should have to ascribe a certain negative value of the energy to a field without gravitation, in such a way (comp. § 57) that the energy in the shell between the spheres described round the origin with radii
r
{\displaystyle r}
r
+
d
r
{\displaystyle r+dr}
−
4
π
c
ϰ
d
r
{\displaystyle -{\frac {4\pi c}{\varkappa }}dr}
The density of the energy in the ordinary sense of the word would be inversely proportional to
r
2
{\displaystyle r^{2}}
It is hardly necessary to remark that, using rectangular coordinates we find a value zero for the same case of a field without gravitation. The normal values of
g
a
b
{\displaystyle g_{ab}}
t
4
′
4
{\displaystyle {\mathfrak {t}}_{4}^{'4}}
g
11
=
−
(
1
+
λ
)
+
x
1
2
r
2
(
λ
−
μ
)
,
e
t
c
.
g
12
=
x
1
x
2
r
2
(
λ
−
μ
)
,
e
t
c
.
g
14
=
g
24
=
g
34
=
0
,
g
44
=
c
2
(
1
+
ν
)
}
{\displaystyle \left.{\begin{array}{l}g_{11}=-(1+\lambda )+{\frac {x_{1}^{2}}{r^{2}}}(\lambda -\mu ),\ etc.\\\\g_{12}={\frac {x_{1}x_{2}}{r^{2}}}(\lambda -\mu ),\ etc.\\\\g_{14}=g_{24}=g_{34}=0,\ g_{44}=c^{2}(1+\nu )\end{array}}\right\}}
(110)
By (109) and (110) we find[ 1]
↑ Of the laborious calculation it may be remarked here only that it is convenient to write the values (110) in the form
g
11
=
−
1
+
α
+
∂
2
β
∂
x
1
2
,
e
t
c
.
{\displaystyle g_{11}=-1+\alpha +{\frac {\partial ^{2}\beta }{\partial x_{1}^{2}}},\ etc.}
g
12
=
∂
2
β
∂
x
1
∂
x
2
,
e
t
c
.
{\displaystyle g_{12}={\frac {\partial ^{2}\beta }{\partial x_{1}\partial x_{2}}},\ etc.}
where
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
r
{\displaystyle r}
t
4
′
4
=
c
2
ϰ
{
−
1
2
∑
(
a
)
(
∂
α
∂
x
a
)
2
+
∑
(
a
)
∂
ν
∂
x
a
∂
α
∂
x
a
+
+
1
4
∑
(
a
i
k
)
[
∂
3
β
∂
x
a
∂
x
i
2
∂
3
β
∂
x
a
∂
x
k
2
−
(
∂
3
β
∂
x
a
∂
x
i
∂
x
k
)
2
]
}
(
a
,
i
,
k
=
1
,
2
,
3
)
{\displaystyle {\begin{array}{l}{\mathfrak {t}}_{4}^{'4}={\frac {c}{2\varkappa }}\left\{-{\frac {1}{2}}\sum (a)\left({\frac {\partial \alpha }{\partial x_{a}}}\right)^{2}+\sum (a){\frac {\partial \nu }{\partial x_{a}}}{\frac {\partial \alpha }{\partial x_{a}}}+\right.\\\\\qquad \left.+{\frac {1}{4}}\sum (aik)\left[{\frac {\partial ^{3}\beta }{\partial x_{a}\partial x_{i}^{2}}}{\frac {\partial ^{3}\beta }{\partial x_{a}\partial x_{k}^{2}}}-\left({\frac {\partial ^{3}\beta }{\partial x_{a}\partial x_{i}\partial x_{k}}}\right)^{2}\right]\right\}\\\\\qquad \qquad (a,i,k=1,2,3)\end{array}}}
α
,
β
{\displaystyle \alpha ,\beta }
γ
,
μ
{\displaystyle \gamma ,\mu }
α
+
1
r
β
′
=
−
λ
,
−
1
r
β
′
+
β
″
=
λ
−
μ
{\displaystyle \alpha +{\frac {1}{r}}\beta '=-\lambda ,\ -{\frac {1}{r}}\beta '+\beta ''=\lambda -\mu }
and the equality
α
′
=
ν
′
{\displaystyle \alpha '=\nu '}
![{\displaystyle {\begin{array}{l}{\mathfrak {t}}_{4}^{'4}={\frac {c}{2\varkappa }}\left\{-{\frac {1}{2}}\sum (a)\left({\frac {\partial \alpha }{\partial x_{a}}}\right)^{2}+\sum (a){\frac {\partial \nu }{\partial x_{a}}}{\frac {\partial \alpha }{\partial x_{a}}}+\right.\\\\\qquad \left.+{\frac {1}{4}}\sum (aik)\left[{\frac {\partial ^{3}\beta }{\partial x_{a}\partial x_{i}^{2}}}{\frac {\partial ^{3}\beta }{\partial x_{a}\partial x_{k}^{2}}}-\left({\frac {\partial ^{3}\beta }{\partial x_{a}\partial x_{i}\partial x_{k}}}\right)^{2}\right]\right\}\\\\\qquad \qquad (a,i,k=1,2,3)\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0c1b647eb57f81283195f8ac2edf1c7267fb95)
