Page:PoincareDynamiqueJuillet.djvu/39

1° The condition (1) shall not be altered by transformations of the Lorentz group.

2° The components X₁, Y₁, Z₁ will be affected by the Lorentz transformations the same way as electromagnetic forces designated by the same letters, that is to say, according to equations (11^bis) of § 1.

3° When two bodies are at rest, we will fall back to the ordinary law of attraction.

It is important to note that in the latter case, the relation (1) disappears, because time does not play any role if the two bodies are at rest.

The problem thus posed is obviously undetermined. We will thus seek to satisfy as many as possible other additional conditions:

4° Astronomical observations do not appear to show significant derogation to Newton's law, we will choose the solution that deviates the least of this law, for low velocities of two bodies.

5° We will endeavor to arrange that T is always negative; if indeed it is conceived that the effect of gravitation takes a certain time to be propagated, it would be more difficult to understand how this effect could depend on the position not yet attained by the attracting body.

There is one case where the indeterminacy of the problem disappears; it is where the two bodies are at rest relative to each other, that is to say that:

$\xi =\xi _{1},\ \eta =\eta _{1},\ \zeta =\zeta _{1}\,$;

this is the case we will consider first, assuming that these velocities are constant, so that the two bodies are drawn into a common translational motion, rectilinear and uniform.

We can assume that the axis of x has been taken parallel to the translation, so that η = ζ = 0, and we take ε = -ξ.

If in these circumstances we apply the Lorentz transformation, after the transformation the two bodies are at rest and we have:

$\xi '=\eta '=\zeta '=0\,$

Then the components x'₀, Y'₀, Z'₀ must conform to Newton's law and we will have a constant factor:

(3)	${\begin{cases}X_{1}^{\prime }=-{\frac {x^{\prime }}{r^{\prime 3}}},\quad Y_{1}^{\prime }=-{\frac {y^{\prime }}{r^{\prime 3}}},\quad Z_{1}^{\prime }=-{\frac {z^{\prime }}{r^{\prime 3}}},\\\\r^{\prime 2}=x^{\prime 2}=y^{\prime 2}+z^{\prime 2}.\end{cases}}$

But we have, according to § 1:

$x^{\prime }=k(x+\epsilon t),\quad y^{\prime }=y,\quad z^{\prime }=z,\quad t^{\prime }=k(t+\epsilon x),$
${\frac {\rho ^{\prime }}{\rho }}=k(1+\xi \epsilon )=k\left(1-\epsilon ^{2}\right)={\frac {1}{k}},\quad \sum X_{1}\xi =-X_{1}\epsilon ,$
$X_{1}^{\prime }=k{\frac {\rho }{\rho ^{\prime }}}\left(X_{1}+\epsilon \sum X_{1}\xi \right)=k^{2}X_{1}\left(1-\epsilon ^{2}\right)=X_{1},$
$Y_{1}^{\prime }={\frac {\rho }{\rho ^{\prime }}}Y_{1}=kY_{1},$
$Z_{1}^{\prime }=kZ_{1}.$