C. THE THEORY OF THE GRAVITATION-FIELD

|gμαgαν| = | δνμ| = 1

So that it follows (17) that |gμν| |gμν| = 1.

Invariant of volume.

We see first the transformation law for the determinantg= |gμν|. According to (11)

Formula.

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

Formula.

Formula."(A)."

"(A)."

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

dτ′ = ∫dx₁dx₂dx₃dx₄

is according to the wellknown law of Jacobi.

Formula."(B)."

"(B)."

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

(18) = √gdτ′ = √gdτ.

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 thatds²can always, according to (1) be expressed in real magnitudesdX₁ ...dXh. If we calldτ₀ the “natural” volume elementdX₁dX₂dX₃dX₄ 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 thatgcan never change its sign; we would, according to the special relativity theory assume thatghas 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,

Formula."(19)."

"(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=gμσAσA =gμνAμν

Aμ=gμσAσ

A =gμνAμν

We would point out specially the following combinations:

Aμν=gμαgνβAαβAμν=gμαgνβAαβ

Aμν=gμαgνβAαβ

(complement to the co-variant or contravariant tensors)

and Bμν=gμνgαβAαβ

We can call Bμνthe reduced tensor related to Aμν.

Similarly

Bμν=gμνgαβAαβ.

It is to be remarked thatgμνis no other than the “complement” ofgμνfor we have,—

gμαgνβgαβ=gμαδνα=gμν.

As the “line element”dsis 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 ∫dsis an extremum (geodetic line),i.e., one which has got a significance independent of the choice of co-ordinates.

Its equation is

(20) δ{ ∫P₂P₁ds} = 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-ordinatesxν; 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-ordinatesxνare 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{ ∫δωdλ = 0(20a) { λ₁{{ ω² =gμν(dxμ/dλ)(dxν/dλ)

{ λ₃

{ ∫δωdλ = 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)}

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{ ∫dλkσδxσ= 0(20b) { λ₁{{ wherekσ= (d/dλ){(gμν/ω) · (dxμ/dλ)} - (1/(2ω))(∂gμν/∂xσ× (dxμ/dλ) · (dxν/dλ).

{ λ₃{ ∫dλkσδxσ= 0(20b) { λ₁{{ wherekσ= (d/dλ){(gμν/ω) · (dxμ/dλ)} - (1/(2ω))(∂gμν/∂xσ

{ λ₃

{ ∫dλkσδxσ= 0

(20b) { λ₁

{

{ wherekσ= (d/dλ){(gμν/ω) · (dxμ/dλ)} - (1/(2ω))(∂gμν/∂xσ

× (dxμ/dλ) · (dxν/dλ).

From which it follows, since the choice of δνσis perfectly arbitrary thatkσ’sshould vanish. Then

(20c)kσ= 0 (σ = 1, 2, 3, 4)

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

Formula.

Or by merely changing the notation suitably,

Formula."20d"

"20d"

where we have put, following Christoffel,

Formula."21"

"21"

Multiply finally (20d) withgστ(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—

Formula.

Here we have put, following Christoffel,

Formula.

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-distancess, measured from a fixed point on the curve, and if further φ, be an invariant space function, thendφ/dsis also an invariant. The proof follows from the fact thatdφ as well asds, are both invariants

Since

Formula.

so that

Formula.

is also an invariant for all curves which go out from a point in the continuum,i.e., for any choice of the vectordxμ. 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/∂staken along any curve is likewise an invariant.

Substituting the value of ψ, we get

Formula.

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 replacingd²xν/ds²according to (22)

Formula.

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

Formula.

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/dsis thus a four-vector, with an arbitrary ratio of components, so that it follows from the results of §7 that

Formula."25"

"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/∂xμwe can get by differentiation a co-variant tensor of 2nd rank

Formula."26"

"26"

We call the tensor Aμνthe “extension” of the tensor Aμ. Then we can easily show that this combination also leads to a tensor, when the vector Aμis 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 :—

Formula.

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, Aμis a four-vector whose components are any given functions ofxν, we have, (with reference to the chosen co-ordinate system) only to put

ψ(1)= A₁ φ(1)=x₁ψ(2)= A₂ φ(2)=x₂ψ(3)= A₃ φ(3)=x₃ψ(4)= A₄ φ(4)=x₄.

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 Sμis equal to Aμ.

In order to prove then that Aμνis a tensor when on the right side of (26) we substitute any co-variant four-vector for Aμwe 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

Formula.

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

Formula.

Thus we get the desired proof for the four-vector, ψ ∂φ/∂xμand hence for any four-vectors Aμas 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 AμBν.

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

Formula.

are tensors. Through outer multiplication of the first with Bνand the 2nd with Aμwe get tensors of the third rank. Their addition gives the tensor of the third rank

Formula."(27)"

"(27)"

where Aμνis put = AμBν. 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 AμBνbut 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 Aμνσthe 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=gμνg dgμν= -gμνgdgμν.

The last form follows from the first when we remember that

gμνgμ′ν= δμ′μ, and thereforegμνgμν= 4,consequentlygμνdgμν+gμνdgμν= 0.

gμνgμ′ν= δμ′μ, and thereforegμνgμν= 4,

consequentlygμνdgμν+gμνdgμν= 0.

From (28), it follows that

Formula."(29)"

"(29)"

Again, sincegμνgνσ= δνμ, we have, by differentiation,

Formula.

By mixed multiplication withgστandgνλrespectively we obtain (changing the mode of writing the indices).

dgμν= -gμαgνβdgαβ∂gμν/∂xσ= -gμαgνβdgαβ

dgμν= -gμαgνβdgαβ

∂gμν/∂xσ= -gμαgνβdgαβ

and

(32)dgμν= -gμαgνβdgαβ∂gμν/∂xσ= -gμαgνβ∂gαβ/∂xσ.

(32)dgμν= -gμαgνβdgαβ

(32)

dgμν= -gμαgνβdgαβ

∂gμν/∂xσ= -gμαgνβ∂gαβ/∂xσ.

The expression (31) allows a transformation which we shall often use; according to (21)

Formula."(33)"

"(33)"

If we substitute this in the second of the formula (31), we get, remembering (23),

Formula."(34)"

"(34)"

By substituting the right-hand side of (34) in (29), we get

Formula."(29a)"

"(29a)"

Divergence of the contravariant four-vector.

Let us multiply (26) with the contravariant fundamental tensorgμν(inner multiplication), then by a transformation of the first member, the right-hand side takes the form

Formula."(A)"

"(A)"

According to (31) and (29), the last member can take the form

Formula."(B)"

"(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

gμνAμ= Aν,

where Aνas well as Aμare vectors which can be arbitrarily chosen, we obtain finally

Formula.

This scalar is theDivergenceof 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- Aνμis an antisymmetrical tensor built up in a very simple manner. We obtain

∂Aμ∂Aν(36) Bμν= --------- - -------∂xν∂xμ

∂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

(37) Bμνσ= Aμνσ+ Aνσμ+ Aσμν

∂Aμν∂Aνσ∂Aσμ= --------- + ---------- + ---------∂xσ∂xμ∂xν

∂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 bygμαgνβ(mixed multiplication), then a tensor is obtained. The first member of the right hand side of (27) can be written in the form

Formula.

If we replacegμαgνβAμνσby Aσαβ,gμαgνβAμνby Aαβand replace in the transformed first member

∂gνβ/∂xσand ∂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

Formula."(38)"

"(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

Formula."(39)"

"(39)"

By the reduction of (38) with reference to the indices β and σ(inner multiplication with δβσ), we get a contravariant four-vector

Formula.

On the account of the symmetry of

Formula.

with reference to the indices β and κ, the third member of the right hand side vanishes when Aαβis an antisymmetrical tensor, which we assume here; the second member can be transformed according to (29a); we therefore get

Formula."(40)"

"(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)

Formula."(41)"

"(41)"

If we introduce into the last term the contravariant tensor Aρσ=gρτAστ, it takes the form

Formula.

If further Aρσor is symmetrical it is reduced to

Formula.

If instead of Aρσ, we introduce in a similar way the symmetrical co-variant tensor Aρσ=gραgσβAαβ, then owing to (31) the last member can take the form

Formula.

In the symmetrical case treated, (41) can be replaced by either of the forms

Formula."(41a)"

"(41a)"

Formula."(41b)"

"(41b)"

which we shall have to make use of afterwards.

We now seek only those tensors, which can be obtained from the fundamental tensorgμνby differentiation alone. It is found easily. We put in (27) instead of any tensor Aμνthe fundamental tensorgμνand 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)

Formula.

i.e., the extension of a four-vector.

Thus we get (by slightly changing the indices) the tensor of the third rank

Formula.

We use these expressions for the formation of the tensor Aμστ- Aμτσ. Thereby the following terms in Aμστcancel 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

Formula."(43)"

"(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 Aρis an arbitrary four vector, it follows, on account of the result of §7, that Bρμστis 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 whichgμν’sare constants, Bρμστall vanish.

If we choose instead of the original co-ordinate system any new one, so would thegμν’s referred to this last system be no longer constants. The tensor character of Bρμστshows 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-systemgμν’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

Formula."(44)"

"(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, Sμνvanishes of itself so that tensor Bμνreduces 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.

A freely moving body not acted on by external forces moves, according to the special relativity theory, along a straight line and uniformly. This also holds for the generalised relativity theory for any part of the four-dimensional region, in which the co-ordinates K0can be, and are, so chosen thatgμν’s have special constant values of the expression (4).

Let us discuss this motion from the stand-point of any arbitrary co-ordinate-system K₁; it moves with reference to K₁ (as explained in §2) in a gravitational field. The laws of motion with reference to K₁ follow easily from the following consideration. With reference to K₀, the law of motion is a four-dimensional straight line and thus a geodesic. As a geodetic-line is defined independently of the system of co-ordinates, it would also be the law of motion for the motion of the material-point with reference to K₁. If we put

Formula."(45)"

"(45)"

we get the motion of the point with reference to K₁, given by

Formula."(46)"

"(46)"

We now make the very simple assumption that this general covariant system of equations defines also the motion of the point in the gravitational field, when there exists no reference-system K₀, with reference to which the special relativity theory holds throughout a finite region. The assumption seems to us to be all the more legitimate, as (46) contains only the first differentials ofgμν, among which there is no relation in the special case when K₀ exists.

If γμντ’s vanish, the point moves uniformly and in a straight line; these magnitudes therefore determine the deviation from uniformity. They are the components of the gravitational field.

In the following, we differentiate gravitation-field from matter in the sense that everything besides the gravitation-field will be signified as matter; therefore the term includes not only matter in the usual sense, but also the electro-dynamic field. Our next problem is to seek the field-equations of gravitation in the absence of matter. For this we apply the same method as employed in the foregoing paragraph for the deduction of the equations of motion for material points. A special case in which the field-equations sought-for are evidently satisfied is that of the special relativity theory in whichgμν’s have certain constant values. This would be the case in a certain finite region with reference to a definite co-ordinate system K₀. With reference to this system, all the components Bρμστof the Riemann’s Tensor [equation 43] vanish. These vanish then also in the region considered, with reference to every other co-ordinate system.

The equations of the gravitation-field free from matter must thus be in every case satisfied when all Bρμστvanish. But this condition is clearly one which goes too far. For it is clear that the gravitation-field generated by a material point in its own neighbourhood can never be transformedawayby any choice of axes,i.e., it cannot be transformed to a case of constantgμν’s.

Therefore it is clear that, for a gravitational field free from matter, it is desirable that the symmetrical tensors Bμνdeduced from the tensors Bρμστshould vanish. We thus get 10 equations for 10 quantitiesgμνwhich are fulfilled in the special case when Bρμστ’s all vanish.

Remembering (44) we see that in absence of matter the field-equations come out as follows; (when referred to the special co-ordinate-system chosen.)

Formula."(47)"

"(47)"

It can also be shown that the choice of these equations is connected with a minimum of arbitrariness. For besides Bμν, there is no tensor of the second rank, which can be built out ofgμν’s and their derivatives no higher than the second, and which is also linear in them.

It will be shown that the equations arising in a purely mathematical way out of the conditions of the general relativity, together with equations (46), give us the Newtonian law of attraction as a first approximation, and lead in the second approximation to the explanation of the perihelion-motion of mercury discovered by Leverrier (the residual effect which could not be accounted for by the consideration of all sorts of disturbing factors). My view is that these are convincing proofs of the physical correctness of my theory.

In order to show that the field equations correspond to the laws of impulse and energy, it is most convenient to write it in the following Hamiltonian form:—

(47a)δ∫ Hdτ = 0H =gμνγαμβγβνα√(-g) = 1

(47a)

δ∫ Hdτ = 0

H =gμνγαμβγβνα

√(-g) = 1

Here the variations vanish at the limits of the finite four-dimensional integration-space considered.

It is first necessary to show that the form (47a) is equivalent to equations (47). For this purpose, let us consider H as a function ofgμνandgμνσ(= ∂gμν/∂xσ)

We have at first

δH = ΓαμβΓβναδgμν+ 2gμνΓαμβδΓβνα= - ΓαμβΓβναδgμν+ 2Γαμβδ(gμνΓβνα).

δH = ΓαμβΓβναδgμν+ 2gμνΓαμβδΓβνα

= - ΓαμβΓβναδgμν+ 2Γαμβδ(gμνΓβνα).

But

Formula.

The terms arising out of the two last terms within the round bracket are of different signs, and change into one another by the interchange of the indices μ and β. They cancel each other in the expression for δH, when they are multiplied by Γμβα, which is symmetrical with respect to μ and β, so that only the first member of the bracket remains for our consideration. Remembering (31), we thus have:—

δH = -ΓμβαΓναβδgμν+ Γμβαδgαμβ

Therefore

(48)∂H/∂gμν= -ΓμβαΓναβ∂H/∂gσμν= Γμνσ

(48)∂H/∂gμν= -ΓμβαΓναβ

(48)

∂H/∂gμν= -ΓμβαΓναβ

∂H/∂gσμν= Γμνσ

If we now carry out the variations in (47a), we obtain the system of equations

(47b) ∂/∂xα( ∂H/∂gαμν) - ∂H/∂gμν= 0,

which, owing to the relations (48), coincide with (47), as was required to be proved.

If (47b) is multiplied bygσμν, since

∂gσμν/∂xα= ∂gαμν/∂xσ

and consequently

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