Chapter VII: Appendix: Mechanics and the Relativity-Postulate (2)

I shall now describe _the ponderomotive force which is exerted by one moving electron upon another moving electron_. Let us suppose that the world-line of a second point-electron passes through the world-point P₁. Let us determine P, Q, _r_ as before, construct the middle-point M of the hyperbola of curvature at P, and finally the normal MN upon a line through P which is parallel to QP₁. With P as the initial point, we shall establish a system of reference in the following way: the _t_-axis will be laid along PQ, the _x_-axis in the direction of QP₁. The _y_-axis in the direction of MN, then the _z_-axis is automatically determined, as it is normal to the _x_-, _y_-, _z_-axes. Let [:_x_], [:_y_], [:_z_], [:_t_] be the acceleration-vector at P, [._x_]₁, [._y_]₁ [._z_]₁, [._t_]₁ be the velocity-vector at P₁. Then the force-vector exerted by the first election _e_, (moving in any possible manner) upon the second election _e_, (likewise moving in any possible manner) at P₁ is represented by

-_e e₁_([._t₁_] - [._x₁_]/_c_)F,

_For the components F_{x}, F_{y}, F_{z}, F_{t} of the vector F the following three relations hold_:—

_c_F_{_t_} - F_{_x_} = 1/_r²_, F_{_y_} = [:_y_]/(_c²__r_), F_{_z_} =
0,

_and fourthly this vector F is normal to the velocity-vector_ P₁, _and through this circumstance alone, its dependence on this last velocity-vector arises_.

If we compare with this expression the previous formulæ[35] giving the elementary law about the ponderomotive action of moving electric charges upon each other, then we cannot but admit, that the relations which occur here reveal the inner essence of full simplicity first in four dimensions; but in three dimensions, they have very complicated projections.

In the mechanics reformed according to the world-postulate, the disharmonies which have disturbed the relations between Newtonian mechanics and modern electrodynamics automatically disappear. I shall now consider the position of the Newtonian law of attraction to this postulate. I will assume that two point-masses _m_ and _m₁_ describe their world-lines; a moving force-vector is exercised by _m_ upon _m₁_, and the expression is just the same as in the case of the electron, only we have to write +_mm₁_ instead -_ee₁_. We shall consider only the special case in which the acceleration-vector of _m_ is always zero: then _t_ may be introduced in such a manner that _m_ may be regarded as fixed, the motion of _m_ is now subjected to the moving-force vector of _m_ alone. If we now modify this given vector by writing -([^.]1/√(1-(_v_²/_c²_)) instead of [._t_] ([._t_] = 1 up to magnitudes of the order (1[^.]/_c²_)), then it appears that Kepler’s laws hold good for the position (_x₁_, _y₁_, _z₁_), of _m₁_ at any time, only in place of the time _t₁_, we have to write the proper time τ₁ of _m₁_. On the basis of this simple remark, it can be seen that the proposed law of attraction in combination with new mechanics is not less suited for the explanation of astronomical phenomena than the Newtonian law of attraction in combination with Newtonian mechanics.

Also the fundamental equations for electro-magnetic processes in moving bodies are in accordance with the world-postulate. I shall also show on a later occasion that the deduction of these equations, as taught by Lorentz, are by no means to be given up.

The fact that the world-postulate holds without exception is, I believe, the true essence of an electromagnetic picture of the world; the idea first occurred to Lorentz, its essence was first picked out by Einstein, and is now gradually fully manifest. In course of time, the mathematical consequences will be gradually deduced, and enough suggestions will be forthcoming for the experimental verification of the postulate; in this way even those, who find it uncongenial, or even painful to give up the old, time-honoured concepts, will be reconciled to the new ideas of time and space,—in the prospect that they will lead to pre-established harmony between pure mathematics and physics.

Footnote 30:

Planck, Zur Dynamik bewegter systeme, Ann. d. physik, Bd. 26, 1908, p.
1.

Footnote 31:

H. Minkowski; the passage refers to paper (2) of the present edition.

Footnote 32:

Minkowski—Mechanics, appendix, page 65 of paper (2). Planck—Verh. d.
D. P. G. Vol. 4, 1906, p. 136.

Footnote 33:

Schütz, Gött. Nachr. 1897, p. 110.

Footnote 34:

Lienard, L’Eclairage électrique T. 16, 1896, p. 53. Wiechert, Ann. d.
Physik, Vol. 4.

Footnote 35:

K. Schwarzschild. Gött-Nachr. 1903. H. A. Lorentz, Enzyklopädie der
Math. Wissenschaften V. Art 14, p. 199.

The Foundation of the Generalised Theory of Relativity
By A. Einstein.
From Annalen der Physik 4.49.1916.

The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as “the theory of Relativity;” this latter theory I differentiate from the former “Special Relativity theory,” and suppose it to be known. The generalization of the Relativity theory has been made much easier through the form given to the special Relativity theory by Minkowski, which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building up of the theory. The mathematical apparatus useful for the general relativity theory, lay already complete in the “Absolute Differential Calculus,” which were based on the researches of Gauss, Riemann and Christoffel on the non-Euclidean manifold, and which have been shaped into a system by Ricci and Levi-civita, and already applied to the problems of theoretical physics. I have in part B of this communication developed in the simplest and clearest manner, all the supposed mathematical auxiliaries, not known to Physicists, which will be useful for our purpose, so that, a study of the mathematical literature is not necessary for an understanding of this paper. Finally in this place I thank my friend Grossmann, by whose help I was not only spared the study of the mathematical literature pertinent to this subject, but who also aided me in the researches on the field equations of gravitation.

A
Principal considerations about the Postulate of Relativity.

§ 1. Remarks on the Special Relativity Theory.

The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics.

If a co-ordinate system K be so chosen that when referred to it, the physical laws hold in their simplest forms these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to an uniform translational motion relative to K. We call this postulate “The Special Relativity Principle.” By the word special, it is signified that the principle is limited to the case, when K′ has _uniform translatory_ motion with reference to K, but the equivalence of K and K′ does not extend to the case of non-uniform motion of K′ relative to K.

The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorenz-transformation, with all the relations between moving rigid bodies and clocks.

The modification which the theory of space and time has undergone through the special relativity theory, is indeed a profound one, but a weightier point remains untouched. According to the special relativity theory, the theorems of geometry are to be looked upon as the laws about any possible relative positions of solid bodies at rest, and more generally the theorems of kinematics, as theorems which describe the relation between measurable bodies and clocks. Consider two material points of a solid body at rest; then according to these conceptions there corresponds to these points a wholly definite extent of length, independent of kind, position, orientation and time of the body.

Similarly let us consider two positions of the pointers of a clock which is at rest with reference to a co-ordinate system; then to these positions, there always corresponds, a time-interval of a definite length, independent of time and place. It would be soon shown that the general relativity theory can not hold fast to this simple physical significance of space and time.

§ 2. About the reasons which explain the extension of the
relativity-postulate.

To the classical mechanics (no less than) to the special relativity theory, is attached an episteomological defect, which was perhaps first cleanly pointed out by E. Mach. We shall illustrate it by the following example; Let two fluid bodies of equal kind and magnitude swim freely in space at such a great distance from one another (and from all other masses) that only that sort of gravitational forces are to be taken into account which the parts of any of these bodies exert upon each other. The distance of the bodies from one another is invariable. The relative motion of the different parts of each body is not to occur. But each mass is seen to rotate by an observer at rest relative to the other mass round the connecting line of the masses with a constant angular velocity (definite relative motion for both the masses). Now let us think that the surfaces of both the bodies (S₁ and S₂) are measured with the help of measuring rods (relatively at rest); it is then found that the surface of S₁ is a sphere and the surface of the other is an ellipsoid of rotation. We now ask, why is this difference between the two bodies? An answer to this question can only then be regarded as satisfactory from the episteomological standpoint when the thing adduced as the cause is an observable fact of experience. The law of causality has the sense of a definite statement about the world of experience only when observable facts alone appear as causes and effects.

The Newtonian mechanics does not give to this question any satisfactory answer. For example, it says:—The laws of mechanics hold true for a space R₁ relative to which the body S₁ is at rest, not however for a space relative to which S₂ is at rest.

The Galiliean space, which is here introduced is however only a purely imaginary cause, not an observable thing. It is thus clear that the Newtonian mechanics does not, in the case treated here, actually fulfil the requirements of causality, but produces on the mind a fictitious complacency, in that it makes responsible a _wholly imaginary cause_ R₁ for the different behaviours of the bodies S₁ and S₂ which are actually observable.

A satisfactory explanation to the question put forward above can only be thus given:—that the physical system composed of S₁ and S₂ shows for itself alone no conceivable cause to which the different behaviour of S₁ and S₂ can be attributed. The cause must thus lie outside the system. We are therefore led to the conception that the general laws of motion which determine specially the forms of S₁ and S₂ must be of such a kind, that the mechanical behaviour of S₁ and S₂ must be essentially conditioned by the distant masses, which we had not brought into the system considered. These distant masses, (and their relative motion as regards the bodies under consideration) are then to be looked upon as the seat of the principal observable causes for the different behaviours of the bodies under consideration. They take the place of the imaginary cause R₁. Among all the conceivable spaces R₁ and R₂ moving in any manner relative to one another, there is a priori, no one set which can be regarded as affording greater advantages, against which the objection which was already raised from the standpoint of the theory of knowledge cannot be again revived. The laws of physics must be so constituted that they should remain valid for any system of co-ordinates moving in any manner. We thus arrive at an extension of the relativity postulate.

Besides this momentous episteomological argument, there is also a well-known physical fact which speaks in favour of an extension of the relativity theory. Let there be a Galiliean co-ordinate system K relative to which (at least in the four-dimensional region considered) a mass at a sufficient distance from other masses move uniformly in a line. Let K′ be a second co-ordinate system which has a uniformly accelerated motion relative to K. Relative to K′ any mass at a sufficiently great distance experiences an accelerated motion such that its acceleration and the direction of acceleration is independent of its material composition and its physical conditions.

Can any observer, at rest relative to K′, then conclude that he is in an actually accelerated reference-system? This is to be answered in the negative; the above-named behaviour of the freely moving masses relative to K′ can be explained in as good a manner in the following way. The reference-system K′ has no acceleration. In the space-time region considered there is a gravitation-field which generates the accelerated motion relative to K′.

This conception is feasible, because to us the experience of the existence of a field of force (namely the gravitation field) has shown that it possesses the remarkable property of imparting the same acceleration to all bodies. The mechanical behaviour of the bodies relative to K′ is the same as experience would expect of them with reference to systems which we assume from habit as stationary; thus it explains why from the physical stand-point it can be assumed that the systems K and K′ can both with the same legitimacy be taken as at rest, that is, they will be equivalent as systems of reference for a description of physical phenomena.

From these discussions we see, that the working out of the general relativity theory must, at the same time, lead to a theory of gravitation; for we can “create” a gravitational field by a simple variation of the co-ordinate system. Also we see immediately that the principle of the constancy of light-velocity must be modified, for we recognise easily that the path of a ray of light with reference to K′ must be, in general, curved, when light travels with a definite and constant velocity in a straight line with reference to K.

§ 3. The time-space continuum. Requirements of the general Co-variance
for the equations expressing the laws of Nature in general.

In the classical mechanics as well as in the special relativity theory, the co-ordinates of time and space have an immediate physical significance; when we say that any arbitrary point has _x₁_ as its X₁ co-ordinate, it signifies that the projection of the point-event on the X₁-axis _ascertained_ by means of a solid rod according to the rules of Euclidean Geometry is reached when a definite measuring rod, the unit rod, can be carried _x₁_ times from the origin of co-ordinates along the X₁ axis. A point having _x₄_ = _t₁_ as the X₄ co-ordinate signifies that a unit clock which is adjusted to be at rest relative to the system of co-ordinates, and coinciding in its spatial position with the point-event and set according to some definite standard has gone over _x₄_ = _t_ periods before the occurrence of the point-event.

This conception of time and space is continually present in the mind of the physicist, though often in an unconscious way, as is clearly recognised from the role which this conception has played in physical measurements. This conception must also appear to the reader to be lying at the basis of the second consideration of the last paragraph and imparting a sense to these conceptions. But we wish to show that we are to abandon it and in general to replace it by more general conceptions in order to be able to work out thoroughly the postulate of general relativity,—the case of special relativity appearing as a limiting case when there is no gravitation.

We introduce in a space, which is free from Gravitation-field, a Galiliean Co-ordinate System K (_x_, _y_, _z_, _t_) and also, another system K′ (_x′_ _y′_ _z′_ _t′_) rotating uniformly relative to K. The origin of both the systems as well as their _z_-axes might continue to coincide. We will show that for a space-time measurement in the system K′, the above established rules for the physical significance of time and space can not be maintained. On grounds of symmetry it is clear that a circle round the origin in the XY plane of K, can also be looked upon as a circle in the plane (X′, Y′) of K′. Let us now think of measuring the circumference and the diameter of these circles, with a unit measuring rod (infinitely small compared with the radius) and take the quotient of both the results of measurement. If this experiment be carried out with a measuring rod at rest relatively to the Galiliean system K we would get π, as the quotient. The result of measurement with a rod relatively at rest as regards K′ would be a number which is greater than π. This can be seen easily when we regard the whole measurement-process from the system K and remember that the rod placed on the periphery suffers a Lorenz-contraction, not however when the rod is placed along the radius. Euclidean Geometry therefore does not hold for the system K′; the above fixed conceptions of co-ordinates which assume the validity of Euclidean Geometry fail with regard to the system K′. We cannot similarly introduce in K′ a time corresponding to physical requirements, which will be shown by all similarly prepared clocks at rest relative to the system K′. In order to see this we suppose that two similarly made clocks are arranged one at the centre and one at the periphery of the circle, and considered from the stationary system K. According to the well-known results of the special relativity theory it follows—(as viewed from K)—that the clock placed at the periphery will go slower than the second one which is at rest. The observer at the common origin of co-ordinates who is able to see the clock at the periphery by means of light will see the clock at the periphery going slower than the clock beside him. Since he cannot allow the velocity of light to depend explicitly upon the time in the way under consideration he will interpret his observation by saying that the clock on the periphery actually goes slower than the clock at the origin. He cannot therefore do otherwise than define time in such a way that the rate of going of a clock depends on its position.

We therefore arrive at this result. In the general relativity theory time and space magnitudes cannot be so defined that the difference in spatial co-ordinates can be immediately measured by the unit-measuring rod, and time-like co-ordinate difference with the aid of a normal clock.

The means hitherto at our disposal, for placing our co-ordinate system in the time-space continuum, in a definite way, therefore completely fail and it appears that there is no other way which will enable us to fit the co-ordinate system to the four-dimensional world in such a way, that by it we can expect to get a specially simple formulation of the laws of Nature. So that nothing remains for us but to regard all conceivable co-ordinate systems as equally suitable for the description of natural phenomena. This amounts to the following law:—

_That in general, Laws of Nature are expressed by means of equations which are valid for all co-ordinate systems, that is, which are covariant for all possible transformations._ It is clear that a physics which satisfies this postulate will be unobjectionable from the standpoint of the general relativity postulate. Because among all substitutions there are, in every case, contained those, which correspond to all relative motions of the co-ordinate system (in three dimensions). This condition of general covariance which takes away the last remnants of physical objectivity from space and time, is a natural requirement, as seen from the following considerations. All our _well-substantiated_ space-time propositions amount to the determination of space-time coincidences. If, for example, the event consisted in the motion of material points, then, for this last case, nothing else are really observable except the encounters between two or more of these material points. The results of our measurements are nothing else than well-proved theorems about such coincidences of material points, of our measuring rods with other material points, coincidences between the hands of a clock, dial-marks and point-events occurring at the same position and at the same time.

The introduction of a system of co-ordinates serves no other purpose than an easy description of totality of such coincidences. We fit to the world our space-time variables (_x₁_ _x₂_ _x₃_ _x₄_) such that to any and every point-event corresponds a system of values of (_x₁_ _x₂_ _x₃_ _x₄_). Two coincident point-events correspond to the same value of the variables (_x₁_ _x₂_ _x₃_ _x₄_); _i.e._, the coincidence is characterised by the equality of the co-ordinates. If we now introduce any four functions (_x′₁_ _x′₂_ _x′₃_ _x′₄_) as co-ordinates, so that there is an unique correspondence between them, the equality of all the four co-ordinates in the new system will still be the expression of the space-time coincidence of two material points. As the purpose of all physical laws is to allow us to remember such coincidences, there is a priori no reason present, to prefer a certain co-ordinate system to another; _i.e._, we get the condition of general covariance.

§ 4. Relation of four co-ordinates to spatial and time-like
measurements.

_Analytical expression for the Gravitation-field._

I am not trying in this communication to deduce the general Relativity-theory as the simplest logical system possible, with a minimum of axioms. But it is my chief aim to develop the theory in such a manner that the reader perceives the psychological naturalness of the way proposed, and the fundamental assumptions appear to be most reasonable according to the light of experience. In this sense, we shall now introduce the following supposition; that for an infinitely small four-dimensional region, the relativity theory is valid in the special sense when the axes are suitably chosen.

The nature of acceleration of an infinitely small (positional) co-ordinate system is hereby to be so chosen, that the gravitational field does not appear; this is possible for an infinitely small region. X₁, X₂, X₃ are the spatial co-ordinates; X₄ is the corresponding time-co-ordinate measured by some suitable measuring clock. These co-ordinates have, with a given orientation of the system, an immediate physical significance in the sense of the special relativity theory (when we take a rigid rod as our unit of measure). The expression

(1) _ds²_ = - _d_X₁² - _d_X₂² - _d_X₃² + _d_X₄²

had then, according to the special relativity theory, a value which may be obtained by space-time measurement, and which is independent of the orientation of the local co-ordinate system. Let us take _ds_ as the magnitude of the line-element belonging to two infinitely near points in the four-dimensional region. If _ds²_ belonging to the element (_d_X₁, _d_X₂, _d_X₃, _d_X₄) be positive we call it with Minkowski, time-like, and in the contrary case space-like.

To the line-element considered, _i.e._, to both the infinitely near point-events belong also definite differentials _dx₁_, _dx₂_, _dx₃_, _dx₄_, of the four-dimensional co-ordinates of any chosen system of reference. If there be also a local system of the above kind given for the case under consideration, _d_X’s would then be represented by definite linear homogeneous expressions of the form

(2) _d_X_{ν} = σ_{σ}_a__{νσ}_dx__{σ}

If we substitute the expression in (1) we get

(3) _ds²_ = σ_{στ}_g__{στ}_dx__{σ}_dx__{τ}

where _g__{στ} will be functions of _x__{σ}, but will no longer depend upon the orientation and motion of the ‘local’ co-ordinates; for _ds²_ is a definite magnitude belonging to two point-events infinitely near in space and time and can be got by measurements with rods and clocks. The _g__{τσ}’s are here to be so chosen, that _g__{τσ} = _g__{στ}; the summation is to be extended over all values of σ and τ, so that the sum is to be extended, over 4 × 4 terms, of which 12 are equal in pairs.

From the method adopted here, the case of the usual relativity theory comes out when owing to the special behaviour of _g__{στ} in a _finite_ region it is possible to choose the system of co-ordinates in such a way that _g__{στ} assumes constant values—

{ -1, 0, 0, 0
(4) { 0 -1 0 0
{ 0 0 -1 0
{ 0 0 0 +1

We would afterwards see that the choice of such a system of co-ordinates for a finite region is in general not possible.

From the considerations in § 2 and § 3 it is clear, that from the physical stand-point the quantities _g__{στ} are to be looked upon as magnitudes which describe the gravitation-field with reference to the chosen system of axes. We assume firstly, that in a certain four-dimensional region considered, the special relativity theory is true for some particular choice of co-ordinates. The _g__{στ}’s then have the values given in (4). A free material point moves with reference to such a system uniformly in a straight-line. If we now introduce, by any substitution, the space-time co-ordinates _x₁_..._x₄_ then in the new system _g__{μν}’s are no longer constants, but functions of space and time. At the same time, the motion of a free point-mass in the new co-ordinates, will appear as curvilinear, and not uniform, in which the law of motion, will be _independent of the nature of the moving mass-points_. We can thus signify this motion as one under the influence of a gravitation field. We see that the appearance of a gravitation-field is connected with space-time variability of _g__{στ}’s. In the general case, we can not by any suitable choice of axes, make special relativity theory valid throughout any finite region. We thus deduce the conception that _g__{στ}’s describe the gravitational field. According to the general relativity theory, gravitation thus plays an exceptional rôle as distinguished from the others, specially the electromagnetic forces, in as much as the 10 functions _g__{στ} representing gravitation, define immediately the metrical properties of the four-dimensional region.

B
Mathematical Auxiliaries for Establishing the General Covariant
Equations.

We have seen before that the general relativity-postulate leads to the condition that the system of equations for Physics, must be co-variants for any possible substitution of co-ordinates _x₁_, ... _x₄_; we have now to see how such general co-variant equations can be obtained. We shall now turn our attention to these purely mathematical propositions. It will be shown that in the solution, the invariant _ds_, given in equation (3) plays a fundamental rôle, which we, following Gauss’s Theory of Surfaces, style as the line-element.

The fundamental idea of the general co-variant theory is this:—With reference to any co-ordinate system, let certain things (tensors) be defined by a number of functions of co-ordinates which are called the components of the tensor. There are now certain rules according to which the components can be calculated in a new system of co-ordinates, when these are known for the original system, and when the transformation connecting the two systems is known. The things herefrom designated as “Tensors” have further the property that the transformation equation of their components are linear and homogeneous; so that all the components in the new system vanish if they are all zero in the original system. Thus a law of Nature can be formulated by putting all the components of a tensor equal to zero so that it is a general co-variant equation; thus while we seek the laws of formation of the tensors, we also reach the means of establishing general co-variant laws.

5. Contra-variant and co-variant Four-vector.

Contra-variant Four-vector. The line-element is defined by the four components _dx__{ν}, whose transformation law is expressed by the equation

(5) $$ dx'_{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial x_{\nu}} dx_{\nu} $$

The _dx′__{σ}’_s_ are expressed as linear and homogeneous function of _dx__{ν}’_s_; we can look upon the differentials of the co-ordinates as the components of a tensor, which we designate specially as a contravariant Four-vector. Everything which is defined by Four quantities A^{σ}, with reference to a co-ordinate system, and transforms according to the same law,

(5a)

$$ A^{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial x_{\nu}} A^{\nu} $$

we may call a contra-variant Four-vector. From (5. a), it follows at once that the sums (A^{σ} ± B^{σ}) are also components of a four-vector, when A^{σ} and B^{σ} are so; corresponding relations hold also for all systems afterwards introduced as “tensors” (Rule of addition and subtraction of Tensors).

_Co-variant Four-vector._

We call four quantities A_{ν} as the components of a covariant four-vector, when for any choice of the contra-variant four vector B^{ν} (6) ∑_{ν} A_{ν} B^{ν} = _Invariant_. From this definition follows the law of transformation of the co-variant four-vectors. If we substitute in the right hand side of the equation

∑_{σ} A′_{σ} B^{σ′} = ∑_{ν} A_{ν} B^{ν}.

the expressions

$$ \sum_{\sigma} \frac{\partial x_{\nu}}{\partial x_{\sigma'}} B^{\sigma'} $$

for B^{ν} following from the inversion of the equation (5a) we get

$$ \sum_{\sigma} B^{\sigma'} \sum_{\nu} \frac{\partial x_{\nu}}{\partial x_{\sigma'}} A_{\nu} = \sum_{\sigma} B^{\sigma'} A'_{\sigma} $$

As in the above equation B^{σ′} are independent of one another and perfectly arbitrary, it follows that the transformation law is:—

$$ A'_{\sigma} = \sum \frac{\partial x_{\nu}}{\partial x_{\sigma'}} A_{\nu} $$

_Remarks on the simplification of the mode of writing the expressions._ A glance at the equations of this paragraph will show that the indices which appear twice within the sign of summation [for example ν in (5)] are those over which the summation is to be made and that only over the indices which appear twice. It is therefore possible, without loss of clearness, to leave off the summation sign; so that we introduce the rule: wherever the index in any term of an expression appears twice, it is to be summed over all of them except when it is not expressedly said to the contrary.

The difference between the co-variant and the contra-variant four-vector lies in the transformation laws [(7) and (5)]. Both the quantities are tensors according to the above general remarks; in it lies its significance. In accordance with Ricci and Levi-civita, the contravariants and co-variants are designated by the over and under indices.

§ 6. Tensors of the second and higher ranks.

Contravariant tensor:—If we now calculate all the 16 products A^{μν} of the components A^{μ} B^{ν}, of two contravariant four-vectors

(8) A^{μν} = A^{μ}B^{ν}

A^{μν}, will according to (8) and (5 a) satisfy the following transformation law.

(9)

$$ A^{\sigma \tau'} = \frac{\partial x'_{\sigma}}{\partial x_{\mu}} \frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} $$

We call a thing which, with reference to any reference system is defined by 16 quantities and fulfils the transformation relation (9), a contravariant tensor of the second rank. Not every such tensor can be built from two four-vectors, (according to 8). But it is easy to show that any 16 quantities A^{μν}, can be represented as the sum of A^{μ}B^{ν} of properly chosen four pairs of four-vectors. From it, we can prove in the simplest way all laws which hold true for the tensor of the second rank defined through (9), by proving it only for the special tensor of the type (8).

_Contravariant Tensor of any rank_:—It is clear that corresponding to (8) and (9), we can define contravariant tensors of the 3rd and higher ranks, with 4³, etc. components. Thus it is clear from (8) and (9) that in this sense, we can look upon contravariant four-vectors, as contravariant tensors of the first rank.

_Co-variant tensor._

If on the other hand, we take the 16 products A_{μν} of the components of two co-variant four-vectors A_{μ} and B_{ν},

(10) A_{μν} = A_{μ} B_{ν}.

for them holds the transformation law

(11)

$$ A^{\sigma \tau'} = \frac{\partial x'_{\mu}}{\partial x_{\sigma'}} \frac{\partial x'_{\nu}}{\partial x_{\tau'}} A^{\mu \nu} $$

By means of these transformation laws, the co-variant tensor of the second rank is defined. All re-marks which we have already made concerning the contravariant tensors, hold also for co-variant tensors.

_Remark_:—

It is convenient to treat the scalar Invariant either as a contravariant or a co-variant tensor of zero rank.

_Mixed tensor._ We can also define a tensor of the second rank of the type

(12) A_{μ}^{ν} = A_{μ}B^{ν}

which is co-variant with reference to μ and contravariant with reference to ν. Its transformation law is

(13)

$$ A^{\tau'}_{\sigma} = \frac{\partial x_{\tau'}}{\partial x_{\beta}} \frac{\partial \alpha}{\partial x_{\sigma'}} A^{\beta}_{\alpha} $$

Naturally there are mixed tensors with any number of co-variant indices, and with any number of contra-variant indices. The co-variant and contra-variant tensors can be looked upon as special cases of mixed tensors.

_Symmetrical tensors_:—

A contravariant or a co-variant tensor of the second or higher rank is called symmetrical when any two components obtained by the mutual interchange of two indices are equal. The tensor A^{μν} or A_{μν} is symmetrical, when we have for any combination of indices

(14) A^{μν} = A^{νμ}

or

(14a) A_{μν} = A_{νμ}.

It must be proved that a symmetry so defined is a property independent of the system of reference. It follows in fact from (9) remembering (14)

$$ A^{\sigma \tau'} = \frac{\partial x_{\sigma'}}{\partial x_{\mu}} \frac{\partial x'_{\tau}}{\partial x_{\nu}} A^{\mu \nu} = \frac{\partial x_{\sigma'}}{\partial x_{\mu}} \frac{\partial x_{\tau'}}{\partial x_{\nu}} A^{\nu \mu} = A^{\tau \sigma'} $$

_Antisymmetrical tensor._

A contravariant or co-variant tensor of the 2nd, 3rd or 4th rank is called _antisymmetrical_ when the two components got by mutually interchanging any two indices are equal and opposite. The tensor or A^{μν} or A_{μν} is thus antisymmetrical when we have

(15) A^{μν} = -A^{νμ}

or

(15a) A_{μν} = -A_{νμ}.

Of the 16 components A^{μν}, the four components A^{μμ} vanish, the rest are equal and opposite in pairs; so that there are only 6 numerically different components present (Six-vector).

Thus we also see that the antisymmetrical tensor A^{μνσ} (3rd rank) has only 4 components numerically different, and the antisymmetrical tensor A^{μνστ} only one. Symmetrical tensors of ranks higher than the fourth, do not exist in a continuum of 4 dimensions.

§ 7. Multiplication of Tensors.

_Outer multiplication of Tensors_:—We get from the components of a tensor of rank _z_, and another of a rank _z′_, the components of a tensor of rank (_z_ + _z′_) for which we multiply all the components of the first with all the components of the second in pairs. For example, we obtain the tensor Τ from the tensors A and B of different kinds:—

Τ_{μνσ} = A_{μν}B_{σ},

Τ^{αβγδ} = A^{αβ}B^{γδ},

Τ_{αβ}^{γδ} = A_{αβ}B^{γδ}.

The proof of the tensor character of Τ, follows immediately from the expressions (8), (10) or (12), or the transformation equations (9), (11), (13); equations (8), (10) and (12) are themselves examples of the outer multiplication of tensors of the first rank.

_Reduction in rank of a mixed Tensor._

From every mixed tensor we can get a tensor which is two ranks lower, when we put an index of co-variant character equal to an index of the contravariant character and sum according to these indices (Reduction). We get for example, out of the mixed tensor of the fourth rank A_{αβ}^{γδ}, the mixed tensor of the second rank

A_{β}^{δ} = A_{αβ}^{αδ} = (∑_{α} A_{αβ}^{αδ})

and from it again by “reduction” the tensor of the zero rank

A = A_{β}^{β} = A_{αβ}^{αβ}.

The proof that the result of reduction retains a truly tensorial character, follows either from the representation of tensor according to the generalisation of (12) in combination with (6) or out of the generalisation of (13).

_Inner and mixed multiplication of Tensors._

This consists in the combination of outer multiplication with reduction. Examples:—From the co-variant tensor of the second rank A_{μν} and the contravariant tensor of the first rank B^{σ} we get by outer multiplication the mixed tensor

D^{σ}_{μν} = A_{μν} B^{σ} .

Through reduction according to indices ν and σ (_i.e._, putting ν = σ), the co-variant four vector

D_{μ} = D^{ν}_{μν} = A_{μν} B^{ν} is generated.

These we denote as the inner product of the tensor A_{μν} and B^{σ}. Similarly we get from the tensors A_{μν} and B^{στ} through outer multiplication and two-fold reduction the inner product A_{μν} B^{μν}. Through outer multiplication and one-fold reduction we get out of A_{μν} and B^{στ}, the mixed tensor of the second rank D^{τ}_{μ} = A_{μν} B^{τν}. We can fitly call this operation a mixed one; for it is outer with reference to the indices μ and τ and inner with respect to the indices ν and σ.

We now prove a law, which will be often applicable for proving the tensor-character of certain quantities. According to the above representation, A_{μν} B^{μν} is a scalar, when A_{μν} and B^{στ} are tensors. We also remark that when A_{μν} B^{μν} is an invariant for every choice of the tensor B^{μν}, then A_{μν} has a tensorial character.

Proof:—According to the above assumption, for any substitution we have

A_{στ′} B^{στ′} = A_{μν} B^{μν}.

From the inversion of (9) we have however

$$ B_{\mu \nu} = \frac{\partial x_{\mu}}{\partial x_{\sigma'}} \frac{\partial x_{\nu}}{\partial \tau'} B^{\sigma \tau'} $$

Substitution of this for B^{μν} in the above equation gives

$$ (A_{\sigma \tau'} - \frac{\partial x_{\mu}}{\partial x_{\sigma'}} \frac{\partial x_{\nu}}{\partial x_{\tau'}}) B^{\sigma \tau'} = 0 $$

This can be true, for any choice of B^{στ′} only when the term within the bracket vanishes. From which by referring to (11), the theorem at once follows. This law correspondingly holds for tensors of any rank and character. The proof is quite similar. The law can also be put in the following form. If B^{μ} and C^{ν} are any two vectors, and if for every choice of them the inner product A_{μν} B^{μ} C^{ν} is a scalar, then A_{μν} is a co-variant tensor. The last law holds even when there is the more special formulation, that with any arbitrary choice of the four-vector B^{μ} alone the scalar product A_{μν} B^{μ} B^{ν} is a scalar, in which case we have the additional condition that A_{μν} satisfies the symmetry condition. According to the method given above, we prove the tensor character of (A_{μν} + A_{νμ}), from which on account of symmetry follows the tensor-character of A_{μν}. This law can easily be generalized in the case of co-variant and contravariant tensors of any rank.

Finally, from what has been proved, we can deduce the following law which can be easily generalized for any kind of tensor: If the quantities A_{μν} B^{ν} form a tensor of the first rank, when B^{ν} is any arbitrarily chosen four-vector, then A_{μν} is a tensor of the second rank. If for example, C^{μ} is any four-vector, then owing to the tensor character of A_{μν} B^{ν}, the inner product A_{μν} C^{μ} B^{ν} is a scalar, both the four-vectors C^{μ} and B^{ν} being arbitrarily chosen. Hence the proposition follows at once.

A few words about the Fundamental Tensor _g__{μν}.

The co-variant fundamental tensor—In the invariant expression of the square of the linear element

_ds²_ = _g__{μν} _dx__{μ} _dx__{ν}

_dx__{μ} plays the rôle of any arbitrarily chosen contravariant vector, since further _g__{μν} = _g__{νμ}, it follows from the considerations of the last paragraph that _g__{μν} is a symmetrical co-variant tensor of the second rank. We call it the “fundamental tensor.” Afterwards we shall deduce some properties of this tensor, which will also be true for any tensor of the second rank. But the special rôle of the fundamental tensor in our Theory, which has its physical basis on the particularly exceptional character of gravitation makes it clear that those relations are to be developed which will be required only in the case of the fundamental tensor.

_The co-variant fundamental tensor._

If we form from the determinant scheme | _g__{μν} | the minors of _g__{μν} and divide them by the determinant _g_ = | _g__{μν} | we get certain quantities _g_^{μν} = _g_^{νμ}, which as we shall prove generates a contravariant tensor.

According to the well-known law of Determinants

(16) _g__{μσ} _g_^{νσ} = δ_{μ}^{ν}

where δ_{μ}^{ν} is 1, or 0, according as μ = ν or not. Instead of the above expression for _ds²_, we can also write

_g__{μσ} δ_{ν}^{σ} _dx__{μ} _dx__{ν}

or according to (16) also in the form

_g__{μσ} _g__{ντ} _g_^{στ} _dx__{μ} _dx__{ν}

Now according to the rules of multiplication, of the fore-going paragraph, the magnitudes

_d_ξ_{σ} = _g__{μσ} _dx__{μ}

forms a co-variant four-vector, and in fact (on account of the arbitrary choice of _dx__{μ}) any arbitrary four-vector.

If we introduce it in our expression, we get

_ds²_ = _g_^{στ} _d_ξ_{σ} _d_ξ_{τ}.

For any choice of the vectors _d_ξ_{σ} _d_ξ_{τ} this is scalar, and _g_^{στ}, according to its definition is a symmetrical thing in σ and τ, so it follows from the above results, that _g_^{στ} is a contravariant tensor. Out of (16) it also follows that δ^{ν}_{μ} is a tensor which we may call the mixed fundamental tensor.

_Determinant of the fundamental tensor._

According to the law of multiplication of determinants, we have

| _g__{μα} _g_^{αν} | = | _g__{μα} | | _g_^{αν} |

On the other hand we have

| _g__{μα} _g_^{αν} | = | δ^{ν}_{μ} | = 1

So that it follows (17) that | _g__{μν} | | _g_^{μν} | = 1.

_Invariant of volume._

We see first the transformation law for the determinant _g_ = | _g__{μν} |. According to (11)

$$ g' = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} \frac{\partial x_{\nu}}{\partial x_{\tau'}} g_{\mu u} | $$

From this by applying the law of multiplication twice, we obtain

$$ g' = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} | | \frac{\partial x_{\nu}}{\partial x_{\tau'}} | | g_{\mu \nu} | = | \frac{\partial x_{\mu}}{\alpha_{\sigma'}} | g $$

or

(A)

$$ \sqrt{g'} = | \frac{\partial x_{\mu}}{\partial x_{\sigma'}} | \sqrt{g} $$

On the other hand the law of transformation of the volume element

_d_τ′ = ∫ _dx₁_ _dx₂_ _dx₃_ _dx₄_

is according to the wellknown law of Jacobi.

(B) $$ d\tau' = | \frac{dx'_{\sigma}}{dx_{\mu}} | d\tau $$

by multiplication of the two last equations (A) and (B) we get

(18) = √_g_ _d_τ′ = √_g_ _d_τ.

Instead of √_g_, we shall afterwards introduce √(-_g_) which has a real value on account of the hyperbolic character of the time-space continuum. The invariant √(-_g_)_d_τ, is equal in magnitude to the four-dimensional volume-element measured with solid rods and clocks, in accordance with the special relativity theory.

_Remarks on the character of the space-time continuum_—Our assumption that in an infinitely small region the special relativity theory holds, leads us to conclude that _ds²_ can always, according to (1) be expressed in real magnitudes _d_X₁ ... _d_X_{_h_}. If we call _d_τ₀ the “_natural_” volume element _d_X₁ _d_X₂ _d_X₃ _d_X₄ we have thus (18a) _d_τ₀ = √(_g_)_i_τ.

Should √(-_g_) vanish at any point of the four-dimensional continuum it would signify that to a finite co-ordinate volume at the place corresponds an infinitely small “natural volume.” This can never be the case; so that _g_ can never change its sign; we would, according to the special relativity theory assume that _g_ has a finite negative value. It is a hypothesis about the physical nature of the continuum considered, and also a pre-established rule for the choice of co-ordinates.

If however (-_g_) remains positive and finite, it is clear that the choice of co-ordinates can be so made that this quantity becomes equal to one. We would afterwards see that such a limitation of the choice of co-ordinates would produce a significant simplification in expressions for laws of nature.

In place of (18) it follows then simply that

_d_τ′ = _d_

from this it follows, remembering the law of Jacobi,

(19)

$$ | \frac{\partial x'_{\sigma}}{dx_{\mu}} | = 1 $$

With this choice of co-ordinates, only substitutions with determinant 1 are allowable.

It would however be erroneous to think that this step signifies a partial renunciation of the general relativity postulate. We do not seek those laws of nature which are co-variants with regard to the transformations having the determinant 1, but we ask: what are the general co-variant laws of nature? First we get the law, and then we simplify its expression by a special choice of the system of reference.

_Building up of new tensors with the help of the fundamental tensor._

Through inner, outer and mixed multiplications of a tensor with the fundamental tensor, tensors of other kinds and of other ranks can be formed.

Example:—

A^{μ} = _g_^{μσ} A_{σ}

A = _g__{μν} A^{μν}

We would point out specially the following combinations:

A^{μν} = _g_^{μα} _g_^{νβ} A_{αβ}

A_{μν} = _g__{μα} _g__{νβ} A^{αβ}

(complement to the co-variant or contravariant tensors)

and B_{μν} = _g__{μν} _g_^{αβ} A_{αβ}

We can call B_{μν} the reduced tensor related to A_{μν}.

Similarly

B^{μν} = _g_^{μν}_g__{αβ}A^{αβ}.

It is to be remarked that _g_^{μν} is no other than the “complement” of _g__{μν} for we have,—

_g_^{μα}_g_^{νβ}_g__{αβ} = _g__{μα}δ^{ν}_{α} = _g_^{μν}.

§ 9. Equation of the geodetic line (or of point-motion).

As the “line element” _ds_ is a definite magnitude independent of the co-ordinate system, we have also between two points P₁ and P₂ of a four dimensional continuum a line for which ∫_ds_ is an extremum (geodetic line), _i.e._, one which has got a significance independent of the choice of co-ordinates.

Its equation is

(20) δ{ ∫^{P₂}_{P₁} _ds_ } = 0

From this equation, we can in a wellknown way deduce 4 total differential equations which define the geodetic line; this deduction is given here for the sake of completeness.

Let λ, be a function of the co-ordinates _x__{ν}; this defines a series of surfaces which cut the geodetic line sought-for as well as all neighbouring lines from P₁ to P₂. We can suppose that all such curves are given when the value of its co-ordinates _x__{ν} are given in terms of λ. The sign δ corresponds to a passage from a point of the geodetic curve sought-for to a point of the contiguous curve, both lying on the same surface λ.

Then (20) can be replaced by

{ λ₃
{ ∫δω _d_λ = 0
(20a) { λ₁
{
{ ω² = _g__{μν}(_dx__{μ}/_d_λ)(_dx__{ν}/_d_λ)

But

δω = (1/ω){½(∂_g__{μν}/∂_x__{σ}) · (_dx__{μ}/_d_λ) · (_dx__{ν}/_d_λ)
· δ_x__{σ}
+ _g__{μν}(_dx__{μ}/_d_λ)δ(_dx__{ν}/_d_λ)}

So we get by the substitution of δω in (20a), remembering that

δ(_dx__{ν}/_d_λ) = (_d_/_d_λ)(δ_x__{ν})

after partial integration,

{ λ₃
{ ∫ _d_λ _k__{σ} δ_x__{σ} = 0
(20b) { λ₁
{
{ where _k__{σ} = (_d_/_d_λ){(_g__{μν}/ω) · (_dx__{μ}/_d_λ)}
- (1/(2ω))(∂_g__{μν}/∂_x__{σ}

× (_dx__{μ}/_d_λ) · (_dx__{ν}/_d_λ).

From which it follows, since the choice of δν_{σ} is perfectly arbitrary that _k__{σ}_’s_ should vanish. Then

(20c) _k__{σ} = 0 (σ = 1, 2, 3, 4)

are the equations of geodetic line; since along the geodetic line considered we have _ds_ ≠ 0, we can choose the parameter λ, as the length of the arc measured along the geodetic line. Then _w_ = 1, and we would get in place of (20c)

$$ g_{\mu\nu} \frac{\partial^2 x_{\mu}}{\partial s^2} + \frac{\partial g_{\mu\nu}}{\partial x_{\sigma}} \frac{\partial x_{\sigma}}{\partial s} \frac{\partial x_{\mu}}{\partial s} - \frac{1}{2} \frac{\partial g_{\mu\sigma}}{\partial x_{\nu}} \frac{\partial x_{\mu}}{\partial s} \frac{\partial x_{\sigma}}{\partial s} = 0 $$

Or by merely changing the notation suitably,

(20d) $$ g_{\alpha\sigma} \frac{d^2 x_{\alpha}}{ds^2} + \begin{bmatrix}\mu\nu\\\sigma\end{bmatrix} \frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} = 0 $$

where we have put, following Christoffel,

(21)

$$ \begin{bmatrix}\mu\nu\\\sigma\end{bmatrix} = \frac{1}{2} \begin{bmatrix}\frac{\partial g_{\mu\sigma}}{\partial x_{\nu}} + \frac{\partial g_{\nu\sigma}}{\partial x_{\mu}} - \frac{\partial g_{\mu\nu}}{\partial \sigma}\end{bmatrix} $$

Multiply finally (20d) with _g_^{στ} (outer multiplication with reference to τ, and inner with respect to σ) we get at last the final form of the equation of the geodetic line—

$$ \frac{d^2 x_{\tau}}{ds^2} + \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} = 0 $$

Here we have put, following Christoffel,

$$ \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} = g^{\tau\alpha} \begin{bmatrix}\mu\nu\\\alpha\end{bmatrix} $$

§ 10. Formation of Tensors through Differentiation.

Relying on the equation of the geodetic line, we can now easily deduce laws according to which new tensors can be formed from given tensors by differentiation. For this purpose, we would first establish the general co-variant differential equations. We achieve this through a repeated application of the following simple law. If a certain curve be given in our continuum whose points are characterised by the arc-distances _s_, measured from a fixed point on the curve, and if further φ, be an invariant space function, then _d_φ/_ds_ is also an invariant. The proof follows from the fact that _d_φ as well as _ds_, are both invariants

Since

$$ \frac{d \phi}{ds} = \frac{\partial \phi}{\partial x_{\mu}} \frac{\partial x_{\mu}}{\partial s} $$

so that

$$ \psi = \frac{\partial \phi}{\partial x_{\mu}} \frac{dx_{\mu}}{ds} $$

is also an invariant for all curves which go out from a point in the continuum, _i.e._, for any choice of the vector _dx__{μ}. From which follows immediately that

A_{μ} = ∂φ/∂_x__{μ}

is a co-variant four-vector (gradient of φ).

According to our law, the differential-quotient χ = ∂ψ/∂_s_ taken along any curve is likewise an invariant.

Substituting the value of ψ, we get

$$ \chi = \frac{\partial^2 \phi}{\partial x_{\mu} \partial x_{\nu}} \frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} + \frac{\partial \phi}{\partial x_{\mu}} \frac{d^2 x_{\mu}}{ds^2} $$

Here however we can not at once deduce the existence of any tensor. If we however take that the curves along which we are differentiating are geodesics, we get from it by replacing _d²__x__{ν}/_ds²_ according to (22)

$$ \chi = \begin{bmatrix}\frac{\partial^2 \phi}{\partial x_{\mu}\partial x_{\nu}} - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \frac{\partial \phi}{\partial x_{\tau}} \end{bmatrix} \frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} $$

From the interchangeability of the differentiation with regard to μ and ν, and also according to (23) and (21) we see that the bracket

$$ \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} $$

is symmetrical with respect to μ and ν.

As we can draw a geodetic line in any direction from any point in the continuum, ∂_x__{μ}/_ds_ is thus a four-vector, with an arbitrary ratio of components, so that it follows from the results of §7 that

(25)

$$ A_{\mu\nu} = \frac{\partial^2 \phi}{\partial x_{\mu} \partial x_{\nu}} - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \frac{\partial \phi}{\partial x_{\tau}} $$

is a co-variant tensor of the second rank. We have thus got the result that out of the co-variant tensor of the first rank A_{μ} = ∂φ/∂_x__{μ} we can get by differentiation a co-variant tensor of 2nd rank

(26)

$$ A_{\mu\nu} = \frac{\partial A_{\mu}}{\partial x_{\nu}} - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} A_{\tau} $$

We call the tensor A_{μν} the “extension” of the tensor A_{μ}. Then we can easily show that this combination also leads to a tensor, when the vector A_{μ} is not representable as a gradient. In order to see this we first remark that ψ (_d_φ/∂_x__{μ}) is a co-variant four-vector when ψ and φ are scalars. This is also the case for a sum of four such terms :—

$$ S_{\mu} = \psi^{(1)} \frac{\partial \phi^{(1)}}{\partial x_{\mu}} + ... + \psi^{(4)} \frac{\partial \phi^{(4)}}{\partial x_{\mu}} $$

when ψ^{(1)}, φ^{(1)} ... ψ^{(4)}, φ^{(4)} are scalars. Now it is however clear that every co-variant four-vector is representable in the form of S_{μ}.

If for example, A_{μ} is a four-vector whose components are any given functions of _x__{ν}, we have, (with reference to the chosen co-ordinate system) only to put

ψ^{(1)} = A₁ φ^{(1)} = _x₁_

ψ^{(2)} = A₂ φ^{(2)} = _x₂_

ψ^{(3)} = A₃ φ^{(3)} = _x₃_

ψ^{(4)} = A₄ φ^{(4)} = _x₄_.

in order to arrive at the result that S_{μ} is equal to A_{μ}.

In order to prove then that A_{μν} is a tensor when on the right side of (26) we substitute any co-variant four-vector for A_{μ} we have only to show that this is true for the four-vector S_{μ}. For this latter case, however, a glance on the right hand side of (26) will show that we have only to bring forth the proof for the case when

A_{μ} = ψ ∂φ/∂_x__{μ}.

Now the right hand side of (25) multiplied by ψ is

$$ \psi \frac{\partial^2 \phi}{\partial x_{\mu} \partial x_{\nu}} - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} \psi \frac{\partial \phi}{\partial x_{\tau}} $$

which has a tensor character. Similarly, (∂φ/∂_x__{μ}) (∂φ/∂_x__{ν}) is also a tensor (outer product of two four-vectors).

Through addition follows the tensor character of

$$ \frac{\partial}{\partial x_{\nu}} (\psi \frac{\partial \phi}{\partial x_{\mu}}) - \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} (\psi \frac{\partial \phi}{\partial x_{\tau}}) $$

Thus we get the desired proof for the four-vector, ψ ∂φ/∂_x__{μ} and hence for any four-vectors A_{μ} as shown above.

With the help of the extension of the four-vector, we can easily define “extension” of a co-variant tensor of any rank. This is a generalisation of the extension of the four-vector. We confine ourselves to the case of the extension of the tensors of the 2nd rank for which the law of formation can be clearly seen.

As already remarked every co-variant tensor of the 2nd rank can be represented as a sum of the tensors of the type A_{μ} B_{ν}.

It would therefore be sufficient to deduce the expression of extension, for one such special tensor. According to (26) we have the expressions

$$ \frac{\partial A_{\mu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau} $$

$$ \frac{\partial B_{\nu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} B_{\tau} $$

are tensors. Through outer multiplication of the first with B_{ν} and the 2nd with A_{μ} we get tensors of the third rank. Their addition gives the tensor of the third rank

(27)

$$ A_{\mu\nu\sigma} = \frac{\partial A_{\mu\nu}}{\partial x_{\sigma}} - \begin{Bmatrix}\sigma\mu\\\tau\end{Bmatrix} A_{\tau\nu} - \begin{Bmatrix}\sigma\nu\\\tau\end{Bmatrix} A_{\mu\tau} $$

where A_{μν} is put = A_{μ} B_{ν}. The right hand side of (27) is linear and homogeneous with reference to A_{μν}, and its first differential co-efficient, so that this law of formation leads to a tensor not only in the case of a tensor of the type A_{μ} B_{ν} but also in the case of a summation for all such tensors, _i.e._, in the case of any co-variant tensor of the second rank. We call A_{μνσ} the extension of the tensor A_{μν}. It is clear that (26) and (24) are only special cases of (27) (extension of the tensors of the first and zero rank). In general we can get all special laws of formation of tensors from (27) combined with tensor multiplication.

Some special cases of Particular Importance.

_A few auxiliary lemmas concerning the fundamental tensor._ We shall first deduce some of the lemmas much used afterwards. According to the law of differentiation of determinants, we have

(28) _dg_ = _g_^{μν} _g dg__{μν} = -_g__{μν} _gdg_^{μν}.

The last form follows from the first when we remember that

_g__{μν} _g_^{μ′ν} = δ^{μ′}_{μ} , and therefore _g__{μν}_g_^{μν} =
4,

consequently _g__{μν}_dg_^{μν} + _g_^{μν} _dg__{μν} = 0.