Can we ascribe gravity to a certain state of the aether?

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

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 easy 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.

See the second of the above mentioned papers.