§ 32. In the two preceding papers[20] we have tried so far as possible to present the fundamental principles of the new gravitation theory in a simple form.

We shall now show how Einstein’s differential equations for the gravitation field can be derived from Hamilton’s principle. In this connexion we shall also have to consider the energy, the stresses, momenta and energy-currents in that field.

We shall again introduce the quantities � � � {\displaystyle g_{ab}} formerly used and we shall also use the “inverse” system of quantities for which we shall now write � � � {\displaystyle g^{ab}}. It is found useful to introduce besides these the quantities

� � �

− � � � � {\displaystyle {\mathfrak {g}}^{ab}={\sqrt {-g}}g^{ab}}

Differential coefficients of all these variables with respect to the coordinates will be represented by the indices belonging to these latter, e.g.

� � � , �

∂ � � � ∂ � � ,

� � � , � �

∂ 2 � � � ∂ � � ∂ � � {\displaystyle g_{ab,p}={\frac {\partial g_{ab}}{\partial x_{p}}},\ g_{ab,pq}={\frac {\partial ^{2}g_{ab}}{\partial x_{q}\partial x_{p}}}}

We shall use Christoffel’s symbols

[ � � � ]

1 2 ( � � � , � + � � � , � − � � � , � ) {\displaystyle \left[{\begin{array}{c}ab\c\end{array}}\right]={\frac {1}{2}}\left(g_{ac,b}+g_{bc,a}-g_{ab,c}\right)}

and Riemann’s symbol

( � � , � � )

1 2 ( � � � , � � + � � � , � � − � � � , � � − � � � , � � ) + + ∑ ( � � ) � � � { [ � � � ] [ � � � ] − [ � � � ] [ � � � ] } {\displaystyle {\begin{array}{l}(ik,lm)={\frac {1}{2}}\left(g_{im,lk}+g_{kl,im}-g_{il,km}-g_{km,il}\right)+\\\qquad +\sum (ab)g^{ab}\left{\left[{\begin{array}{c}im\a\end{array}}\right]\left[{\begin{array}{c}kl\b\end{array}}\right]-\left[{\begin{array}{c}il\a\end{array}}\right]\left[{\begin{array}{c}km\b\end{array}}\right]\right}\end{array}}}

Further we put

� � �

∑ ( � � ) � � � ( � � , � � ) {\displaystyle G_{im}=\sum (kl)g^{kl}(ik,lm)} (40) �

∑ ( � � ) � � � � � � {\displaystyle G=\sum (im)g^{im}G_{im}} (41) This latter quantity is a measure for the curvature of the field-figure. The principal function of the gravitation field is ​ 1 2 � ∫ � � � {\displaystyle {\frac {1}{2\varkappa }}\int QdS}

where

�

− � � {\displaystyle Q={\sqrt {-g}}G}

In the integral � � {\displaystyle dS}, the element of the field-figure, is expressed in � {\displaystyle x}-units. The integration has to be extended over the domain within a certain closed surface �{\displaystyle \sigma }; �{\displaystyle \varkappa } is a positive constant.

§ 33. When we pass from the system of coordinates � 1 , … � 4 {\displaystyle x_{1},\dots x_{4}} to another, the value of � {\displaystyle G} proves to remain unaltered; it is a scalar quantity. This may be verified by first proving that the quantities � � , � � {\displaystyle ik,lm} form a covariant tensor of the fourth order[21]. Next, � � � {\displaystyle g^{kl}} being a contravariant tensor of the second order[22], we can deduce from (40) that ( � � � ) {\displaystyle \left(G_{im}\right)} is a covariant tensor of the same order[23]. According to (41) � {\displaystyle G} is then a scalar. The same is true[24] for � � � {\displaystyle QdS}.

We remark that � � �

� � � {\displaystyle g_{ba}=g_{ab}}[25] and � � � , � �

� � � , � � {\displaystyle g_{ab,fe}=g_{ab,ef}}. We shall suppose � {\displaystyle Q} to be written in such a way that its form is not altered by interchanging � � � {\displaystyle g_{ba}} and � � � {\displaystyle g_{ab}} or � � � , � � {\displaystyle g_{ab,fe}} and � � � , � � {\displaystyle g_{ab,ef}}. If originally this condition is not fulfilled it is easy to pass to a “symmetrical” form of this kind.

It is clear that � {\displaystyle Q} may also be expressed in the quantities � � � {\displaystyle g_{ab}} and their first and second derivatives and in the same way in the � � � {\displaystyle {\mathfrak {g}}_{ab}} and first and second derivatives of these quantities.

If the necessary substitutions are executed with due care, these new forms of � {\displaystyle Q} will also be symmetrical.

§ 34. We shall first express the quantity � {\displaystyle Q} in the � � � {\displaystyle g_{ab}}’s and their ​derivatives and we shall determine the variation it undergoes by arbitrarily chosen variations � � � � {\displaystyle \delta g_{ab}}, these latter being continuous functions of the coordinates. We have evidently

� �

∑ ( � � ) ∂ � ∂ � � � � � � � + ∑ ( � � � ) ∂ � ∂ � � � , � � � � � , � + ∑ ( � � � � ) � � ∂ � � � , � � � � � � , � � {\displaystyle \delta Q=\sum (ab){\frac {\partial Q}{\partial g_{ab}}}\delta g_{ab}+\sum (abe){\frac {\partial Q}{\partial g_{ab,e}}}\delta g_{ab,e}+\sum (abef){\frac {\delta Q}{\partial g_{ab,ef}}}\delta g_{ab,ef}}

By means of the equations

� � � � , � �

∂ ∂ � � � � � � , � {\displaystyle \delta g_{ab,ef}={\frac {\partial }{\partial x_{f}}}\delta g_{ab,e}} and � � � � , �

∂ ∂ � � � � � � {\displaystyle \delta g_{ab,e}={\frac {\partial }{\partial x_{e}}}\delta g_{ab}}

this may be decomposed into two parts

� �

� 1 � + � 2 � {\displaystyle dQ=\delta _{1}Q+\delta _{2}Q} (42) namely

� 1 �

∑ ( � � ) { ∂ � ∂ � � � − ∑ ( � ) ∂ ∂ � � ∂ � ∂ � � � , � + ∑ ( � � ) ∂ 2 ∂ � � ∂ � � ∂ � ∂ � � � , � � } � � � � {\displaystyle \delta {1}Q=\sum (ab)\left{{\frac {\partial Q}{\partial g{ab}}}-\sum (e){\frac {\partial }{\partial x_{e}}}{\frac {\partial Q}{\partial g_{ab,e}}}+\sum (ef){\frac {\partial ^{2}}{\partial x_{e}\partial x_{f}}}{\frac {\partial Q}{\partial g_{ab,ef}}}\right}\delta g_{ab}} (43) � 2 �

∑ ( � � � ) ∂ � ∂ � � ( ∂ � ∂ � � � , � � � � � ) + ∑ ( � � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , � � � � � � , � ) − − ∑ ( � � � � ) ∂ ∂ � � { ∂ ∂ � � ( ∂ � ∂ � � � , � � ) � � � � } {\displaystyle {\begin{array}{c}\delta {2}Q=\sum (abe){\frac {\partial Q}{\partial x{e}}}\left({\frac {\partial Q}{\partial g_{ab,e}}}\delta g_{ab}\right)+\sum (abef){\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,ef}}}\delta g_{ab,e}\right)-\\-\sum (abef){\frac {\partial }{\partial x_{e}}}\left{{\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,ef}}}\right)\delta g_{ab}\right}\end{array}}} (44) The last equation shows that

∫ � 2 � � �

0 {\displaystyle \int \delta {2}QdS=0} (45) if the variations � � � � {\displaystyle \delta g{ab}} and their first derivatives vanish at the boundary of the domain of integration.

§ 35. Equations of the same form may also be found if � {\displaystyle Q} is expressed in one of the two other ways mentioned in § 33. If e.g. we work with the quantities � � � {\displaystyle {\mathfrak {g}}^{ab}} we shall find

( � � )

( � 1 � ) + ( � 2 � ) {\displaystyle (\delta Q)=\left(\delta _{1}Q\right)+\left(\delta _{2}Q\right)}

where ( � 1 � ) {\displaystyle \left(\delta {1}Q\right)} and ( � 2 � ) {\displaystyle \left(\delta {2}Q\right)} are directly found from (43) and (44) by replacing � � � {\displaystyle g{ab}}, � � � , � {\displaystyle g{ab,e}}, � � � , � � {\displaystyle g_{ab,ef}}, � � � � {\displaystyle \delta g_{ab}} and � � � � , � {\displaystyle \delta g_{ab,e}} etc. by � � � {\displaystyle {\mathfrak {g}}^{ab}}, � � � , � {\displaystyle {\mathfrak {g}}^{ab,e}} etc. If the variations chosen in the two cases correspond to each other we shall have of course

( � � )

� � {\displaystyle (dQ)=\delta Q}

Moreover we can show that the equalities

( � 1 � )

� 1 � ,

( � 2 � )

� 2 � {\displaystyle \left(\delta _{1}Q\right)=\delta _{1}Q,\ \left(\delta _{2}Q\right)=\delta _{2}Q}

exist separately.[26]

​The decomposition of � � {\displaystyle \delta Q} into two parts is therefore the same, whether we use � � � , � � � {\displaystyle g_{ab},g^{ab}} or � � � {\displaystyle {\mathfrak {g}}^{ab}}.

It is further of importance that when the system of coordinates is changed, not only � � � � {\displaystyle \delta QdS} is an invariant, but that this is also the case with � 1 � � � {\displaystyle \delta _{1}QdS} and � 2 � � � {\displaystyle \delta _{2}QdS} separately.[27]

We have therefore

� 1 � ′ − � ′

� 1 � − � {\displaystyle {\frac {\delta _{1}Q’}{\sqrt {-g’}}}={\frac {\delta _{1}Q}{\sqrt {-g}}}} (46)

§ 36. For the calculation of � 1 � {\displaystyle \delta _{1}Q} we shall suppose � {\displaystyle Q} to be expressed in the quantities � � � {\displaystyle {\mathfrak {g}}^{ab}} and their derivatives. Therefore (comp. (43))

� 1 �

∑ ( � � ) � � � � � � � {\displaystyle \delta {1}Q=\sum (ab)M{ab}d{\mathfrak {g}}^{ab}} (47) if we put

� � �

∂ � ∂ � � � − ∑ ( � ) ∂ ∂ � � ∂ � ∂ � � � , � + ∑ ( � � ) ∂ ∂ � � ∂ � � ∂ � ∂ � � � , � � {\displaystyle M_{ab}={\frac {\partial Q}{\partial {\mathfrak {g}}^{ab}}}-\sum (e){\frac {\partial }{\partial x_{e}}}{\frac {\partial Q}{\partial {\mathfrak {g}}^{ab,e}}}+\sum (ef){\frac {\partial }{\partial x_{e}\partial x_{f}}}{\frac {\partial Q}{\partial {\mathfrak {g}}^{ab,ef}}}}

Now we can show that the quantities � � � {\displaystyle M_{ab}} are exactly the quantities � � � {\displaystyle G_{ab}} defined by (40). To this effect we may use the following considerations.

We know that ( 1 − � � � � ) {\displaystyle \left({\tfrac {1}{\sqrt {-g}}}{\mathfrak {g}}^{ab}\right)} is a contravariant tensor of the second ​order. From this we can deduce that ( 1 − � � � � � ) {\displaystyle \left({\frac {1}{\sqrt {-g}}}\delta {\mathfrak {g}}^{ab}\right)} is also such a tensor.

Writing for it � � � {\displaystyle \epsilon ^{ab}} we find according to (46) and (47) that

∑ ( � � ) � � � � � � {\displaystyle \sum (ab)M_{ab}\epsilon ^{ab}}

is a scalar for every choice of ( � � � ) {\displaystyle \left(\epsilon ^{ab}\right)}.

This involves that ( � � � ) {\displaystyle \left(M_{ab}\right)} is a covariant tensor of the second order and as the same is true for ( � � � ) {\displaystyle \left(G_{ab}\right)} we must prove the equation

� � �

� � � {\displaystyle M_{ab}=G_{ab}}

only for one special choice of coordinates.

§ 37. Now this choice can be made in such a way that at the point � {\displaystyle P} of the field-figure � 11

� 22

� 33

− 1 {\displaystyle g_{11}=g_{22}=g_{33}=-1}, � 44

1 {\displaystyle g_{44}=+1}, � � �

0 {\displaystyle g_{ab}=0} for � ≠ � {\displaystyle a\neq b} and that moreover all first derivatives � � � , � {\displaystyle g_{ab,e}} vanish. If then the values � � � {\displaystyle g_{ab}} at a point � {\displaystyle Q} near � {\displaystyle P} are developed in series of ascending powers of the differences of coordinates � � ( � ) − � � ( � ) {\displaystyle x_{a}(Q)-x_{a}(P)} the terms directly following the constant ones will be of the second order. It is with these terms that we are concerned in the calculation both of � � � {\displaystyle M_{ab}} and of � � � {\displaystyle G_{ab}} for the point � {\displaystyle P}. As in the results the coefficients of these terms occur to the first power only, it is sufficient to show that each of the above mentioned terms separately contributes the same value to � � � {\displaystyle M_{ab}} and to � � � {\displaystyle G_{ab}}.

From these considerations we may conclude that

� 1 �

∑ ( � � ) � � � � � � � {\displaystyle \delta {1}Q=\sum (ab)G{ab}\delta {\mathfrak {g}}^{ab}} (48) Expressions containing instead of � � � � {\displaystyle \delta {\mathfrak {g}}^{ab}} either the variations � � � � {\displaystyle \delta g^{ab}} or � � � � {\displaystyle \delta g_{ab}} might be derived from this by using the relations between the different variations. Of these we shall only mention the formula

� � � �

1 − � � � � � − � � � 2 − � ∑ ( � � ) � � � � � � � {\displaystyle \delta g^{ab}={\frac {1}{\sqrt {-g}}}\delta {\mathfrak {g}}^{ab}-{\frac {g^{ab}}{2{\sqrt {-g}}}}\sum (cd)g_{cd}\delta {\mathfrak {g}}^{cd}} (49)

§ 38. In connexion with what precedes we here insert a consideration the purpose of which will be evident later on. Let the infinitely small quantity �{\displaystyle \xi } be an arbitrarily chosen continuous function of the coordinates and let the variations � � � � {\displaystyle \delta g_{ab}} be defined by the condition that at some point � {\displaystyle P} the quantities � � � {\displaystyle g_{ab}} have after the change the values which existed before the change at the point � {\displaystyle Q}, to which � {\displaystyle P} is shifted when � ℎ {\displaystyle x_{h}} is diminished by �{\displaystyle \xi }, while the three other coordinates are left constant. Then we have

� � � �

− � � � , ℎ �{\displaystyle \delta g_{ab}=-g_{ab,h}\xi }

and similar formulae for the variations � � � � {\displaystyle \delta {\mathfrak {g}}^{ab}}.

​If for � 1 � {\displaystyle \delta _{1}Q} and � 2 � {\displaystyle \delta _{2}Q} the expressions (48) and (44) are taken, the equation

� � − � 2 �

� 1 � {\displaystyle dQ-\delta _{2}Q=\delta _{1}Q} (50) is an identity for every choice of the variations.

It will likewise be so in the special case considered and we shall also come to an identity if in (50) the terms with the derivatives of �{\displaystyle \xi } are omitted while those with �{\displaystyle \xi } itself are preserved.

When this is done � � {\displaystyle \delta Q} reduces to

− ∂ � ∂ � ℎ �{\displaystyle -{\frac {\partial Q}{\partial x_{h}}}\xi }

and, taking into consideration (44) and (48), we find after division by �{\displaystyle \xi }

− ∂ � ∂ � ℎ + ∑ ( � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , � � � � , ℎ ) + ∑ ( � � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , � � � � � , � ℎ ) − − ∑ ( � � � � ) ∂ ∂ � � { ∂ ∂ � � ( ∂ � ∂ � � � , � � ) � � � , ℎ }

− ∑ ( � � ) � � � � � � , ℎ {\displaystyle {\begin{array}{c}-{\frac {\partial Q}{\partial x_{h}}}+\sum (abe){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,e}}}g_{ab,h}\right)+\sum (abef){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,fh}\right)-\\-\sum (abef){\frac {\partial }{\partial x_{e}}}\left{{\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,h}\right}=-\sum (ab)G_{ab}{\mathfrak {g}}^{ab,h}\end{array}}} (51) In the second term of (44) we have interchanged here the indices � {\displaystyle e} and � {\displaystyle f}.

If for shortness’ sake we put, for � ≠ ℎ {\displaystyle e\neq h}

� ℎ �

∑ ( � � ) ∂ � ∂ � � � , � � � � , ℎ + ∑ ( � � � ) ∂ � ∂ � � � , � � � � � , � ℎ − ∑ ( � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , � � ) � � � , ℎ {\displaystyle {\mathfrak {s}}{h}^{e}=\sum (ab){\frac {\partial Q}{\partial g{ab,e}}}g_{ab,h}+\sum (abf){\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,fh}-\sum (abf){\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,h}} (52) and for �

ℎ {\displaystyle e=h}

� ℎ ℎ

− � + ∑ ( � � ) ∂ � ∂ � � � , ℎ � � � , ℎ + ∑ ( � � � ) ∂ � ∂ � � � , � ℎ � � � , � ℎ − ∑ ( � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , ℎ � ) � � � , ℎ {\displaystyle {\mathfrak {s}}{h}^{h}=-Q+\sum (ab){\frac {\partial Q}{\partial g{ab,h}}}g_{ab,h}+\sum (abf){\frac {\partial Q}{\partial g_{ab,fh}}}g_{ab,fh}-\sum (abf){\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,hf}}}\right)g_{ab,h}} (53) we may write

∑ ( � ) ∂ � ℎ � ∂ � �

− ∑ ( � � ) � � � � � � , ℎ {\displaystyle \sum (e){\frac {\partial {\mathfrak {s}}{h}^{e}}{\partial x{e}}}=-\sum (ab)G_{ab}{\mathfrak {g}}^{ab,h}} (54) The set of quantities � ℎ � {\displaystyle {\mathfrak {s}}_{h}^{e}} will be called the complex � {\displaystyle {\mathfrak {s}}} and the set of the four quantities which stand on the left hand side of (54) in the cases ℎ

1 , 2 , 3 , 4 {\displaystyle h=1,2,3,4}, the divergency of the complex.[28] It will be denoted by � � � � {\displaystyle div{\mathfrak {s}}} and each of the four quantities separately by � � � ℎ � {\displaystyle div_{h}{\mathfrak {s}}}.

The equation therefore becomes

� � � ℎ �

− ∑ ( � � ) � � � � � � , ℎ {\displaystyle div_{h}{\mathfrak {s}}=-\sum (ab)G_{ab}{\mathfrak {g}}^{ab,h}} (55) ​If we take other coordinates the right hand side of this equation is transformed according to a formula which can be found easily. Hence we can also write down the transformation formula for the left hand side. It is as follows � � � ℎ ′ � ′

� ∑ ( � ) � � ℎ � � � � � − � ∑ ( � ) � � ℎ ∂ � ∂ � � + 2 � ∑ ( � � � ) � � ℎ , � � � � � � � {\displaystyle div’{h}{\mathfrak {s}}’=p\sum (m)p{mh}div_{m}{\mathfrak {s}}-Q\sum (a)p_{ah}{\frac {\partial p}{\partial x_{a}}}+2p\sum (abc)p_{ah,c}{\mathfrak {g}}^{bc}G_{ab}} (56)

§ 39. We shall now consider a second complex � 0 {\displaystyle {\mathfrak {s}}_{0}}, the components of which are defined by

� 0 ℎ �

− � ∑ ( � ) � � � � � ℎ + 2 ∑ ( � ) � � � � � ℎ {\displaystyle {\mathfrak {s}}{0h}^{e}=-G\sum (a){\mathfrak {g}}^{ae}g{ah}+2\sum (a){\mathfrak {g}}^{ae}G_{ah}} (57) Taking also the divergency of this complex we find that the difference

� � � ℎ ′ � 0 ′ − � ∑ ( � ) � � ℎ � � � � � 0 {\displaystyle div’{h}{\mathfrak {s}}’{0}-p\sum (m)p_{mh}div_{m}{\mathfrak {s}}_{0}}

has just the value which we can deduce from (56) for the corresponding difference

� � � ℎ ′ � ′ − � ∑ ( � ) � � ℎ � � � � � {\displaystyle div’{h}{\mathfrak {s}}’-p\sum (m)p{mh}div_{m}{\mathfrak {s}}}

It is thus seen that

� � � ℎ ′ � ′ − � � � ℎ ′ � 0 ′

� ∑ ( � ) � � ℎ ( � � � � � − � � � � � 0 ) {\displaystyle div’{h}{\mathfrak {s}}’-div’{h}{\mathfrak {s}}’{0}=p\sum (m)p{mh}\left(div_{m}{\mathfrak {s}}-div_{m}{\mathfrak {s}}_{0}\right)}

and that we have therefore

� � � �

� � � � 0 {\displaystyle div{\mathfrak {s}}=div{\mathfrak {s}}_{0}} (58) for all systems of coordinates as soon as this is the case for one system.

Now a direct calculation starting from (52), (53) and (57) teaches us that the terms with the highest derivatives of the quantities � � � {\displaystyle g_{ab}}, (viz. those of the third order) are the same in � � � ℎ � {\displaystyle div_{h}{\mathfrak {s}}} and � � � ℎ � 0 {\displaystyle div_{h}{\mathfrak {s}}{0}}. Further it is evident that in the system of coordinates introduced in § 37 these terms with the third derivatives are the only ones. This proves the general validity of equation (58). It is especially to be noticed that if � {\displaystyle {\mathfrak {s}}} and � 0 {\displaystyle {\mathfrak {s}}{0}} are determined by (52), (53) and (57) and if the function defined in § 32 is taken for � {\displaystyle G}, the relation is an identity.

§ 40. We shall now derive the differential equations for the gravitation field, first for the case of an electromagnetic system.[29] For the part of the principal function belonging to it we write

∫ L � � {\displaystyle \int \mathrm {L} dS}

where L {\displaystyle \mathrm {L} } is defined by (35) (1915). From L {\displaystyle \mathrm {L} } we can derive the stresses, the momenta, the energy-current and the energy of the ​electromagnetic system; for this purpose we must use the equations (45) and (46) (1915) or in Einstein’s notation, which we shall follow here,[30]

� � �

− L + ∑ � ≠ � ( � ) � � � ∗ � � ′ � ′ {\displaystyle {\mathfrak {T}}_{c}^{c}=-\mathrm {L} +\sum \limits _{a\neq c}(a)\psi _{ac}^{*}\psi _{a’c’}} (59) and for � ≠ � {\displaystyle b\neq c}

� � �

∑ � ≠ � ( � ) � � � ∗ � � ′ � ′ {\displaystyle {\mathfrak {T}}_{c}^{b}=\sum \limits _{a\neq c}(a)\psi _{ac}^{*}\psi {a’c’}} (60) The set of quantities � � � {\displaystyle {\mathfrak {T}}{c}^{b}} might be called the stress-energy-complex (comp. § 38). As for a change of the system of coordinates the transformation formulae for � {\displaystyle {\mathfrak {T}}} are similar to those by which tensors are defined, we can also speak of the stress-energy-tensor. We have namely

1 − � ′ � � ′ �

1 − � ∑ ( � � ) � � � � � � � � � {\displaystyle {\frac {1}{\sqrt {-g’}}}{\mathfrak {T}}{c}^{‘b}={\frac {1}{\sqrt {-g}}}\sum (kl)p{kc}\pi lb{\mathfrak {T}}_{k}^{l}}

§ 41. The equations for the gravitation field are now obtained (comp. §§ 13 and 14, 1915) from the condition that

� � ∫ L � � + 1 2 � � ∫ � � �

0 {\displaystyle \delta {\psi }\int \mathrm {L} dS+{\frac {1}{2\varkappa }}\delta \int QdS=0} (61) for all variations � � � � {\displaystyle \delta g{ab}} which vanish at the boundary of the field of integration together with their first derivatives. The index �{\displaystyle \psi } in the first term indicates that in the variation of L {\displaystyle \mathrm {L} } the quantities � � � {\displaystyle \psi _{ab}} must be kept constant.

If we suppose L {\displaystyle \mathrm {L} } to be expressed in the quantities � � � {\displaystyle g^{ab}} and if (42), (45) and (48) are taken into consideration, we find from (61) that at each point of the field-figure

∑ ( � � ) ( ∂ L ∂ � � � ) � � � � � + 1 2 � ∑ ( � � ) � � � � � � �

0 {\displaystyle \sum (ab)\left({\frac {\partial \mathrm {L} }{\partial g^{ab}}}\right){\psi }\delta g^{ab}+{\frac {1}{2\varkappa }}\sum (ab)G{ab}\delta {\mathfrak {g}}^{ab}=0} (62) If now in the first term we put ​

( ∂ L ∂ � � � ) �

1 2 − � � � � {\displaystyle \left({\frac {\partial \mathrm {L} }{\partial g^{ab}}}\right){\psi }={\frac {1}{2}}{\sqrt {-g}}T{ab}} (63) and if for ∂ � � � {\displaystyle \partial g^{ab}} the value (49) is substituted, this term becomes

1 2 ∑ ( � � ) � � � ∂ � � � − 1 4 ∑ ( � � � � ) � � � � � � � � � � � � � {\displaystyle {\frac {1}{2}}\sum (ab)T_{ab}\partial {\mathfrak {g}}^{ab}-{\frac {1}{4}}\sum (abcd)g^{ab}g_{cd}T_{ab}\delta {\mathfrak {g}}^{cd}}

or if in the latter summation � , � {\displaystyle a,b} is interchanged with � , � {\displaystyle c,d} and if the quantity

�

∑ ( � � ) � � � � � � {\displaystyle T=\sum (cd)g^{cd}T_{cd}} (64) is introduced,

1 2 ∑ ( � � ) ( � � � − 1 2 � � � � ) � � � � {\displaystyle {\frac {1}{2}}\sum (ab)\left(T_{ab}-{\frac {1}{2}}g_{ab}T\right)\delta {\mathfrak {g}}^{ab}}

Finally, putting equal to zero the coefficient of each � � � � {\displaystyle \delta {\mathfrak {g}}^{ab}} we find from (62) the differential equation required

� � �

− � ( � � � − 1 2 � � � � ) {\displaystyle G_{ab}=-\varkappa \left(T_{ab}-{\frac {1}{2}}g_{ab}T\right)} (65) This is of the same form as Einstein’s field equations, but to see that the formulae really correspond to each other it remains to show that the quantities � � � {\displaystyle T_{ab}} and � � � {\displaystyle {\mathfrak {T}}_{c}^{b}} defined by (63), f59) and (60) are connected by Einstein’s formulae

� � �

− � ∑ ( � ) � � � � � � {\displaystyle {\mathfrak {T}}{c}^{b}={\sqrt {-g}}\sum (a)g^{ab}T{ac}} (66) We must have therefore

2 ∑ ( � ) � � � ( ∂ L ∂ � � � ) �

− L + ∑ � ≠ � ( � ) � � � ∗ � � ′ � ′ {\displaystyle 2\sum (a)g^{ac}\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }=-\mathrm {L} +\sum \limits _{a\neq c}(a)\psi _{ac}^{*}\psi _{a’c’}} (67) and for � ≠ � {\displaystyle b\neq c}

2 ∑ ( � ) � � � ( ∂ L ∂ � � � ) �

∑ � ≠ � ( � ) � � � ∗ � � ′ � ′ {\displaystyle 2\sum (a)g^{ab}\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{\psi }=\sum \limits _{a\neq c}(a)\psi _{ab}^{*}\psi _{a’c’}} (68)

§ 42. This can be tested in the following way. The function L {\displaystyle \mathrm {L} } (comp. § 9, 1915) is a homogeneous quadratic function of the � � � {\displaystyle \psi {ab}}’s and when differentiated with respect to these variables it gives the quantities � ¯ � � {\displaystyle {\bar {\psi }}{ab}}. It may therefore also be regarded as a homogeneous quadratic function of the � ¯ � � {\displaystyle {\bar {\psi }}_{ab}}. From (35), (29) and (32)[31], 1915 we find therefore

�

1 8 − � ∑ ( � � � � ) ( � � � � � � − � � � � � � ) � ¯ � � � ¯ � � {\displaystyle L={\frac {1}{8}}{\sqrt {-g}}\sum (pqrs)\left(g^{pr}g^{qs}-g^{qr}g^{ps}\right){\bar {\psi }}{pq}{\bar {\psi }}{rs}} (69) Now we can also differentiate with respect to the � � � {\displaystyle g^{ab}}’s, while not the � � � {\displaystyle \psi {ab}}’s but the quantities � ¯ � � {\displaystyle {\bar {\psi }}{ab}} are kept constant, and we have e.g.

( ∂ L ∂ � � � ) �

− ( ∂ L ∂ � � � ) �{\displaystyle \left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right){\psi }=-\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right){\psi }} (70) According to (69) one part of the latter differential coefficient is ​obtained by differentiating the factor − � {\displaystyle {\sqrt {-g}}} only and the other part by keeping this factor constant.

For the calculation of the first of these parts we can use the relation

∂ log ⁡ ( − � ) ∂ � � �

− 1 2 � � � {\displaystyle {\frac {\partial \log \left({\sqrt {-g}}\right)}{\partial g^{ac}}}=-{\frac {1}{2}}g_{ac}} and for the second part we find

1 2 − � ∑ ( � � ) � � � � ¯ � � � ¯ � � {\displaystyle {\frac {1}{2}}{\sqrt {-g}}\sum (pq)g^{pq}{\bar {\psi }}{ap}{\bar {\psi }}{cq}}

If (32) 1915 is used (67) and (68) finally become

∑ ( � ) � � � � ¯ � � + ∑ � ≠ � ( � ) � � � ∗ � � ′ � ′

2 L ∑ ( � ) � ¯ � � � � � + ∑ � ≠ � ( � ) � � � ∗ � � ′ � ′

0 {\displaystyle {\begin{array}{c}\sum (q)\psi {cq}{\bar {\psi }}{cq}+\sum \limits _{a\neq c}(a)\psi _{ac}^{}\psi {a’c’}=2\mathrm {L} \\\sum (q){\bar {\psi }}{cq}\psi _{bq}+\sum \limits _{a\neq c}(a)\psi _{ab}^{}\psi _{a’c’}=0\end{array}}}

These equations are really fulfilled. This is evident from � � �

0 {\displaystyle \psi _{aa}=0}, � ¯ � �

0 {\displaystyle {\bar {\psi }}_{aa}=0}, � � �

− � � � {\displaystyle \psi _{ba}=-\psi _{ab}} and � ¯ � �

− � ¯ � � {\displaystyle {\bar {\psi }}{ba}=-{\bar {\psi }}{ab}}, besides, the meaning of � � � ∗{\displaystyle \psi _{ab}^{*}} (§ 11, 1915) and equation (35) 1915 must be taken into consideration.

§ 43. In nearly the same way we can treat the gravitation field of a system of incoherent material points; here the quantities � � {\displaystyle w_{a}} and � � {\displaystyle u_{a}} (§§ 4 and 5, 1915) play a similar part as � � � {\displaystyle \psi {ab}} and � ¯ � � {\displaystyle {\bar {\psi }}{ab}} in what precedes. To consider a more general case we can suppose “molecular forces” to act between the material points (which we assume to be equal to each other); in such a way that in ordinary mechanics we should ascribe to the system a potential energy depending on the density only. Conforming to this we shall add to the Lagrangian function L {\displaystyle \mathrm {L} } (§ 4, 1915) a term which is some function of the density of the matter at the point � {\displaystyle P} of the field-figure, such as that density is when by a transformation the matter at that point has been brought to rest. This can also be expressed as follows. Let � �{\displaystyle d\sigma } be an infinitely small three-dimensional extension expressed in natural units, which at the point � {\displaystyle P} is perpendicular to the world-line passing through that point, and � ¯ � �{\displaystyle {\bar {\varrho }}d\sigma } the number of points where � �{\displaystyle d\sigma } intersects world-lines. The contribution of an element of the field-figure to the principal function will then be found by multiplying the magnitude of that element expressed in natural units by a function of � ¯{\displaystyle {\bar {\varrho }}}. Further calculation teaches us that the term to be added to L {\displaystyle \mathrm {L} } must have the form

− � � ( � − � ) {\displaystyle {\sqrt {-g}}\varphi \left({\frac {P}{\sqrt {-g}}}\right)} (71) ​where � {\displaystyle P} is given by (15) 1915. As the Lagrangian function defined by (11) 1915 equally falls under this form and also the sum of this function and the new term, the expression (71) may be regarded as the total function L {\displaystyle \mathrm {L} }. The function �{\displaystyle \varphi } may be left indeterminate. If now with this function the calculations of §§ 5 and 6, 1915 are repeated, we find the components of the stress-energy-tensor of the matter. The equations for the gravitation field again take the form (65). � � � {\displaystyle T_{ab}} is defined by an equation of the form (63), where on the left hand side we must differentiate while the � � {\displaystyle w_{a}}’s are kept constant. Relation (66) can again be verified without difficulty.

We shall not, however, dwell upon this, as the following considerations are more general and apply e.g. also to systems of material points that are anisotropic as regards the configuration and the molecular actions.

§ 44. At any point � {\displaystyle P} of the field-figure the Lagrangian function L {\displaystyle \mathrm {L} } will evidently be determined by the course and the mutual situation of the world-lines of the material points in the neighbourhood of � {\displaystyle P}. This leads to the assumption that for constant � � � {\displaystyle g_{ab}}’s the variation � L {\displaystyle \delta \mathrm {L} } is a homogeneous linear function of the virtual displacements � � � {\displaystyle \delta x_{a}} of the material points and of the differential coefficients

∂ � � � ∂ � � {\displaystyle {\frac {\partial \delta x_{a}}{\partial x_{b}}}}

these last quantities evidently determining the deformation of an infinitesimal part of the figure formed by the world-lines[32].

The calculation becomes most simple if we put

L

− � � {\displaystyle \mathrm {L} ={\sqrt {-g}}H} (72) and for constant � � � {\displaystyle g_{ab}}’s

� �

∑ ( � ) � � � � � + ∑ ( � � ) � � � ∂ � � � ∂ � � {\displaystyle \delta H=\sum (a)U_{a}\delta x_{a}+\sum (ab)V_{a}^{b}{\frac {\partial \delta x_{a}}{\partial x_{b}}}} (73) Considerations corresponding exactly to those mentioned in §§ 4 — 6, 1915, now lead to the equations of motion and to the following expressions for the components of the stress-energy-tensor

� � �

− L − − � � � � {\displaystyle {\mathfrak {T}}{c}^{c}=-\mathrm {L} -{\sqrt {-g}}V{c}^{c}} (74) and for � ≠ � {\displaystyle b\neq c}

� � �

− − � � � � {\displaystyle {\mathfrak {T}}{c}^{b}=-{\sqrt {-g}}V{c}^{b}} (75) The differential equations again take the form (65) if the quantities � � � {\displaystyle T_{ab}} are defined by ​ ( ∂ L ∂ � � � ) �

1 2 − � � � � {\displaystyle \left({\frac {\partial \mathrm {L} }{\partial g^{ab}}}\right){x}={\frac {1}{2}}{\sqrt {-g}}T{ab}}

in the differentiation on the left hand side the coordinates of the material points are kept constant. To show that � � � {\displaystyle T_{ab}} and � � � {\displaystyle {\mathfrak {T}}_{c}^{b}} satisfy equation (66) we must now show that

− L − − � � � �

2 ∑ ( � ) � � � ( ∂ � ∂ � � � ) � {\displaystyle -\mathrm {L} -{\sqrt {-g}}V_{c}^{c}=2\sum (a)g^{ac}\left({\frac {\partial L}{\partial g^{ac}}}\right)_{x}}

and for � ≠ � {\displaystyle b\neq c}

− − � � � �

2 ∑ ( � ) � � � ( ∂ L ∂ � � � ) � {\displaystyle -{\sqrt {-g}}V_{c}^{b}=2\sum (a)g^{ab}\left({\frac {\partial \mathrm {L} }{\partial g^{ac}}}\right)_{x}}

If here the value (72) is substituted for L {\displaystyle \mathrm {L} } and if (70) is taken into account, these equations say that for all values of � {\displaystyle b} and � {\displaystyle c} we must have

2 ∑ ( � ) � � � ( ∂ � ∂ � � � ) � + � � �

0 {\displaystyle 2\sum (a)g^{ab}\left({\frac {\partial H}{\partial g^{ac}}}\right){x}+V{c}^{b}=0} (76) Now this relation immediately follows from a condition, to which L {\displaystyle \mathrm {L} } must be subjected at any rate, viz. that L � � {\displaystyle \mathrm {L} dS} is a scalar quantity. This involves that in a definite case we must find for � {\displaystyle H} always the same value whatever be the choice of coordinates.

§ 45. Let us suppose that instead of only one coordinate � � {\displaystyle x_{c}} a new one � � ′ {\displaystyle x’{c}} has been introduced, which differs infinitely little from � � {\displaystyle x{c}}, with the restriction that if

� � ′

� � + � � {\displaystyle x’{c}=x{c}+\xi _{c}}

the term � � {\displaystyle \xi {c}} depends on the coordinate � � {\displaystyle x{b}} only and is zero at the point in question of the field-figure. The quantities � � � {\displaystyle g^{ab}} then take other values and in the new system of coordinates the world-lines of the material points will have a slightly changed course.

By each of these circumstances separately � {\displaystyle H} would change, but all together must leave it unaltered. As to the first change we remark that, according to the transformation formula for � � � {\displaystyle g^{ab}}, the variation � � � � {\displaystyle \delta g^{ab}} vanishes when the two indices are different from � {\displaystyle c}, while

� � � �

2 � � � ∂ � � ∂ � � {\displaystyle \delta g^{cc}=2g^{cb}{\frac {\partial \xi {c}}{\partial x{b}}}}

and for � ≠ � {\displaystyle a\neq c}

� � � �

2 � � �

� � � ∂ � � ∂ � � {\displaystyle \delta g^{ac}=2g^{ca}=g^{ab}{\frac {\partial \xi {c}}{\partial x{b}}}}

The change of � {\displaystyle H} due to these variations is

2 ∂ � � ∂ � � ∑ ( � ) � � � ( ∂ � ∂ � � � ) � {\displaystyle 2{\frac {\partial \xi {c}}{\partial x{b}}}\sum (a)g^{ab}\left({\frac {\partial H}{\partial g^{ac}}}\right)_{x}}

​Further, in the new system of coordinates the figure formed by the world-lines differs from that figure in the old system by the variation � � �

� � {\displaystyle \delta x_{c}=\xi {c}} which is a function of � � {\displaystyle x{b}} only. Therefore according to (73) the second variation of � {\displaystyle H} is � � �

∂ � � ∂ � � {\displaystyle V_{c}^{b}={\frac {\partial \xi {c}}{\partial x{b}}}}

By putting equal to zero the sum of this expression and the preceding one we obtain (76).

§ 46. We have thus deduced for some cases the equations of the gravitation field from the variation theorem. Probably this can also be done for thermodynamic systems, if the Lagrangian function is properly chosen in connexion with the thermodynamic functions, entropy and free energy. But as soon as we are concerned with irreversible phenomena, when e.g. the energy-current consists in a conduction of heat, the variation principle cannot be applied. We shall then be obliged to take Einstein’s field-equations as our point of departure, unless, considering the motions of the individual atoms or molecules, we succeed in treating these by means of the generalized principle of Hamilton.

§ 47. Finally we shall consider the stresses, the energy etc. which belong to the gravitation field itself. The results will be the same for all the systems treated above, but we shall confine ourselves to the case of §§ 44 and 45. We suppose certain external forces � � {\displaystyle K_{a}} to act on the material points, though we shall see that strictly speaking this is not allowed.

For any displacements � � � {\displaystyle \delta x_{a}} of the matter and variations of the gravitation field we first have the equation which summarizes what we found above

� L + 1 2 � � � + ∑ ( � ) � � ∗ � � �

− � ∑ ( � ) � � � � � + + ∑ ( � � ) ∂ ∂ � � ( − � � � � � � � ) − ∑ ( � � ) ∂ ∂ � � ( − � � � � ) � � � + + ∑ ( � � ) ( ∂ � ∂ � � � ) � � � � � + 1 2 � � 1 � + 1 2 � � 2 � + ∑ ( � ) � � � � � . {\displaystyle {\begin{array}{l}\delta \mathrm {L} +{\frac {1}{2\varkappa }}\delta Q+\sum (a)K_{a}^{*}\delta x_{a}={\sqrt {-g}}\sum (a)U_{a}\delta x_{a}+\\\qquad +\sum (ab){\frac {\partial }{\partial x_{b}}}\left({\sqrt {-g}}V_{a}^{b}\delta x_{a}\right)-\sum (ab){\frac {\partial }{\partial x_{b}}}\left({\sqrt {-g}}V_{a}^{b}\right)\delta x_{a}+\\\qquad \qquad +\sum (ab)\left({\frac {\partial L}{\partial g^{ab}}}\right)_{x}\delta g^{ab}+{\frac {1}{2\varkappa }}\delta {1}Q+{\frac {1}{2\varkappa }}\delta {2}Q+\sum (a)K{a}\delta x{a}.\end{array}}}

In virtue of the equations of motion of the matter, the terms with � � � {\displaystyle \delta x_{a}} cancel each other on the right hand side and similarly, on account of the equations of the gravitation field, the terms with � � � � {\displaystyle \delta g^{ab}} and � 1 � {\displaystyle \delta _{1}Q}. Thus we can write[33] ​

∑ ( � ) � � � � �

− � � + ∑ ( � � ) ∂ ∂ � � ( − � � � � � � � ) − 1 2 � ( � � − � 2 � ) {\displaystyle \sum (a)K_{a}\delta x_{a}=-\delta L+\sum (ae){\frac {\partial }{\partial x_{e}}}\left({\sqrt {-g}}V_{a}^{e}\delta x_{a}\right)-{\frac {1}{2\varkappa }}\left(\delta Q-\delta {2}Q\right)} (77) Let us now suppose that only the coordinate � ℎ {\displaystyle x{h}} undergoes an infinitely small change, which has the same value at all points of the field-figure. Let at the same time the system of values � � � {\displaystyle g_{ab}} be shifted everywhere in the direction of � ℎ {\displaystyle x_{h}} over the distance � � ℎ {\displaystyle \delta x_{h}}. The left hand side of the equation then becomes � ℎ � � ℎ {\displaystyle K_{h}\delta x_{h}} and we have on the right hand side

� L

− ∂ L ∂ � ℎ � � ℎ ,

� �

− ∂ � ∂ � ℎ � � ℎ {\displaystyle \delta \mathrm {L} =-{\frac {\partial \mathrm {L} }{\partial x_{h}}}\delta x_{h},\ dQ=-{\frac {\partial Q}{\partial x_{h}}}\delta x_{h}}

After dividing the equation by � � ℎ {\displaystyle \delta x_{h}} we may thus, according to (74) and (75), write

− ∑ ( � ) ∂ T ℎ � ∂ � �

− � � � ℎ � {\displaystyle -\sum (e){\frac {\partial \mathrm {T} h^{e}}{\partial x_{e}}}=-div_{h}{\mathfrak {T}}}

By the same division we obtain from � � − � 2 � {\displaystyle \delta Q-\delta _{2}Q} the expression occurring on the left hand side of (51), which we have represented by

∑ ( � ) ∂ � ℎ � ∂ � �

� � � ℎ � {\displaystyle \sum (e){\frac {\partial {\mathfrak {s}}{h}^{e}}{\partial x{e}}}=div_{h}{\mathfrak {s}}}

where the complex � {\displaystyle {\mathfrak {s}}} is defined by (52) and (53). If therefore we introduce a new complex � {\displaystyle {\mathfrak {t}}} which differs from � {\displaystyle {\mathfrak {s}}} only by the factor 1 2 �{\displaystyle {\tfrac {1}{2\varkappa }}}, so that

� ℎ �

1 2 � � ℎ � {\displaystyle {\mathfrak {t}}{h}^{e}={\frac {1}{2\varkappa }}{\mathfrak {s}}{h}^{e}} (78) we find

� ℎ

− � � � ℎ � − � � � ℎ � {\displaystyle K_{h}=-div_{h}{\mathfrak {T}}-div_{h}{\mathfrak {t}}} (79) The form of this equation leads us to consider � {\displaystyle {\mathfrak {t}}} as the stress-energy-complex of the gravitation field, just as � {\displaystyle {\mathfrak {T}}} is the stress-energy-tensor for the matter. We need not further explain that for the case � ℎ

0 {\displaystyle K_{h}=0} the four equations contained in (79) express the conservation of momentum and of energy for the total system, matter and gravitation field taken together.

§ 48. To learn something about the nature of the stress-energy-complex � {\displaystyle {\mathfrak {t}}} we shall consider the stationary gravitation field caused by a quantity of matter without motion and distributed symmetrically around a point � {\displaystyle O}. In this problem it is convenient to introduce for the three space coordinates � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}}, ( � 4 {\displaystyle x_{4}} will represent the time) “polar” coordinates. By � 3 {\displaystyle x_{3}} we shall therefore denote a quantity � {\displaystyle r} ​which is a measure for the “distance” to the centre. As to � 1 {\displaystyle x_{1}} and � 2 {\displaystyle x_{2}}, we shall put � 1

cos ⁡ �{\displaystyle x_{1}=\cos \vartheta }, � 2

�{\displaystyle x_{2}=\varphi }, after first having introduced polar coordinates � , �{\displaystyle \vartheta ,\varphi } (in such a way that the rectangular coordinates are � cos ⁡ �{\displaystyle r\cos \vartheta }, � sin ⁡ � cos ⁡ �{\displaystyle r\sin \vartheta \cos \varphi }, � sin ⁡ � sin ⁡ �{\displaystyle r\sin \vartheta \sin \varphi }). It can be proved that, because of the symmetry about the centre, � � �

0 {\displaystyle g_{ab}=0} for � ≠ � {\displaystyle a\neq b}, while we may put for the quantities � � � {\displaystyle g_{aa}}

� 11

− � 1 − � 1 2 ,

� 22

− � ( 1 − � 1 2 ) ,

� 33

− � ,

� 44

� {\displaystyle g_{11}=-{\frac {u}{1-x_{1}^{2}}},\ g_{22}=-u\left(1-x_{1}^{2}\right),\ g_{33}=-v,\ g_{44}=w} (80) where � , � , � {\displaystyle u,v,w} are certain functions of � {\displaystyle r}. Ditferentiations of these functions will be represented by accents. We now find that of the complex � {\displaystyle {\mathfrak {t}}} only the components � 1 1 {\displaystyle {\mathfrak {t}}{1}^{1}}, � 3 3 {\displaystyle {\mathfrak {t}}{3}^{3}} and � 4 4 {\displaystyle {\mathfrak {t}}_{4}^{4}} are different from zero. The expressions found for them may be further simplified by properly choosing � {\displaystyle r}. If the distance to the centre is measured by the time the light requires to be propagated from to the point in question, we have �

� {\displaystyle w=v}. One then finds

� 1 1

1 2 � ( − � ′ 2 2 � + 2 � ″ − � � ′ 2 � 2 + � � ″ � ) , � 3 3

1 2 � ( − 2 � + � ′ 2 2 � + � � ′ � ) , � 4 4

1 2 � ( − 2 � − � ′ 2 2 � + 2 � ″ + � � ″ � ) , } {\displaystyle \left.{\begin{array}{l}{\mathfrak {t}}{1}^{1}={\frac {1}{2\varkappa }}\left(-{\frac {u’^{2}}{2u}}+2u’’-{\frac {uv’^{2}}{v^{2}}}+{\frac {uv’’}{v}}\right),\\{\mathfrak {t}}{3}^{3}={\frac {1}{2\varkappa }}\left(-2v+{\frac {u’^{2}}{2u}}+{\frac {uv’}{v}}\right),\\{\mathfrak {t}}_{4}^{4}={\frac {1}{2\varkappa }}\left(-2v-{\frac {u’^{2}}{2u}}+2u’’+{\frac {uv’’}{v}}\right),\end{array}}\right}} (81)

§ 49. We must assume that in the gravitation fields really existing the quantities � � � {\displaystyle g_{ab}} have values differing very little from those which belong to a field without gravitation. In this latter we should have

�

� 3 ,

�

�

1 , {\displaystyle u=r^{3},\ v=w=1,}

and thus we put now

�

� 2 ( 1 + � ) ,

�

�

1 + �{\displaystyle u=r^{2}(1+\mu ),\ v=w=1+\nu }

where the quantities �{\displaystyle \mu } and �{\displaystyle \nu } which depend on � {\displaystyle r} are infinitely small, say of the first order, and their derivatives too. Neglecting quantities of the second order we find from (81)

� 1 1

1 2 � ( 2 + 2 � + 6 � � ′ + 2 � 2 � ″ + � 2 � ″ ) , � 3 3

1 � ( � − � + � � ′ + � � ′ ) , � 4 4

1 2 � ( 2 � − 2 � + 6 � � ′ + 2 � 2 � ″ + � 2 � ″ ) , {\displaystyle {\begin{array}{l}{\mathfrak {t}}{1}^{1}={\frac {1}{2\varkappa }}\left(2+2\mu +6r\mu ‘+2r^{2}\mu ‘’+r^{2}\nu ‘’\right),\\{\mathfrak {t}}{3}^{3}={\frac {1}{\varkappa }}\left(\mu -\nu +r\mu ‘+r\nu ‘\right),\\{\mathfrak {t}}_{4}^{4}={\frac {1}{2\varkappa }}\left(2\mu -2\nu +6r\mu ‘+2r^{2}\mu ‘’+r^{2}\nu ‘’\right),\end{array}}}

For our degree of approximation we may suppose that of the quantities � � � {\displaystyle T_{ab}} only � 44 {\displaystyle T_{44}} differs from 0. If we put ​

� 44

�{\displaystyle T_{44}=\varrho } (82) a quantity which depends on � {\displaystyle r} and which we shall assume to be zero outside a certain sphere, we find from the field equations

�

� { − 2 � ∫ 0 � � � � ∫ 0 � � 2 � � � − 1 � ∫ 0 � � 2 � � � + ∫ ∞ � � � � � } , �

� { − 1 � ∫ 0 � � 2 � � � + ∫ ∞ � � � � � } {\displaystyle {\begin{array}{c}\mu =\varkappa \left{-{\frac {2}{r}}\int \limits _{0}^{r}{\frac {dr}{r}}\int \limits _{0}^{r}r^{2}\varrho dr-{\frac {1}{r}}\int \limits _{0}^{r}r^{2}\varrho dr+\int \limits _{\infty }^{r}r\varrho dr\right},\\\nu =\varkappa \left{-{\frac {1}{r}}\int \limits _{0}^{r}r^{2}\varrho dr+\int \limits _{\infty }^{r}r\varrho dr\right}\end{array}}}

We thus obtain

� 1 1

1 � + ∫ ∞ � � � � � − 1 � ∫ 0 � � 2 � � � − 1 2 � 2 � , {\displaystyle {\mathfrak {t}}_{1}^{1}={\frac {1}{\varkappa }}+\int \limits _{\infty }^{r}r\varrho dr-{\frac {1}{r}}\int \limits _{0}^{r}r^{2}\varrho dr-{\frac {1}{2}}r^{2}\varrho ,} (83) � 3 3

0 ,

� 4 4

− 1 2 � 2 �{\displaystyle {\mathfrak {t}}{3}^{3}=0,\ {\mathfrak {t}}{4}^{4}=-{\frac {1}{2}}r^{2}\varrho } (84)

§ 50. If first we leave aside the first term of � 1 1 {\displaystyle {\mathfrak {t}}{1}^{1}}, which would also exist if no attracting matter were present, it is remarkable that the gravitation constant �{\displaystyle \varkappa } does not occur in the stress � 1 1 {\displaystyle {\mathfrak {t}}{1}^{1}} nor in the energy � 4 4 {\displaystyle {\mathfrak {t}}{4}^{4}}; the same would have been found if we had used other coordinates. This constitutes an important difference between Einstein’s theory and other theories in which attracting or repulsing forces are reduced to “field actions”. The pulsating spheres of Bjerknes e.g. are subjected to forces which, for a given motion, are proportional to the density of the fluid in which they are imbedded; and the changes of pressure and the energy in that fluid are likewise proportional to this density. In this case we shall therefore ascribe to the stress-energy-complex values proportional to the intensity of the actions which we want to explain. In Einstein’s theory such a proportionality does not exist. The value of � 4 4 {\displaystyle {\mathfrak {t}}{4}^{4}} is of the same order of magnitude as � 4 4 {\displaystyle {\mathfrak {T}}_{4}^{4}} in the matter. To our degree of approximation we find namely from (82) � 4 4

� 2 �{\displaystyle {\mathfrak {T}}_{4}^{4}=r^{2}\varrho }.

§ 51. If we had not worked with polar coordinates but with rectangular coordinates we should have had to put for the field without gravitation � 11

� 22

� 33

− 1 {\displaystyle g_{11}=g_{22}=g_{33}=-1}, � 44

1 {\displaystyle g_{44}=1}, � � �

0 {\displaystyle g_{ab}=0} for � ≠ � {\displaystyle a\neq b}. Then we should have found zero for all the components of the complex. In the system of coordinates used above we found for the field without gravitation � 1 1

1 �{\displaystyle {\mathfrak {t}}{1}^{1}={\tfrac {1}{\varkappa }}}; this is due to the complex � {\displaystyle {\mathfrak {t}}} being no tensor. If it were, the quantities � � � {\displaystyle {\mathfrak {t}}{a}^{b}} would be zero in every system of coordinates if they had that value in one system.

​It is also remarkable that in real eases the first term in (83) can be much larger than the following ones. If we consider e. g. a point � {\displaystyle P} outside the attracting sphere, we can prove that the ratio of the first term to the third is of the same order as the ratio of the square of the velocity of light to the square of the velocity with which a material point can describe a circular orbit passing through � {\displaystyle P}.

The following must also be noticed. In the system of polar coordinates used above there will exist in the field without gravitation the stress � 1 1

1 �{\displaystyle {\mathfrak {t}}_{1}^{1}={\tfrac {1}{\varkappa }}}. If a stress of this magnitude were produced by means of actions which give rise to a stress-energy-tensor, the passage to rectangular coordinates would give us a stress which becomes infinite at the point � {\displaystyle O}. In those coordinates we should namely have

� 1 ′ 1

sin 2 ⁡ � � 2 ⋅ 1 �{\displaystyle {\mathfrak {t}}_{1}^{‘1}={\frac {\sin ^{2}\vartheta }{r^{2}}}\cdot {\frac {1}{\varkappa }}}

§ 52. Evidently it would be more satisfactory if we could ascribe a stress-energy-tensor to the gravitation field. Now this can really be done. Indeed, the quantities � 0 ℎ � {\displaystyle {\mathfrak {s}}_{0h}^{e}} determined by (57) form a tensor and according to (58), (79) may be replaced by

� ℎ

− � � � ℎ � − � � � ℎ � 0 {\displaystyle K_{h}=-div_{h}{\mathfrak {T}}-div_{h}{\mathfrak {t}}{0}} (85) if � 0 {\displaystyle {\mathfrak {t}}{0}} is defined by a relation similar to (78), viz.

� 0 ℎ �

1 2 � � 0 ℎ � {\displaystyle {\mathfrak {t}}{0h}^{e}={\frac {1}{2\varkappa }}{\mathfrak {s}}{0h}^{e}} (86) Equation (85) shows that, just as well as � ℎ � {\displaystyle {\mathfrak {t}}{h}^{c}}, we may consider the quantities � 0 ℎ � {\displaystyle {\mathfrak {t}}{0h}^{e}} as the stresses etc. in the gravitation field. This way of interpretation is very simple. With a view to (41) we can namely derive from the equations for the gravitation field (65)

�

� � {\displaystyle G=\varkappa T}

and

� � �

− 1 � ( � � � − 1 2 � � � � ) {\displaystyle T_{ab}=-{\frac {1}{\varkappa }}\left(G_{ab}-{\frac {1}{2}}g_{ab}G\right)}

Further we find from (66)

� ℎ �

1 2 � � ∑ ( � ) � � � � � ℎ − 1 � ∑ ( � ) � � � � � ℎ {\displaystyle {\mathfrak {T}}{h}^{e}={\frac {1}{2\varkappa }}G\sum (a){\mathfrak {g}}^{ae}g{ah}-{\frac {1}{\varkappa }}\sum (a){\mathfrak {g}}^{ae}G_{ah}}

and from (57) and (86)

� 0 ℎ �

− � ℎ � {\displaystyle {\mathfrak {t}}{0h}^{e}=-{\mathfrak {T}}{h}^{e}} (87) At every point of the field-figure the components of the stress-energy-tensor of the gravitation field would therefore be equal to ​the corresponding quantities for the matter or the electro-magnetic system with the opposite sign. It is obvious that by this the condition of the conservation of momentum and energy for the whole system would be immediately fulfilled. It was in fact this circumstance that made me think of the tensor � 0

− � {\displaystyle {\mathfrak {t}}{0}=-{\mathfrak {T}}}. The way in which � 0 {\displaystyle {\mathfrak {s}}{0}} was introduced in §§ 38 and 39 has only been chosen in order to lay stress on (58) being an identity, so that equation (85) is but another form of (79).

At first sight the relations (87) and the conception to which they have led, may look somewhat startling. According to it we should have to imagine that behind the directly observable world with its stresses, energy etc. there is hidden the gravitation field with stresses, energy etc. that are everywhere equal and opposite to the former; evidently this is in agreement with the interchange of momentum and energy which accompanies the action of gravitation. On the way of a light-beam e.g. there would be everywhere in the gravitation field an energy current equal and opposite to the one existing in the beam. If we remember that this hidden energy-current can be fully described mathematically by the quantities � � � {\displaystyle g_{ab}} and that only the interchange just mentioned makes it perceptible to us, this mode of viewing the phenomena does not seem unacceptable. At all events we are forcibly led to it if we want to preserve the advantage of a stress-energy-tensor also for the gravitation field. It can namely be shown that a tensor which is transformed in the same way as the tensor � 0 {\displaystyle {\mathfrak {t}}{0}} defined by (57) and (86) and which in every system of coordinates has the same divergency as the latter, must coincide with � 0 {\displaystyle {\mathfrak {t}}{0}}.

Finally we may remark that (78), (86), (58), (87) give

� � �

�

� � �

� 0

− � � �

� {\displaystyle div\ {\mathfrak {t}}=div\ {\mathfrak {t}}_{0}=-div\ {\mathfrak {T}}}

so that we have, both from (79) and from (85), � ℎ

0 {\displaystyle K_{h}=0}.

The question is this, that, so long as the gravitation field is considered as given, we may introduce “external” forces, but that in the equations for the gravitation field itself we must also take into consideration the stress-energy-tensor of the system by which those forces are exerted. ​ IV.

(Communicated in the meeting of October 28, 1916).

§ 53. The expressions for the stress-energy-components of the gravitation field found in the preceding paper call for some further remarks. If by � ℎ � {\displaystyle \delta _{h}^{e}} we denote a quantity having the value 1 for �

ℎ {\displaystyle e=h} and being 0 for � ≠ ℎ {\displaystyle e\neq h}, those expressions can be written in the form (comp. equations (52) and (78))

� ℎ �

1 2 � { − � ℎ � � + ∑ ( � � ) ∂ � ∂ � � � , � � � � , ℎ + ∑ ( � � � ) ∂ � ∂ � � � , � � � � � , � ℎ − ∑ ( � � � ) ∂ � ∂ � � ( ∂ � ∂ � � � , � � ) � � � , ℎ } {\displaystyle {\mathfrak {t}}{h}^{e}={\frac {1}{2\varkappa }}\left{-\delta {h}^{e}Q+\sum (ab){\frac {\partial Q}{\partial g{ab,e}}}g{ab,h}+\sum (abf){\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,fh}-\sum (abf){\frac {\partial Q}{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,ef}}}\right)g_{ab,h}\right}} (88) They contain the first and second derivatives of the quantities � � � {\displaystyle g_{ab}}. Einstein on the contrary has given values for the stress-energy-components which contain the first derivatives only and which therefore are in many respects much more fit for application.

It will now be shown how we can also find formulae without second derivatives, if we start from (88).

§ 54. For this purpose we shall consider the complex � {\displaystyle {\mathfrak {u}}} defined by

� ℎ �

1 2 � { � ℎ � � − ∑ ( � � � ) ∂ ∂ � ℎ ( ∂ � ∂ � � � , � � � � � , � ) } {\displaystyle {\mathfrak {u}}{h}^{e}={\frac {1}{2\varkappa }}\left{\delta {h}^{e}Q-\sum (abf){\frac {\partial }{\partial x{h}}}\left({\frac {\partial Q}{\partial g{ab,fe}}}g_{ab,f}\right)\right}} (89) and we shall seek its divergency.

We have

( � � �

� ) ℎ

∑ ( � ) ∂ � � ∂ � �

1 2 � { ∂ � ∂ � ℎ − ∑ ( � � � � ) � 2 ∂ � � ∂ � ℎ ( ∂ � ∂ � � � , � � � � � , � ) } {\displaystyle (div\ {\mathfrak {u}}){h}=\sum (e){\frac {\partial {\mathfrak {u}}^{e}}{\partial x{e}}}={\frac {1}{2\varkappa }}\left{{\frac {\partial Q}{\partial x_{h}}}-\sum (abfe){\frac {Q^{2}}{\partial x_{e}\partial x_{h}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,f}\right)\right}}

or

( � � �

� ) ℎ

1 2 � ∂ � ∂ � ℎ {\displaystyle (div\ {\mathfrak {u}}){h}={\frac {1}{2\varkappa }}{\frac {\partial R}{\partial x{h}}}} (90) if we put

�

� − ∑ ( � � � � ) ∂ � ∂ � � ( ∂ � ∂ � � � , � � � � � , � ) {\displaystyle R=Q-\sum (abfe){\frac {\partial Q}{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,f}\right)} (91) Now �

− � � {\displaystyle Q={\sqrt {-g}}G} can be divided into two parts, the first of which � 1 {\displaystyle Q_{1}} contains differential coefficients of the quantities � � � {\displaystyle g_{ab}} of the first order only, while the second � 2 {\displaystyle Q_{2}} is a homogeneous linear function ​of the second derivatives of those quantities. This latter involves that, if we replace (91) by

�

� 1 + � 2 − ∑ ( � � � � ) ( ∂ � ∂ � � � , � � � � � , � � ) − ∑ ( � � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , � � ) � � � , � {\displaystyle R=Q_{1}+Q_{2}-\sum (abfe)\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,fe}\right)-\sum (abfe){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,f}}

the second and the third term annul each other. Thus

, � {\displaystyle R=Q_{1}-\sum (abfe){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,f}} (92) If now we define a complex � {\displaystyle {\mathfrak {v}}} by the equation

� ℎ �

− 1 2 � � ℎ � � {\displaystyle {\mathfrak {v}}_{h}^{e}=-{\frac {1}{2\varkappa }}\delta _{h}^{e}R} (93) we have

( � � �

� ) ℎ

− 1 2 � ∂ � ∂ � ℎ {\displaystyle (div\ {\mathfrak {v}}){h}=-{\frac {1}{2\varkappa }}{\frac {\partial R}{\partial x{h}}}} (94) If finally we put

� ′

� + � + � {\displaystyle {\mathfrak {t’=t+u+v}}}

we infer from (90) and (94)

� � �

� ′

� � �

� {\displaystyle div\ {\mathfrak {t}}’=div\ {\mathfrak {t}}} (95) and from (88), (89), (93) and (92)

� ℎ } {\displaystyle {\begin{array}{c}{\mathfrak {t}}{h}^{‘h}={\frac {1}{2\varkappa }}\left{-Q{1}+\sum (ab){\frac {\partial Q}{\partial g_{ab,h}}}g_{ab,h}-\sum (abf){\frac {\partial }{\partial x_{h}}}\left({\frac {\partial Q}{\partial g_{ab,fh}}}\right)g_{ab,f}-\right.\\\left.\sum (abf){\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,hf}}}\right)g_{ab,h}+\sum (abfe){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,f}\right}\end{array}}} (96) and for

Formula (95) shows that the quantities � ℎ ′ � {\displaystyle {\mathfrak {t}}{h}^{’e}} can be taken just as well as the expressions (88) for the stress-energy-components and we see from (96) and (97) that these new expressions contain only the first derivatives of the coefficients � � � {\displaystyle g{ab}}; they are homogeneous quadratic functions of these differential coefficients.

This becomes clear when we remember that � 1 {\displaystyle Q_{1}} is a function of this kind and that only � 1 {\displaystyle Q_{1}} contributes something to the second term of (96) and the first of (97); further that the derivatives of � {\displaystyle Q} occurring in the following terms contain only the quantities � � � {\displaystyle g_{ab}} and not their derivatives.

§ 55. Einstein’s stress-energy-components have a form widely different from that of the above mentioned ones. They are ​ � ( � ) ℎ �

1 2 � � ℎ � ∑ ( � � � � ) � � � Γ � � � Γ � � � − 1 � ∑ ( � � � ) � � � Γ � � � Γ � ℎ � {\displaystyle {\mathfrak {t}}_{(E)h}^{e}={\frac {1}{2\varkappa }}\delta _{h}^{e}\sum (abcf)g^{ab}\Gamma _{ac}^{f}\Gamma _{bf}^{c}-{\frac {1}{\varkappa }}\sum (abc)g^{ab}\Gamma _{ac}^{e}\Gamma _{bh}^{c}}

where for the sake of simplicity it has been assumed that − �

1 {\displaystyle {\sqrt {-g}}=1}. Further we have

Γ � � �

− { � � � }

− ∑ ( � ) � � � [ � � � ] {\displaystyle \Gamma _{ab}^{c}=-\left{{\begin{array}{c}ab\c\end{array}}\right}=-\sum (e)g^{ce}\left[{\begin{array}{c}ab\e\end{array}}\right]}

If now our formulae (96) and (97) are likewise simplified by the assumption − �

1 {\displaystyle {\sqrt {-g}}=1} (so that � {\displaystyle Q} becomes equal to � {\displaystyle G}), we may expect that � ′ {\displaystyle {\mathfrak {t}}’} will become identical with � ( � ) {\displaystyle {\mathfrak {t}}_{(E)}}. This is really so in the case � � �

0 {\displaystyle g_{ab}=0} for � ≠ � {\displaystyle a\neq b}; by which it seems very probable that the agreement will exist in general.

In the preceding paper it was shown already that the stress-energy-components � ℎ � {\displaystyle {\mathfrak {t}}{h}^{e}} do not form a “tensor”, but what was called a “complex”. The same may be said of the quantities � ℎ ′ � {\displaystyle {\mathfrak {t}}{h}^{’e}} defined by (96) and (97) and of the expressions given by Einstein. If we want a stress-energy-tensor, there are only left the quantities � 0 ℎ � {\displaystyle {\mathfrak {t}}{0h}^{e}} defined by (86) and (57), the values of which are always equal and opposite to the corresponding stress-energy-components � ℎ � {\displaystyle {\mathfrak {T}}{h}^{e}} for the matter or the electromagnetic field.

It must be noticed that the four equations

∑ ( � ) ∂ ∂ � � ( � ℎ � + � ( � ) ℎ � )

0 {\displaystyle \sum (e){\frac {\partial }{\partial x_{e}}}\left({\mathfrak {T}}{h}^{e}+{\mathfrak {T}}{(g)h}^{e}\right)=0}

always express the same relations, whether we choose � 0 ℎ � ,

� ℎ � ,

� ℎ ′ � {\displaystyle {\mathfrak {t}}{0h}^{e},\ {\mathfrak {t}}{h}^{e},\ {\mathfrak {t}}{h}^{’e}} or � ( � ) ℎ � {\displaystyle {\mathfrak {t}}{(E)h}^{e}} as stress-energy-components � ( � ) ℎ � {\displaystyle {\mathfrak {T}}{(g)h}^{e}} of the gravitation field. If however in a definite case we want to use the equations in order to calculate how the momentum and the energy of the matter and the electromagnetic field change by the gravitational actions, it is best to use � ℎ ′ � {\displaystyle {\mathfrak {t}}{h}^{’e}} or � ( � ) ℎ � {\displaystyle {\mathfrak {t}}{(E)h}^{e}}, just because these quantities are homogeneous quadratic functions of the derivatives � � � , � {\displaystyle g{ab,c}}.

Experience namely teaches us that the gravitation fields occurring in nature may be regarded as feeble, in this sense that the values of the � � � {\displaystyle g_{ab}}’s are little different from those which might be assumed if no gravitation field existed. For these latter values, which will be called the “normal” ones, we may write in orthogonal coordinates

� 11

� 22

� 32

− 1 ,

� 44

� 2 ,

� � �

0 , for � ≠ � {\displaystyle g_{11}=g_{22}=g_{32}=-1,\ g_{44}=c^{2},\ g_{ab}=0,\quad {\textrm {for}}\quad a\neq b} (98) In a first approximation, which most times will be sufficient, the deviations of the values of the � � � {\displaystyle g_{ab}}’s from these normal ones may be taken proportional to the gravitation constant �{\displaystyle \varkappa }. This factor also appears in the differential coefficients � � � , � {\displaystyle g_{ab,c}}; hence, according to the character of the functions � ℎ ′ � {\displaystyle {\mathfrak {t}}_{h}^{’e}} mentioned above (and on account ​of the factor 1 �{\displaystyle {\tfrac {1}{\varkappa }}} in (96) and (97)) these functions become proportional to �{\displaystyle \varkappa }, so that in a feeble gravitation field they have low values.

§ 56. Because of the complicated form of equations (96) and (97), we shall confine ourselves to the calculation for some cases of � 4 ′ 4 {\displaystyle {\mathfrak {t}}{4}^{‘4}}, i.e. of the energy per unit of volume. This calculation is considerably simplified if we consider stationary fields only. Then all differential coefficients with respect to � 4 {\displaystyle x{4}} vanish, so that we have according to (96)

� 4 ′ 4

1 2 � { − � 1 + ∑ ( � � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , � � ) � � � , � } {\displaystyle {\mathfrak {t}}{4}^{‘4}={\frac {1}{2\varkappa }}\left{-Q{1}+\sum (abfe){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,f}\right}} (99) We shall work out the calculation, first for a field without gravitation and secondly for the case of an attracting spherical body in which the matter is distributed symmetrically round the centre.

If there is no gravitation field we may take for the quantities � � � {\displaystyle g_{ab}} the “normal” values. For the case of orthogonal coordinates these are given by (98). When we want to use the polar coordinates introduced into § 48 we have the corresponding formulae

� 11

− � 2 1 − � 1 2 ,

� 22

− � 2 ( 1 − � 1 2 ) ,

� 33

− 1 ,

� 44

� 2 , � � �

0 , for � ≠ � } {\displaystyle \left.{\begin{array}{c}g_{11}=-{\frac {r^{2}}{1-x_{1}^{2}}},\ g_{22}=-r^{2}\left(1-x_{1}^{2}\right),\ g_{33}=-1,\ g_{44}=c^{2},\\g_{ab}=0,\quad {\textrm {for}}\quad a\neq b\end{array}}\right}} (100) If, using polar coordinates, we have to do with an attracting sphere and if we take its centre as origin, we may put

� 11

− � 1 − � 1 2 ,

� 22

− ( 1 − � 1 2 ) � ,

� 33

− � ,

� 44

� , {\displaystyle g_{11}=-{\frac {u}{1-x_{1}^{2}}},\ g_{22}=-\left(1-x_{1}^{2}\right)u,\ g_{33}=-v,\ g_{44}=w,} (101) where � , � , � {\displaystyle u,v,w} are functions of � {\displaystyle r}. The � � � {\displaystyle g_{ab}}’s which belong to an orthogonal system of coordinates may be expressed in the same functions.

These � � � {\displaystyle g_{ab}}’s are

� 11

− � � 2 + � 1 2 � 2 ( � � 2 − � ) ,

� � � . � 12

� 1 � 2 � 2 ( � � 2 − � ) ,

� � � . � 14

� 24

� 34

0 ,

� 44

� . {\displaystyle {\begin{array}{l}g_{11}=-{\frac {u}{r^{2}}}+{\frac {x_{1}^{2}}{r^{2}}}\left({\frac {u}{r^{2}}}-v\right),\ etc.\\g_{12}={\frac {x_{1}x_{2}}{r^{2}}}\left({\frac {u}{r^{2}}}-v\right),\ etc.\\g_{14}=g_{24}=g_{34}=0,\ g_{44}=w.\end{array}}}

The “etc.” means that for � 22 , � 33 {\displaystyle g_{22},g_{33}} we have similar expressions as for � 11 {\displaystyle g_{11}} and for � 23 , � 31 {\displaystyle g_{23},g_{31}} similar ones as for � 12 {\displaystyle g_{12}}.

§ 57. In order to deduce the differential equations determining � , � , � {\displaystyle u,v,w} we may arbitrarily use rectangular or polar coordinates; the latter however are here to be preferred. If differentiations ​with respect to � {\displaystyle r} are indicated by accents, we have according to (40) and (101)

� 11

1 1 − � 1 2 ( − 1 + � ″ 2 � − � ′ � ′ 4 � 2 + � ′ � ′ 4 � � ) , � 22

( 1 − � 1 2 ) ( − 1 + � ″ 2 � − � ′ � ′ 4 � 2 + � ′ � ′ 4 � � ) , � 33

� ″ � − � ′ 2 2 � 2 − � ′ � ′ 2 � � − � ′ � ′ 4 � � + � ″ 2 � − � ′ 2 4 � 2 , � 44

− � ′ � ′ 2 � � + � ′ � ′ 4 � 2 − � ″ 2 � + � ′ 2 4 � � , {\displaystyle {\begin{array}{l}G_{11}={\frac {1}{1-x_{1}^{2}}}\left(-1+{\frac {u’’}{2v}}-{\frac {u’v’}{4v^{2}}}+{\frac {u’w’}{4vw}}\right),\\G_{22}=\left(1-x_{1}^{2}\right)\left(-1+{\frac {u’’}{2v}}-{\frac {u’v’}{4v^{2}}}+{\frac {u’w’}{4vw}}\right),\\G_{33}={\frac {u’’}{u}}-{\frac {u’^{2}}{2u^{2}}}-{\frac {u’v’}{2uv}}-{\frac {v’w’}{4vw}}+{\frac {w’’}{2w}}-{\frac {w’^{2}}{4w^{2}}},\\G_{44}=-{\frac {u’w’}{2uv}}+{\frac {v’w’}{4v^{2}}}-{\frac {w’’}{2v}}+{\frac {w’^{2}}{4vw}},\end{array}}}

� � �

0 {\displaystyle G_{ab}=0} for � ≠ � {\displaystyle a\neq b}

So we have found the left hand sides of the field equations (65). Before considering these equations more closely we shall introduce the simplification that the � � � {\displaystyle g_{ab}}’s, are very little different from the normal values (100). For these latter we have

�

� 2 ,

�

1 ,

�

� 2 {\displaystyle u=r^{2},\ v=1,\ w=c^{2}} (102) and therefore we now put

�

� 2 ( 1 + � ) ,

�

1 + � ,

�

� 2 ( 1 + � ) {\displaystyle u=r^{2}(1+\lambda ),\ v=1+\mu ,\ w=c^{2}(1+\nu )} (103) The quantities � , � , �{\displaystyle \lambda ,\mu ,\nu }, which depend on r, will be regarded as infinitely small of the first order and in the field equations we shall neglect quantities of second and higher orders.

Then we may write for � 11 {\displaystyle G_{11}} etc.

� 11

1 1 − � 1 2 ( � + 2 � � ′ + 1 2 � 2 � ″ − � − 1 2 � � ′ + 1 2 � � ′ ) � 22

1 − � 1 2 ( � + 2 � � ′ + 1 2 � 2 � ″ − � − 1 2 � � ′ + 1 2 � � ′ ) � 33

2 � � ′ + � ″ − 1 � � ′ + 1 2 � ″ , � 44

− � 2 ( 1 � � ′ + 1 2 � ″ ) {\displaystyle {\begin{array}{l}G_{11}={\frac {1}{1-x_{1}^{2}}}\left(\lambda +2r\lambda ‘+{\frac {1}{2}}r^{2}\lambda ‘’-\mu -{\frac {1}{2}}r\mu ‘+{\frac {1}{2}}r\nu ‘\right)\\G_{22}=1-x_{1}^{2}\left(\lambda +2r\lambda ‘+{\frac {1}{2}}r^{2}\lambda ‘’-\mu -{\frac {1}{2}}r\mu ‘+{\frac {1}{2}}r\nu ‘\right)\\G_{33}={\frac {2}{r}}\lambda ‘+\lambda ‘’-{\frac {1}{r}}\mu ‘+{\frac {1}{2}}\nu ‘’,\\G_{44}=-c^{2}\left({\frac {1}{r}}\nu ‘+{\frac {1}{2}}\nu ‘’\right)\end{array}}}

On the right hand-sides of the field equations (65) we may take for � � � {\displaystyle g_{ab}} the normal value; moreover we shall take for � � � {\displaystyle T_{ab}} and � {\displaystyle T} the values which hold for a system of incoherent material points. We may do so if we assume no other internal stresses but those caused by the mutual attractions; these stresses may be neglected in the present approximation.

As we supposed the attracting matter to be at rest we have according to (10), (16) and (15) (1915) � 1

� 2

� 3

0 {\displaystyle w_{1}=w_{2}=w_{3}=0}, � 4

�{\displaystyle w_{4}=\varrho }, � 1

� 2

� 3

0 {\displaystyle u_{1}=u_{2}=u_{3}=0}, � 4

� 2 �{\displaystyle u_{4}=c^{2}\varrho }, �

� �{\displaystyle P=c\varrho }.

In the notations we are now using we have further, according to (23) (1915), ​ � ℎ �

� ℎ � � � {\displaystyle {\mathfrak {T}}{h}^{e}={\frac {u{h}w_{e}}{P}}}

so that of the stress-energy-components of the matter only one is different from zero, namely

� 4 4

� �{\displaystyle {\mathfrak {T}}_{4}^{4}=c\varrho }

Further (66) involves that, also of the quantities � � � {\displaystyle T_{ab}}, only one, namely � 44 {\displaystyle T_{44}}, is not equal to zero. As we may put − �

� � 2 {\displaystyle {\sqrt {-g}}=cr^{2}} we have namely

� 44

� 2 � 2 � ,

�

1 � 2 �{\displaystyle T_{44}={\frac {c^{2}}{r^{2}}}\varrho ,\ T={\frac {1}{r^{2}}}\varrho }

Finally we are led to the three differential equations

�

2 � � ′ + 1 2 � 2 � ″ − � − 1 2 � � ′ + 1 2 � � ′

− 1 2 � �{\displaystyle \lambda =2r\lambda ‘+{\frac {1}{2}}r^{2}\lambda ‘’-\mu -{\frac {1}{2}}r\mu ‘+{\frac {1}{2}}r\nu ‘=-{\frac {1}{2}}\varkappa \varrho } (104) 2 � � ′ + � 2 � ″ − � � ′ + 1 2 � � ″

− 1 2 � �{\displaystyle 2r\lambda ‘+r^{2}\lambda ‘’-r\mu ‘+{\frac {1}{2}}r\nu ‘’=-{\frac {1}{2}}\varkappa \varrho } (105) � � ′ + 1 2 � 2 � ″

1 2 � �{\displaystyle r\nu ‘+{\frac {1}{2}}r^{2}\nu ‘’={\frac {1}{2}}\varkappa \varrho } (106) It may be remarked that � � � 1 � � 2 � � 3 {\displaystyle \varrho dx_{1}dx_{2}dx_{3}}, represents the “mass” present in the element of volume � � 1 � � 2 � � 3 {\displaystyle dx_{1}dx_{2}dx_{3}}. Because of the meaning of � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}} (§ 48) the mass in the shell between spheres with radii � {\displaystyle r} and � + � � {\displaystyle r+dr} is found when � � � 1 � � 2 � � 3 {\displaystyle \varrho dx_{1}dx_{2}dx_{3}} is integrated with respect to � 1 {\displaystyle x_{1}} between the limits —1 and +1 and with respect to � 2 {\displaystyle x_{2}} between 0 and 2 �{\displaystyle 2\pi }. As �{\displaystyle \varrho } depends on � {\displaystyle r} only, this latter mass becomes 4 � � � � {\displaystyle 4\pi \varrho dr}, so that �{\displaystyle \varrho } is connected with the “density” in the ordinary sense of the word, which will be called � ¯{\displaystyle {\overline {\varrho }}}, by the equation

�

� 2 � ¯{\displaystyle \varrho =r^{2}{\overline {\varrho }}}

The differential equations also hold outside the sphere if �{\displaystyle \varrho } is put equal to zero. We can first imagine �{\displaystyle \varrho } to change gradually to near the surface and then treat the abrupt change as a limiting case.

In all the preceding considerations we have tacitly supposed the second derivatives of the quantities � � � {\displaystyle g_{ab}} to have everywhere finite values. Therefore �{\displaystyle \nu } and � ′ {\displaystyle \nu ‘} will be continuous at the surface, even in the case of an abrupt change.

§ 58. Equation (106) gives

� ′

� � 2 ∫ 0 � �

� � {\displaystyle \nu ‘={\frac {\varkappa }{r^{2}}}\int \limits _{0}^{r}\varrho \ dr} (107) where the integration constant is determined by the consideration that for �

0 {\displaystyle r=0} all the quantities � � � {\displaystyle g_{ab}} and their derivatives must be finite, so that for �

0 {\displaystyle r=0} the product � 2 � ′ {\displaystyle r^{2}\nu ‘} must be zero. As it is natural to suppose that at an infinite distance �{\displaystyle \nu } vanishes, we find further ​

�

� ∫ ∞ � � � � 2 ∫ 0 � � � � {\displaystyle \nu =\varkappa \int \limits _{\infty }^{r}{\frac {dr}{r^{2}}}\int \limits _{0}^{r}\varrho dr} (108) The quantities �{\displaystyle \lambda } and �{\displaystyle \mu } on the contrary are not completely determined by the differential equations. If namely equations (105) and (106) are added to (104) after having been multiplied by − 1 2 {\displaystyle -{\tfrac {1}{2}}} and + 1 2 {\displaystyle +{\tfrac {1}{2}}} respectively, we find

� + � � ′ − � + � � ′

0 {\displaystyle \lambda +r\lambda ‘-\mu +r\nu ‘=0} (109) and it is clear that (104) and (105) are satisfied as soon as this is the case with this condition (109) and with (106). So we have only to attend to (108) and (109). The indefiniteness remaining in �{\displaystyle \lambda } and �{\displaystyle \mu } is inevitable on account of the covariancy of the field equations. It does not give rise to any difficulties.

Equation (107) teaches us that near the centre

� ′

1 3 � � 0 ¯ � {\displaystyle \nu ‘={\frac {1}{3}}\varkappa {\overline {\varrho _{0}}}r}

if � 0 ¯{\displaystyle {\overline {\varrho _{0}}}} is the density at the centre, whereas from (108) we find a finite value for �{\displaystyle \nu } itself. This confirms what has been said above about the values at the centre. We shall assume that at that point � , �{\displaystyle \lambda ,\mu } and their derivatives have likewise finite values. Moreover we suppose (and this agrees with (109)) that � , � , � ′ {\displaystyle \lambda ,\mu ,\lambda ‘} and � ′ {\displaystyle \mu ‘} are continuous at the surface of the sphere.

If � {\displaystyle a} is the radius of the sphere we find from (108) for an external point

�

− � � ∫ 0 � � � � {\displaystyle \nu =-{\frac {\varkappa }{r}}\int \limits _{0}^{a}\varrho dr}

Without contradicting (109) we may assume that at a great distance from the centre �{\displaystyle \lambda } and �{\displaystyle \mu } are likewise proportional to 1 � {\displaystyle {\tfrac {1}{r}}}, so that � ′ {\displaystyle \lambda ‘} and � ′ {\displaystyle \mu ‘} decrease proportionally to 1 � 2 {\displaystyle {\tfrac {1}{r^{2}}}}.

§ 59. We can now continue the calculation of � 4 ′ 4 {\displaystyle {\mathfrak {t}}_{4}^{‘4}} (§ 56). Substituting (101) in (99) and using polar coordinates we find

� 4 ′ 4

− 1 2 � � � � ( 1 2 � ′ 2 � 2 + � ′ � ′ � � ) {\displaystyle {\mathfrak {t}}_{4}^{‘4}=-{\frac {1}{2\varkappa }}u{\sqrt {\frac {w}{v}}}\left({\frac {1}{2}}{\frac {u’^{2}}{u^{2}}}+{\frac {u’w’}{uw}}\right)}

whence by substituting (102) we derive for a field without gravitation

� 4 ′ 4

− � �{\displaystyle {\mathfrak {t}}_{4}^{‘4}=-{\frac {c}{\varkappa }}}

This equation shows that, working with polar coordinates, we ​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 � {\displaystyle r} and � + � � {\displaystyle r+dr} becomes

− 4 � � � � � {\displaystyle -{\frac {4\pi c}{\varkappa }}dr}

The density of the energy in the ordinary sense of the word would be inversely proportional to � 2 {\displaystyle r^{2}}, so that it would become infinite at the centre.

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 � � � {\displaystyle g_{ab}} are then constants and their derivatives vanish.

§ 60. Using rectangular coordinates we shall now indicate the form of � 4 ′ 4 {\displaystyle {\mathfrak {t}}_{4}^{‘4}} for the field of a spherical body, with the approximation specified in § 57. Thus we put

� 11

− ( 1 + � ) + � 1 2 � 2 ( � − � ) ,

� � � . � 12

� 1 � 2 � 2 ( � − � ) ,

� � � . � 14

� 24

� 34

0 ,

� 44

� 2 ( 1 + � ) } {\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[34] ​

� 4 ′ 4

� 2 � { � ′ 2 + 1 � ( � − � ) [ 1 � ( � − � ) + 2 ( � ′ − � ′ ) ] } {\displaystyle {\mathfrak {t}}{4}^{‘4}={\frac {c}{2\varkappa }}\left{\nu ‘^{2}+{\frac {1}{r}}(\lambda -\mu )\left[{\frac {1}{r}}(\lambda -\mu )+2(\lambda ‘-\mu ‘)\right]\right}} (111) Thus we see (comp. § 58) that at a distance from the attracting sphere � 4 ′ 4 {\displaystyle {\mathfrak {t}}{4}^{‘4}} decreases proportionally to 1 � 4 {\displaystyle {\tfrac {1}{r^{4}}}}. Further it is to be noticed that on account of the indefiniteness pointed out in § 58, there remains some uncertainty as to the distribution of the energy over the space, but that nevertheless the total energy of the gravitation field

�

4 � ∫ 0 ∞ � 4 ′ 4 � 2 � � {\displaystyle E=4\pi \int \limits {0}^{\infty }{\mathfrak {t}}{4}^{‘4}r^{2}dr}

has a definite value.

Indeed, by the integration the last terra of (111) vanishes. After multiplication by � 2 {\displaystyle r^{2}} this term becomes namely

( � − � ) 2 + 2 � ( � − � ) ( � ′ − � ′ )

� � � [ � ( � − � ) 2 ] {\displaystyle (\lambda -\mu )^{2}+2r(\lambda -\mu )(\lambda ‘-\mu ‘)={\frac {d}{dr}}\left[r(\lambda -\mu )^{2}\right]}

The integral of this expression is 0 because (comp. §§ 57 and 58) � ( � − � ) 2 {\displaystyle r(\lambda -\mu )^{2}} is continuous at the surface of the sphere and vanishes both for �

0 {\displaystyle r=0} and for �

∞{\displaystyle r=\infty }.

We have thus

�

� � � ∫ 0 ∞ � ′ 2 � 2 � � {\displaystyle E={\frac {\pi c}{\varkappa }}\int \limits _{0}^{\infty }\nu ‘^{2}r^{2}dr} (112) where the value (107) can be substituted for � ′ {\displaystyle \nu ‘}. If e.g. the density � ¯{\displaystyle {\overline {\varrho }}} is everywhere the same all over the sphere, we have at an internal point

� ′

1 3 � � ¯ � {\displaystyle \nu ‘={\frac {1}{3}}\varkappa {\overline {\varrho }}r}

and at an external point

� ′

1 3 � � ¯ � 3 � 2 {\displaystyle \nu ‘={\frac {1}{3}}\varkappa {\overline {\varrho }}{\frac {a^{3}}{r^{2}}}}

From this we find

�

2 15 � � � � ¯ 2 � 5 {\displaystyle E={\frac {2}{15}}\pi c\varkappa {\overline {\varrho }}^{2}a^{5}}

§ 61. The general equation (99) found for � 4 ′ 4 {\displaystyle {\mathfrak {t}}_{4}^{‘4}} can be transformed in a simple way. We have namely

∑ ( � � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , � � ) � � � , �

∑ ( � � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , � � � � � , � ) − − ∑ ( � � � � ) ∂ � ∂ � � � , � � � � � , � � {\displaystyle {\begin{array}{c}\sum (abfe){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,f}=\sum (abfe){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,f}\right)-\\-\sum (abfe){\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,fe}\end{array}}}

and we may write − � 2 {\displaystyle -Q_{2}} (§ 54) for the last term. Hence ​

� 4 ′ 4

1 2 � { − � + ∑ ( � � � � ) ∂ ∂ � � ( ∂ � ∂ � � � , � � � � � , � ) } {\displaystyle {\mathfrak {t}}{4}^{‘4}={\frac {1}{2\varkappa }}\left{-Q+\sum (abfe){\frac {\partial }{\partial x{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,f}\right)\right}} (113) where we must give the values 1, 2, 3 to � {\displaystyle e} and � {\displaystyle f}.

The gravitation energy lying within a closed surface consists therefore of two parts, the first of which is

� 1

− 1 2 � ∫ �

� � 1 � � 2 � � 3 {\displaystyle E_{1}=-{\frac {1}{2\varkappa }}\int Q\ dx_{1}dx_{2}dx_{3}} (114) while the second can be represented by surface integrals. If namely � 1 , � 2 , � 3 {\displaystyle q_{1},q_{2},q_{3}} are the direction constants of the normal drawn outward

� 2

1 2 � ∑ ( � � � � ) ∂ � ∂ � � � , � � � � � , � � � � �{\displaystyle E_{2}={\frac {1}{2\varkappa }}\sum (abfe){\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,f}q_{e}d\sigma } (115) In the case of the infinitely feeble gravitation field represented by � , � , �{\displaystyle \lambda ,\mu ,\nu } (§ 57) both expressions � 1 {\displaystyle E_{1}} and � 2 {\displaystyle E_{2}} contain quantities of the first order, but it can easily be verified that these cancel each other in the sum, so that, as we knew already, the total energy is of the second order.

From �

− � � {\displaystyle Q={\sqrt {-g}}G} and the equations of § 32 we find namely

∂ � ∂ � � � , � �

1 2 − � ( 2 � � � � � � − � � � � � � − � � � � � � ) {\displaystyle {\frac {\partial Q}{\partial g_{ab,fe}}}={\frac {1}{2}}{\sqrt {-g}}\left(2g^{ab}g^{fe}-g^{bf}g^{ae}-g^{af}g^{be}\right)} (116) so that we can write

� 2

1 4 � ∫ − � ∑ ( � � � � ) ( 2 � � � � � � − � � � � � � − � � � � � � ) � � � , � � � � �{\displaystyle E_{2}={\frac {1}{4\varkappa }}\int {\sqrt {-g}}\sum (abfe)\left(2g^{ab}g^{fe}-g^{bf}g^{ae}-g^{af}g^{be}\right)g_{ab,f}q_{e}d\sigma } The factor � � � , � {\displaystyle g_{ab,f}} is of the first order. Thus, if we confine ourselves to that order, we may take for all the other quantities these normal values. Many of these are zero and we find

� 2

− � 2 � ∑ ( � � ) ∫ � � � ( � � � , � − � � � , � ) � � � �{\displaystyle E_{2}=-{\frac {c}{2\varkappa }}\sum (ae)\int g^{aa}\left(g_{aa,e}-g_{ae,a}\right)q_{e}d\sigma } (117) Here we must take �

1 , 2 , 3 , 4 {\displaystyle a=1,2,3,4}; �

1 , 2 , 3 {\displaystyle e=1,2,3}, while we remark that for �

� {\displaystyle a=e} the expression between brackets vanishes. For �

4 {\displaystyle a=4} the integral becomes ∫ ∂ � ∂ � � � � � �{\displaystyle \int {\frac {\partial \nu }{\partial x_{e}}}q_{e}d\sigma } do, which after summation with respect to � {\displaystyle e} gives

∫ ∂ � ∂ � � �{\displaystyle \int {\frac {\partial \nu }{\partial n}}d\sigma } (118) � {\displaystyle n} representing the normal to the surface. If � {\displaystyle a} and � {\displaystyle e} differ from each other, while neither of them is equal to 4, we can deduce from (110) and (109)

� � � , � − � � � , �

∂ � ∂ � � {\displaystyle g_{aa,e}-g_{ae,a}={\frac {\partial \nu }{\partial x_{e}}}}

​Each value of � {\displaystyle e} occurring twice, i.e. combined with the two values different from � {\displaystyle e} which � {\displaystyle a} can take, we have in addition to (118) − 2 ∫ ∂ � ∂ � � �{\displaystyle -2\int {\frac {\partial \nu }{\partial n}}d\sigma }

so that (117) becomes

� 2

� 2 � ∫ ∂ � ∂ � � �{\displaystyle E_{2}={\frac {c}{2\varkappa }}\int {\frac {\partial \nu }{\partial n}}d\sigma }

As now outside the sphere

�

− � � ∫ 0 � �

� � {\displaystyle \nu =-{\frac {\varkappa }{r}}\int \limits _{0}^{a}\varrho \ dr}

we have for every closed surface that does not surround the sphere � 2

0 {\displaystyle E_{2}=0}, but for every surface that does

� 2

2 � � ∫ 0 � �

� � {\displaystyle E_{2}=2\pi c\int \limits {0}^{a}\varrho \ dr} (119) As to � 1 {\displaystyle E{1}} we remark that substituting (65) in (41) and taking into consideration (64) we find,

�

� � ,

�

� − � � {\displaystyle G=\varkappa T,\ Q=\varkappa {\sqrt {-g}}T} (120) From this we conclude that � 1 {\displaystyle E_{1}} is zero if there is no matter inside the surface �{\displaystyle \sigma }. In order to determine � 1 {\displaystyle E_{1}} in the opposite case, we remember that � {\displaystyle G} is independent of the choice of coordinates. To calculate this quantity we may therefore use the value of � {\displaystyle T} indicated in § 56, which is sufficient to calculate � 1 {\displaystyle E_{1}} as far as the terms of the first order. We have therefore

�

� � 2 �{\displaystyle G={\frac {\varkappa }{r^{2}}}\varrho }

and if, using further on rectangular coordinates, we take for − � {\displaystyle {\sqrt {-g}}} the normal value � {\displaystyle c},

�

� � � 2 �{\displaystyle Q={\frac {c\varkappa }{r^{2}}}\varrho }

From this we find by substitution in (114) for the case of the closed surface a surrounding the sphere

� 1

− 2 � � ∫ 0 � �

� � {\displaystyle E_{1}=-2\pi c\int \limits _{0}^{a}\varrho \ dr}

This equation together with (119) shows that in (113) when integrated over the whole space the terms of the first order really cancel each other. In order to calculate those of the second order ​and thus to derive the result (112) from (113), we should have to determine the quantity � {\displaystyle T} (comp. 120)), accurately to the order �{\displaystyle \varkappa }. The surface integrals in (115) too would have to be considered more closely. We shall not however dwell upon this.

§ 62. From the expression for � 4 ′ 4 {\displaystyle {\mathfrak {t}}_{4}^{‘4}} given in (113) and the value

�

� 1 + � 2 {\displaystyle E=E_{1}+E_{2}}

derived from it, it can be inferred that, though � ′ {\displaystyle {\mathfrak {t}}’} is no tensor, we yet may change a good deal in the system of coordinates in which the phenomena are described, without altering the value of the total energy. Let us suppose e.g. that � 4 {\displaystyle x_{4}} is left unchanged but that, instead of the rectangular coordinates � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}} hitherto used, other quantities � 1 ′ , � 2 ′ , � 3 ′ {\displaystyle x_{1}’,x_{2}’,x_{3}’} are introduced, which are some continuous function of � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}}, with the restriction that � 1 ′

� 1 , � 2 ′

� 2 , � 3 ′

� 3 {\displaystyle x’{1}=x{1},x’{2}=x{2},x’{3}=x{3}} outside a certain closed surface surrounding the attracting matter at a sufficient distance. If we use these new coordinates, we shall have to introduce other quantities � � � ′ {\displaystyle g’{ab}} instead of � � � {\displaystyle g{ab}} however outside the closed surface the quantities � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}} and their derivatives do not change, the value of � 1 {\displaystyle E_{1}} will approach the same limit as when we used the coordinates � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}}, if the surface �{\displaystyle \sigma } for which it is calculated expands indefinitely. The value which we find for � 1 {\displaystyle E_{1}} after the transformation of coordinates will also be the same as before. Indeed, if � �{\displaystyle d\tau } is an element of volume expressed in � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}}-units and � � ′ {\displaystyle d\tau ‘} the same element expressed in � 1 ′ , � 2 ′ , � 3 ′ {\displaystyle x_{1}’,x_{2}’,x_{3}’}-units, while � ′ {\displaystyle Q’} represents the new value of � {\displaystyle Q}, we have

� � �

� ′ � � ′ {\displaystyle Qd\tau =Q’d\tau ‘}

It is clear that the total energy will also remain unchanged if � 1 ′ , � 2 ′ , � 3 ′ {\displaystyle x_{1}’,x_{2}’,x_{3}’} differ from � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}} at all points, provided only that these differences decrease so rapidly with increasing distance from the attracting body, that they have no influence on the limit of the expression (115).

The result which we have now found admits of another interpretation. In the mode of description which we first followed (using � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}}), �{\displaystyle \varrho }[35]) and � � � {\displaystyle g_{ab}} are certain functions of � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}}; in the new one � ′ {\displaystyle \varrho ‘}, � � � ′ {\displaystyle g’{ab}} are certain other functions of � 1 ′ , � 2 ′ , � 3 ′ {\displaystyle x{1}’,x_{2}’,x_{3}’}. If now, without leaving the system of coordinates � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}}, we ascribe to the density and to the gravitation potentials values which depend on � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}}, in the same way as � ′ {\displaystyle \varrho ‘}, � � � ′ {\displaystyle g’{ab}} depended on � 1 ′ , � 2 ′ , � 3 ′ {\displaystyle x{1}’,x_{2}’,x_{3}’} just now, we shall obtain a new system (consisting of the attracting body and the gravitation field) which is different from the original system ​because other functions of the coordinates occur in it, but which nevertheless no observation will be able to discern from it, the indefiniteness which is a necessary consequence of the covariancy of the field equations, again presenting itself.

What has been said shows that the total gravitation energy in this new system will have the same value as in the original one, as has been found already in § 60 with the restrictions then introduced.

§ 63. If � ′ {\displaystyle {\mathfrak {t}}’} were a tensor, we should have for all substitutions the transformation formulae given at the end of § 40. In reality this is not the case now, but from (96) and (97) we can still deduce that those formulae hold for linear substitutions. They may likewise be applied to the stress-energy-components of the matter or of an electromagnetic system. Hence, if � � � {\displaystyle {\mathfrak {T}}_{a}^{b}} represents the total stress-energy-components, i. e. quantities in which the corresponding components for the gravitation field, the matter and the electromagnetic field are taken together, we have for any linear transformation

1 − � ′ � � ′ �

1 − � ∑ ( � � ) � � � � � � � � � {\displaystyle {\frac {1}{\sqrt {-g’}}}{\mathfrak {T}}{c}^{‘b}={\frac {1}{\sqrt {-g}}}\sum (kl)p{kc}\pi {lb}{\mathfrak {T}}{k}^{l}} (121) We shall apply this to the case of a relativity transformation, which can be represented by the equations

� 1 ′

� � 1 + � � � 4 ,

� 2 ′

� 2 ,

� 3 ′

� 3 ,

� 4 ′

� � 4 + � � � 1 {\displaystyle x’{1}=ax{1}+bcx_{4},\ x’{2}=x{2},\ x’{3}=x{3},\ x’{4}=ax{4}+{\frac {b}{c}}x_{1}} (122) with the relation

� 2 − � 2

1 {\displaystyle a^{2}-b^{2}=1} (123) In doing so we shall assume that the system, when described in the rectangular coordinates � 1 , � 2 , � 3 {\displaystyle x_{1},x_{2},x_{3}} and with respect to the time � 4 {\displaystyle x_{4}}, is in a stationary state and at rest.

Then we derive from (97)[36] ​ � 1 ′ 4

� 2 ′ 4

� 3 ′ 4

0 ;

� 4 ′ 1

� 4 ′ 2

� 4 ′ 3

0 {\displaystyle {\mathfrak {t}}{1}^{‘4}={\mathfrak {t}}{2}^{‘4}={\mathfrak {t}}{3}^{‘4}=0;\ {\mathfrak {t}}{4}^{‘1}={\mathfrak {t}}{4}^{‘2}={\mathfrak {t}}{4}^{‘3}=0}

which means that in the system ( � 1 , � 2 , � 3 , � 4 ) {\displaystyle \left(x_{1},x_{2},x_{3},x_{4}\right)} there are neither momenta nor energy currents in the gravitation field.

We may assume the same for the matter, so that we have for the total stress-energy-components in the system ( � 1 , � 2 , � 3 , � 4 ) {\displaystyle \left(x_{1},x_{2},x_{3},x_{4}\right)}

� 1 4

� 2 4

� 3 4

0 ;

� 4 1

� 4 2

� 4 3

0 {\displaystyle {\mathfrak {T}}{1}^{4}={\mathfrak {T}}{2}^{4}={\mathfrak {T}}{3}^{4}=0;\ {\mathfrak {T}}{4}^{1}={\mathfrak {T}}{4}^{2}={\mathfrak {T}}{4}^{3}=0}

Let us now consider especially the components � 1 ′ 4 , � 4 ′ 1 {\displaystyle {\mathfrak {T}}{1}^{‘4},{\mathfrak {T}}{4}^{‘1}} and � 4 ′ 4 {\displaystyle {\mathfrak {T}}{4}^{‘4}} in the system ( � 1 ′ , � 2 ′ , � 3 ′ , � 4 ′ ) {\displaystyle \left(x’{1},x’{2},x’{3},x’_{4}\right)} For these we find from (121) and (122)

� 1 ′ 4

� � � � 1 1 − � � � � 4 4 ,

� 4 ′ 1

− � � �

� 1 1 + � � �

� 4 4 {\displaystyle {\mathfrak {T}}{1}^{‘4}={\frac {ab}{c}}{\mathfrak {T}}{1}^{1}-{\frac {ab}{c}}{\mathfrak {T}}{4}^{4},\ {\mathfrak {T}}{4}^{‘1}=-abc\ {\mathfrak {T}}{1}^{1}+abc\ {\mathfrak {T}}{4}^{4}} (124) � 4 ′ 4

− � 2 � 1 1 + � 2 � 4 4 {\displaystyle {\mathfrak {T}}{4}^{‘4}=-b^{2}{\mathfrak {T}}{1}^{1}+a^{2}{\mathfrak {T}}{4}^{4}} (125) It is thus seen in the first place that between the momentum in the direction of � 1 ( − � 1 ′ 4 ) {\displaystyle x{1}\left(-{\mathfrak {T}}{1}^{‘4}\right)} and the energy-current in that direction ( � 4 ′ 1 ) {\displaystyle \left({\mathfrak {T}}{4}^{‘1}\right)} there exists the relation

� 4 ′ 1

− � 2 � 1 ′ 4 {\displaystyle {\mathfrak {T}}{4}^{‘1}=-c^{2}{\mathfrak {T}}{1}^{‘4}}

well known from the theory of relativity.

Further we have for the total energy in the system ( � 1 ′ , � 2 ′ , � 3 ′ , � 4 ′ ) {\displaystyle \left(x’{1},x’{2},x’{3},x’{4}\right)}

� ′

∫ � 4 ′ 4 � � 1 ′ � � 2 ′ � � 3 ′ {\displaystyle E’=\int {\mathfrak {T}}{4}^{‘4}dx’{1}dx’{2}dx’{3}}

where the integration has to be performed for a definite value of the time � 4 ′ {\displaystyle x’_{4}}. On account of (122) we may write for this

� ′

1 � ∫ � 4 ′ 4 � � 1 � � 2 � � 3 {\displaystyle E’={\frac {1}{a}}\int {\mathfrak {T}}{4}^{‘4}dx{}{1}dx{}{2}dx{}{3}}

where we have to keep in view a definite value of the time � 4 {\displaystyle x_{4}}.

If the value (125) is substituted here and if we take into consideration that, the state being stationary in the system ( � 1 , � 2 , � 3 , � 4 ) {\displaystyle \left(x_{1},x_{2},x_{3},x_{4}\right)},

∫ � 1 1 � � 1 � � 2 � � 3

0 {\displaystyle \int {\mathfrak {T}}{1}^{1}dx{}{1}dx{}{2}dx{}{3}=0}

we have

� ′

� � {\displaystyle E’=aE}

if � {\displaystyle E} is the energy ascribed to the system in the coordinates ( � 1 , � 2 , � 3 , � 4 ) {\displaystyle \left(x_{1},x_{2},x_{3},x_{4}\right)}. By integration of the first of the expressions (124) we find in the same way for the total momentum in the direction of � 1 {\displaystyle x_{1}}

� ′

� � � {\displaystyle G’={\frac {b}{c}}E}

​§ 64. Equations (122) show that in the coordinates ( � 1 ′ , � 2 ′ , � 3 ′ , � 4 ′ ) {\displaystyle \left(x’{1},x’{2},x’{3},x’{4}\right)} the system has a velocity of translation � � � {\displaystyle {\tfrac {bc}{a}}} in the direction of � 1 ′ {\displaystyle x’_{1}}. If this velocity is denoted by � {\displaystyle v}, we have according to (123)

�

1 1 − � 2 � 2 {\displaystyle a={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}

If therefore we put

�

� � 2 {\displaystyle M={\frac {E}{c^{2}}}}

we find

� ′

� � 2 1 − � 2 � 2 ,

� ′

� � 1 − � 2 � 2 {\displaystyle E’={\frac {Mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\ G’={\frac {Mv}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}} (126) When the system moves as a whole we may therefore ascribe to it an energy and a momentum which depend on the velocity of translation in the way known from the theory of relativity. The quantity � {\displaystyle M}, to which the energy of the gravitation field also contributes a certain part, may be called the “mass” of the system. From what has been said in § 62 it follows that within certain limits it depends on the way in which the system and the gravitation field are described.

It must be remarked however that, if for the gravitation field we had chosen the stress-energy-tensor � 0 {\displaystyle {\mathfrak {t}}_{0}} (§ 52), the total energy of the system even when in motion would be zero. The same would be true of the total momentum and we should have to put �

0 {\displaystyle M=0}.

At first sight it may seem strange that we may arbitrarily ascribe to the moving system the momentum determined by (126) or a momentum 0; one might be inclined to think that, when a definite system of coordinates has been chosen, the momentum must have a definite value, which might be determined by an experiment in which the system is brought to rest by “external” forces. We must remember however (comp. § 52) that in the theory of gravitation we may introduce no “external” forces without considering also the material system � ′ {\displaystyle S’} in which they originate. This system � ′ {\displaystyle S’} together with the system � {\displaystyle S} with which we were originally concerned, will form an entity, in which there is a gravitation field, part of which is due to � ′ {\displaystyle S’} (and a part also to the simultaneous existence of � {\displaystyle S} and � ′ {\displaystyle S’}). There is no doubt that we may apply the above considerations to the total system ( � , � ′ {\displaystyle S,S’}) without being led into contradiction with any observation.

A. Einstein, Zur allgemeinen Relativitätstheorie, Berliner Sitzungsberichte 1915, pp. 778 799; Die Feldgleichungen der Gravitation, ibid. 1915, p. 844. D. Hilbert, Die Grundlagen der Physik I, Göttinger Nachrichten, Math.-phys. Klasse, Nov. 1915. It will be known that in the theory of relativity Minkowski was the first who used this geometric representation in an extension of four dimensions. The name “world-line” has been borrowed from him. For the sake of simplicity we shall imagine the two motions not to be disturbed by this coincidence, so that e.g. two material points penetrate each other or pass each other at an extremely small distance without any mutual influence. In a correspondence I had with him. In other terms, that the data procured by astronomical observations can be extended arbitrarily and unboundedly. A “surface” determined by one equation between the coordinates is a three-dimensional extension. It will cause no confusion if sometimes we apply the name of “plane” to certain two-dimensional extensions, if we speak e.g. of the “plane” determined by two line-elements. This corresponds to the negative value which (1) gives for � � 2 {\displaystyle ds^{2}}. For a radius-vector on the asymptotic cone we may take either of these values; this makes no difference, as the numerical value of a line-element in the direction of such a radius-vector becomes 0 in both cases. This agrees with the value of the Lagrangian function, which is to be found e.g. in my paper on “Hamilton’s principle in Einstein’s theory of gravitation.” These Proceedings 19 (1916). p. 751. If, according to circumstances, different signs arc given to R {\displaystyle \mathrm {R} }, the angle whose sine occurs in the formula for the area of a parallelogram must be understood to be positive in one case and negative in the other. From § 10 it follows that if the length of a vector A {\displaystyle \mathrm {A} } that is represented by a line (§ 17) coincides with a radius-vector of the conjugate indicatrix, it is always represented by an imaginary number. We may however obtain a vector which in natural units is represented by a real number e.g. by 1 (§ 13) if we multiply the vector A {\displaystyle \mathrm {A} } by an imaginary factor, which means that its components and also those of a vector product in which it occurs are multiplied by that factor. In the above considerations difficulties might arise if the vector N {\displaystyle \mathrm {N} } lay on the asymptotic cone of the indicatrix, our definition of a vector of the value 1 would then fail (comp. note 2, p. 1345). With a view to this we can choose the form of the extension Ω{\displaystyle \Omega } (§ 13) in such a way that this case does not occur, a restriction leading to a boundary with sharp edges. Zittingsverslag Akad. Amsterdam, 23 (1915), p. 1073; translated in Proceedings Amsterdam, 19 (1910), p. 751. Further on this last paper will be cited by l. c. Comp. § 7, l. c. For the infinitesimal quantities � � {\displaystyle x_{a}} occurring in (19) we have namely (comp. (30)) � � ′

∑ ( � ) � � � � � {\displaystyle x’_{a}=\sum (b)\pi {ba}x{b}}

and taking into consideration (19) and (20), i e.

� �

∑ ( � ) � � � � � ,

� �

∑ ( � ) � � � � � {\displaystyle \xi {a}=\sum (b)g{ab}x_{b},\ x_{a}=\sum (b)\gamma _{ba}\xi _{b}}

and formula (7) l. c, we may write (comp. note 2, p. 758, l. c.)

� � ′

∑ ( � ) � � � ′ � � ′

∑ ( � � � � ) � � � � � � � � � � � � � �

= ∑ ( � � ) � � � � � � � �

∑ ( � � � ) � � � � � � � � � � �

∑ ( � ) � � � � � {\displaystyle {\begin{array}{l}\xi ‘{a}=\sum (b)g’{ab}x’{b}=\sum (bcde)p{ca}p_{db}\pi {eb}g{cd}x_{e}=\\\qquad =\sum (cd)p_{ca}g_{cd}x_{d}=\sum (cdf)p_{ca}g_{cd}\gamma _{fd}\xi {f}=\sum (c)p{ca}\xi _{c}\end{array}}}

Put Ξ � � Ξ � � �

� � � {\displaystyle \Xi _{a}^{I}\Xi _{b}^{II}=\vartheta _{ab}}. Then we have � � � ′

Ξ � ′ � Ξ � ′ � �

∑ ( � � ) � � � � � � Ξ � � Ξ � � �

∑ ( � � ) � � � � � � � � � {\displaystyle \vartheta ‘_{ab}=\Xi {a}^{‘I}\Xi {b}^{‘II}=\sum (cd)p{ca}p{db}\Xi {c}^{I}\Xi {d}^{II}=\sum (cd)p{ca}p{db}\vartheta _{cd}}

and similar formulae for the other three parts of (25).

Comp. (28) l. c. Published September 1916, a revision having been found desirable. See Proceedings Vol. XIX, p. 1341 and 1354. Namely: � ′ � �

∑ ( � � ) � � � � � � � � � {\displaystyle g’^{lk}=\sum (ab)\pi _{ak}\pi _{bl}g^{ab}}

The symbol ( � � � ) {\displaystyle \left(g^{kl}\right)} denotes the complex of all the quantities � � � {\displaystyle g^{kl}}.

Namely: � � � ′

∑ ( � � ) � � � � � � � � � {\displaystyle G’{im}=\sum (ab)p{ai}p_{bm}G_{ab}}

On account of the relation − � ′ � � ′

− � � � {\displaystyle {\sqrt {-g’}}dS’={\sqrt {-g}}dS}

Similarly: � � �

� � � ,

� � �

� � � {\displaystyle g^{ba}=g^{ab},\ {\mathfrak {g}}^{ba}={\mathfrak {g}}^{ab}}

This means that the transformation formulae for these quantities have the form ( � � , � � ) ′

∑ ( � � � � ) � � � � � � � � � � � � ( � � , � � ) {\displaystyle (ik,lm)’=\sum (abce)p_{ai}p_{bk}p_{cl}p_{em}(ab,ce)}

See for the notations used here and for some others to be used later on my communication in Zittingsverslag Akad Amsterdam 23 (1915), p. 1073 (translated in Proceedings Amsterdam 19 (1916), p. 751). In referring to the equations and the articles of this paper I shall add the indication 1915.

Suppose that at the boundary of the domain of integration � � � �

0 {\displaystyle \delta g_{ab}=0} and � � � � , �

0 {\displaystyle \delta g_{ab,e}=0}. Then we have also � � � �

0 {\displaystyle \delta {\mathfrak {g}}^{ab}=0} and � � � � , �

0 {\displaystyle \delta {\mathfrak {g}}^{ab,e}=0}, so that ∫ ( � 2 � ) � �

0 ,

∫ � 2 � � �

0 {\displaystyle \int \left(\delta _{2}Q\right)dS=0,\ \int \delta _{2}QdS=0}

and from

∫ ( � � ) � �

∫ � �

� � {\displaystyle \int (\delta Q)dS=\int \delta Q\ dS}

we infer

∫ ( � 1 � ) � �

∫ � 1 �

� � {\displaystyle \int (\delta _{1}Q)dS=\int \delta _{1}Q\ dS}

As this must hold for every choice of the variations � � � � {\displaystyle \delta g_{ab}} (by which choice the variations � � � � {\displaystyle \delta {\mathfrak {g}}_{ab}} are determined too) we must have at each point of the field-figure

( � 1 � )

� 1 � {\displaystyle (\delta _{1}Q)=\delta _{1}Q}

This may be made clear by a reasoning similar to that used in the preceding note. We again suppose � � � � {\displaystyle \delta g_{ab}} and � � � � , � {\displaystyle \delta g_{ab,e}} to be zero at the boundary of the domain of integration. Then � � � � ′ {\displaystyle \delta g’{ab}} and � � � � , � ′ {\displaystyle \delta g’{ab,e}} vanish too at the boundary, so that ∫ � 2 � ′

� � ′

0 ,

∫ � 2 �

� �

0 {\displaystyle \int \delta _{2}Q’\ dS’=0,\ \int \delta _{2}Q\ dS=0}

From

∫ � � ′

� � ′

∫ � �

� � {\displaystyle \int \delta Q’\ dS’=\int \delta Q\ dS}

we may therefore conclude that

∫ � 1 � ′

� � ′

∫ � 1 �

� � {\displaystyle \int \delta _{1}Q’\ dS’=\int \delta _{1}Q\ dS}

As this must hold for arbitrarily chosen variations � � � � {\displaystyle \delta g_{ab}} we have the equation

� 1 � ′

� � ′

� 1 �

� � {\displaystyle \delta _{1}Q’\ dS’=\delta _{1}Q\ dS}

Einstein uses the word “divergency” in a somewhat different sense. It seemed desirable however to have a name for the left hand side of (54) and it was difficult to find a better one. This has also been done by de Donder, Zittingsverslag Akad. Amsterdam, 35 (1916), p. 153. The notations � � � , � � � ¯{\displaystyle \psi _{ab},{\overline {\psi _{ab}}}} and � � � ∗{\displaystyle \psi _{ab}^{*}} (see (27), (29) and § 11, 1915), will however be preserved though they do not correspond to those of Einstein. As to formulae (59) and (60) it is to be understood that if � {\displaystyle p} and � {\displaystyle q} are two of the numbers 1, 2, 3, 4, � ′ {\displaystyle p’} and � ′ {\displaystyle q’} denote the other two in such a way that the order �

�

� ′

� ′ {\displaystyle p\ q\ p’\ q’} is obtained from 1 2 3 4 by an even number of permutations of two ciphers. If � 1 , � 2 , � 3 , � 4 {\displaystyle x_{1},x_{2},x_{3},x_{4}} are replaced by � , � , � , � {\displaystyle x,y,z,t} and if for the stresses the usual notations � � , � � {\displaystyle X_{x},X_{y}}, etc., are used (so that e.g. for a surface element � �{\displaystyle d\sigma } perpendicular to the axis of � , � � {\displaystyle x,X_{x}} is the first component of the force per unit of surface which the part of the system situated on the positive side of � �{\displaystyle d\tau } exerts on the opposite part) then � 1 1

� � , � 1 2

� � {\displaystyle {\mathfrak {T}}{1}^{1}=X{x},{\mathfrak {T}}{1}^{2}=X{y}}, etc. Further − � 1 4 , − � 2 4 , − � 3 4 {\displaystyle -{\mathfrak {T}}{1}^{4},-{\mathfrak {T}}{2}^{4},-{\mathfrak {T}}{3}^{4}} are the components of the momentum per unit of volume and � 4 1 , � 4 2 , � 4 3 {\displaystyle {\mathfrak {T}}{4}^{1},{\mathfrak {T}}{4}^{2},{\mathfrak {T}}{4}^{3}} the components of the energy-current. Finally � 4 4 {\displaystyle {\mathfrak {T}}_{4}^{4}} is the energy per unit of volume. The quantities � � � {\displaystyle \gamma _{ab}} in that equation are the same as those which are now denoted by � � � {\displaystyle g^{ab}}. In the cases considered in § 43, � L {\displaystyle \delta \mathrm {L} } can indeed be represented in this way. To make the notation agree with that of § 38 � {\displaystyle b} has been replaced by � {\displaystyle e}. Of the laborious calculation it may be remarked here only that it is convenient to write the values (110) in the form � 11

− 1 + � + ∂ 2 � ∂ � 1 2 ,

� � � . {\displaystyle g_{11}=-1+\alpha +{\frac {\partial ^{2}\beta }{\partial x_{1}^{2}}},\ etc.}

� 12

∂ 2 � ∂ � 1 ∂ � 2 ,

� � � . {\displaystyle g_{12}={\frac {\partial ^{2}\beta }{\partial x_{1}\partial x_{2}}},\ etc.}

where �{\displaystyle \alpha } and �{\displaystyle \beta } are infinitesimal functions of � {\displaystyle r}. We then find

which reduces to (111) if the relations between � , �{\displaystyle \alpha ,\beta } and � , �{\displaystyle \gamma ,\mu }, viz.

and the equality � ′

� ′ {\displaystyle \alpha ‘=\nu ‘} involved in (109) are taken into consideration.

By �{\displaystyle \varrho } we mean here what was denoted by � ¯{\displaystyle {\bar {\varrho }}} in § 56. We have � 14

� 24

� 34

0 {\displaystyle g_{14}=g_{24}=g_{34}=0}, while all the other quantities gab are independent of � 4 {\displaystyle x_{4}}. Thus we can say that the quantities � � � {\displaystyle g_{ab}} and � � � , � {\displaystyle g_{ab,c}} are equal to zero when among their indices the number 4 occurs an odd number of times. The same may be said of � � � {\displaystyle g^{ab}}, � � � , � {\displaystyle g^{ab,c}}, ∂ � ∂ � � � , � � {\displaystyle {\tfrac {\partial Q}{\partial g_{ab,cd}}}} (according to (116)), ∂ ∂ � � ( ∂ � ∂ � � � , � � ) {\displaystyle {\tfrac {\partial }{\partial x_{k}}}\left({\tfrac {\partial Q}{\partial g_{ab,cd}}}\right)} and also of products of two or more of such quantities. As in the last two terms of (97) the indices � , � {\displaystyle a,b} and � {\displaystyle f} occur twice, these terms will vanish when only one of the indices � {\displaystyle e} and ℎ {\displaystyle h} has the value 4. As to the first term of (97) we remark that, according to the formulae of § 32, each of the indices � , � {\displaystyle a,b} and � {\displaystyle e} occurs only once in the differential coefficient of � {\displaystyle Q} with respect to � � � , � {\displaystyle g_{ab,e}}, while other indices are repeated. As to the number of times which � , ℎ {\displaystyle e,h} and the other indices occur we can therefore say the same of the first term of (97) as of the other terms. The first term also is therefore zero, if no more than one of the two indices � {\displaystyle e} and ℎ {\displaystyle h} has the value 4. That � 4 ′ � {\displaystyle t{}_{4}^{’e}} vanishes for � ≠ 4 {\displaystyle e\neq 4} is seen immediately.