The Principle of Relativity

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

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

If we substitute the expression in (1) we get

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—

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

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,

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

the expressions

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

As in the above equation B^σ′ 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 A^μν of the components A^μ B^ν, of two contravariant four-vectors

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

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^μB^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,

for them holds the transformation law

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

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

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

or

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 A^μν or A_μν is thus antisymmetrical when we have

or

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:—

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

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

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

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

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

From the inversion of (9) we have however

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

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

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

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

or according to (16) also in the form

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

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

For any choice of the vectors dξ_σ dξ_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

On the other hand we have