(66) K_{σ} = ∂τ_{σ}^ν/∂_x__{ν} - ½ _g_^{τμ} ∂_g__{μν}/∂_x__{σ}
τ_{τ}^ν
where
(66a) τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{αβ}.
On account of (30) the equation (66) becomes equivalent to (57) and
(57a) when K_{σ} vanishes. Thus τ_{σ}^ν’s are the energy-components of
the electro-magnetic field. With the help of (61) and (64) we can easily
show that the energy-components of the electro-magnetic field, in the
case of the special relativity theory, give rise to the well-known
Maxwell-Poynting expressions.
We have now deduced the most general laws which the gravitation-field
and matter satisfy when we use a co-ordinate system for which √(-_g_) =
1. Thereby we achieve an important simplification in all our formulas
and calculations, without renouncing the conditions of general
covariance, as we have obtained the equations through a specialisation
of the co-ordinate system from the general covariant-equations. Still
the question is not without formal interest, whether, when the
energy-components of the gravitation-field and matter is defined in a
generalised manner without any specialisation of co-ordinates, the laws
of conservation have the form of the equation (56), and the
field-equations of gravitation hold in the form (52) or (52a); such that
on the left-hand side, we have a divergence in the usual sense, and on
the right-hand side, the sum of the energy-components of matter and
gravitation. I have found out that this is indeed the case. But I am of
opinion that the communication of my rather comprehensive work on this
subject will not pay, for nothing essentially new comes out of it.
E. §21. Newton’s theory as a first approximation.
We have already mentioned several times that the special relativity
theory is to be looked upon as a special case of the general, in which
_g__{μν}’s have constant values (4). This signifies, according to what
has been said before, a total neglect of the influence of gravitation.
We get one important approximation if we consider the case when
_g__{μν}’s differ from (4) only by small magnitudes (compared to 1)
where we can neglect small quantities of the second and higher orders
(first aspect of the approximation.)
Further it should be assumed that within the space-time region
considered, _g__{μν}’s at infinite distances (using the word infinite in
a spatial sense) can, by a suitable choice of co-ordinates, tend to the
limiting values (4); _i.e._, we consider only those gravitational fields
which can be regarded as produced by masses distributed over finite
regions.
We can assume that this approximation should lead to Newton’s theory.
For it however, it is necessary to treat the fundamental equations from
another point of view. Let us consider the motion of a particle
according to the equation (46). In the case of the special relativity
theory, the components
_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
can take any values. This signifies that any velocity
_v_ = √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²)
can appear which is less than the velocity of light in vacuum (_v_ < 1).
If we finally limit ourselves to the consideration of the case when _v_
is small compared to the velocity of light, it signifies that the
components
_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
can be treated as small quantities, whereas _dx₄_/_ds_ is equal to 1, up
to the second-order magnitudes (the second point of view for
approximation).
Now we see that, according to the first view of approximation, the
magnitudes γ_{μν}^τ’s are all small quantities of at least the first
order. A glance at (46) will also show, that in this equation according
to the second view of approximation, we are only to take into account
those terms for which μ = ν = 4.
By limiting ourselves only to terms of the lowest order we get instead
of (46), first, the equations:—
_d²__x__{τ}/_dt²_ = Γ₄₄^τ, where _ds_ = _dx₄_ = _dt_,
or by limiting ourselves only to those terms which according to the
first stand-point are approximations of the first order,
It must be admitted, that this introduction of the energy-tensor of
matter cannot be justified by means of the Relativity-Postulate alone;
for we have in the foregoing analysis deduced it from the condition that
the energy of the gravitation-field should exert gravitating action in
the same way as every other kind of energy. The strongest ground for the
choice of the above equation however lies in this, that they lead, as
their consequences, to equations expressing the conservation of the
components of total energy (the impulses and the energy) which exactly
correspond to the equations (49) and (49a). This shall be shown
afterwards.
§17. The laws of conservation in the general case.
The equations (52) can be easily so transformed that the second member
on the right-hand side vanishes. We reduce (52) with reference to the
indices μ and σ and subtract the equation so obtained after
multiplication with ½ δ_{μ}^σ from (52).
We obtain,
(52a) ∂/∂_x__{α}(_g_^{σβ} Γ_{μβ}^α - ½ δ_{μ}^σ _g_^{λβ} Γ_{λβ}^α)
= -κ(_t__{μ}^σ + T_{μ}^σ)
we operate on it by ∂/∂_x__{σ}. Now,
∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}Γ_{μβ}^α)
= -½ ∂²/∂_x__{α}∂_x__{σ} [_g_^{σβ} _g_^{αλ}(∂_g__{μλ}/∂_x__{β}
+ ∂_g__{βλ}/∂_x__{μ} - ∂_g__{μβ}/∂_x__{λ})].
The first and the third member of the round bracket lead to expressions
which cancel one another, as can be easily seen by interchanging the
summation-indices α, and σ, on the one hand, and β and λ, on the other.
The second term can be transformed according to (31). So that we get,
(54) ∂²/∂_x__{α}∂_x__{σ} (_g_^{σβ}γ_{μβ}^α)
= ½ ∂³_g_^{αβ}/∂_x__{σ}∂_x__{β}∂_x__{μ}
The second member of the expression on the left-hand side of (52a) leads
first to
- ½ ∂²/∂_x__{α}∂_x__{μ} (_g_^{λβ}Γ_{λβ}^α) or
to 1/4 ∂²/∂_x__{α}∂_x__{μ} [_g_^{λβ}_g_^{αδ}( ∂_g__{δλ}/∂_x__{β}
+ ∂_g__{δβ}/∂_x__{λ} - ∂_g__{λβ}/∂_x__{δ})].
The expression arising out of the last member within the round bracket
vanishes according to (29) on account of the choice of axes. The two
others can be taken together and give us on account of (31), the
expression
-½ ∂³_g_^{αβ}/∂_x__{α}∂_x__{β}∂_x__{μ}