# On Einstein's Theory of gravitation

"On Einstein's Theory of gravitation." By Prof. H. A. Lorentz.

I.

(Communicated in the meeting of February 26, 1916).

§ 1. In pursuance of his important researches on gravitation Einstein has recently attained the aim which he had constantly kept in view; he has succeeded in establishing equations whose form is not changed by an arbitrarily chosen change of the system of coordinates[1]. Shortly afterwards, working out an idea that had been expressed already in one of Einstein's papers, Hilbert[2] has shown the use that may be made of a variation law that may be regarded as Hamilton's principle in a suitably generalized form. By these results the "general theory of relativity" may be said to have taken a definitive form, though much remains still to be done in further developing it and in applying it to special problems. It will also be desirable to present the fundamental ideas in a form as simple as possible.

In this communication it will be shown that a four-dimensional geometric representation may be of much use for this latter purpose; by means of it we shall be able to indicate for a system containing a number of material points and an electromagnetic field (or eventually only one of these) the quantity ${\displaystyle H}$, which occurs in the variation theorem, and which we may call the principal function. This quantity consists of three parts, of which the first relates to the material points, the second to the electromagnetic field and the third to the gravitation field itself.

As to the material points, it will be assumed that the only connexion between them is that which results from their mutual gravitational attraction.

§ 2. We shall be concerned with a four-dimensional extension ${\displaystyle R_{4}}$, in which "space" and "time" are combined, so that each point ${\displaystyle P}$ in it indicates a definite place ${\displaystyle A}$ and at the same time a definite moment of time ${\displaystyle t}$. If we say that ${\displaystyle P}$ refers to a material point we mean that at the time ${\displaystyle t}$ this point is found at the place ${\displaystyle A}$. In the course of time the material point is represented every moment by a new point ${\displaystyle P}$; all these points lie on the "world-line", which represents the state of motion (or eventually the state of rest) of the material point[3]. In the same sense we may speak of the world-line of a propagated light-vibration. An intersection of two world-lines means that the two objects to which they belong meet at a certain moment, that a "coincidence" takes place[4]. Now Einstein has made the striking remark[5] that the only thing we can learn from our observations and with which our theories are essentially concerned, is the existence of these coincidences. Let us suppose e.g. that we have observed an occultation of a star by the moon or rather the reappearance of a star at the moon's border. Then the world-line of a certain light-vibration starting from a point on the world-line of the star has in its further course intersected the world-line of a point of the border of the moon and finally that of the observer's eye. A similar remark may be made when the moment of reappearance is read on a clock. Let us suppose that the light-vibration itself lights the dial-plate, reaching it when the hand is at the point ${\displaystyle a}$; then we may say that three world-lines, viz. that of the light-vibration, that of the hand and that of the point ${\displaystyle a}$ intersect.

§ 3. We may imagine that, in order to investigate a gravitation field as e.g. that of the sun, a great number of material points, moving in all directions and with different velocities, are thrown into it, that light-beams are also made to traverse the field and that all coincidences are noted[6]. It would be possible to represent the results of these observations by world-lines in a four-dimensional figure — let us say in a "field-figure" — the lines being drawn in such a way that each observed coincidence is represented by an intersection of two lines and that the points of intersection of one line with a number of the others succeed each other in the right order.

Now, as we have to attend only to the intersections, we have a great degree of liberty in the construction of the "field-figure". If, independently of each other, two persons were to describe the same observations, their figures would probably look quite different and if these figures were deformed in an arbitrary way, without break of continuity, they would not cease to serve the purpose.

After having constructed a field-figure ${\displaystyle F}$ we may introduce "coordinates", by which we mean that to each point ${\displaystyle P}$ we ascribe four numbers ${\displaystyle x_{1},x_{2},x_{3},x_{4}}$, in such a way that along any line in the field-figure these numbers change continuously and that never two different points get the same four numbers. Having done this we may for each point ${\displaystyle P}$ seek a point ${\displaystyle P'}$ in a four-dimensional extension ${\displaystyle R'_{4}}$, in which the numbers ${\displaystyle x_{1},\dots x_{4}}$ ascribed to ${\displaystyle P}$ are the Cartesian coordinates of the point ${\displaystyle P'}$. In this way we obtain in ${\displaystyle R'_{4}}$ a figure ${\displaystyle F'}$, which just as well as ${\displaystyle F}$ can serve as field-figure and which of course may be quite different according to the choice of the numbers ${\displaystyle x_{1},\dots x_{4}}$, that have been ascribed to the points of ${\displaystyle F}$.

If now it is true that the coincidences only are of importance it must be possible to express the fundamental laws of the phenomena by geometric considerations referring to the field-figure, in such a way that this mode of expression is the same for all possible field-figures; from our point of view all these figures can be considered as being the same. In such a geometric treatment the introduction of coordinates will be of secondary importance; with a single exception (§ 13) it only serves for short calculations which we have to intercalate (for the proof of certain geometric propositions) and for establishing the final equations, which have to be used for the solution of special problems. In the discussion of the general principles coordinates play no part; and it is thus seen that the formulation of these principles can take place in the same way whatever be our choice of coordinates. So we are sure beforehand of the general covariancy of the equations that was postulated by Einstein.

§ 4. Einstein ascribes to a line-element ${\displaystyle PQ}$ in the field-figure a length ${\displaystyle ds}$ defined by the equation

 ${\displaystyle {\begin{array}{c}ds^{2}=\sum (ab)g_{ab}dx_{a}dx_{b}\\\\\left(g_{ab}=g_{ba}\right)\end{array}}}$ (1)

Here ${\displaystyle dx_{1},\dots dx_{4}}$ are the changes of the coordinates when we pass from ${\displaystyle P}$ to ${\displaystyle Q}$, while the coefficients ${\displaystyle g_{ab}}$ depend in one way or another on the coordinates. The gravitation field is known when these 10 quantities are given as functions of ${\displaystyle x_{1},\dots x_{4}}$. Here it must be remarked that in all real cases the coordinates can be chosen in such a way that for one point arbitrarily chosen (1) becomes

${\displaystyle ds^{2}=-dx_{1}^{2}-dx_{2}^{2}-dx_{3}^{2}+dx_{4}^{2}}$

This requires that the determinant ${\displaystyle g}$ of the coefficients of (1) be always negative. The minor of this determinant corresponding to the coefficient ${\displaystyle g_{ab}}$ will be denoted by ${\displaystyle G_{ab}}$.

Around each point ${\displaystyle P}$ of the field-figure as a centre we may now construct an infinitesimal surface[7], which, when ${\displaystyle P}$ is chosen as origin of coordinates, is determined by the equation

 ${\displaystyle \sum (ab)g_{ab}x_{a}x_{b}=\epsilon ^{2}}$ (2)

where ${\displaystyle \epsilon }$ is an infinitely small positive constant which we shall fix once for all. This surface, which we shall call the indicatrix, is a hyperboloid with one real axis and three imaginary ones. We shall also introduce the surface determined by the equation

 ${\displaystyle \sum (ab)g_{ab}x_{a}x_{b}=-\epsilon ^{2}}$ (3)

which differs from (2) only by the sign of ${\displaystyle \epsilon ^{2}}$. We shall call this the conjugate indicatrix. It is to be understood that the indicatrices and conjugate indicatrices take part in the changes to which the field-figure may be subjected. As these surfaces are infinitely small, they always remain hyperboloids of the said kind. The gravitation field will now be determined by these indicatrices, which we can imagine to have been constructed in the field-figure without the introduction of coordinates. When we have occasion to use these latter, we shall so choose them that the "axes" ${\displaystyle x_{1},x_{2},x_{3}}$ intersect the conjugate indicatrix constructed around their starting point, while the indicatrix itself is intersected by the axis ${\displaystyle x_{4}}$. This involves that the coefficients ${\displaystyle g_{11},g_{22},g_{33}}$ are negative and that ${\displaystyle g_{44}}$ is positive.

§ 5. The indicatrices will give us the units in which we shall express the length of lines in the field-figure and the magnitude of two-, three or four-dimensional extensions. When we use these units we shall say that the quantities in question are expressed in natural measure.

In the case of a line-element ${\displaystyle PQ}$ the unit might simply be the radius-vector in the direction ${\displaystyle PQ}$ of the indicatrix or the conjugate indicatrix described about ${\displaystyle P}$. It is however desirable to distinguish the two cases that ${\displaystyle PQ}$ intersects the indicatrix itself or the conjugate indicatrix. In the latter case we shall ascribe an imaginary length to the line-element[8]. Besides, by taking as unit not the radius-vector itself but a length proportional to it, the numerical value of a line-element may be made to be independent of the choice of the quantity ${\displaystyle \epsilon }$.

These considerations lead us to define the length that will be ascribed to line-elements by the assumption that each radius-vector of the indicatrix has in natural measure the length ${\displaystyle \epsilon }$, while each radius-vector of the conjugate indicatrix has the length ${\displaystyle i\epsilon }$.[9]

It will now be clear that the length of an arbitrary line in the field-figure can be found by integration, each of its elements being measured by means of the indicatrix or the conjugate indicatrix belonging to the position of the element. In virtue of our definitions a deformation of the field-figure will not change the length of lines expressed in natural measure and a geodetic line will remain a geodetic line.

§ 6. We are now in a position to indicate the first part ${\displaystyle H_{1}}$ of the principal function (§ 1). Let ${\displaystyle \sigma }$ be a closed surface in the field-figure and let us confine ourselves to the principal function so far as it belongs to the space ${\displaystyle \Omega }$ enclosed by that surface. Then the quantity ${\displaystyle H_{1}}$ is the sum, taken with the negative sign, of the lengths of all world-lines of material points so far as they lie within ${\displaystyle \Omega }$, each length multiplied by a constant ${\displaystyle m}$, characteristic of the point in question and to be called its mass.[10]

It must be remarked that the elements of the world-lines of material points intersect the corresponding indicatrices themselves. The lengths of these lines are therefore real positive quantities.

A deformation of the field-figure leaves ${\displaystyle H_{1}}$ unchanged.

§ 7. We shall now pass on to the part of the principal function belonging to the gravitation field. The mathematical expression for this part was communicated to me by Einstein in our correspondence. It is also to be found in Hilbert's paper in which it is remarked that the quantity in question may be regarded as the measure of the curvature of the four-dimensional extension to which (1) relates. Here we have to speak only of the interpretation of this quantity. To find this the following geometrical considerations may be used.

Let ${\displaystyle PQ}$ and ${\displaystyle PR}$ be two line-elements starting from a point ${\displaystyle P}$ of the field-figure, ${\displaystyle QR}$ the line-element joining the extremities ${\displaystyle Q}$ and ${\displaystyle R}$. If then the lengths of these elements in natural measure are

${\displaystyle PQ=ds',\ PR=ds'',\ QR=ds}$

we define the angle ${\displaystyle (s',s'')}$ between ${\displaystyle PQ}$ and ${\displaystyle PR}$ by the well known trigonometric formula

 ${\displaystyle {\begin{array}{c}ds^{2}=ds'^{2}+ds''^{2}-2ds'ds''\cos(s',s'')\\\\\cos(s',s'')={\frac {ds'^{2}+ds''^{2}-ds^{2}}{2ds'ds''}}\end{array}}}$ (4)

from which one can derive

 ${\displaystyle \cos(s',s'')=\sum (ab)g_{ab}{\frac {dx'_{a}}{ds'}}{\frac {dx''_{b}}{ds''}}}$ (5)

By means of this formula we are able to determine the angle between any two intersecting lines. Of course the two other angles of the triangle ${\displaystyle PQR}$ can be calculated in the same way.

Now two cases must be distinguished.

a. The plane of the triangle ${\displaystyle PQR}$ cuts the conjugate indicatrix, but not the indicatrix itself. Then the three sides have positive imaginary values. Moreover each of them proves to be smaller than the sum of the others, from which one finds that the angles have real values and that their sum is ${\displaystyle \pi }$.

b. The plane PQR cuts both the indicatrix and the conjugate indicatrix. In this case different positions of the triangle are still possible. We can however confine ourselves to triangles the three sides of which are real. These are really possible, for in the plane of a hyperbola we can draw triangles the sides of which are parallel to radius-vectors drawn from the centre to points of the curve (and not of the conjugate hyperbola).

By a closer consideration of the triangles now in question it is found however that by the choice of our "natural" units one side is necessarily longer than the sum of the other two. Formula (4) then shows that the cosines of the angles are real quantities, greater than 1 in absolute value, two of them being positive, and the third negative. We must therefore ascribe to the angles imaginary or complex values. If for ${\displaystyle p>+1}$ we put

${\displaystyle \arccos p=i\log \left(p+{\sqrt {p^{2}-1}}\right)}$

and

${\displaystyle \arccos(-p)=\pi -\arccos p}$

we find for the three angles expressions of the form

${\displaystyle i\alpha ,\ i\beta }$ and ${\displaystyle \pi -i(\alpha +\beta )}$

so that the sum is again ${\displaystyle \pi }$.

From the cosine calculated by (4) or (5) the sine can be derived by means of the formula

${\displaystyle \sin \varphi ={\sqrt {1-\cos ^{2}\varphi }}}$

where for the case ${\displaystyle \cos ^{2}\varphi >1}$ we can confine ourselves to the value

${\displaystyle \sin \varphi =i{\sqrt {\cos ^{2}\varphi -1}}}$

with the positive sign.

It deserves special notice that two conjugate radius-vectors of the indicatrix and the conjugate indicatrix are perpendicular to each other and that a deformation of the field-figure does not change the angle between two intersecting lines determined according to our definitions.

§ 8. Before proceeding further we must now indicate the natural units (§ 5) for two-, three-, or four-dimensional extensions in the field-figure. Like the unit of length, these are defined for each point separately, so that the numerical value of a finite extension is found by dividing it into infinitely small parts.

A two-dimensional extension cuts the conjugate indicatrix in an ellipse, or the indicatrix itself and the conjugate indicatrix in two conjugate hyperbolae. In both cases we derive our unit from the area of a parallelogram described on conjugate radius-vectors.

A three-dimensional extension cuts the conjugate indicatrix in an ellipsoid, or the indicatrix and its conjugate in two conjugate hyperboloids. Now our unit will be derived from the volume of a parallelepiped described on three conjugate radius-vectors.

In a similar way the magnitude of four-dimensional extensions will be determined by comparison with a parallelepiped the edges of which are four conjugate radius-vectors of the indicatrix and the conjugate indicatrix.

It must here be kept in mind that, according to well known theorems, the area of the parallelogram and the volume of the parallelepipeds in question are independent of the special choice of the conjugate radius-vectors.

We shall further specify the units in such a way (comp. § 5) that the numerical magnitude of a parallelogram or a parallelepiped described on conjugate radius-vectors is found by multiplying the numbers by which the edges are expressed in natural measure.

From what has been said it follows that the area of the parallelogram described on two line-elements is given by the product of the lengths of these elements and the sine of the enclosed angle. Similarly the area of an infinitely small triangle is determined by half the product of two sides and the sine of the angle between them.

We need hardly add that the numerical value of any two-, three- or four-dimensional domain expressed in natural measure is not changed by a deformation of the field-figure.

§ 9. Let, at any point ${\displaystyle P}$ of the field-figure, 1, 2, 3, 4 be four arbitrarily chosen conjugate radius-vectors of the indicatrix. Two of these determine an infinitely small part ${\displaystyle V}$ of a two-dimensional extension. We may prolong this part to finite distances from ${\displaystyle P}$ by drawing from this point geodetic lines whose initial directions lie in the plane ${\displaystyle V}$. In this way we obtain six two-dimensional extensions (1,2), (2,3), (3,1), (1,4), (2,4) and (3,4). Let us now consider in one of these e. g. (${\displaystyle a,b}$) an infinitesimal triangle near the point ${\displaystyle P}$, the sides of which are geodetic lines (viz. geodetic lines in (${\displaystyle a,b}$)). If in calculating the angles of this triangle we go to quantities of the second order with respect to the sides and to the distances from ${\displaystyle P}$, the sum ${\displaystyle s}$ of the angles proves to have no longer the value ${\displaystyle \pi }$ (comp. § 7). The "excess" ${\displaystyle e=s-\pi }$ is proportional to the area ${\displaystyle \Delta }$ of the triangle, independently of the length of the sides, of their ratios and of the position of the triangle in the extension (${\displaystyle a,b}$). For the three extensions (1,2) (2,3), (3,1), which do not intersect the indicatrix itself but the conjugate indicatrix, this proposition follows from a well-known theorem of Gauss in the theory of curvature of surfaces; for the other three (1,4), (2,4), (3,4), which cut the indicatrix itself, the proof can be given by direct calculation. The considerations necessary for this, and some other calculations with which we shall be concerned further on will be communicated in a later paper.

In considering the three last-mentioned extensions I have confined myself to triangles with real sides (§ 7, b).

The quotient

${\displaystyle {\frac {e}{\Delta }}=K_{ab}}$

is now for each extension a definite number, which we may consider as a measure of the curvature of the two-dimensional extension (${\displaystyle a,b}$); the sum ${\displaystyle K}$ of the six numbers ${\displaystyle K_{ab}}$ may be called the curvature of the field-figure at the point ${\displaystyle P}$ in question. This quantity is the same that has been introduced by Hilbert; this results from the calculation of its value, which at the same time shows ${\displaystyle K}$ to be independent of the special choice of the directions 1, 2, 3, 4 introduced in the beginning of this §.

The numbers ${\displaystyle K_{ab}}$ all real and have a meaning that can be indicated without the introduction of coordinates; moreover their sum ${\displaystyle K}$ is not changed by a deformation of the field-figure.

If now ${\displaystyle d\Omega }$ is an element of the four-dimensional extension of the field-figure, expressed in natural measure, the part of the principal function belonging to the gravitation field is

 ${\displaystyle H_{3}={\frac {i}{\varkappa }}\int Kd\Omega }$ (6)

where the integration is extended to the domain considered (§ 6) while ${\displaystyle \varkappa }$ is the gravitation constant. ${\displaystyle H_{3}}$ too is not changed by a deformation of the field-figure.

The factor ${\displaystyle i}$ has been introduced in order to obtain a real value for ${\displaystyle H_{3}}$, the element ${\displaystyle d\Omega }$ being represented in natural measure by a negative imaginary number (§ 8).

§ 10. What we have to say of the electromagnetic field must be preceded by some considerations belonging to what may be called the "vector theory" of the field-figure.

A line-element ${\displaystyle PQ}$, taken in a definite, direction (indicated by the order of the letters), may be called a vector. Such vectors can be compounded or decomposed by means of parallelograms or parallelepipeds. Especially, when coordinates ${\displaystyle x_{1},\dots x_{4}}$ have been chosen, a vector may be resolved into four components which have the directions of the coordinates, viz. such directions that a shift along the first e.g. changes ${\displaystyle x_{1}}$, while ${\displaystyle x_{2},x_{3},x_{4}}$ remain constant. The four components in question are determined by the differentials ${\displaystyle dx_{1},\dots dx_{4}}$ corresponding to ${\displaystyle PQ}$. We shall say that by these they are expressed in "${\displaystyle x}$-measure". Their values in natural measure are found by multiplying ${\displaystyle dx_{1},\dots dx_{4}}$ by certain factors. If we keep in mind that the radius-vectors of the e conjugate indicatrix and the indicatrix in the directions of the axes are expressed in "${\displaystyle x}$ measure" by

${\displaystyle {\frac {\epsilon }{\sqrt {-g_{11}}}},\ {\frac {\epsilon }{\sqrt {-g_{22}}}},\ {\frac {\epsilon }{\sqrt {-g_{33}}}},\ {\frac {\epsilon }{\sqrt {g_{44}}}},}$

and in natural units by

${\displaystyle i\epsilon ,\ i\epsilon ,\ i\epsilon ,\ \epsilon }$

we find for the reducing factors

 ${\displaystyle l_{1}=i{\sqrt {-g_{11}}},\ l_{2}=i{\sqrt {-g_{22}}},\ l_{3}=i{\sqrt {-g_{33}}},\ l_{4}=i{\sqrt {g_{44}}}.}$ (7)

In the language of vector-analysis the vector obtained by the composition of two or more vectors is also called the sum of these vectors.

We shall also speak of finite vectors, i.e. of directed quantities which can be represented on an infinitely reduced scale by line-elements in the field-figure. If ${\displaystyle \omega }$ is the constant "reduction factor" chosen for this purpose, a vector ${\displaystyle \mathrm {A} }$ will be represented by a line-element ${\displaystyle \omega \mathrm {A} }$, the direction of which is also ascribed to ${\displaystyle \omega \mathrm {A} }$. It will now be evident that two finite vectors, as well as two infinitely small ones, determine an infinitesimal two dimensional extension and that finite vectors can be compounded and resolved by means of parallelograms and parallelepipeds. Also that we may speak of the "magnitude" of such figures, that e.g. the rule given in § 8 applies to the parallelogram described on two vectors.

The components of a vector in the directions of the coordinates expressed in ${\displaystyle x}$-measure will be called ${\displaystyle X_{1},X_{2},X_{3},X_{4}}$. This means that ${\displaystyle \omega X_{1},\dots \omega X_{4}}$ are equal to the differentials ${\displaystyle dx_{1},\dots dx_{4}}$ corresponding to the infinitely small vector ${\displaystyle \omega \mathrm {A} }$.

If we want to know the components of ${\displaystyle \mathrm {A} }$ in natural units we must multiply ${\displaystyle X_{1},\dots X_{4}}$ by the factors (7).

§ 11. Two vectors ${\displaystyle \mathrm {A} }$ and ${\displaystyle \mathrm {B} }$ starting from a point ${\displaystyle P}$ of the field-figure and lying in a plane ${\displaystyle V}$, determine what we shall call a rotation ${\displaystyle \mathrm {R} }$ in that plane. We ascribe to it the direction indicated by the order ${\displaystyle \mathrm {AB} }$ and a value given by the parallelogram described on ${\displaystyle \mathrm {A} }$ and ${\displaystyle \mathrm {B} }$ and expressed in natural measure[11]. This involves that the same rotation may be represented in many different ways by two vectors in the plane ${\displaystyle V}$.

For the rotation ${\displaystyle \mathrm {R} }$ we shall also use the symbol ${\displaystyle [\mathrm {A\cdot B]} }$.

By the vector product ${\displaystyle [\mathrm {A\cdot B\cdot C]} }$ of three vectors ${\displaystyle \mathrm {A,B,C} }$ at a point of the field-figure and not lying in one plane we shall understand a vector ${\displaystyle \mathrm {D} }$ the direction of which is conjugate with each of the three vectors (and therefore with the three-dimensional extension ${\displaystyle \mathrm {A,B,C} }$), the direction of ${\displaystyle \mathrm {D} }$ corresponding to those of ${\displaystyle \mathrm {A,B} }$ and ${\displaystyle \mathrm {C} }$ in a way presently to be indicated, while the magnitude of ${\displaystyle \mathrm {D} }$, expressed in natural measure, is equal to that of the parallelepiped described on ${\displaystyle \mathrm {A} }$, ${\displaystyle \mathrm {B} }$ and ${\displaystyle \mathrm {C} }$ and expressed in the same measure. This definition involves that the value is ascribed to the vector product of three vectors lying in one and the same plane.

A further statement about the direction of ${\displaystyle \mathrm {D} }$ is necessary because two opposite directions are conjugate with ${\displaystyle \mathrm {A,B,C} }$. For one set of three directions ${\displaystyle \mathrm {A_{0},B_{0},C_{0}} }$ we shall choose arbitrarily which of its two conjugate directions will be said to correspond to it. If this is the direction ${\displaystyle \mathrm {D} _{0}}$, then the direction ${\displaystyle \mathrm {D} }$ corresponding to ${\displaystyle \mathrm {A,B,C} }$ will be determined by the rule that ${\displaystyle \mathrm {D} _{0}}$, passes into ${\displaystyle \mathrm {D} }$ by a gradual passage of the first three vectors from ${\displaystyle \mathrm {A_{0},B_{0},C_{0}} }$ into ${\displaystyle \mathrm {A,B,C} }$, this latter passage being effected in such a way that during the change the vectors never come to lie in one plane.

The vector product ${\displaystyle [\mathrm {A\cdot B\cdot C]} }$ takes the opposite direction when one of the vectors is reversed as well as when two of them are interchanged. We must therefore always attend to the order of the symbols in ${\displaystyle [\mathrm {A\cdot B\cdot C]} }$.

The vector product possesses the distributive property with respect to each of the three vectors, so that e.g. if ${\displaystyle \mathrm {A} _{1}}$ and ${\displaystyle \mathrm {A} _{2}}$ are vectors,

${\displaystyle \left[\left(\mathrm {A} _{1}+\mathrm {A} _{2}\right)\cdot \mathrm {B\cdot C} \right]=\mathrm {\left[A_{1}\cdot B\cdot C\right]+\left[A_{2}\cdot B\cdot C\right]} }$

From this we can infer that ${\displaystyle [\mathrm {A\cdot B\cdot C]} }$ depends only on ${\displaystyle \mathrm {C} }$ and the rotation ${\displaystyle \mathrm {R} }$ determined by ${\displaystyle \mathrm {A} }$ and ${\displaystyle \mathrm {B} }$. For this reason we write for the vector product also ${\displaystyle [\mathrm {R\cdot C]} }$; in calculating it we are free to replace the rotation ${\displaystyle \mathrm {R} }$ by any two vectors by means of which it can be represented.

If ${\displaystyle \mathrm {R} }$, ${\displaystyle \mathrm {R} _{1}}$ and ${\displaystyle \mathrm {R} _{2}}$ are rotations in the same plane, such that the value and direction of ${\displaystyle \mathrm {R} }$ are found by adding ${\displaystyle \mathrm {R} _{1}}$ and ${\displaystyle \mathrm {R} _{2}}$ algebraically, we have, in virtue of the distributive property

${\displaystyle [\mathrm {R_{1}\cdot C]} +[\mathrm {R_{2}\cdot C]} =[\mathrm {R\cdot C]} }$

§ 12. In what precedes we were concerned with the volumes of parallelepipeds expressed in natural units. When we have introduced coordinates ${\displaystyle x_{1},\dots x_{4}}$ we may also express these volumes in the "${\displaystyle x}$-units" corresponding to the coordinates chosen.

Let us consider e.g. the three-dimensional extension ${\displaystyle x_{4}=const.}$, which cuts the conjugate indicatrix in the ellipsoid

${\displaystyle g_{11}x_{1}^{2}+g_{22}x_{2}^{2}+g_{33}x_{3}^{2}+2g_{12}x_{1}x_{2}+2g_{23}x_{2}x_{3}+2g_{31}x_{3}x_{1}=-\epsilon ^{2}}$

If we agree that in ${\displaystyle x}$-measure spaces in this extension will be represented by positive numbers and that a parallelepiped with the positive edges ${\displaystyle dx_{1},dx_{2},dx_{3}}$ will have the volume ${\displaystyle dx_{1}\ dx_{2}\ dx_{3}}$, we find for that of the parallelepiped on three conjugate radius-vectors

${\displaystyle {\frac {\epsilon ^{3}}{\sqrt {-G_{44}}}}}$

where it has been taken into consideration that ${\displaystyle G_{44}}$ is negative.

The volume of the same parallelepiped being expressed in natural measure by — ${\displaystyle -i\epsilon ^{3}}$ (§ 8), we have to multiply by

 ${\displaystyle l_{123}=-i{\sqrt {-G_{44}}}\,}$ (8)

if we want to pass from the expression in ${\displaystyle x}$-measure to that in natural measure.

For the extension ${\displaystyle \left(x_{2},x_{3},x_{4}\right)}$, i.e. ${\displaystyle x_{1}=0}$ the corresponding factor is

 ${\displaystyle l_{234}=-{\sqrt {G_{11}}}}$ (9)

§ 13. In the theory of electromagnetic phenomena we are concerned in the first place with the electric charge and the convection current. So far as these quantities belong to a definite element ${\displaystyle d\Omega }$ of the field-figure they may be combined into

${\displaystyle \mathrm {q} d\Omega }$

where ${\displaystyle \mathrm {q} }$ is a vector which we may call the current vector. When it is resolved into four components having the directions of the axes, the first three components determine the convection current, while the fourth component gives the density of the electric charge.

As to the electric and the magnetic force, these two taken together can be represented at each point of the field-figure by two rotations

${\displaystyle \mathrm {R} _{e}}$ and ${\displaystyle \mathrm {R} _{h}}$

in definite, mutually conjugate two-dimensional extensions. These quantities are closely connected with the current vector, for after having introduced coordinates ${\displaystyle x_{1},\dots x_{4}}$ we have for each closed surface ${\displaystyle \sigma }$ the vector equation

 ${\displaystyle \int \left\{\left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]\right\}_{x}d\sigma =i\int \{\mathrm {q} \}_{x}d\Omega }$ (10)

where the second integral has to be taken over the domain ${\displaystyle \Omega }$ enclosed by ${\displaystyle \sigma }$. On the left hand side ${\displaystyle d\sigma }$ represents a three-dimensional surface-element expressed in natural units and ${\displaystyle \mathrm {N} }$ a vector of the magnitude 1 in natural measure conjugate with or perpendicular to that element (§ 7) and directed towards the outside of the domain ${\displaystyle \Omega }$. The index ${\displaystyle x}$ shows that the vector ${\displaystyle \left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]}$ must be expressed in ${\displaystyle x}$-measure. At each point of the surface we must resolve the vector along the four directions of the coordinates, express each component in ${\displaystyle x}$-measure (§10) and finally, after multiplication by ${\displaystyle d\sigma }$, we must add algebraically all ${\displaystyle x_{1}}$-components; similarly all ${\displaystyle x_{2}}$-components and so on.

It must be expressly remarked that if an equation like (10) in which we are concerned with the composition of vectors at different points of the field-figure, shall have a definite meaning we must know which components are to be considered as having the same direction, so that they can be added. This has been determined by the introduction of coordinates.

On the right hand side of the equation the index ${\displaystyle x}$ means that the vector ${\displaystyle \mathrm {q} }$ must be expressed in ${\displaystyle x}$-measure and the factor ${\displaystyle i}$ had to be introduced because ${\displaystyle d\Omega }$ is imaginary.

One can prove that equation (10) is equivalent to the differential equations which in Einstein's theory serve for the same purpose and further that when the equation holds for one choice of coordinates it will also be true for any other choice.

§ 14. The proof for these assertions must be deferred to the second part of this communication. For the present we shall only add that the part of the principal function referring to the electromagnetic field is given by

${\displaystyle H_{2}=i\int {\frac {1}{2}}\left(\mathrm {R} _{e}^{2}+\mathrm {R} _{h}^{2}\right)d\Omega }$

where ${\displaystyle \mathrm {R} _{e}}$ and ${\displaystyle \mathrm {R} _{h}}$ are, expressed in natural units, the two rotations that are characteristic of the field. Like the two other parts of the principal function, ${\displaystyle H_{2}}$ is not changed by a deformation of the field-figure. In this statement it is to be understood that the parallelograms by which ${\displaystyle \mathrm {R} _{e}}$ and ${\displaystyle \mathrm {R} _{h}}$ are represented take part in the deformation.

Some remarks on the way in which, starting from the principal function, we may obtain the fundamental equations of the theory must also be deferred. I shall conclude now by remarking that, as an immediate consequence of Hamilton's principle, the world-line of a material point which is acted on only by a given gravitation field, will be a geodetic line, and that the equations which determine the gravitation field caused by material and electromagnetic systems will be found by the consideration of infinitely small variations of the indicatrices, by which the numerical values of all quantities that are measured by means of these surfaces will be changed.

II.

(Communicated in the meeting of March 25, 1916).

§ 15. In the first part of this communication the connexion between the electric and the magnetic force on one hand and the charge and the convection current on the other was expressed by the equation

 ${\displaystyle \int \left\{\left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]\right\}_{x}d\sigma =i\int \{\mathrm {q} \}_{x}d\Omega }$ (10)

which has been discussed in § 13. It will now be shown that this formula is equivalent to the differential equations by which the connexion in question is expressed in the theory of Einstein. For this purpose some further geometrical considerations must first be developed. They refer to the special case that the quantities ${\displaystyle g_{ab}}$, have the same values at every point of the field-figure.

If this condition is fulfilled, considerations which generally may be applied to infinitesimal extensions only are valid for finite extensions too.

§ 16. The factor required, in the measurement of four-dimensional domains, for the passage from ${\displaystyle x}$-units to natural units has now the same value at every point of the field-figure. Similarly, when any one-, two- or three-dimensional extension in the field-figure that is determined by linear equations ("linear extensions") is considered, the factor by means of which the said passage may be effected for parts of that extension, will be the same for all those parts. Moreover the factor in question will be the same for two "parallel" extensions of this kind, i.e. for two extensions the determining equations of which can be written in such a way that the coefficients of ${\displaystyle x_{1},\dots x_{4}}$ are the same in them.

It is obvious that linear one-dimensional extensions can be called "straight lines", also it will be clear what is to be understood by a "prism" (or "cylinder"). This latter is bounded by two mutually parallel linear three-dimensional extensions ${\displaystyle \sigma _{1}}$ and ${\displaystyle \sigma _{s}}$ and by a lateral surface which may be extended indefinitely to both sides and in which mutually parallel straight lines ("generating lines") can be drawn.

We need not dwell upon the elementary properties of the prism.

§ 17. A vector may now be represented by a straight line of finite length; the quantities ${\displaystyle X_{1},\dots X_{4}}$, which have been introduced in § 10, are the changes of the coordinates caused by a displacement along that line. The magnitude of the vector, expressed in natural units, will be denoted by ${\displaystyle S}$. It is given by a formula similar to (1), viz. by

 ${\displaystyle S^{2}=\sum (ab)g_{ab}X_{a}X_{b}}$ (11)

A vector may be regarded as being the same everywhere in the field-figure, if ${\displaystyle X_{1},\dots X_{4}}$ have constant values. In the same way a rotation ${\displaystyle \mathrm {R} }$ (§ 11) may be said to be the same everywhere, if it can be represented by two vectors of this kind.

If from a point ${\displaystyle P}$ two vectors ${\displaystyle PQ}$ and ${\displaystyle PR}$ issue, denoted by ${\displaystyle X'_{1},\dots X'_{4}}$, ${\displaystyle S'}$ and ${\displaystyle X''_{1},\dots X''_{4}}$, ${\displaystyle S''}$ resp., the angle between them (comp. (5)) is defined by

 ${\displaystyle S'S''\cos(S',S'')=\sum (ab)g_{ab}X'_{a}X''_{b}}$ (12)

We remark here that ${\displaystyle X'_{a},\ X''_{b}}$ are real, positive or negative quantities and that ${\displaystyle S'}$ and ${\displaystyle S''}$ are expressed in the way indicated in § 5 ("absolute" values). It is to be understood that ${\displaystyle S}$ does not change when the signs of ${\displaystyle X_{1},\dots X_{4}}$ are reversed at the same time.

If ${\displaystyle S'''}$ is the value of the vector ${\displaystyle RQ}$ and if the angle between this vector and ${\displaystyle RP}$ is denoted by (${\displaystyle S'',S'''}$), it follows further from (11) and (12) that

${\displaystyle S''=S'\cos(S',S'')+S'''\cos(S'',S''')}$

In the special case of a right angle ${\displaystyle R}$ we have

${\displaystyle S''=S'\cos(S',S'')}$

an equation expressing the connexion between a vector ${\displaystyle PQ}$ and its "projection" on a line ${\displaystyle PR}$. The angle (${\displaystyle S',S''}$) is the angle between the vector and its projection, both reckoned from the same point ${\displaystyle P}$.

§ 18. Let us now return to the prism ${\displaystyle R}$ mentioned in § 16. From a point ${\displaystyle A_{2}}$ of the boundary of the "upper face"${\displaystyle \sigma _{2}}$, we can draw a line perpendicular to ${\displaystyle \sigma _{2}}$ and ${\displaystyle \sigma _{1}}$. Let ${\displaystyle B_{1}}$ be the point, where it cuts thus last, plane, the "base", and ${\displaystyle A_{1}}$ the point where this plane is encountered by the generating line through ${\displaystyle A_{2}}$. If then ${\displaystyle \angle A_{1}A_{2}B_{1}=\vartheta }$, we have

 ${\displaystyle {\overline {A_{2}B_{1}}}={\overline {A_{2}A_{1}}}\cos \vartheta }$ (13)

The strokes over the letters indicate the absolute values of the distances ${\displaystyle A_{2}B_{1}}$ and ${\displaystyle A_{2}A_{1}}$.

It can be shown (§ 8) that, all quantities being expressed in natural units, the "volume" of the prism ${\displaystyle P}$ is found by taking the product of the numerical values of the base ${\displaystyle \sigma _{1}}$ and the "height" ${\displaystyle A_{2}B_{1}}$.

Let now linear three-dimensional extensions perpendicular to ${\displaystyle A_{1}A_{2}}$ be made to pass through ${\displaystyle A_{1}}$ and ${\displaystyle A_{2}}$. From these extensions the lateral boundary of the prism cuts the parts ${\displaystyle \sigma '_{1}}$ and ${\displaystyle \sigma '_{2}}$ and these parts, together with the lateral surface, enclose a new prism ${\displaystyle P'}$, the volume of which is equal to that of ${\displaystyle P}$. As now the volume of ${\displaystyle P'}$ is given by the product of ${\displaystyle {\overline {A_{2}A_{1}}}}$ and ${\displaystyle \sigma '_{1}}$, we have with regard to (13)

${\displaystyle \sigma '_{1}=\sigma {}_{1}\cos \vartheta }$

If now we remember that, if a vector perpendicular to ${\displaystyle \sigma _{1}}$ is projected on the generating line, the ratio between the projection and the vector itself (viz. between their absolute values) is given by ${\displaystyle \cos \vartheta }$ and that a connexion similar to that which was found above between a normal section ${\displaystyle \sigma '_{1}}$ of the prism and ${\displaystyle \sigma _{1}}$, also exists between ${\displaystyle \sigma '_{1}}$ and any other oblique section, we easily find the following theorem:

Let ${\displaystyle \sigma }$ and ${\displaystyle {\bar {\sigma }}}$ be two arbitrarily chosen linear three-dimensional sections of the prism, ${\displaystyle \mathrm {N} }$ and ${\displaystyle {\bar {\mathrm {N} }}}$ two vectors, perpendicular to ${\displaystyle \sigma }$ and ${\displaystyle {\bar {\sigma }}}$ resp. and of the same length, ${\displaystyle S}$ and ${\displaystyle {\bar {S}}}$ the absolute values of the projections of ${\displaystyle \mathrm {N} }$ and ${\displaystyle {\bar {\mathrm {N} }}}$ on a generating line. Then we have

 ${\displaystyle S\sigma ={\bar {S}}{\bar {\sigma }}}$ (14)

§ 19. After these preliminaries we can show that the left hand side of (10) is equal to 0, if the numbers ${\displaystyle g_{ab}}$ are constants and if moreover both the rotation ${\displaystyle \mathrm {R} _{e}}$ and the rotation ${\displaystyle \mathrm {R} _{h}}$ are everywhere the same. For the two parts of the integral the proof may be given in the same way, so that it suffices to consider the expression

 ${\displaystyle \int \left[\mathrm {R} _{e}\cdot \mathrm {N} \right]_{x}d\sigma }$ (15)
Let ${\displaystyle X_{1},\dots X_{4}}$ be the components of the vector ${\displaystyle \mathrm {N} }$, expressed in ${\displaystyle x}$-units. From the distributive property of the vector product it then follows that each of the four components of

${\displaystyle \left[\mathrm {R} _{e}\cdot \mathrm {N} \right]_{x}}$

is a homogeneous linear function of ${\displaystyle X_{1},\dots X_{4}}$. Under the special assumptions specified at the beginning of this § these are every where, the same functions. Let us thus consider a definite component of (15) e.g. that which corresponds to the direction of the coordinate ${\displaystyle x_{a}}$. We can represent it by an expression of the form

${\displaystyle \int \left(\alpha _{1}X_{1}+\dots +\alpha _{4}X_{4}\right)d\sigma }$

where ${\displaystyle \alpha _{1},\dots \alpha _{4}}$ are constants. It will therefore be sufficient to prove that the four integrals

 ${\displaystyle \int X_{1}d\sigma \dots \int X_{4}d\sigma }$ (16)

vanish.

In order to calculate ${\displaystyle \int X_{1}d\sigma }$ we consider an infinitely small prism, the edges of which have the direction ${\displaystyle x_{1}}$. This prism cuts from the boundary surface ${\displaystyle \sigma }$ two elements ${\displaystyle d\sigma }$ and ${\displaystyle {\overline {d\sigma }}}$. Proceeding along a generating line in the direction of the positive ${\displaystyle x_{1}}$ we shall enter the extension ${\displaystyle \Omega }$ bounded by ${\displaystyle \sigma }$ through one of these elements and leave it through the other. Now the vectors perpendicular to ${\displaystyle \sigma }$, which occur in (15) and which we shall denote by ${\displaystyle \mathrm {N} }$ and ${\displaystyle {\bar {\mathrm {N} }}}$ for the two elements, have the same value.[12] If, therefore, ${\displaystyle S}$ and ${\displaystyle {\bar {S}}}$ are the absolute values of the projections of ${\displaystyle \mathrm {N} }$ and ${\displaystyle {\bar {\mathrm {N} }}}$ on a line in the direction ${\displaystyle x_{1}}$, we have according to (14)

 ${\displaystyle Sd\sigma ={\bar {S}}{\overline {d\sigma }}}$ (17)

Let first the four directions of coordinates be perpendicular to one another. Then the components of the vector obtained by projecting ${\displaystyle \mathrm {N} }$ on the above mentioned line are ${\displaystyle X_{1},0,0,0}$ and similarly those of the projection of ${\displaystyle {\bar {\mathrm {N} }}:{\bar {X}}_{1},0,0,0}$. But as, proceeding in the direction of ${\displaystyle x_{1}}$ we enter ${\displaystyle \Omega }$ through one element and leave it through the other, while ${\displaystyle \mathrm {N} }$ and ${\displaystyle {\bar {\mathrm {N} }}}$ are both directed outward, ${\displaystyle X_{1}}$ and ${\displaystyle {\overline {X_{1}}}}$, must have opposite signs. So we have

${\displaystyle S:{\bar {S}}=X_{1}:-{\bar {X}}_{1}}$

and because of (17) we may now conclude that the elements ${\displaystyle X_{1}d\sigma }$ and ${\displaystyle {\overline {X_{1}}}{\overline {d\sigma }}}$ in the first of the integrals (16) annul each other. It will be clear now that the whole integral vanishes and that similar considerations may be applied to the other three.

So we have proved that under the special assumptions made the left hand side of (10) will vanish in the special case that the directions of the coordinates are perpendicular to each other. This conclusion likewise holds for an other set of coordinates if only the assumption made at the beginning of this § is fulfilled. This is obvious, as we can pass from mutually perpendicular coordinates ${\displaystyle x_{1},\dots x_{4}}$ to arbitrarily chosen other ones ${\displaystyle x'_{1},\dots x'_{4}}$ which fulfil this latter condition by linear transformation formulae with constant coefficients. The ${\displaystyle x}$- and the ${\displaystyle x'}$-components of the vector

${\displaystyle \left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]}$

are then connected by homogeneous linear formulae with coefficients which have the same value at all points of the surface ${\displaystyle \sigma }$. Hence if, as has been shown above, the four ${\displaystyle x}$-components of the vector

${\displaystyle \int \left\{\left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]\right\}d\sigma }$

vanish, the four ${\displaystyle x'}$-components are now seen to do so likewise.[13]

§ 20. The above considerations were intended to prepare a corollary which will be of use in the treatment of the integral on the left hand side of (10), if we now leave the special assumptions made above and suppose the quantities ${\displaystyle g_{ab}}$ to be functions of the coordinates while also the rotations ${\displaystyle \mathrm {R} _{e}}$ and ${\displaystyle \mathrm {R} _{h}}$ may change from point to point.

This corollary may be formulated as follows: If all dimensions of the limiting surface ${\displaystyle \sigma }$ are infinitely small of the first order, the integral

${\displaystyle \int \left\{\left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]\right\}_{x}d\sigma }$

will be of the fourth order.

In order to make this clear let us suppose that in the calculation of the integral we confine ourselves to quantities of the third order. The surface ${\displaystyle \sigma }$ being already of that order we may then omit all infinitesimal values in the quantities by which ${\displaystyle d\sigma }$ is multiplied; we may therefore neglect the infinitesimal changes of the quantities ${\displaystyle g_{ab}}$ over the extension considered, and also those of ${\displaystyle \mathrm {R} _{e}}$ and ${\displaystyle \mathrm {R} _{h}}$. By this we just come to the case considered in § 19. Thus it is evident, that as regards quantities of the third order the first part of (10) is 0. From this it follows that in reality it is at least of the fourth order.

§ 21. Let us now return to the general case that the extension ${\displaystyle \Omega }$ to which equation (10) refers, has finite dimensions. If by a surface ${\displaystyle {\bar {\sigma }}}$ this extension is divided into two extensions ${\displaystyle \Omega _{1}}$ and ${\displaystyle \Omega _{2}}$, the quantities on the two sides in (10) each consist of two parts referring to these extensions. For the right hand side this is immediately clear and as to the quantity on the left hand side, it follows from the consideration that the contributions of a to the integrals over the boundaries of ${\displaystyle \Omega _{1}}$ and ${\displaystyle \Omega _{2}}$ are equal with opposite signs. In the two cases namely we must take for ${\displaystyle \mathrm {N} }$ equal but opposite vectors.

Also, if the extension ${\displaystyle \Omega }$ is divided into an arbitrary number of parts, each term in (10) will be the sum of a number of integrals, each relating to one of these parts.

By surfaces with the equations ${\displaystyle x_{1}=\mathrm {const.} ,\dots x_{4}=\mathrm {const} .}$ we can divide the extension ${\displaystyle \Omega }$ into elements which we shall denote by ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$. As a rule there will be left near the surface ${\displaystyle \sigma }$ certain infinitely small extensions of a different form. From the preceding § it is evident that, in the calculation of the integrals, these latter extensions may be neglected and that only the extensions ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$ have to be considered. From this we can conclude that equation (10) is valid for any finite extension, as soon at it holds for each of the elements ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$.

§ 22. We shall now show what equation (10) becomes for one element ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$. Besides the infinitesimal quantities ${\displaystyle x_{1},\dots x_{4}}$, occurring in the equation

${\displaystyle F=\sum (ab)g_{ab}x_{a}x_{b}=\epsilon ^{2}}$

of the indicatrix we introduce four other quantities ${\displaystyle \xi _{1},\dots \xi _{4}}$, which we define by

 ${\displaystyle \xi _{a}={\frac {1}{2}}{\frac {\partial F}{\partial x_{a}}}}$ (18)

or

 ${\displaystyle \left.{\begin{array}{c}\xi _{1}=g_{11}x_{1}+g_{12}x_{2}+\dots +g_{14}x_{4}\\\cdots \cdots \cdots \cdots \cdots \cdots \\\cdots \cdots \cdots \cdots \cdots \cdots \\\xi _{4}=g_{41}x_{1}+g_{42}x_{2}+\dots +g_{44}x_{4}\end{array}}\right\}}$ (19)

with the equalities ${\displaystyle g_{ba}=g_{ab}}$.

To each of these quantities corresponds a definite direction, viz. that in which we have to proceed in order to make the considered quantity change in positive sense while the other three remain constant. If we denote these directions by ${\displaystyle 1^{*},2^{*},3^{*},4^{*}}$ and in the same way the directions of the coordinates ${\displaystyle x_{1},x_{2},x_{3}x_{4}}$ by 1, 2, 3, 4, it is evident that ${\displaystyle 1^{*}}$ is conjugate with 2, 3 and 4, ${\displaystyle 2^{*}}$ with 3, 1 and 4, and so on; inversely 1 with ${\displaystyle 2^{*},3^{*},4^{*}}$; 2 with ${\displaystyle 3^{*},1^{*},4^{*}}$, and so on. From what has been said above about the algebraic signs of ${\displaystyle g_{11},g_{22},g_{33},g_{44}}$ it follows further that, if directions opposite to 1, ${\displaystyle 1^{*}}$ etc. are denoted by — 1, ${\displaystyle -1^{*}}$ etc., the directions — 1 and ${\displaystyle 1^{*}}$ will point to the same side of an extension ${\displaystyle x_{1}=\mathrm {const} .}$. The same may be said of the directions —2 and${\displaystyle 2^{*}}$ or —3 and ${\displaystyle 3^{*}}$ with respect to extensions ${\displaystyle x_{2}=\mathrm {const} .}$, or ${\displaystyle x_{3}=\mathrm {const} .}$, while with respect to an extension ${\displaystyle x_{4}=\mathrm {const} .}$, the directions 4 and ${\displaystyle 4^{*}}$ point to the same side.

Finally, we shall fix (§11) as far as is necessary, which direction corresponds to three others. For that purpose we shall imagine the directions of coordinates ${\displaystyle 1,\dots 4}$ to pass into mutually conjugate directions, which will also be called ${\displaystyle $1,\dots 4}$$, by gradual changes, in such a way that never three of them come to lie in one plane. We shall agree that after this change —4 corresponds to 1, 2, 3.

Let ${\displaystyle a,b,c,d}$ be the numbers 1, 2, 3, 4 in an order obtained from the natural one by an even number of permutations. Then the rule of § 11 teaches us that the direction ${\displaystyle -d}$ corresponds to ${\displaystyle a,b,c}$. It is clear that this would be the ease with ${\displaystyle d}$, if ${\displaystyle a,b,c,d}$ were obtained from 1, 2, 3, 4 by an odd number of permutations. If further it is kept in mind that, always in the new case, the directions ${\displaystyle 1^{*},2^{*},3^{*},4^{*}}$ coincide with —1, —2, —3, 4, we come to the conclusion that the directions 1, 2, 3 and 4 correspond to the sets ${\displaystyle 2^{*},3^{*},4^{*};3^{*},1^{*},4^{*};1^{*},2^{*},4^{*}}$ and ${\displaystyle 1^{*},2^{*},3^{*}}$ respectively. The rule of gradual change (§11) involves that this holds also for the original case, in which 1, 2, 3, 4 were not yet mutually conjugate.

This is all that has to be said about the relations between the different directions. It must only be kept in mind, that whenever two of the first three directions are interchanged, the fourth must be reversed.

§ 23. In the neighbourhood of a point ${\displaystyle P}$ of the field-figure we may introduce as coordinates instead of ${\displaystyle x_{1},\dots x_{4}}$ the quantities ${\displaystyle \xi _{1},\dots \xi _{4}}$ defined by (19). Line-elements or finite vectors can be resolved in the directions of these coordinates, i.e. in the directions ${\displaystyle 1^{*},2^{*},3^{*},4^{*}}$. Their components and the magnitudes of different extensions can now be expressed in ${\displaystyle \xi }$-nits in the same way as formerly in ${\displaystyle x}$-units. So the volume of a three-dimensional parallelepiped with the positive edges ${\displaystyle d\xi _{1},d\xi _{2},d\xi _{3}}$ is represented by the product ${\displaystyle d\xi _{1}d\xi _{2}d\xi _{3}}$.

Solving ${\displaystyle x_{1},\dots x_{4}}$ from (19) we obtain expressions of the form

 ${\displaystyle \left.{\begin{array}{c}x_{1}=\gamma _{11}\xi _{1}+\gamma _{21}\xi _{2}+\dots +\gamma _{41}\xi _{4}\\\cdots \cdots \cdots \cdots \cdots \cdots \\\cdots \cdots \cdots \cdots \cdots \cdots \\x_{4}=\gamma _{14}\xi _{1}+\gamma _{24}\xi _{2}+\dots +\gamma _{44}\xi _{4}\\\gamma _{ba}=\gamma _{ab}\end{array}}\right\}}$ (20)

If we use the coordinates ${\displaystyle \xi }$ the coefficients ${\displaystyle \gamma _{ab}}$ play the same part as the coefficients ${\displaystyle g_{ab}}$ when the coordinates ${\displaystyle x}$ are used. According to (18) and (20) we have namely

${\displaystyle F=\sum (a)\xi _{a}x_{a}=\sum (ab)\gamma _{ab}\xi _{a}\xi _{b}}$

so that the equation of the indicatrix may be written

${\displaystyle \sum (ab)\gamma _{ba}\xi _{a}\xi _{b}=\epsilon ^{2}}$

§ 24. Let the rotations ${\displaystyle \mathrm {R} _{e}}$ and ${\displaystyle \mathrm {R} _{h}}$ of which we spoke in § 13 be defined by the vectors ${\displaystyle \mathrm {A^{I},A^{II}} }$ and ${\displaystyle \mathrm {A^{III},A^{IV}} }$ respectively, the resultants of the vectors ${\displaystyle \mathrm {A_{1^{*}}^{I},\dots A_{4^{*}}^{I}} }$, etc. in the directions ${\displaystyle 1^{*},\dots 4^{*}}$. Then, according to the properties of the vector product that were discussed in § 11,

${\displaystyle {\begin{array}{ll}\left[\mathrm {R} _{e}\cdot \mathrm {N} \right]&=\left[\mathrm {\left(A_{1^{*}}^{I}+\dots +A_{4^{*}}^{I}\right)\cdot \left(A_{1^{*}}^{II}+\dots +A_{4^{*}}^{II}\right)\cdot N} \right]\\&=\sum ({\overline {ab}})\left\{\left[\mathrm {A} _{a^{*}}^{I},\ \mathrm {A} _{b^{*}}^{II}\cdot \mathrm {N} \right]-\left[\mathrm {A} _{a^{*}}^{II},\ \mathrm {A} _{b^{*}}^{I}\cdot \mathrm {N} \right]\right\}\end{array}}}$

where the stroke over ${\displaystyle ab}$ indicates that each combination of two different numbers ${\displaystyle a,b}$ contributes one term to the sum. For the vector product ${\displaystyle \left[\mathrm {R} _{h}\cdot \mathrm {N} \right]}$ we have a similar equation. Now two or more rotations in one and the same plane, e.g. in the plane ${\displaystyle a^{*}b^{*}}$, may be replaced by one rotation, which can be represented by means of two vectors with arbitrarily chosen directions in that plane, e.g. the directions ${\displaystyle a^{*}}$ and ${\displaystyle b^{*}}$. We may therefore introduce two vectors ${\displaystyle \mathrm {B} _{a^{*}}}$ and ${\displaystyle \mathrm {B} _{b^{*}}}$ directed along ${\displaystyle a^{*}}$ and ${\displaystyle b^{*}}$ resp., so that

 ${\displaystyle \left[\mathrm {B} _{a^{*}}\cdot \mathrm {B} _{b^{*}}\right]=\left[\mathrm {A} _{a^{*}}^{I}\cdot \mathrm {A} _{b^{*}}^{II}\right]-\left[\mathrm {A} _{a^{*}}^{II}\cdot \mathrm {A} _{b^{*}}^{I}\right]+\left[\mathrm {A} _{a^{*}}^{III}\cdot \mathrm {A} _{b^{*}}^{IV}\right]-\left[\mathrm {A} _{a^{*}}^{IV}\cdot \mathrm {A} _{b^{*}}^{III}\right]}$ (21)

Then we must substitute in (10)

 ${\displaystyle \left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]=\sum ({\overline {ab}})\left[\mathrm {B} _{a^{*}}\cdot \mathrm {B} _{b^{*}}\cdot \mathrm {N} \right]}$ (22)

Here it must be remarked that the magnitude and the sense of one of the vectors ${\displaystyle \mathrm {B} }$ may be chosen arbitrarily; when this has been done, the other vector is perfectly determined.

In the following calculations the vector ${\displaystyle \mathrm {N} }$ has one of the directions ${\displaystyle 1^{*},\dots 4^{*}}$. As this is also the case with the vectors ${\displaystyle \mathrm {B} _{a^{*}}}$ and ${\displaystyle \mathrm {B} _{b^{*}}}$, the vector product occurring in (22) can easily be expressed in ${\displaystyle \xi }$-units. After that we may pass to natural units and finally, as is necessary for the substitution in (10), to ${\displaystyle x}$-units.

In order to pass from ${\displaystyle \xi }$-units to natural units we have to multiply a vector in the direction ${\displaystyle a^{*}}$ by a certain coefficient ${\displaystyle \lambda _{a}}$, and a part of the extension ${\displaystyle a^{*},b^{*},c^{*}}$ by a coefficient ${\displaystyle \lambda _{abc}}$. These coefficients correspond to ${\displaystyle l_{a}}$ (§ 10) and ${\displaystyle l_{abc}}$ (§ 12). The factors ${\displaystyle \lambda _{abc}}$ e.g. can be expressed by means of the minors ${\displaystyle \Gamma _{ab}}$ of the determinant ${\displaystyle \gamma }$ of the quantities ${\displaystyle \gamma _{ab}}$. If this is worked out and if the equations

${\displaystyle \gamma _{ab}={\frac {G_{ab}}{g}},\ g_{ab}={\frac {\Gamma _{ab}}{\gamma }},\ g\gamma =1}$

are taken into consideration, we obtain the following corollary, which we shall soon use:

Let ${\displaystyle a,b,c,d}$ and also ${\displaystyle a',b',c',d'}$ be the numbers 1, 2, 3, 4 in any order, ${\displaystyle a'}$ being not the same as ${\displaystyle a}$, then we have, if none of the two numbers ${\displaystyle \alpha }$ and ${\displaystyle \alpha '}$ is 4,

 ${\displaystyle {\frac {l_{bcd}\lambda b'c'd'}{l_{a'}\lambda _{a}}}=-1}$ (23)

and if one of the two is 4

 ${\displaystyle {\frac {l_{bcd}\lambda b'c'd'}{l_{a'}\lambda _{a}}}=1}$ (24)

§ 25. We shall now suppose (comp. § 24) that in ${\displaystyle \xi }$-units the vector ${\displaystyle \mathrm {B} _{a^{*}}}$ has the value +1, and we shall write ${\displaystyle \chi _{ab}}$ for the value that must then be given to ${\displaystyle \mathrm {B} _{b^{*}}}$. If the ${\displaystyle \xi }$-components of the vectors ${\displaystyle \mathrm {A^{I}} }$ etc. are denoted by ${\displaystyle \Xi _{1}^{I},\dots \Xi _{4}^{I}}$ etc., we find from (21)

 ${\displaystyle \chi _{ab}=\left(\Xi _{a}^{I}\Xi _{b}^{II}-\Xi _{a}^{II}\Xi _{b}^{I}\right)+\left(\Xi _{a}^{III}\Xi _{b}^{IV}-\Xi _{a}^{IV}\Xi _{b}^{III}\right)}$ (25)

This formula involves that

 ${\displaystyle \chi _{ba}=-\chi _{ab}}$ (26)

It may be remarked that ${\displaystyle \chi _{ba}}$ is the value that must be given to the vector ${\displaystyle \mathrm {B} _{a^{*}}}$ if ${\displaystyle \mathrm {B} _{b^{*}}}$ is taken to be 1.

The quantities ${\displaystyle \chi _{ab}}$ may be said to represent the rotations ${\displaystyle \left[\mathrm {B} _{a^{*}}\cdot \mathrm {B} _{b^{*}}\right]}$.

At the end of our calculations we shall introduce instead of ${\displaystyle \chi _{ab}}$ the quantities t${\displaystyle \psi _{ab}}$ defined by

 ${\displaystyle \psi _{ab}=\chi _{a'b'}(a\mp b),\ \psi _{aa}=0}$ (27)

In the first of these equations ${\displaystyle a,b,a',b'}$ are supposed to be the numbers 1, 2, 3, 4, in an order obtained from 1, 2, 3, 4 by an even number of permutations.

§ 26. We have now to calculate the left hand side of equation (10) for the case that ${\displaystyle \sigma }$ is the surface of an element ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$. For this purpose we shall each time take together two opposite sides, calculating for each pair the contributions due to the different terms on the right hand side of (22), or as we may say to the different rotations ${\displaystyle \chi _{ab}}$. It is convenient now to denote by ${\displaystyle a,b,c}$ the numbers 1, 2, 3 either in this order or in any other derived from it by a cyclic permutation, while the ${\displaystyle x}$-components of the vector we are calculating and which stands on the left hand side of (10) will be represented by ${\displaystyle X_{1},\dots X_{4}}$.

a. Let us first consider that one of the sides ${\displaystyle \left(dx_{a},dx_{b},dx_{c}\right)}$ which faces towards the side of the positive ${\displaystyle x_{4}}$. The vector ${\displaystyle \mathrm {N} }$ drawn outward has the direction ${\displaystyle 4^{*}}$ and in ${\displaystyle \xi }$-units the magnitude ${\displaystyle {\frac {1}{\lambda _{4}}}}$. As the direction ${\displaystyle c}$ corresponds to ${\displaystyle a^{*},b^{*},4^{*}}$, the rotation ${\displaystyle \chi _{ab}}$ gives with ${\displaystyle \mathrm {N} }$ a vector product represented by a vector in the direction ${\displaystyle c}$. The magnitude of this vector is in ${\displaystyle \xi }$-units

${\displaystyle {\frac {1}{\lambda _{4}}}\chi _{ab}}$

and in natural units

${\displaystyle {\frac {\chi _{ab4}}{\lambda _{4}}}\chi _{ab}}$

This must be multiplied by ${\displaystyle l_{abc}dx_{a}dx_{b}dx_{c}}$, the magnitude of the side under consideration in natural units, and finally by ${\displaystyle {\tfrac {1}{l_{c}}}}$ to express the vector product in ${\displaystyle x}$-units. Because of (24) we may write for the result

${\displaystyle \chi _{abc}dx_{a}dx_{b}dx_{c}=\psi _{c4}dx_{a}dx_{b}dx_{c}}$

The opposite side gives a similar result with the opposite sign (${\displaystyle \mathrm {N} }$ having for that side the direction ${\displaystyle -4^{*}}$), so that together the sides contribute the term

${\displaystyle {\frac {\partial \psi _{c4}}{\partial x_{4}}}dW}$

to the component ${\displaystyle X_{c}}$. For shortness sake we have put here

${\displaystyle dx_{1}dx_{2}dx_{3}dx_{4}=dW}$

Finally we may take, ${\displaystyle c=1,2,3}$.

b. Secondly we consider a side ${\displaystyle \left(dx_{a},dx_{b},dx_{4}\right)}$ facing towards the positive ${\displaystyle x_{c}}$. The vector ${\displaystyle \mathrm {N} }$ has now the direction ${\displaystyle -c^{*}}$. We consider the vector products of this vector with the rotations ${\displaystyle \chi _{b4}}$, ${\displaystyle \chi _{4a}}$ and ${\displaystyle \chi _{ba}}$, which vector products have the directions ${\displaystyle a,b}$ and 4. A calculation exactly similar to the one we performed just now gives the contributions to ${\displaystyle X_{a},X_{b},X_{4}}$. For these we thus find the products of ${\displaystyle dx_{a}dx_{b}dx_{4}}$ by

${\displaystyle {\begin{array}{c}{\frac {l_{ab4}\lambda _{bc4}}{l_{a}\lambda _{c}}}\chi _{b4}=\chi _{4b}=\psi _{ac},\\\\{\frac {l_{ab4}\lambda _{ac4}}{l_{b}\lambda _{c}}}\chi _{4a}=\chi _{a4}=\psi _{bc},\\\\{\frac {l_{ab4}\lambda _{abc}}{l_{4}\lambda _{c}}}\chi _{ba}=\chi _{ba}=\psi _{4c}.\end{array}}}$

Taking also into consideration the opposite side ${\displaystyle \left(dx_{a},dx_{b},dx_{4}\right)}$ we find for ${\displaystyle X_{a},X_{b},X_{4}}$ the contributions

${\displaystyle {\frac {\partial \psi _{ac}}{\partial x_{c}}}dW,\ {\frac {\partial \psi _{bc}}{\partial x_{c}}}dW,\ {\frac {\partial \psi _{4c}}{\partial x_{c}}}dW.}$

This may be applied to each of the three pairs of sides not yet mentioned under ${\displaystyle a}$; we have only to take for ${\displaystyle c}$ successively 1, 2, 3.

Summing up what has been said in this § we may say: the components of the vector on the left hand side of (10) are

${\displaystyle X_{a}=\sum (b){\frac {\partial \psi _{ab}}{\partial x_{b}}}dW}$

§ 27. For the components of the vector occurring on the right hand side of (10) we may write

${\displaystyle i\mathrm {q} _{a}d\Omega }$

if ${\displaystyle \mathrm {q} _{a}}$ is the component of the vector ${\displaystyle \mathrm {q} }$ in the direction ${\displaystyle x_{a}}$ expressed in ${\displaystyle x}$-units, while ${\displaystyle d\Omega }$ represents the magnitude of the element ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$ in natural units. This magnitude is

${\displaystyle -i{\sqrt {-g}}dW}$

so that by putting

 ${\displaystyle {\sqrt {-g}}\mathrm {q} _{a}=w_{a}}$ (28)

we find for equation (10)

 ${\displaystyle \sum (b){\frac {\partial \psi _{ab}}{\partial x_{b}}}=w_{a}}$ (29)

The four relations contained in this equation have the same form as those expressed by formula (25) in my paper of last year[14]. We shall now show that the two sets of equations correspond in all respects. For this purpose it will be shown that the transformation formulae formerly deduced for ${\displaystyle w_{a}}$ and ${\displaystyle \psi _{ac}}$ follow from the way in which these quantities have been now defined. The notations from the former paper will again be used and we shall suppose the transformation determinant ${\displaystyle p}$ to be positive.

§ 28. Between the differentials of the original coordinates ${\displaystyle x_{a}}$ and the new coordinates ${\displaystyle x'_{a}}$ which we are going to introduce we have the relations

 ${\displaystyle dx'_{a}=\sum (b)\pi _{ba}dx_{b}}$ (30)

and formulae of the same form (comp. § 10) may be written down for the components of a vector expressed in ${\displaystyle x}$-measure. As the quantities ${\displaystyle \mathrm {q} _{a}}$ constitute a vector and as

${\displaystyle {\sqrt {-g'}}=p{\sqrt {-g}}}$

we have according to (28)[15]

${\displaystyle {\frac {1}{\sqrt {-g'}}}w'_{a}={\frac {1}{\sqrt {-g}}}\sum (b)\pi _{ba}w_{b}}$

or

${\displaystyle w'_{a}=p\sum (b)\pi _{ba}w_{b}}$

Further we have for the infinitely small quantities ${\displaystyle \xi _{a}}$[16] defined by (19)

${\displaystyle \xi '_{a}=\sum (b)p_{ba}\xi _{b}}$

and in agreement with this for the components of a vector expressed in ${\displaystyle \xi }$-units

${\displaystyle \Xi '_{a}=\sum (b)p_{ba}\Xi _{b}}$

so that we find from (25)[17]

${\displaystyle \chi '_{ab}=\sum (cd)p_{ca}p_{db}\chi _{cd}}$

Interchanging here ${\displaystyle c}$ and ${\displaystyle d}$, we obtain

${\displaystyle \chi '_{ab}=\sum (cd)p_{da}p_{cb}\chi _{dc}=-\sum (cd)p_{da}p_{cb}\chi _{cd}}$

and

 ${\displaystyle \chi '_{ab}={\frac {1}{2}}\sum (cd)\left(p_{ca}p_{db}-p_{da}p_{cb}\right)\chi _{cd}}$ (31)

The quantity between brackets on the right hand side is a second order minor of the determinant ${\displaystyle p}$ and as is well known this minor is related to a similar minor of the determinant of the coefficients ${\displaystyle \pi _{ab}}$. If ${\displaystyle a'b'}$ corresponds to ${\displaystyle ab}$ in the way mentioned in § 25, and ${\displaystyle c'd'}$ in the same way to ${\displaystyle cd}$, we have

${\displaystyle p_{ca}d_{db}-p_{da}p_{ch}=p\left(\pi _{c'a'}\pi _{d'b'}-\pi _{d'a'}\pi _{c'b'}\right)}$

so that (31) becomes

${\displaystyle \chi '_{ab}={\frac {1}{2}}p\sum (cd)\left(\pi _{c'a'}\pi _{d'b'}-\pi _{d'a'}\pi _{c'b'}\right)\chi _{cd}}$

According to (27) this becomes

${\displaystyle \psi '_{a'b'}={\frac {1}{2}}p\sum (cd)\left(\pi _{c'a'}\pi _{d'b'}-\pi _{d'a'}\pi _{c'b'}\right)\psi _{c'd'}}$

for which we may write

${\displaystyle \psi '_{ab}={\frac {1}{2}}p\sum (cd)\left(\pi _{ca}\pi _{db}-\pi _{da}\pi _{cb}\right)\psi _{cd}}$

Interchanging ${\displaystyle c}$ and ${\displaystyle d}$ in the second of the two parts into which the sum on the right hand side can be decomposed, and taking into consideration that

${\displaystyle \psi _{dc}=-\psi _{cd}}$

as is evident from (26) and (27), we find[18]

${\displaystyle \psi '_{ab}=p\sum (cd)\pi _{ca}\pi _{db}\psi _{cd}}$

§ 29. Finally it can be proved that if equation (10) holds for one system of coordinates ${\displaystyle x_{1},\dots x_{4}}$, it will also be true for every other system ${\displaystyle x'_{1},\dots x'_{4}}$, so that

 ${\displaystyle \int \left\{\left[\mathrm {R} _{e}\cdot \mathrm {N} \right]+\left[\mathrm {R} _{h}\cdot \mathrm {N} \right]\right\}_{x'}d\sigma =i\int \{\mathrm {q} \}_{x'}d\Omega }$ (32)

To show this we shall first assume that the extension ${\displaystyle \Omega }$, which is understood to be the same in the two cases, is the element ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$.

For the four equations taken together in (10) we may then write

 ${\displaystyle \int u_{1}d\sigma =v_{1}d\Omega ,\dots \int u_{4}d\sigma =v_{4}d\Omega }$ (33)

and in the same way for the four equations (32)

 ${\displaystyle \int u'_{1}d\sigma =v'_{1}d\Omega ,\dots \int u'_{4}d\sigma =v'_{4}d\Omega }$ (34)

We have now to deduce these last equations from (33). In doing so we must keep in mind that ${\displaystyle u_{1},\dots u_{4}}$ are the ${\displaystyle x}$-components and ${\displaystyle u'_{1},\dots u'_{4}}$ the ${\displaystyle x}$-components of one definite vector and that the same may be said of ${\displaystyle v_{1},\dots v_{4}}$ and ${\displaystyle v'_{1},\dots v'_{4}}$.

Hence, at a definite point (comp. (30))

 ${\displaystyle v'_{a}=\sum (b)\pi _{ba}v_{b}}$ (35)

We shall particularly denote by ${\displaystyle \pi _{ba}}$ the values of these quantities belonging to the angle ${\displaystyle P}$ from which the edges ${\displaystyle dx_{1},\dots dx_{4}}$ issue in positive directions. To the right hand sides of the equations (34) we may apply transformation (35) with these values of ${\displaystyle \pi _{ba}}$, ${\displaystyle d\Omega }$-being infinitely small of the fourth order and it being allowed to confine ourselves to quantities of this order.

On the left hand sides of (34), however, we must take into consideration, the surface being of the third order, that the values of ${\displaystyle \pi _{ba}}$ change from point to point. Let ${\displaystyle \mathrm {x} _{1},\dots \mathrm {x} _{4}}$ be the changes which ${\displaystyle x_{1},\dots x_{4}}$ undergo when we pass from ${\displaystyle P}$ to any other point of the surface. Then we must write for the value of the coefficient at this last point

${\displaystyle \pi _{ba}+\sum (c){\frac {d\pi _{ba}}{dx_{c}}}\mathrm {x} _{c}}$

We thus have

${\displaystyle \int u'_{a}d\sigma =\sum (b)\pi _{ba}\int u_{b}d\sigma +\sum (b)\int u_{b}\sum (c){\frac {\partial \pi {}_{ba}}{\partial x_{c}}}\mathrm {x} _{c}d\sigma }$

It will be shown presently that the last term vanishes. This being proved, it is clear that the relations (34) follow from (33); indeed, multiplying equations (33) by${\displaystyle \pi _{1a},\dots \pi _{4a}}$ respectively and adding them we find

${\displaystyle \int u'_{a}d\sigma =v'_{a}d\Omega }$

§ 30. The proof for

 ${\displaystyle \sum (b)\int u_{b}\sum (c){\frac {\partial \pi {}_{ba}}{\partial x_{c}}}\mathrm {x} _{c}d\sigma =0}$ (36)

rests on the relations

 ${\displaystyle {\frac {\partial \pi {}_{ba}}{\partial x_{c}}}={\frac {\partial \pi {}_{ea}}{\partial x_{b}}}}$ (37)

${\displaystyle \pi {}_{ba}={\frac {\partial x'_{b}}{\partial x_{b}}},\ \pi {}_{ea}={\frac {\partial x'_{a}}{\partial x_{e}}},}$

The integral which occurs in (36) differs from

 ${\displaystyle \int u_{b}d\sigma }$ (38)

by the infinitely small factor under the sign of integration

${\displaystyle \sum (c){\frac {\partial \pi {}_{ba}}{\partial x_{c}}}\mathrm {x} _{c}}$

Now we have calculated in § 26 integrals like (38) by taking together each time two opposite sides, one of which ${\displaystyle \Sigma _{1}}$ passes through ${\displaystyle P}$ while the second ${\displaystyle \Sigma _{2}}$ is obtained from the first by a shift in the direction of one of the coordinates e. g. of ${\displaystyle x_{e}}$ over the distance ${\displaystyle dx_{e}}$. We had then to keep in mind that for the two sides the values of ${\displaystyle u_{b}}$, which have opposite signs, are a little different; and it was precisely this difference that was of importance. In the calculation of the integral

 ${\displaystyle \int u_{b}\sum (c){\frac {\partial \pi {}_{ba}}{\partial x_{c}}}\mathrm {x} _{c}d\sigma }$ (39)

however it may be neglected. Hence, when we express the components ${\displaystyle u_{b}}$ in terms of the quantities ${\displaystyle \psi _{ab}}$, we may give to these latter the values which they have at the point ${\displaystyle P}$.

Let us consider two sides situated at the ends of the edges ${\displaystyle dx_{e}}$ and whose magnitude we may therefore express in ${\displaystyle x}$-units ${\displaystyle dx_{j}dx_{k}dx_{l}}$ if ${\displaystyle j,k,l}$ are the numbers which are left of 1, 2, 3, 4 when the number ${\displaystyle e}$ is omitted. For the part contributed to (38) by the side ${\displaystyle \Sigma _{2}}$ we found in § 26

${\displaystyle \psi {}_{be}dx_{j}dx_{k}dx_{l}}$

We now find for the part of (39) due to the two sides

${\displaystyle \psi {}_{be}\sum (c){\frac {\partial \pi {}_{ba}}{\partial x_{c}}}\left[\int \limits _{2}\mathrm {x} _{c}d\sigma -\int \limits _{1}\mathrm {x} _{c}d\sigma \right]}$

where the first integral relates to ${\displaystyle \Sigma _{2}}$ and the second to ${\displaystyle \Sigma _{1}}$. It is clear that but one value of ${\displaystyle c}$, viz. ${\displaystyle e}$ has to be considered. As everywhere in ${\displaystyle \Sigma _{1}:\mathrm {x} _{c}=0}$ and everywhere in ${\displaystyle \Sigma _{2}:\mathrm {x} _{c}=dx_{e}}$ it is further evident that the above expression becomes

${\displaystyle \psi {}_{eb}{\frac {\partial \pi {}_{ba}}{\partial x_{c}}}dW}$

This is one part contributed to the expression (36). A second part, the origin of which will be immediately understood, is found by interchanging ${\displaystyle b}$ and ${\displaystyle e}$. With a view to (37) and because of

${\displaystyle \psi {}_{eb}=-\psi {}_{be}}$

we have for each term of (36) another by which it is cancelled. This is what had to be proved.

§ 31. Now that we have shown that equation (32) holds for each element ${\displaystyle \left(dx_{1},\dots dx_{4}\right)}$ we may conclude by the considerations of § 21 that this is equally true for any arbitrarily chosen magnitude and shape of the extension ${\displaystyle \Omega }$. In particular the equation may be applied to an element ${\displaystyle \left(dx'_{1},\dots dx'_{4}\right)}$ and by considerations exactly similar to those presented in § 26 we see that in the new coordinates as well as in the original ones we have equations of the form (29).

Whatever be our choice of the coordinates the part of the principal function indicated in § 14 can therefore be derived for a given current vector ${\displaystyle \mathrm {q} }$.

In a sequel to this paper some conclusions that may be drawn from Hamilton's principle will be considered.

III.

(Communicated in the meeting of April 1916.)[19]

§ 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

${\displaystyle g^{ab}={\sqrt {-g}}g^{ab}}$

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

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

We shall use Christoffel's symbols

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

and Riemann's symbol

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

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

${\displaystyle {\frac {1}{2\varkappa }}\int QdS}$

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 ${\displaystyle x_{1},\dots x_{4}}$ 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

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

namely

 ${\displaystyle \delta _{1}Q=\sum (ab)\left\{{\frac {\partial Q}{\partial g_{ab}}}-\sum (e){\frac {\partial }{\partial x_{e}}}{\frac {\partial Q}{\partial g_{ab,e}}}+\sum (ef){\frac {\partial ^{2}}{\partial x_{e}\partial x_{f}}}{\frac {\partial Q}{\partial g_{ab,ef}}}\right\}\delta g_{ab}}$ (43)
 ${\displaystyle {\begin{array}{c}\delta _{2}Q=\sum (abe){\frac {\partial Q}{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,e}}}\delta g_{ab}\right)+\sum (abef){\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,ef}}}\delta g_{ab,e}\right)-\\\\-\sum (abef){\frac {\partial }{\partial x_{e}}}\left\{{\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,ef}}}\right)\delta g_{ab}\right\}\end{array}}}$ (44)

The last equation shows that

 ${\displaystyle \int \delta _{2}QdS=0}$ (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

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

where ${\displaystyle \left(\delta _{1}Q\right)}$ and ${\displaystyle \left(\delta _{2}Q\right)}$ 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

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

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 ${\displaystyle \delta _{1}QdS}$ and ${\displaystyle \delta _{2}QdS}$ separately.[27]

We have therefore

 ${\displaystyle {\frac {\delta _{1}Q'}{\sqrt {-g'}}}={\frac {\delta _{1}Q}{\sqrt {-g}}}}$ (46)

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

 ${\displaystyle \delta _{1}Q=\sum (ab)M_{ab}d{\mathfrak {g}}^{ab}}$ (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 ${\displaystyle \left({\tfrac {1}{\sqrt {-g}}}{\mathfrak {g}}^{ab}\right)}$ is a contravariant tensor of the second order. From this we can deduce that ${\displaystyle \left({\frac {1}{\sqrt {-g}}}\delta {\mathfrak {g}}^{ab}\right)}$ 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 ${\displaystyle g_{11}=g_{22}=g_{33}=-1}$, ${\displaystyle g_{44}=+1}$, ${\displaystyle g_{ab}=0}$ 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

 ${\displaystyle \delta _{1}Q=\sum (ab)G_{ab}\delta {\mathfrak {g}}^{ab}}$ (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

 ${\displaystyle \delta g^{ab}={\frac {1}{\sqrt {-g}}}\delta {\mathfrak {g}}^{ab}-{\frac {g^{ab}}{2{\sqrt {-g}}}}\sum (cd)g_{cd}\delta {\mathfrak {g}}^{cd}}$ (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 ${\displaystyle x_{h}}$ is diminished by ${\displaystyle \xi }$, while the three other coordinates are left constant. Then we have

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

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

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

 ${\displaystyle dQ-\delta _{2}Q=\delta _{1}Q}$ (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

${\displaystyle -{\frac {\partial Q}{\partial x_{h}}}\xi }$

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

 ${\displaystyle {\begin{array}{c}-{\frac {\partial Q}{\partial x_{h}}}+\sum (abe){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,e}}}g_{ab,h}\right)+\sum (abef){\frac {\partial }{\partial x_{e}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,fh}\right)-\\\\-\sum (abef){\frac {\partial }{\partial x_{e}}}\left\{{\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,h}\right\}=-\sum (ab)G_{ab}{\mathfrak {g}}^{ab,h}\end{array}}}$ (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 ${\displaystyle e\neq h}$

 ${\displaystyle {\mathfrak {s}}_{h}^{e}=\sum (ab){\frac {\partial Q}{\partial g_{ab,e}}}g_{ab,h}+\sum (abf){\frac {\partial Q}{\partial g_{ab,fe}}}g_{ab,fh}-\sum (abf){\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,fe}}}\right)g_{ab,h}}$ (52)

and for ${\displaystyle e=h}$

 ${\displaystyle {\mathfrak {s}}_{h}^{h}=-Q+\sum (ab){\frac {\partial Q}{\partial g_{ab,h}}}g_{ab,h}+\sum (abf){\frac {\partial Q}{\partial g_{ab,fh}}}g_{ab,fh}-\sum (abf){\frac {\partial }{\partial x_{f}}}\left({\frac {\partial Q}{\partial g_{ab,hf}}}\right)g_{ab,h}}$ (53)

we may write

 ${\displaystyle \sum (e){\frac {\partial {\mathfrak {s}}_{h}^{e}}{\partial x_{e}}}=-\sum (ab)G_{ab}{\mathfrak {g}}^{ab,h}}$ (54)

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

The equation therefore becomes

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

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

 ${\displaystyle {\mathfrak {s}}_{0h}^{e}=-G\sum (a){\mathfrak {g}}^{ae}g_{ah}+2\sum (a){\mathfrak {g}}^{ae}G_{ah}}$ (57)

Taking also the divergency of this complex we find that the difference

${\displaystyle div'_{h}{\mathfrak {s}}'_{0}-p\sum (m)p_{mh}div_{m}{\mathfrak {s}}_{0}}$

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

${\displaystyle div'_{h}{\mathfrak {s}}'-p\sum (m)p_{mh}div_{m}{\mathfrak {s}}}$

It is thus seen that

${\displaystyle div'_{h}{\mathfrak {s}}'-div'_{h}{\mathfrak {s}}'_{0}=p\sum (m)p_{mh}\left(div_{m}{\mathfrak {s}}-div_{m}{\mathfrak {s}}_{0}\right)}$

and that we have therefore

 ${\displaystyle div{\mathfrak {s}}=div{\mathfrak {s}}_{0}}$ (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 ${\displaystyle div_{h}{\mathfrak {s}}}$ and ${\displaystyle div_{h}{\mathfrak {s}}_{0}}$. 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 ${\displaystyle {\mathfrak {s}}_{0}}$ 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

${\displaystyle \int \mathrm {L} dS}$

where ${\displaystyle \mathrm {L} }$ is defined by (35) (1915). From ${\displaystyle \mathrm {L} }$ 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]

 ${\displaystyle {\mathfrak {T}}_{c}^{c}=-\mathrm {L} +\sum \limits _{a\neq c}(a)\psi _{ac}^{*}\psi _{a'c'}}$ (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

${\displaystyle {\frac {1}{\sqrt {-g'}}}{\mathfrak {T}}_{c}^{'b}={\frac {1}{\sqrt {-g}}}\sum (kl)p_{kc}\pi lb{\mathfrak {T}}_{k}^{l}}$

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

 ${\displaystyle \delta _{\psi }\int \mathrm {L} dS+{\frac {1}{2\varkappa }}\delta \int QdS=0}$ (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