§13. Equation of motion of a material point in a gravitation-field.
Expression for the field-components of gravitation.
A freely moving body not acted on by external forces moves, according to
the special relativity theory, along a straight line and uniformly. This
also holds for the generalised relativity theory for any part of the
four-dimensional region, in which the co-ordinates K_{0} can be, and
are, so chosen that _g__{μν}’s have special constant values of the
expression (4).
Let us discuss this motion from the stand-point of any arbitrary
co-ordinate-system K₁; it moves with reference to K₁ (as explained in
§2) in a gravitational field. The laws of motion with reference to K₁
follow easily from the following consideration. With reference to K₀,
the law of motion is a four-dimensional straight line and thus a
geodesic. As a geodetic-line is defined independently of the system of
co-ordinates, it would also be the law of motion for the motion of the
material-point with reference to K₁. If we put
(45) $$ \Gamma^{\tau}_{\mu\nu} = - \begin{Bmatrix}\mu & &
\nu\ \tau&\end{Bmatrix} $$
we get the motion of the point with reference to K₁, given by
(46) $$ \frac{d^2 x_{\tau}}{ds^2} = \Gamma^{\tau}_{\mu\nu}
\frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} $$
We now make the very simple assumption that this general covariant
system of equations defines also the motion of the point in the
gravitational field, when there exists no reference-system K₀, with
reference to which the special relativity theory holds throughout a
finite region. The assumption seems to us to be all the more legitimate,
as (46) contains only the first differentials of _g__{μν}, among which
there is no relation in the special case when K₀ exists.
If γ_{μν}^{τ}’s vanish, the point moves uniformly and in a straight
line; these magnitudes therefore determine the deviation from
uniformity. They are the components of the gravitational field.
§14. The Field-equation of Gravitation in the absence of matter.
In the following, we differentiate gravitation-field from matter in the
sense that everything besides the gravitation-field will be signified as
matter; therefore the term includes not only matter in the usual sense,
but also the electro-dynamic field. Our next problem is to seek the
field-equations of gravitation in the absence of matter. For this we
apply the same method as employed in the foregoing paragraph for the
deduction of the equations of motion for material points. A special case
in which the field-equations sought-for are evidently satisfied is that
of the special relativity theory in which _g__{μν}’s have certain
constant values. This would be the case in a certain finite region with
reference to a definite co-ordinate system K₀. With reference to this
system, all the components B^{ρ}_{μστ} of the Riemann’s Tensor [equation
43] vanish. These vanish then also in the region considered, with
reference to every other co-ordinate system.
The equations of the gravitation-field free from matter must thus be in
every case satisfied when all B^{ρ}_{μστ} vanish. But this condition is
clearly one which goes too far. For it is clear that the
gravitation-field generated by a material point in its own neighbourhood
can never be transformed _away_ by any choice of axes, _i.e._, it cannot
be transformed to a case of constant _g__{μν}’s.
Therefore it is clear that, for a gravitational field free from matter,
it is desirable that the symmetrical tensors B_{μν} deduced from the
tensors B^{ρ}_{μστ} should vanish. We thus get 10 equations for 10
quantities _g__{μν} which are fulfilled in the special case when
B^{ρ}_{μστ}’s all vanish.
Remembering (44) we see that in absence of matter the field-equations
come out as follows; (when referred to the special co-ordinate-system
chosen.)
(47) $$ \frac{\partial \Gamma^{\alpha}_{\mu\nu}}{\partial x_{\alpha}} +
\Gamma^{\alpha}_{\mu\beta} \Gamma^{\beta}_{\mu\alpha} = 0 $$
$$ \sqrt{-g} = 1 $$
$$ \Gamma^{\alpha}_{\mu\nu} = - \begin{Bmatrix}\mu & &
\nu\ \alpha&\end{Bmatrix} $$
It can also be shown that the choice of these equations is connected
with a minimum of arbitrariness. For besides B_{μν}, there is no tensor
of the second rank, which can be built out of _g__{μν}’s and their
derivatives no higher than the second, and which is also linear in them.
It will be shown that the equations arising in a purely mathematical way
out of the conditions of the general relativity, together with equations
(46), give us the Newtonian law of attraction as a first approximation,
and lead in the second approximation to the explanation of the
perihelion-motion of mercury discovered by Leverrier (the residual
effect which could not be accounted for by the consideration of all
sorts of disturbing factors). My view is that these are convincing
proofs of the physical correctness of my theory.
§15. Hamiltonian Function for the Gravitation-field.
Laws of Impulse and Energy.
In order to show that the field equations correspond to the laws of
impulse and energy, it is most convenient to write it in the following
Hamiltonian form:—
(47a)
δ∫ H_d_τ = 0