IV
In order to demonstrate that the assumption of the group Gc for the physical laws does not possibly lead to any contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group. The revision has already been successfully made in the case of “Thermodynamics and Radiation,”[[30]] for “Electromagnetic phenomena”,[[31]] and finally for “Mechanics with the maintenance of the idea of mass.”
For this last mentioned province of physics, the question may be asked: if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-point (x, y, z, t), where the velocity-vector is ([.x], [.y], [.z], [.t]), then how are we to regard this force when the system of reference is changed in any possible manner? Now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields, where the group Gc is undoubtedly permissible. These theorems lead us to the following simple rule; if the system of reference be changed in any way, then the supposed force is to be put as a force in the new space-coordinates in such a manner, that the corresponding vector with the components
[.t]X, [.t]Y, [.t]Z, [.t]T,
where T = 1/c² ([.x]/[.t] X + [.y]/[.t] Y + [.z]/[.t] Z) = 1/c²
(the rate of
which work is done at the world-point), remains unaltered.
This vector is always normal to the velocity-vector at P. Such a force-vector, representing a force at P, may be called a moving force-vector at P.
Now the world-line passing through P will be described by a substantial point with the constant mechanical mass m. Let us call m-times the velocity-vector at P as the impulse-vector, and m-times the acceleration-vector at P as the force-vector of motion, at P. According to these definitions, the following law tells us how the motion of a point-mass takes place under any moving force-vector[[32]]:
The force-vector of motion is equal to the moving force-vector.
This enunciation comprises four equations for the components in the four directions, of which the fourth can be deduced from the first three, because both of the above-mentioned vectors are perpendicular to the velocity-vector. From the definition of T, we see that the fourth simply expresses the “Energy-law.” Accordingly c²-times the component of the impulse-vector in the direction of the t-axis is to be defined as the kinetic-energy of the point-mass. The expression for this is
mc² dt/dτ = mc² /√(1 - v²/c²)
i.e., if we deduct from this the additive constant mc², we obtain the expression ½ mv² of Newtonian-mechanics up to magnitudes of the order of 1/c². Hence it appears that the energy depends upon the system of reference. But since the t-axis can be laid in the direction of any time-like axis, therefore the energy-law comprises, for any possible system of reference, the whole system of equations of motion. This fact retains its significance even in the limiting case c = ∞, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schütz.[[33]]
From the very beginning, we can establish the ratio between the units of time and space in such a manner, that the velocity of light becomes unity. If we now write √-1 t = l, in the place of l, then the differential expression
dτ² = -(dx² + dy² + dz² + dl²),
becomes symmetrical in (x, y, r, l); this symmetry then enters into each law, which does not contradict the world-postulate. We can clothe the “essential nature of this postulate in the mystical, but mathematically significant formula
3·10⁵ km = √-1 Sec.
V
The advantages arising from the formulation of the world-postulate are illustrated by nothing so strikingly as by the expressions which tell us about the reactions exerted by a point-charge moving in any manner according to the Maxwell-Lorentz theory.
Let us conceive of the world-line of such an electron with the charge (e), and let us introduce upon it the “Proper-time” τ reckoned from any possible initial point. In order to obtain the field caused by the electron at any world-point P₁ let us construct the fore-cone belonging to P₁ (vide fig. 4). Clearly this cuts the unlimited world-line of the electron at a single point P, because these directions are all time-like vectors. At P, let us draw the tangent to the world-line, and let us draw from P₁ the normal to this tangent. Let r be the measure of P₁Q. According to the definition of a fore-cone, r/e is to be reckoned as the measure of PQ. Now at the world-point P₁, the vector-potential of the field excited by e is represented by the vector in direction PQ, having the magnitude e/cr, in its three space components along the x-, y-, z-axes; the scalar-potential is represented by the component along the t-axis. This is the elementary law found out by A. Lienard, and E. Wiechert.[[34]]
If the field caused by the electron be described in the above-mentioned way, then it will appear that the division of the field into electric and magnetic forces is a relative one, and depends upon the time-axis assumed; the two forces considered together bears some analogy to the force-screw in mechanics; the analogy is, however, imperfect.
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 Fx, Fy, Fz, Ft of the vector F the following three relations hold:—
cFt - Fx = 1/r², Fy = [:y]/(c²r), Fz = 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-(v²/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.
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 = - dX₁² - dX₂² - dX₃² + dX₄²
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 (dX₁, dX₂, dX₃, dX₄) 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, dX’s would then be represented by definite linear homogeneous expressions of the form
(2) dXν = σσaνσdxσ
If we substitute the expression in (1) we get
(3) ds² = σστgστdxσdxτ
where gστ will be functions of xσ, 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 gτσ’s are here to be so chosen, that gτσ = gστ; 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 gστ in a finite region it is possible to choose the system of co-ordinates in such a way that gστ 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 gστ 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 gστ’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 gμν’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 gστ’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 gστ’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 gστ 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 dxν, whose transformation law is expressed by the equation
"(5)."
The dx′σ’s are expressed as linear and homogeneous function of dxν’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 Aσ, with reference to a co-ordinate system, and transforms according to the same law,
"(5a)."
we may call a contra-variant Four-vector. From (5. a), it follows at once that the sums (Aσ ± Bσ) are also components of a four-vector, when Aσ and Bσ 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 Aν as the components of a covariant four-vector, when for any choice of the contra-variant four vector Bν (6) ∑ν Aν Bν = 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 Bσ′ = ∑ν Aν Bν.
the expressions
for Bν following from the inversion of the equation (5a) we get
As in the above equation Bσ′ are independent of one another and perfectly arbitrary, it follows that the transformation law is:—
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 Aμν of the components Aμ Bν, of two contravariant four-vectors
(8) Aμν = AμBν
Aμν, will according to (8) and (5 a) satisfy the following transformation law.
"(9)."
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 Aμν, can be represented as the sum of AμBν 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 Aμν of the components of two co-variant four-vectors Aμ and Bν,
(10) Aμν = Aμ Bν.
for them holds the transformation law
"(11)."
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) Aμν = AμBν
which is co-variant with reference to μ and contravariant with reference to ν. Its transformation law is
"(13)."
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 Aμν or Aμν is symmetrical, when we have for any combination of indices
(14) Aμν = Aνμ
or
(14a) Aμν = Aνμ.
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)
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 Aμν or Aμν is thus antisymmetrical when we have
(15) Aμν = -Aνμ
or
(15a) Aμν = -Aνμ.
Of the 16 components Aμν, the four components Aμμ 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 Aμνσ (3rd rank) has only 4 components numerically different, and the antisymmetrical tensor Aμνστ 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:—
Τμνσ = AμνBσ,
Ταβγδ = AαβBγδ,
Ταβγδ = AαβBγδ.
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 Aαβγδ, the mixed tensor of the second rank
Aβδ = Aαβαδ = (∑α Aαβαδ)
and from it again by “reduction” the tensor of the zero rank
A = Aββ = Aαβαβ.
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 Aμν and the contravariant tensor of the first rank Bσ we get by outer multiplication the mixed tensor
Dσμν = Aμν Bσ .
Through reduction according to indices ν and σ (i.e., putting ν = σ), the co-variant four vector
Dμ = Dνμν = Aμν Bν is generated.
These we denote as the inner product of the tensor Aμν and Bσ. Similarly we get from the tensors Aμν and Bστ through outer multiplication and two-fold reduction the inner product Aμν Bμν. Through outer multiplication and one-fold reduction we get out of Aμν and Bστ, the mixed tensor of the second rank Dτμ = Aμν Bτν. 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, Aμν Bμν is a scalar, when Aμν and Bστ are tensors. We also remark that when Aμν Bμν is an invariant for every choice of the tensor Bμν, then Aμν has a tensorial character.
Proof:—According to the above assumption, for any substitution we have
Aστ′ Bστ′ = Aμν Bμν.
From the inversion of (9) we have however
Substitution of this for Bμν in the above equation gives
This can be true, for any choice of Bστ′ 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 Bμ and Cν are any two vectors, and if for every choice of them the inner product Aμν Bμ Cν is a scalar, then Aμν 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 Bμ alone the scalar product Aμν Bμ Bν is a scalar, in which case we have the additional condition that Aμν satisfies the symmetry condition. According to the method given above, we prove the tensor character of (Aμν + Aνμ), from which on account of symmetry follows the tensor-character of Aμν. 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 Aμν Bν form a tensor of the first rank, when Bν is any arbitrarily chosen four-vector, then Aμν is a tensor of the second rank. If for example, Cμ is any four-vector, then owing to the tensor character of Aμν Bν, the inner product Aμν Cμ Bν is a scalar, both the four-vectors Cμ and Bν being arbitrarily chosen. Hence the proposition follows at once.
A few words about the Fundamental Tensor gμν.
The co-variant fundamental tensor—In the invariant expression of the square of the linear element
ds² = gμν dxμ dxν
dxμ plays the rôle of any arbitrarily chosen contravariant vector, since further gμν = gνμ, it follows from the considerations of the last paragraph that gμν 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 | gμν | the minors of gμν and divide them by the determinant g = | gμν | we get certain quantities gμν = gνμ, which as we shall prove generates a contravariant tensor.
According to the well-known law of Determinants
(16) gμσ gνσ = δμν
where δμν is 1, or 0, according as μ = ν or not. Instead of the above expression for ds², we can also write
gμσ δνσ dxμ dxν
or according to (16) also in the form
gμσ gντ gστ dxμ dxν
Now according to the rules of multiplication, of the fore-going paragraph, the magnitudes
dξσ = gμσ dxμ
forms a co-variant four-vector, and in fact (on account of the arbitrary choice of dxμ) any arbitrary four-vector.
If we introduce it in our expression, we get
ds² = gστ dξσ dξτ.
For any choice of the vectors dξσ dξτ this is scalar, and gστ, according to its definition is a symmetrical thing in σ and τ, so it follows from the above results, that gστ is a contravariant tensor. Out of (16) it also follows that δνμ 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
| gμα gαν | = | gμα | | gαν |
On the other hand we have
| gμα gαν | = | δνμ | = 1
So that it follows (17) that | gμν | | gμν | = 1.
Invariant of volume.
We see first the transformation law for the determinant g = | gμν |. According to (11)
From this by applying the law of multiplication twice, we obtain
or
"(A)."
On the other hand the law of transformation of the volume element
dτ′ = ∫ dx₁ dx₂ dx₃ dx₄
is according to the wellknown law of Jacobi.
"(B)."
by multiplication of the two last equations (A) and (B) we get
(18) = √g dτ′ = √g dτ.
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)dτ, 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 dX₁ ... dXh. If we call dτ₀ the “natural” volume element dX₁ dX₂ dX₃ dX₄ we have thus (18a) dτ₀ = √(g)iτ.
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
dτ′ = d
from this it follows, remembering the law of Jacobi,
"(19)."
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:—
Aμ = gμσ Aσ
A = gμν Aμν
We would point out specially the following combinations:
Aμν = gμα gνβ Aαβ
Aμν = gμα gνβ Aαβ
(complement to the co-variant or contravariant tensors)
and Bμν = gμν gαβ Aαβ
We can call Bμν the reduced tensor related to Aμν.
Similarly
Bμν = gμνgαβAαβ.
It is to be remarked that gμν is no other than the “complement” of gμν for we have,—
gμαgνβgαβ = gμαδνα = gμν.
§ 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{ ∫P₂P₁ 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 xν; 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 xν 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 dλ = 0
(20a) { λ₁
{
{ ω² = gμν(dxμ/dλ)(dxν/dλ)
But
δω = (1/ω){½(∂gμν/∂xσ) · (dxμ/dλ) · (dxν/dλ) · δxσ
+ gμν(dxμ/dλ)δ(dxν/dλ)}
So we get by the substitution of δω in (20a), remembering that
δ(dxν/dλ) = (d/dλ)(δxν)
after partial integration,
{ λ₃
{ ∫ dλ kσ δxσ = 0
(20b) { λ₁
{
{ where kσ = (d/dλ){(gμν/ω) · (dxμ/dλ)} - (1/(2ω))(∂gμν/∂xσ
× (dxμ/dλ) · (dxν/dλ).
From which it follows, since the choice of δνσ is perfectly arbitrary that kσ’s should vanish. Then
(20c) kσ = 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)
Or by merely changing the notation suitably,
"20d"
where we have put, following Christoffel,
"21"
Multiply finally (20d) with gστ (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—
Here we have put, following Christoffel,
§ 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 dφ/ds is also an invariant. The proof follows from the fact that dφ as well as ds, are both invariants
Since
so that
is also an invariant for all curves which go out from a point in the continuum, i.e., for any choice of the vector dxμ. From which follows immediately that
Aμ = ∂φ/∂xμ
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
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 d²xν/ds² according to (22)
From the interchangeability of the differentiation with regard to μ and ν, and also according to (23) and (21) we see that the bracket
is symmetrical with respect to μ and ν.
As we can draw a geodetic line in any direction from any point in the continuum, ∂xμ/ds is thus a four-vector, with an arbitrary ratio of components, so that it follows from the results of §7 that
"25"
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 Aμ = ∂φ/∂xμ we can get by differentiation a co-variant tensor of 2nd rank
"26"
We call the tensor Aμν the “extension” of the tensor Aμ. Then we can easily show that this combination also leads to a tensor, when the vector Aμ is not representable as a gradient. In order to see this we first remark that ψ (dφ/∂xμ) is a co-variant four-vector when ψ and φ are scalars. This is also the case for a sum of four such terms :—
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 Sμ.
If for example, Aμ is a four-vector whose components are any given functions of xν, 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 Sμ is equal to Aμ.
In order to prove then that Aμν is a tensor when on the right side of (26) we substitute any co-variant four-vector for Aμ we have only to show that this is true for the four-vector Sμ. 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
Aμ = ψ ∂φ/∂xμ.
Now the right hand side of (25) multiplied by ψ is
which has a tensor character. Similarly, (∂φ/∂xμ) (∂φ/∂xν) is also a tensor (outer product of two four-vectors).
Through addition follows the tensor character of
Thus we get the desired proof for the four-vector, ψ ∂φ/∂xμ and hence for any four-vectors Aμ 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 Aμ Bν.
It would therefore be sufficient to deduce the expression of extension, for one such special tensor. According to (26) we have the expressions
are tensors. Through outer multiplication of the first with Bν and the 2nd with Aμ we get tensors of the third rank. Their addition gives the tensor of the third rank
"(27)"
where Aμν is put = Aμ Bν. The right hand side of (27) is linear and homogeneous with reference to Aμν, 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 Aμ Bν 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 Aμνσ the extension of the tensor Aμν. 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 = gμν g dgμν = -gμν gdgμν.
The last form follows from the first when we remember that
gμν gμ′ν = δμ′μ , and therefore gμνgμν = 4,
consequently gμνdgμν + gμν dgμν = 0.
From (28), it follows that
"(29)"
Again, since gμν gνσ = δνμ , we have, by differentiation,
By mixed multiplication with gστ and gνλ respectively we obtain (changing the mode of writing the indices).
dgμν = -gμα gνβ dgαβ
∂gμν/∂xσ = -gμα gνβ dgαβ
and
(32)
dgμν = -gμα gνβ dgαβ
∂gμν/∂xσ = -gμα gνβ ∂gαβ/∂xσ.
The expression (31) allows a transformation which we shall often use; according to (21)
"(33)"
If we substitute this in the second of the formula (31), we get, remembering (23),
"(34)"
By substituting the right-hand side of (34) in (29), we get
"(29a)"
Divergence of the contravariant four-vector.
Let us multiply (26) with the contravariant fundamental tensor gμν (inner multiplication), then by a transformation of the first member, the right-hand side takes the form
"(A)"
According to (31) and (29), the last member can take the form
"(B)"
Both the first members of the expression (B), and the second member of the expression (A) cancel each other, since the naming of the summation-indices is immaterial. The last member of (B) can then be united with first of (A). If we put
gμν Aμ = Aν,
where Aν as well as Aμ are vectors which can be arbitrarily chosen, we obtain finally
This scalar is the Divergence of the contravariant four-vector Aν.
Rotation of the (covariant) four-vector.
The second member in (26) is symmetrical in the indices μ, and ν. Hence Aμν - Aνμ is an antisymmetrical tensor built up in a very simple manner. We obtain
∂Aμ ∂Aν
(36) Bμν = --------- - -------
∂xν ∂xμ
Antisymmetrical Extension of a Six-vector.
If we apply the operation (27) on an antisymmetrical tensor of the second rank Aμ{ν²} and form all the equations arising from the cyclic interchange of the indices μ, ν, σ, and add all them, we obtain a tensor of the third rank
(37) Bμνσ = Aμνσ + Aνσμ + Aσμν
∂Aμν ∂Aνσ ∂Aσμ
= --------- + ---------- + ---------
∂xσ ∂xμ ∂xν
from which it is easy to see that the tensor is antisymmetrical.
Divergence of the Six-vector.
If (27) is multiplied by gμα gνβ (mixed multiplication), then a tensor is obtained. The first member of the right hand side of (27) can be written in the form
If we replace gμα gνβ Aμνσ by Aσαβ, gμα gνβ Aμν by Aαβ and replace in the transformed first member
∂gνβ/∂xσ and ∂gμα/∂xσ
with the help of (34), then from the right-hand side of (27) there arises an expression with seven terms, of which four cancel. There remains
"(38)"
This is the expression for the extension of a contravariant tensor of the second rank; extensions can also be formed for corresponding contravariant tensors of higher and lower ranks.
We remark that in the same way, we can also form the extension of a mixed tensor Aμα
"(39)"
By the reduction of (38) with reference to the indices β and σ(inner multiplication with δβσ), we get a contravariant four-vector
On the account of the symmetry of
with reference to the indices β and κ, the third member of the right hand side vanishes when Aαβ is an antisymmetrical tensor, which we assume here; the second member can be transformed according to (29a); we therefore get
"(40)"
This is the expression of the divergence of a contravariant six-vector.
Divergence of the mixed tensor of the second rank.
Let us form the reduction of (39) with reference to the indices α and σ, we obtain remembering (29a)
"(41)"
If we introduce into the last term the contravariant tensor Aρσ = gρτ Aστ, it takes the form
If further Aρσ or is symmetrical it is reduced to
If instead of Aρσ, we introduce in a similar way the symmetrical co-variant tensor Aρσ = gρα gσβ Aαβ, then owing to (31) the last member can take the form
In the symmetrical case treated, (41) can be replaced by either of the forms
"(41a)"
or
"(41b)"
which we shall have to make use of afterwards.
§12. The Riemann-Christoffel Tensor.
We now seek only those tensors, which can be obtained from the fundamental tensor gμν by differentiation alone. It is found easily. We put in (27) instead of any tensor Aμν the fundamental tensor gμν and get from it a new tensor, namely the extension of the fundamental tensor. We can easily convince ourselves that this vanishes identically. We prove it in the following way; we substitute in (27)
i.e., the extension of a four-vector.
Thus we get (by slightly changing the indices) the tensor of the third rank
We use these expressions for the formation of the tensor Aμστ - Aμτσ. Thereby the following terms in Aμστ cancel the corresponding terms in Aμτσ; the first member, the fourth member, as well as the member corresponding to the last term within the square bracket. These are all symmetrical in σ, and τ. The same is true for the sum of the second and third members. We thus get
"(43)"
The essential thing in this result is that on the right hand side of (42) we have only Aρ, but not its differential co-efficients. From the tensor-character of Aμστ - Aμτσ, and from the fact that Aρ is an arbitrary four vector, it follows, on account of the result of §7, that Bρμστ is a tensor (Riemann-Christoffel Tensor).
The mathematical significance of this tensor is as follows; when the continuum is so shaped, that there is a co-ordinate system for which gμν’s are constants, Bρμστ all vanish.
If we choose instead of the original co-ordinate system any new one, so would the gμν’s referred to this last system be no longer constants. The tensor character of Bρμστ shows us, however, that these components vanish collectively also in any other chosen system of reference. The vanishing of the Riemann Tensor is thus a necessary condition that for some choice of the axis-system gμν’s can be taken as constants. In our problem it corresponds to the case when by a suitable choice of the co-ordinate system, the special relativity theory holds throughout any finite region. By the reduction of (43) with reference to indices to τ and ρ, we get the covariant tensor of the second rank
"(44)"
Remarks upon the choice of co-ordinates.—It has already been remarked in §8, with reference to the equation (18a), that the co-ordinates can with advantage be so chosen that √(-g) = 1. A glance at the equations got in the last two paragraphs shows that, through such a choice, the law of formation of the tensors suffers a significant simplification. It is specially true for the tensor Bμν, which plays a fundamental rôle in the theory. By this simplification, Sμν vanishes of itself so that tensor Bμν reduces to Rμν.
I shall give in the following pages all relations in the simplified form, with the above-named specialisation of the co-ordinates. It is then very easy to go back to the general covariant equations, if it appears desirable in any special case.