Part 8

# Considerations on Gravitation September 28, 2021

Every theory of gravitation has to deal with the problem of the influence, exerted on this force by the motion of the heavenly bodies. The solution is easily deduced from our equations; it takes the same form as the corresponding solution for the electromagnetic actions between charged particles.

I shall only treat the case of a body A, revolving around a central body M, this latter having a given constant velocity p. Let r be the line MA, taken in the direction from M towards A, x, y, z the relative coordinates of A with respect to M, w the velocity of A’s motion relatively to M, ϑ {\displaystyle \vartheta } {\displaystyle \vartheta } the angle between w and p, finally p r {\displaystyle p_{r}} {\displaystyle p_{r}} the component of p in the direction of r.

Then, besides the attraction k r 2 {\displaystyle {\frac {k}{r^{2}}}} {\displaystyle {\frac {k}{r^{2}}}}, (14)

which would exist if the bodies were both at rest, A will be subject to the following actions.

1st. A force k ⋅ p 2 2 V 2 ⋅ 1 r 2 {\displaystyle k\cdot {\frac {p^{2}}{2V^{2}}}\cdot {\frac {1}{r^{2}}}} {\displaystyle k\cdot {\frac {p^{2}}{2V^{2}}}\cdot {\frac {1}{r^{2}}}} (15)

in the direction of r.

2nd. A force whose components are − k 2 V 2 ∂ ∂ x ( p r 2 r ) , − k 2 V 2 ∂ ∂ y ( p r 2 r ) , − k 2 V 2 ∂ ∂ z ( p r 2 r ) . {\displaystyle -{\frac {k}{2V^{2}}}{\frac {\partial }{\partial x}}\left({\frac {p_{\mathsf {r}}^{2}}{r}}\right),\quad -{\frac {k}{2V^{2}}}{\frac {\partial }{\partial y}}\left({\frac {p_{\mathsf {r}}^{2}}{r}}\right),\quad -{\frac {k}{2V^{2}}}{\frac {\partial }{\partial z}}\left({\frac {p_{\mathsf {r}}^{2}}{r}}\right).} {\displaystyle -{\frac {k}{2V^{2}}}{\frac {\partial }{\partial x}}\left({\frac {p_{\mathsf {r}}^{2}}{r}}\right),\quad -{\frac {k}{2V^{2}}}{\frac {\partial }{\partial y}}\left({\frac {p_{\mathsf {r}}^{2}}{r}}\right),\quad -{\frac {k}{2V^{2}}}{\frac {\partial }{\partial z}}\left({\frac {p_{\mathsf {r}}^{2}}{r}}\right).}. (16) 3rd. A force − k V 2 p ⋅ 1 r 2 d r d t , {\displaystyle -{\frac {k}{V^{2}}}p\cdot {\frac {1}{r^{2}}}{\frac {dr}{dt}},} {\displaystyle -{\frac {k}{V^{2}}}p\cdot {\frac {1}{r^{2}}}{\frac {dr}{dt}},}, (17)

parallel to the velocity p.

4th. A force k V 2 1 r 2 p w cos ⁡ ϑ {\displaystyle {\frac {k}{V^{2}}}{\frac {1}{r^{2}}}p\ w\ \cos \ \vartheta } {\displaystyle {\frac {k}{V^{2}}}{\frac {1}{r^{2}}}p\ w\ \cos \ \vartheta }, (18)

in the direction of r.

Of these, (15) and (16) depend only on the common velocity p, ( 17) and (18) on the contrary, on p and w conjointly.

It is further to be remarked that the additional forces (15) — (18) are all of the second order with respect to the small quantities

p V {\displaystyle {\frac {p}{V}}} {\displaystyle {\frac {p}{V}}} and w V {\displaystyle {\frac {w}{V}}} {\displaystyle {\frac {w}{V}}}.

In so far, the law expressed by the above formulae presents a certain analog)- with the laws proposed by Weber, Riemann and Clausius for the electromagnetic actions, and applied by some astronomers to the motions of the planets. Like the formulae of Clausius, our equations contain the absolute velocities, i. e. the velocities, relatively to the aether.

There is no doubt but that, in the present state of science, if we wish to try for gravitation a similar law as for electromagnetic forces, the law contained in (15) — (18) is to be preferred to the three other just mentioned laws.

§ 9. The forces (15) — (18) will give rise to small inequalities in the elements of a planetary orbit ; in computing these, we have to take for p the velocity of the Sun’s motion through space. I have calculated the secular variations, using the formulae communicated by Tisserand in his Mécanique céleste. Let a be the mean distance to the sun, e the eccentricity, φ the inclination to the ecliptic, θ the longitude of the ascending node, ϖ {\displaystyle \varpi } {\displaystyle \varpi } the longitude of perihelion, ϰ ′ {\displaystyle \varkappa ‘} {\displaystyle \varkappa ‘} the mean anomaly at time t=0, in this sense that, if n be the mean motion, as determined by a, the mean anomaly at time t is given by

ϰ ′ + ∫ 0 t n d t {\displaystyle \varkappa ‘+\int \limits _{0}^{t}\ n\ dt} {\displaystyle \varkappa ‘+\int \limits _{0}^{t}\ n\ dt}.

Further, let λ, μ, and ν be the direction-cosines of the velocity p with respect to= 1st. the radius vector of the perihelion, 2nd. a direction which is got by giving to that radius vector a rotation of 90°, in the direction of the planet’s revolution, 3rd. the normal to the plane of the orbit, drawn towards the side whence the planet is seen to revolve in the same direction as the hands of a watch.

Put ω = ϖ − θ {\displaystyle \omega =\varpi -\theta } {\displaystyle \omega =\varpi -\theta }, p V = δ {\displaystyle {\frac {p}{V}}=\delta } {\displaystyle {\frac {p}{V}}=\delta } and n a V = δ ′ {\displaystyle {\frac {na}{V}}=\delta ‘} {\displaystyle {\frac {na}{V}}=\delta ‘} (na is the velocity in a circular orbit of radius a).

Then I find for the variations during one revolution Δ a = 0 Δ e = 2 π 1 − e 2 { λ μ δ 2 ( 2 − e 2 ) − 2 1 − e 2 e 3 − λ δ δ ′ 1 − 1 − e 2 e 2 } Δ φ = 2 π 1 − e 2 ν { [ − λ δ 2 cos ⁡ ω + δ ( e δ ′ − μ δ ) sin ⁡ ω ] 1 − 1 − e 2 e 2 + μ δ 2 sin ⁡ ω } Δ θ = − 2 π 1 − e 2 sin ⁡ φ ν { [ λ δ 2 sin ⁡ ω + δ ( e δ ′ − μ δ ) cos ⁡ ω ] 1 − 1 − e 2 e 2 + μ δ 2 cos ⁡ ω } Δ ϖ = π ( μ 2 − λ 2 ) δ 2 ( 2 − e 2 − 2 1 − e 2 ) e 4 + 2 π μ δ δ ′ 1 − e 2 − 1 e 3 − − 2 π t g 1 2 φ 1 − e 2 ν { [ λ δ 2 sin ⁡ ω + δ ( e δ ′ − μ δ ) cos ⁡ ω ] 1 − 1 − e 2 e 2 + μ δ 2 cos ⁡ ω } Δ ϰ ′ = π ( λ 2 − μ 2 ) δ 2 ( 2 + e 2 ) 1 − e 2 − 2 e 4 − 2 π δ 2 − 2 π μ 2 δ 2 − 2 π μ δ δ ′ ( 1 − e 2 ) − 1 − e 2 e 3 . {\displaystyle {\begin{array}{l}\Delta a=0\\\Delta e=2\pi {\sqrt {1-e^{2}}}\left{\lambda \mu \delta ^{2}{\frac {\left(2-e^{2}\right)-2{\sqrt {1-e^{2}}}}{e^{3}}}-\lambda \delta \delta ‘{\frac {1-{\sqrt {1-e^{2}}}}{e^{2}}}\right}\\\Delta \varphi ={\frac {2\pi }{\sqrt {1-e^{2}}}}\nu \left{\left[-\lambda \delta ^{2}\cos \omega +\delta \left(e\delta ‘-\mu \delta \right)\sin \omega \right]{\frac {1-{\sqrt {1-e^{2}}}}{e^{2}}}+\mu \delta ^{2}\sin \omega \right}\\\Delta \theta =-{\frac {2\pi }{{\sqrt {1-e^{2}}}\sin \varphi }}\nu \left{\left[\lambda \delta ^{2}\sin \omega +\delta \left(e\delta ‘-\mu \delta \right)\cos \omega \right]{\frac {1-{\sqrt {1-e^{2}}}}{e^{2}}}+\mu \delta ^{2}\cos \omega \right}\\\Delta \varpi =\pi \left(\mu ^{2}-\lambda ^{2}\right)\delta ^{2}{\frac {\left(2-e^{2}-2{\sqrt {1-e^{2}}}\right)}{e^{4}}}+2\pi \mu \delta \delta ‘{\frac {{\sqrt {1-e^{2}}}-1}{e^{3}}}-\\\qquad -{\frac {2\pi \ tg\ {\frac {1}{2}}\varphi }{\sqrt {1-e^{2}}}}\nu \left{\left[\lambda \delta ^{2}\sin \omega +\delta \left(e\delta ‘-\mu \delta \right)\cos \omega \right]{\frac {1-{\sqrt {1-e^{2}}}}{e^{2}}}+\mu \delta ^{2}\cos \omega \right}\\\Delta \varkappa ‘=\pi \left(\lambda ^{2}-\mu ^{2}\right)\delta ^{2}{\frac {\left(2+e^{2}\right){\sqrt {1-e^{2}}}-2}{e^{4}}}-2\pi \delta ^{2}-2\pi \mu ^{2}\delta ^{2}-2\pi \mu \delta \delta ‘{\frac {\left(1-e^{2}\right)-{\sqrt {1-e^{2}}}}{e^{3}}}.\end{array}}} {\displaystyle {\begin{array}{l}\Delta a=0\\\Delta e=2\pi {\sqrt {1-e^{2}}}\left{\lambda \mu \delta ^{2}{\frac {\left(2-e^{2}\right)-2{\sqrt {1-e^{2}}}}{e^{3}}}-\lambda \delta \delta ‘{\frac {1-{\sqrt {1-e^{2}}}}{e^{2}}}\right}\\\Delta \varphi ={\frac {2\pi }{\sqrt {1-e^{2}}}}\nu \left{\left[-\lambda \delta ^{2}\cos \omega +\delta \left(e\delta ‘-\mu \delta \right)\sin \omega \right]{\frac {1-{\sqrt {1-e^{2}}}}{e^{2}}}+\mu \delta ^{2}\sin \omega \right}\\\Delta \theta =-{\frac {2\pi }{{\sqrt {1-e^{2}}}\sin \varphi }}\nu \left{\left[\lambda \delta ^{2}\sin \omega +\delta \left(e\delta ‘-\mu \delta \right)\cos \omega \right]{\frac {1-{\sqrt {1-e^{2}}}}{e^{2}}}+\mu \delta ^{2}\cos \omega \right}\\\Delta \varpi =\pi \left(\mu ^{2}-\lambda ^{2}\right)\delta ^{2}{\frac {\left(2-e^{2}-2{\sqrt {1-e^{2}}}\right)}{e^{4}}}+2\pi \mu \delta \delta ‘{\frac {{\sqrt {1-e^{2}}}-1}{e^{3}}}-\\\qquad -{\frac {2\pi \ tg\ {\frac {1}{2}}\varphi }{\sqrt {1-e^{2}}}}\nu \left{\left[\lambda \delta ^{2}\sin \omega +\delta \left(e\delta ‘-\mu \delta \right)\cos \omega \right]{\frac {1-{\sqrt {1-e^{2}}}}{e^{2}}}+\mu \delta ^{2}\cos \omega \right}\\\Delta \varkappa ‘=\pi \left(\lambda ^{2}-\mu ^{2}\right)\delta ^{2}{\frac {\left(2+e^{2}\right){\sqrt {1-e^{2}}}-2}{e^{4}}}-2\pi \delta ^{2}-2\pi \mu ^{2}\delta ^{2}-2\pi \mu \delta \delta ‘{\frac {\left(1-e^{2}\right)-{\sqrt {1-e^{2}}}}{e^{3}}}.\end{array}}}

§ 10. I have worked out the case of the planet Mercury, taking 276° and + 34° for the right ascension and declination of the apex of the Sun’s motion. I have got the following results: Δ a = 0 Δ e = 0 , 018 δ 2 + 1 , 38 δ δ ′ Δ φ = 0 , 95 δ 2 + 0 , 28 δ δ ′ Δ θ = 7 , 60 δ 2 − 4 , 26 δ δ ′ Δ ϖ = − 0 , 09 δ 2 + 1 , 95 δ δ ′ Δ ϰ ′ = − 6 , 82 δ 2 − 1 , 93 δ δ ′ {\displaystyle {\begin{array}{ll}\Delta a=&0\\\Delta e=&0,018\ \delta ^{2}+1,38\ \delta \delta ‘\\\Delta \varphi =&0,95\ \delta ^{2}+0,28\ \delta \delta ‘\\\Delta \theta =&7,60\ \delta ^{2}-4,26\ \delta \delta ‘\\\Delta \varpi =-&0,09\ \delta ^{2}+1,95\ \delta \delta ‘\\\Delta \varkappa ‘=-&6,82\ \delta ^{2}-1,93\ \delta \delta ‘\end{array}}} {\displaystyle {\begin{array}{ll}\Delta a=&0\\\Delta e=&0,018\ \delta ^{2}+1,38\ \delta \delta ‘\\\Delta \varphi =&0,95\ \delta ^{2}+0,28\ \delta \delta ‘\\\Delta \theta =&7,60\ \delta ^{2}-4,26\ \delta \delta ‘\\\Delta \varpi =-&0,09\ \delta ^{2}+1,95\ \delta \delta ‘\\\Delta \varkappa ‘=-&6,82\ \delta ^{2}-1,93\ \delta \delta ‘\end{array}}}

Now, δ ′ = 1 , 6 × 10 − 4 {\displaystyle \delta ‘=1,6\ \times \ 10^{-4}} {\displaystyle \delta ‘=1,6\ \times \ 10^{-4}} and, if we put δ = 5 , 3 × 10 − 5 {\displaystyle \delta =5,3\ \times \ 10^{-5}} {\displaystyle \delta =5,3\ \times \ 10^{-5}}, we get Δ e = 117 × 10 − 10 {\displaystyle \Delta e=117\ \times \ 10^{-10}} {\displaystyle \Delta e=117\ \times \ 10^{-10}}, Δ φ = 51 × 10 − 10 , {\displaystyle \Delta \varphi =51\ \times \ 10^{-10},} {\displaystyle \Delta \varphi =51\ \times \ 10^{-10},}

Δ θ = − 137 × 10 − 10 , Δ ϖ = 162 × 10 − 10 , Δ ϰ ′ = 355 × 10 − 10 {\displaystyle \Delta \theta =-137\ \times \ 10^{-10},\ \Delta \varpi =162\ \times \ 10^{-10},\ \Delta \varkappa ‘=355\ \times \ 10^{-10}} {\displaystyle \Delta \theta =-137\ \times \ 10^{-10},\ \Delta \varpi =162\ \times \ 10^{-10},\ \Delta \varkappa ‘=355\ \times \ 10^{-10}}.

The changes that take place in a century are found from these numbers, if we multiply them by 415, and, if the variations of φ, θ, ϖ {\displaystyle \varpi } {\displaystyle \varpi }, and ϰ ′ {\displaystyle \varkappa ‘} {\displaystyle \varkappa ‘} are to be expressed in seconds, we have to introduce the factor 2 , 06 × 10 5 {\displaystyle 2,06\times 10^{5}} {\displaystyle 2,06\times 10^{5}}. The result is, that the changes in φ, θ, ϖ {\displaystyle \varpi } {\displaystyle \varpi }, and ϰ ′ {\displaystyle \varkappa ‘} {\displaystyle \varkappa ‘} amount to a few seconds, and that in e to 0,000005.

Hence we conclude that our modification of Newton’s law cannot account for the observed inequality in the longitude of the perihelion — as Weber’s law can to some extent do — but that, if we do not pretend to explain this inequality by an alteration of the law of attraction, there is nothing against the proposed formulae. Of course it will be necessary to apply them to other heavenly bodies, though it seems scarcely probable that there will be found any case in which the additional terms have an appreciable influence.

The special form of these terms may perhaps be modified. Yet, what has been said is sufficient to show that gravitation may be attributed to actions which are propagated with no greater velocity than that of light.

As is well known, Laplace has been the first to discuss this question of the velocity of propagation of universal attraction, and later astronomers have often treated the same problem. Let a body B be attracted by a body A, moving with the velocity p. Then, if the action is propagated with a finite velocity V, the influence which reaches B at time t, will have been emitted by A at an anterior moment, say t—τ. Let A1 be the position of the acting body at this moment, A2 that at time t. It is an easy matter to calculate the distance between these positions. Now, if the action at time t is calculated, as if A had continued to occupy the position A1, one is led to an influence on the astronomical motions of the order p V {\displaystyle {\tfrac {p}{V}}} {\displaystyle {\tfrac {p}{V}}}; if V were equal to the velocity of light, this influence would be much greater than observations permit us to suppose. If, on the contrary, the terms with p V {\displaystyle {\tfrac {p}{V}}} {\displaystyle {\tfrac {p}{V}}} are to have admissible values, V ought to be many millions of times as great as the velocity of light.

From the considerations in this paper, it appears that this conclusion can be avoided. Changes of state in the aether, satisfying equations of the form (I), are propagated with the velocity V; yet, no quantities of the first order p V {\displaystyle {\tfrac {p}{V}}} {\displaystyle {\tfrac {p}{V}}} or w V {\displaystyle {\tfrac {w}{V}}} {\displaystyle {\tfrac {w}{V}}} (§ 8), but only terms containing p 2 V 2 {\displaystyle {\tfrac {p^{2}}{V^{2}}}} {\displaystyle {\tfrac {p^{2}}{V^{2}}}} and p w V 2 {\displaystyle {\tfrac {pw}{V^{2}}}} {\displaystyle {\tfrac {pw}{V^{2}}}} appear in the results. This is brought about by the peculiar way—determined by the equations—in which moving matter changes the state of the aether; in the above mentioned case the condition of the aether will not be what it would have been, if the acting body were at rest in the position A1.

Lorentz. La théorie electromagnetique de Maxwell et son application aux corps mouvants, Arch. Néerl. XXV, p. 363; Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern. D i v d = ∂ d x ∂ x + ∂ d y ∂ y + ∂ d z ∂ z {\displaystyle Div\ {\mathfrak {d}}={\frac {\partial {\mathfrak {d}}{\mathsf {x}}}{\partial x}}+{\frac {\partial {\mathfrak {d}}{\mathsf {y}}}{\partial y}}+{\frac {\partial {\mathfrak {d}}{\mathsf {z}}}{\partial z}}} {\displaystyle Div\ {\mathfrak {d}}={\frac {\partial {\mathfrak {d}}{\mathsf {x}}}{\partial x}}+{\frac {\partial {\mathfrak {d}}{\mathsf {y}}}{\partial y}}+{\frac {\partial {\mathfrak {d}}{\mathsf {z}}}{\partial z}}} R o t d {\displaystyle Rot\ {\mathfrak {d}}} {\displaystyle Rot\ {\mathfrak {d}}} is a vector, whose components are ∂ d z ∂ y − ∂ d y ∂ z {\displaystyle {\frac {\partial {\mathfrak {d}}{\mathsf {z}}}{\partial y}}-{\frac {\partial {\mathfrak {d}}{\mathsf {y}}}{\partial z}}} {\displaystyle {\frac {\partial {\mathfrak {d}}{\mathsf {z}}}{\partial y}}-{\frac {\partial {\mathfrak {d}}{\mathsf {y}}}{\partial z}}} etc. [ v . H ] {\displaystyle [{\mathfrak {v.H}}]} {\displaystyle [{\mathfrak {v.H}}]} is the vector-product of v {\displaystyle {\mathfrak {v}}} {\displaystyle {\mathfrak {v}}} and H {\displaystyle {\mathfrak {H}}} {\displaystyle {\mathfrak {H}}}. See the second of the above mentioned papers.