As Newton's law is proportional to the experimental mass, we are tempted to conclude that there is some relation between the cause that generates gravitation and the one that generates the additional potential.
§ 9. — Hypotheses on gravitation
Thus Lorentz's theory would completely explain the impossibility to demonstrate absolute motion, if all forces are of electromagnetic origin.
But there are forces which we can not assign an electromagnetic origin, as for example gravitation. It could happen, indeed, that two systems of bodies produce equivalent electromagnetic fields, that is to say, exerting the same action on the electrified bodies and on the currents, and yet these two systems do not exercise the same gravitational action on the Newtonian mass. The gravitational field is thus distinct from the electromagnetic field. Lorentz was thus forced to complete his hypothesis by assuming that forces of any origin, and in particular gravitation, are affected by translation (or, if preferred, by the Lorentz transformation) the same way as electromagnetic forces.
It is now convenient to enter into details and look more closely at this hypothesis. If we want that the Newtonian force is affected in this way by the Lorentz transformation, we can not accept that the force depends only on the relative position of the attracting body and of the body attracted at the instant considered. It will also depend on the velocities of the two bodies. And that's not all: it is natural to assume that the force acting at time t on the attracted body, depends on the position and velocity of this body at the same time t; but it will depend, in addition, on the position and velocity of the attracting body, not at time t, but a moment earlier, as if gravitation needs a certain time to propagate.
Consider therefore the position of the attracted body at the instant t0 and, at this point, x0, y0, z0 are the coordinates, ξ, η, ζ the components of its velocity; consider the other attracting body at the corresponding time t0 + t and, at this point, x0 + x, y0 + y, z0 + z are the coordinates, ξ1, η1, ζ1 the components of its velocity.
We must first have a relationship
| (1) |
to define the time t. This relation will define the law of propagation of the gravitational action (I do not impose on me the condition that the propagation takes place with the same speed in all directions).
Now let X1, Y1, Z1 the 3 components of the action exerted at time t0 on the body; we have to express X1, Y1, Z1 as functions of
| (2) |
What are the conditions to fulfill?
