2 ) / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 - ( r ˙ 2 + r 2 θ ˙ 2 ) d d t ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 3 2 = - 2 G M r ˙ / r 2 ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2
d d t ( r ˙ 2 + r 2 θ ˙ 2 ) / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) = - 2 G M r ˙ / r 2 ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ )
r 2 + r 2 θ ˙ 2 = ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) × ( 2 G M / r + + a ) .
1 - r ˙ 2 / c 2 - r 2 θ ˙ 2 / c 2 = 1 - 2 G M / r c 2 - a / c 2 = ( 1 - 2 G M / r c 2 ) ( 1 - a / c 2 ) = γ / b
1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ
1 - r ˙ 2 / c 2 - r 2 θ ˙ 2 / c 2 .
1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ = γ / b
( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 = γ / b 1 2
1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ = ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 / b 1 2
1 - r ˙ 2 / c 2 - r 2 θ ˙ 2 / c 2 = ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 / b 1 2 .
( 2 r r ˙ θ + r 2 θ ¨ ) / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 - 1 2 r 2 θ ˙ d / d t ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) / × ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 3 2 = G M r ˙ / r 2 c 2 · r 2 θ ˙ / γ ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 = 1 2 d γ / d t [ r 2 θ ˙ / γ ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 ] = 1 2 r 2 θ ˙ d / d t ( 1 - r ˙ 2 / c 2 γ 2 · r 2 θ ˙ 2 / c 2 γ ) / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 3 2 .
F θ = m 0 { ( 2 r ˙ θ ˙ + r θ ¨ ) / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 + r θ ˙ d / d t [ 1 / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 ] } = 0.
( r ¨ - r θ ˙ 2 ) / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 - 1 2 r ˙ d / d t ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) / ( 1 - r ˙ 2 / c γ 2 - r 2 θ ˙ 2 / c 2 γ ) 3 2 = - G M / r 2 ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 1 2 ( 1 + r 2 θ ˙ 2 / c 2 γ ) .
1 2 r ˙ d / d t ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) / ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ ˙ 2 / c 2 γ ) 3 2 ,
G M r ˙ / r 2 c 2 · r ˙ / γ ( 1 - r ˙ 2 / c 2 γ 2 - r 2 θ 2 / c 2 γ ) 1 2 .
F R = m 0 { ( r ¨ - r θ ˙ 2 ) / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 + r ˙ d / d t [ 1 / ( γ - r ˙ 2 / c 2 γ - r 2 θ ˙ 2 / c 2 ) 1 2 ] } = - G M m 0 / r 2 ( γ - r ˙ 2 / c 2 γ · r 2 θ ˙ 2 / c 2 ) 1 2 × ( 1 + r 2 θ ˙ 2 / c 2 γ + r ˙ 2 / c 2 γ ) .
d 2 u d θ 2 + u = - F R m 0 h 2 u 2 ( γ - r ˙ 2 c 2 γ - r 2 θ ˙ 2 c 2 ) 1 2 .
d 2 u d θ 2 + u ( 1 - 6 G 2 M 2 b 2 c 2 h 2 ) = K ¯ .
m = m 0 ( 1 - r ˙ 2 c 2 - r 2 θ ˙ 2 c 2 ) 1 2
d 2 u d θ 2 + u ( 1 - G 2 M 2 c 2 h 2 ) = K ¯
r 2 θ ˙ ( γ - r ˙ 2 c 2 γ - r ˙ 2 θ ˙ 2 c 2 ) 1 2 = h ,
d d t ( r ˙ 2 + r 2 θ ˙ 2 ) ( 1 - r ˙ 2 c 2 γ 2 - r 2 θ ˙ 2 c 2 γ ) 1 2
( γ - r ˙ 2 c 2 γ - r 2 θ ˙ 2 c 2 ) 1 2 = γ b 1 2
γ = f ( G M , r ) r 2 θ ˙ ( γ - r ˙ 2 c 2 γ - r 2 θ ˙ 2 c 2 ) 1 2 = h ( γ - r ˙ 2 c 2 γ - r 2 θ ˙ 2 c 2 ) 1 2 = γ k