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  Finally, there follows from what has been proved, this law, which may also be generalized for any tensors: If for any choice of the four-vector Bv the quantities Aμv Bv form a tensor of the first rank, then Aμv is a tensor of the second rank. For, if Cμ is any four-vector, then on account of the tensor character of Aμv Bv, the inner product AμvBvCμ is a scalar for any choice of the two four-vectors Bv and Cμ. From which the proposition follows.

  § 8. SOME ASPECTS OF THE FUNDAMENTAL TENSOR gμv

  The Covariant Fundamental Tensor.—In the invariant expression for the square of the linear element,

  the part played by the dxμ is that of a contravariant vector which may be chosen at will. Since further, gμv = gvμ, it follows from the considerations of the preceding paragraph that gμv is a covariant tensor of the second rank. We call it the “fundamental tensor.” In what follows we deduce some properties of this tensor which, it is true, apply to any tensor of the second rank. But as the fundamental tensor plays a special part in our theory, which has its physical basis in the peculiar effects of gravitation, it so happens that the relations to be developed are of importance to us only in the case of the fundamental tensor.

  The Contravariant Fundamental Tensor.—If in the determinant formed by the elements gμv, we take the co-factor of each of the gμv and divide it by the determinant g = |gμv|, we obtain certain quantities gμv (= gvμ) which, as we shall demonstrate, form a contravariant tensor.

  By a known property of determinants

  where the symbol denotes 1 or 0, according as μ = v or μ ≠ v. Instead of the above expression for ds2 we may thus write

  or, by (16)

  But, by the multiplication rules of the preceding paragraphs, the quantities

  form a covariant four-vector, and in fact an arbitrary vector, since the dxμ are arbitrary. By introducing this into our expression we obtain

  Since this, with the arbitrary choice of the vector dξσ, is a scalar, and gστ by its definition is symmetrical in the indices σ and τ, it follows from the results of the preceding paragraph that gστ is a contravariant tensor.

  It further follows from (16) that δμ is also a tensor, which we may call the mixed fundamental tensor.

  The Determinant of the Fundamental Tensor.—By the rule for the multiplication of determinants

  On the other hand

  It therefore follows that

  The Volume Scalar.—We seek first the law of transformation of the determinant g = |gμv|. In accordance with (11)

  Hence, by a double application of the rule for the multiplication of determinants, it follows that

  or

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

  is, in accordance with the theorem of Jacobi,

  By multiplication of the last two equations, we obtain

  Instead of we introduce in what follows the quantity , which is always real on account of the hyperbolic character of the space-time continuum. The invariant is equal to the magnitude of the four-dimensional element of volume in the “local” system of reference, as measured with rigid rods and clocks in the sense of the special theory of relativity.

  Note on the Character of the Space-time Continuum.—Our assumption that the special theory of relativity can always be applied to an infinitely small region, implies that ds2 can always be expressed in accordance with (1) by means of real quantities dX1 . . . dX4. If we denote by dτ0 the “natural” element of volume dX1, dX2, dX3, dX4, then

  If were to vanish at a point of the four-dimensional continuum, it would mean that at this point an infinitely small “natural” volume would correspond to a finite volume in the co-ordinates. Let us assume that this is never the case. Then g cannot change sign. We will assume that, in the sense of the special theory of relativity, g always has a finite negative value. This is a hypothesis as to the physical nature of the continuum under consideration, and at the same time a convention as to the choice of co-ordinates.

  But if – g is always finite and positive, it is natural to settle the choice of co-ordinates a posteriori in such a way that this quantity is always equal to unity. We shall see later that by such a restriction of the choice of co-ordinates it is possible to achieve an important simplification of the laws of nature.

  In place of (18), we then have simply dτ′ = dτ, from which, in view of Jacobi’s theorem, it follows that

  Thus, with this choice of co-ordinates, only substitutions for which the determinant is unity are permissible.

  But it would be erroneous to believe that this step indicates a partial abandonment of the general postulate of relativity. We do not ask “What are the laws of nature which are co-variant in face of all substitutions for which the determinant is unity?” but our question is “What are the generally co-variant laws of nature?” It is not until we have formulated these that we simplify their expression by a particular choice of the system of reference.

  The Formation of New Tensors by Means of the Fundamental Tensor.—Inner, outer, and mixed multiplication of a tensor by the fundamental tensor give tensors of different character and rank. For example,

  The following forms may be specially noted:—

  (the “complements” of covariant and contravariant tensors respectively), and

  We call Bμv the reduced tensor associated with Aμv. Similarly,

  It may be noted that gμv is nothing more than the complement of gμv, since

  § 9. THE EQUATION OF THE GEODETIC LINE. THE MOTION OF A PARTICLE

  As the linear element ds is defined independently of the system of coordinates, the line drawn between two points P and P′ of the four-dimensional continuum in such a way that ∫ ds is stationary—a geodetic line—has a meaning which also is independent of the choice of co-ordinates. Its equation is

  Carrying out the variation in the usual way, we obtain from this equation four differential equations which define the geodetic line; this operation will be inserted here for the sake of completeness. Let λ be a function of the co-ordinates xv, and let this define a family of surfaces which intersect the required geodetic line as well as all the lines in immediate proximity to it which are drawn through the points P and P′. Any such line may then be supposed to be given by expressing its co-ordinates xv as functions of λ. Let the symbol δ indicate the transition from a point of the required geodetic to the point corresponding to the same λ on a neighbouring line. Then for (20) we may substitute

  But since

  and

  we obtain from (20a), after a partial integration,

  where

  Since the values of δxσ are arbitrary, it follows from this that

  are the equations of the geodetic line.

  If ds does not vanish along the geodetic line we may choose the “length of the arc” s, measured along the geodetic line, for the parameter λ. Then w = 1, and in place of (20c) we obtain

  or, by a mere change of notation,

  where, following Christoffel, we have written

  Finally, if we multiply (20d) by gστ (outer multiplication with respect to τ, inner with respect to σ), we obtain the equations of the geodetic line in the form

  where, following Christoffel, we have set

  § 10. THE FORMATION OF TENSORS BY DIFFERENTIATION

  With the help of the equation of the geodetic line we can now easily deduce the laws by which new tensors can be formed from old by differentiation. By this means we are able for the first time to formulate generally covariant differential equations. We reach this goal by repeated application of the following simple law:—

  If in our continuum a curve is given, the points of which are specified by the arcual distance s measured from a fixed point on the curve, and if, further, ϕ is an invariant function of space, then dϕ/ds is also an invariant. The proof lies in this, that ds is an invariant as well as dϕ.

  As

  therefore

  is also an invariant, and an invariant for all curves starting from a point
of the continuum, that is, for any choice of the vector dxμ. Hence it immediately follows that

  is a covariant four-vector—the “gradient” of ϕ.

  According to our rule, the differential quotient

  taken on a curve, is similarly an invariant. Inserting the value of ψ, we obtain in the first place

  The existence of a tensor cannot be deduced from this forthwith. But if we may take the curve along which we have differentiated to be a geodetic, we obtain on substitution for d2xv/ds2 from (22),

  Since we may interchange the order of the differentiations, and since by (23) and (21) {μv, τ} is symmetrical in μ and v, it follows that the expression in brackets is symmetrical in μ and v Since a geodetic line can be drawn in any direction from a point of the continuum, and therefore dxμ/ds is a four-vector with the ratio of its components arbitrary, it follows from the results of § 7 that

  is a covariant tensor of the second rank. We have therefore come to this result: from the covariant tensor of the first rank

  we can, by differentiation, form a covariant tensor of the second rank

  We call the tensor Aμv the “extension” (covariant derivative) of the tensor Aμ. In the first place we can readily show that the operation leads to a tensor, even if the vector Aμ cannot be represented as a gradient. To see this, we first observe that

  is a covariant vector, if ψ and ϕ are scalars. The sum of four such terms

  is also a covariant vector, if ψ(1), ϕ(1) . . . ψ(4), ϕ(4) are scalars. But it is clear that any covariant vector can be represented in the form Sμ. For, if Aμ is a vector whose components are any given functions of the xv, we have only to put (in terms of the selected system of co-ordinates)

  in order to ensure that Sμ shall be equal to Aμ.

  Therefore, in order to demonstrate that Aμv is a tensor if any covariant vector is inserted on the right-hand side for Aμ, we only need show that this is so for the vector Sμ. But for this latter purpose it is sufficient, as a glance at the right-hand side of (26) teaches us, to furnish the proof for the case

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

  is a tensor. Similarly

  being the outer product of two vectors, is a tensor. By addition, there follows the tensor character of

  As a glance at (26) will show, this completes the demonstration for the vector

  and consequently, from what has already been proved, for any vector Aμ.

  By means of the extension of the vector, we may easily define the “extension” of a covariant tensor of any rank. This operation is a generalization of the extension of a vector. We restrict ourselves to the case of a tensor of the second rank, since this suffices to give a clear idea of the law of formation.

  As has already been observed, any covariant tensor of the second rank can be represented* as the sum of tensors of the type AμBv. It will therefore be sufficient to deduce the expression for the extension of a tensor of this special type. By (26) the expressions

  are tensors. On outer multiplication of the first by Bv, and of the second by Aμ, we obtain in each case a tensor of the third rank. By adding these, we have the tensor of the third rank

  where we have put Aμv = AμBv. As the right-hand side of (27) is linear and homogeneous in the Aμv and their first derivatives, this law of formation leads to a tensor, not only in the case of a tensor of the type AμBv, but also in the case of a sum of such tensors, i.e. in the case of any covariant tensor of the second rank. We call Aμvσ the extension of the tensor Aμv.

  It is clear that (26) and (24) concern only special cases of extension (the extension of the tensors of rank one and zero respectively).

  In general, all special laws of formation of tensors are included in (27) in combination with the multiplication of tensors.

  § 11. SOME CASES OF SPECIAL IMPORTANCE

  The Fundamental Tensor.—We will first prove some lemmas which will be useful hereafter. By the rule for the differentiation of determinants

  The last member is obtained from the last but one, if we bear in mind that so that gμvgμv = 4, and consequently

  From (28), it follows that

  Further, from it follows on differentiation that

  From these, by mixed multiplication by gστ and gvλ respectively, and a change of notation for the indices, we have

  and

  The relation (31) admits of a transformation, of which we also have frequently to make use. From (21)

  Inserting this in the second formula of (31), we obtain, in view of (23)

  Substituting the right-hand side of (34) in (29), we have

  The “Divergence” of a Contravariant Vector.—If we take the inner product of (26) by the contravariant fundamental tensor gμv, the right-hand side, after a transformation of the first term, assumes the form

  In accordance with (31) and (29), the last term of this expression may be written

  As the symbols of the indices of summation are immaterial, the first two terms of this expression cancel the second of the one above. If we then write gμvAμ = Av, so that Av like Aμ is an arbitrary vector, we finally obtain

  This scalar is the divergence of the contravariant vector Av.

  The “Curl” of a Covariant Vector.—The second term in (26) is symmetrical in the indices μ and v. Therefore Aμv – Avμ is a particularly simply constructed antisymmetrical tensor. We obtain

  Antisymmetrical Extension of a Six-vector.—Applying (27) to an antisymmetrical tensor of the second rank Aμv, forming in addition the two equations which arise through cyclic permutations of the indices, and adding these three equations, we obtain the tensor of the third rank

  which it is easy to prove is antisymmetrical.

  The Divergence of a Six-vector.—Taking the mixed product of (27) by gμαgvβ, we also obtain a tensor. The first term on the right-hand side of (27) may be written in the form

  If we write for gμαgvβAμvσ and Aαβ for gμαgvβAμv, and in the transformed first term replace

  by their values as given by (34), there results from the right-hand side of (27) an expression consisting of seven terms, of which four cancel, and there remains

  This is the expression for the extension of a contravariant tensor of the second rank, and corresponding expressions for the extension of contravariant tensors of higher and lower rank may also be formed.

  We note that in an analogous way we may also form the extension of a mixed tensor:—

  On contracting (38) with respect to the indices β and σ (inner multiplication by ), we obtain the vector

  On account of the symmetry of {βγ, α} with respect to the indices β and γ, the third term on the right-hand side vanishes, if Aαβ is, as we will assume, an antisymmetrical tensor. The second term allows itself to be transformed in accordance with (29a). Thus we obtain

  This is the expression for the divergence of a contravariant six-vector.

  The Divergence of a Mixed Tensor of the Second Rank.—Contracting (39) with respect to the indices α and σ, and taking (29a) into consideration, we obtain

  If we introduce the contravariant tensor in the last term, it assumes the form

  If, further, the tensor Aρσ is symmetrical, this reduces to

  Had we introduced, instead of Aρσ, the covariant tensor Aρσ = gραgσβ Aαβ, which is also symmetrical, the last term, by virtue of (31), would assume the form

  In the case of symmetry in question, (41) may therefore be replaced by the two forms

  which we have to employ later on.

  § 12. THE RIEMANN-CHRISTOFFEL TENSOR

  We now seek the tensor which can be obtained from the fundamental tensor alone, by differentiation. At first sight the solution seems obvious. We place the fundamental tensor of the gμv in (27) instead of any given tensor Aμv, and thus have a new tensor, namely, the extension of the fundamental tensor. But we easily convince ourselves that this extension vanishes identically. We reach our goal, however, in the following way. In (27) place
/>   i.e. the extension of the four-vector Aμ. Then (with a somewhat different naming of the indices) we get the tensor of the third rank

  This expression suggests forming the tensor Aμστ – Aμτσ. For, if we do so, the following terms of the expression for Aμστ cancel those of Aμτσ, the first, the fourth, and the member corresponding to the last term in square brackets; because all these are symmetrical in σ and τ. The same holds good for the sum of the second and third terms. Thus we obtain

  where

  The essential feature of the result is that on the right side of (42) the Aρ occur alone, without their derivatives. From the tensor character of Aμστ – Aμτσ in conjunction with the fact that Aρ is an arbitrary vector, it follows, by reason of § 7, that is a tensor (the Riemann-Christoffel tensor).