unified field theory - 4

MATHEMA TICS: T. Y. THOMAS

ON THE UNIFIED FIELD THEORY. I

By TRAcY YERKES THOMAS

DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY

Communicated September 30, 1930

In a number of notes in the Berlin Sitzungsberichtel followed by a revisedaccount in the Mathematische Annalen,2 Einstein has attempted to de-velop a unified theory of the gravitational and electromagnetic field by in-troducing into the scheme of Riemann geometry the possibility of distantparallelism. According to this view, there exists in each point of the un-derlying continuum of the world of space and time, a local cartesian co-ordinate system in which the Pythagorican theorem is satisfied. Theselocal coordinate systems are determined by four independent vector fieldswith components h?(x) depending on the coordinates x: of the continuum.There is, therefore, associated with each point, a configuration consistingof four independent vectors; it is assumed that these vector configurationsare in parallel orientation in such a way that the arbitrary orientation ofthe configuration at one point determines uniquely the orientation of theconfigurations at all points of the continuum. This affords the possibilityof determining whether or not two vectors at different points are parallel,namely, by comparing their components in the local systems: Two vec-tors are parallel if the corresponding components are equal when referredto a local system of coordinates. It is in the theory of the space of distantparallelism that Einstein has hoped to find his long-sought unification ofelectricity and gravitation.

It is important to determine an exact relation between the coordinateszi of the local system and the coordinates x} of the space of distance parallel-ism. This problem is solved in the present communication in such a waythat certain requirements specified precisely in Sect. 2 are satisfied. Asso defined there is a certain analogy between the local coordinates and thenormal coordinates introduced into the theory of relativity by Birkhoff3and later discussed by the writer.4 The relation between the zt and xicoordinates enables us to construct a set of absolute invariants with respectto transformations of the xs coordinates sufficient for the complete char-acterization of the space of distant parallelism. Equations in the localsystem, in which the coordinates zt are interpretable as coordinates oftime and space, can be transformed directly into equations of general in-variantive character. In view of this property we are led to the construc-tion of a system of wave equations as the equations of the combined gravi-tational and electromagnetic field. This system is composed of 16 equa-tions for the determination of the 16 quantities hi and is closely analogousto the system of 10 equations for the determination of the 10 components

VOL. 16, 1930 761


gas in the original theory of gravitation. It is an interesting fact that thecovariant components h of the fundamental vectors, when considered aselectromagnetic potential vectors, satisfy in the local coordinate systemthe universally recognized laws of Maxwell for the electromagnetic fieldin free space, as a consequence of the field equations.

It is my intention to supplement the results of the present paper by aseries of papers devoted to an existence-theoretic treatment of the fieldequations and related problems.

1. Let us denote by h9(x) for i = 1,2,3,4, the components of four con-travariant vector fields; we observe that the Latin letter i is thus used todenote the vector, and the Greek letter a to denote the component of thevector. In general, we shall adopt this convention of Einstein, i.e., weshall employ a Latin letter for an index which is of invariantive characterwith respect to the arbitrary transformations of the xi coordinates, and aGreek letter in all contrary cases. Departures from this rule, as well asdepartures from the rule that an index which appears twice in a term is tobe summed over the values of its range, will be such as are easily recognizedon observation. By means of the quantities hF(x) we impose on the under-lying continuum its structure as a space of distant parallelism in accordancewith the following

POSTULATES OF SPACE STRUCTURE

A. In any coordinate system (x) there exists a unique set of componentsA, of affine connection, for the determination of the affine properties of thecontinuum.

B. In any coordinate system (x) there exists a unique quadratic differentialform gpdxadx" of signature -2, for the determination of the metric proper-ties of the continuum.

C. At each point P of the continuum there is determined a configurationconsisting offour orthogonal unit vectors issuingfrom P.

D. Corresponding vectors in the configurations, determined at two pointsP and Q of the continuum, are parallel.

E. The components hi of the vectors determining the configuration at anypoint P of the continuum, are analyticfunctions of the xi coordinates.The quadratic differential form in Postulate B will be referred to as the

fundamental form and its coefficients gas as the components of the funda-mental metric tensor. The four unit vectors with components h?'(x),which enter in Postulates C, D and E, will be referred to as the funda-mental vectors. Since these vectors are orthogonal by Postulate C, theexpression gahi'hlk is equal to zero for i # k; the condition that thefundamental vectors are unit vectors, likewise specified by Postulate C,means that for i = k, the expression g.Xhlh' has the value ± 1. Hencewe can write

762 PRoc. N. A. S.


gashhIk = ek(1.1)

where, following Eisenhart,5 the convenient notation ej for + 1 or -1, isintroduced. The fact that the fundamental form has signature -2 byPostulate B leads us to take

e= 1,e2 = e3 = 1;

more precisely, the signature -2 requires that one of the ej have the value+ 1 and that the remaining ej have values -1, the selection of the par-ticular es to which the value + 1 is assigned being an unessential matter ofnotation. From (1.1) we see that the product of the determinants gas|and h, '12 is equal to -1. This shows that the determinant |gB| is nega-tive and leads to the conclusion of the independence of the fundamentalvectors. It is, therefore, possible to deduce from the fundamental vectorsa system of four covariant vectors with components hs(x) uniquely definedby the relations

W'h = b, hahs = k;the components h' will be called the covariant components of the funda-mental vectors to distinguish them from the contravariant componentshS of these vectors. In consequence of the above relations, equations (1.1)can be solved so as to obtain

gap Eeighth'(1.2)i=l

as the equations defining the components of the fundamental metric tensor.The condition that the fundamental vectors be parallel as demanded by

Postulate D has its analytical expression in the equations

x + hi Vs = 0.

This gives

(1.3)

as the equations which define the components A' of affine connection.Let us now suppose the existence of another system of fundamental vec-

tors in the same space of distant parallelism. Let us denote the contra-variant components of these vectors by *h?(x) in the (x) coordinate system,and let us put

bhs= hc' aAL i bx-y

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MATHEMATICS: T. Y. THOMAS

*h = ahA*; (1.4)these equations define the quantities ak as analytic functions of the x'coordinates in consequence of Postulate E. Multiplying both members of(1.4) by *h'h- we obtain the equations

h,, = a.*7'h (1.5)

which represent the transformation induced by (1.4) on the covariantcomponents hL. On account of the uniqueness of determination of thecomponents A;,(x) demanded by Postulate A, we must have

bh~ 6*hsohis ace=*hia h6-

From (1.4) and these latter equations, it readily follows that the quantitiesak are constants. The quantities ak are, however, not arbitrary constants,since they must satisfy a condition of orthogonality, namely,

4

eea' as = ek k5 (1.6)

which is obtained from (1.5) as the direct result of the uniqueness of deter-mination of the fundamental form specified by Postulate B. Taking thedeterminant of both members of (1.6) we find that the determinant ahas the value ± 1. We can therefore define uniquely a set of quantitiesb' by the equations

a' bi = Sk, aj bSt=a

Another form of the conditions (1.6) which is sometimes useful, can be de-rived in the following manner. Multiply both members of (1.6) by b,so as to obtain

emar = elb.,or

a= elembl.When we multiply both members of these latter equations through byelal and sum of the index 1, we find

4

keaka7 = eker (1.7)

The transformation (1.4) in which the coefficients a4 are constants satis-fying (1.6) or (1.7) will be called an orthogonal transformation of the com-

ponents of the fundamental vectors.

764 PROC. N. A. S.


2. The idea of the local coordinate system (z) is inherent in the idea ofthe configuration formed by the four vectors whose components h? aresubject to the orthogonal transformations (1.4), and, in fact, the construc-tion of the local system was not carried beyond this stage by Einstein.Nevertheless a configuration of four vectors is not a coordinate system andwe must face the problem of showing exactly how a system of local co-ordinates z' can be defined. We require of the local system that it satisfycertain conditions which are stated precisely in the following

POSTULATES OF THE LOCAL SYSTEM

A. With each point P of the space of distant parallelism there is asso-ciated a local co6rdinate system (z) having its origin at the point P.

B. The coordinate axes of the local system (z) at the point P are tangentto the fundamental vectors at P, in such a way that the zi axis is tangent to thevector with components h? and has its positive direction along the directionof this vector.

C. The interval ds is given by

ds2 = (dzl) 2-(dz2) - (dz3) 2 - (dz4) 2 (2.1)

at the origin of the local system.D. The paths6 of the space of distant parallelism which pass through the

origin of the local system (z) have the form

=eV' (2.2)

where the e are constants and v is a parameter.The above postulates give a complete geometrical characterization of the

coordinates z2 of the local system. The paths which enter in Postulate Dare, in general, defined as those curves which are given as solutions of theinvariant system of equations

d~x" dx" dx7'd~a + -a=0, (2.3)dv dv dv

where

=-1 (A"a+ Ae); (2.4)

such curves are generated by continuously displacing a vector parallel toitself along its own direction and are analogous to the straight lines ofaffine Euclidean geometry. If we denote the components of affine con-

VOL. 16, 1930 765


nection A; by Ni4 (z), when referred to the local coordinate system, wehave in consequence of Postulate D that

N7k j e = o

along a path through the origin of the local system. It follows imme-diately from these latter equations that the equations

k.k = 0 (2.5)

are satisfied identically in the local system. When we replace the com-ponents 'Xjk in (2.5) by their values in terms of the components A' we ob-tain a system of partial differential equations for the determination of thecoordinates x' in terms of the coordinates zi, namely,

( 82Xe + AO' (2.6)

This system of equations possesses a unique solution x = spa(z) satisfyinga set of initial conditions

x= p (z = 0), (2.7)

* ~~~~~~~azaziPi'(Z' 0), ~~~~(2.8)

where pa and pi' are arbitrary constants.7 By Postulate A condition (2.7)is satisfied, provided that the constants pa denote the coordinates of the

point P; it remains to determinethe values of the constants pi in(2.8). For this purpose we considerthe relation, imposed by Postulate

X B. between the local coordinates/ and the fundamental vectors at the

point P; this is illustrated in theaccompanying figure in which we

ok_Z have indicated only the positive z'and z2 axes and the correspondingtangent vectors with components

el and h2, respectively. In consequence, we see that

(axo) = ahW(p),

where the o, are positive constants. On account of this set of equationsand the condition imposed by Postulate C, it readily follows that the oiare equal to ± 1. Hence as = +1, since these constants are positive,-and the above set of equations becomes

766 PRoc. N. A. S.


(az )zO * (P). (2.9)

The values h?(p) are therefore to be ascribed to the above constants P?.By repeated differentiation of (2.6) and use of the initial conditions (2.7)and (2.9), we determine the successive coefficients H(p) of the power seriesexpansions

a pa + ha (p)zi - - Hia (p)z'z' - - Htjk(P)z ZZk - (2.10)2! 3!

Thus we have thatHew = A 4yhqhj,

where

AN= 2 a + It (2.11)and

H'jk = AzahhJi k,where

Aa= 1 [( + + a -2(AO AO's + Ae7 Acp + AaaAB8Y)(2.12)

etc. The jacobian of (2.10) does not vanish since it is equal to the deter-minant 11i (p)I at the origin of the local system; hence (2.10) possesses aunique inverse. Either the transformation (2.10), or its inverse, gives aunique defination of the local coordinate system.

3. Let us denote by z' the coordinates of the local system determinedby the point P and the components of affine connection A4y(-x) whichresult from the components Ap7(x) by an analytic transformation Tof the Xa coordinates. The relation between the z' and z coordinatesmust then be such that

iZ = 0 (Z = 0) x - = ak (Z" = °)-(3 .1 )

Also this relation must be such as to satisfy the system of equations

(ala-* + VP a-? -z Z- = 0. (3.2)

Hence

i, - (3.3)

VOL. 16, 1930 767


This follows from the fact that (3.3) satisfies (3.2) and the initial con-ditions (3.1), and that (3.2) possesses a unique solution which satisfies theconditions (3.1). We can therefore say that the local coordinates z' remainunchanged when the underlying coordinates xi undergo an arbitrary analytictransformation T. In a similar manner we can show that when the funda-mental vectors undergo an orthogonal transformation (1.4), the local co6rdi-nates iz associated with any point P likewise undergo an orthogonal trans-formation, i.e., a linear homogeneous transformation

|a'=Zk~a: |' (3.4)

which leaves the form4

Z eiz'z~

invariant. The behavior of the coordinates z? of the local system (1) underarbitrary analytic transformations of the xi coordinates, and (2) under or-thogonal transformations of the fundamental vectors as described by theabove italicized statements, is of great importance for the development ofthe theory of relativity.

4. If we transform the components of a tensor to a system of local co-ordinates z? and evaluate at the origin of this system, we obtain a set ofquantities which are of the nature of absolute invariants with respect totransformations of the xt coordinates. To prove this formally we shall findit convenient to denote a set of tensor components T., (x) with respectto the (x) system, by t Ikm.l in the (z) coordinate system, i.e., we shalladopt the convention that in the (z) coordinate system, Latin letters whenenclosed by I |, correspond to indices of covariant or contravariantcharacter. If we put

T'k... tik. ..mi)z = °

thenTk~"m (X) = Tk. .-'m(X

in consequence of (3.3). The explicit expressions for these invariants are

obtained by evaluating at z 0= 0 both members of the set of equations

t k .i= (4.1)

This givesTk-m T ... Ihy . .. h h'a. .. hj,.

Similarly, if we differentiate the components t any number of times andevaluate at the origin of the (z) system, we obtain a set of quantities, namely,

768 PRoc. N. A. S.


each of which is an absolute invariant with respect to analytic transforma-tions of the xe coordinates. On account of this invariant property, thename absolute derivatives will be used to refer to these quantities. To de-rive the explicit formulae for any absolute derivative Tg:]., ... wehave merely to differentiate (4.1) with respect to z... .z and evaluateat the origin of the (z) system. For example, the first absolute derivativeof the covariant vector with components Ta is given by the formula

TkP = (a TaA7) hk 14.

We next consider the absolute derivatives of the fundamental vectors,using the covariant components h, of these vectors as the basis of dis-cussions. Let us denote these components by A' 1 when referred to the(z) coordinate system. Then

Alj1 = 14 . (4.2)

In general, the absolute derivatives are defined by the equations

1,k... m = (4)Z= .3)

We note that

A'j,= h' hj= 5j (z' = 0); (4.4)also that

hj',k- hLl|J (4.5)

and

|~k - [ 821L' -h A -^ e 4LiX~X ?,x6 ox (4-6)

Aajp - h'AaB-y hjahk'h7 |

The special formulae (4.5) and (4.6) will be important in our later work.By (4.3) the absolute derivative M.k... .m is symmetric in the indices

k. . .m. Hence, these quantities satisfy the identities

VOL. 16, 1930 769


|h;k...m = h>p... (4.7)

where p... q denotes any permutation of the indices k... m. To deriveother identities satisfied by the h>,k. . .m we observe that the equations

(6AbI\) zjzk = 0 (4-8)%zk!

which are equivalent to (2.5), are satisfied identically in the (z) coordinatesystem. By repeated differentiation of (4.8) and evaluation at the originof the (z) system, we obtain

S (h1,k ...m) = |(4.9)

where the symbol S is used to stand for the summation of all the terms ob-tainable from the one in parenthesis by permutating the indices jk. . mcyclically. As special cases of the identities (4.9) let us observe that

hjk + hk = 0 (4.10)and

hj,kz + hk,lj + hjk = 0. (4.11)The identities (4.7) and (4.9) constitute a complete set of identities of theabsolute derivatives hi,*... , in the sense that any other identity satis-fied by these quantities is derivable from these identities.8When we make an orthogonal transformation (1.4) of the components of

the fundamental vectors, the components A!j, go over into a set of com-ponents *Aiji referred to the (z*) system, which are related to the ATjj by

*Aji, aiP = AP aj'; (4.12)

this follows from (1.5), (3.4), and (4.2). Differentiating both members of(4.12) with respect to zk... z and evaluating at the origin of the localsystem, we obtain

*hM*..." at = hr ...t a.... .a (4.13)

We express this result by saying that the absolute derivatives I,k*...constitute the components of a tensor with respect to orthogonal transforma-tions of thefundamental vectors. A similar discussion can, of course, be madeon the basis of the contravariants components h? of the fundamental vec-tors; also equations similar to (4.13) give the transformation of the com-ponents T*.-.-. or more generally of the components Tk.M,p.. , in-duced by the transformation (1.4) of the fundamental vectors.9

5. It is the sense of the local coordinate system (z) that the coordinate

.70 PRoc. N. A. S.


z' is interpretable as the time coordinate and Z2, Z3, Z4, as rectangular car-tesian coordinates of space. Let us, therefore, put

z1 = t, Z2 = X, z3 = y, z4 =Z

in accordance with the unusual designations. More precisely, we shouldsay that z1 has the significance of a time coordinate and Z2, Z3, Z4, the sig-mnificance of rectangular cartesian coordinates x, y, z at the origin of co-ordinates, i.e., in the infinitesimal neighborhood of the origin, since theinterval ds has the exact form (2.1) only at the origin of the local system.In conformity with this interpretation of the coordinates of the local system,we now impose on the space of distant parallelism, the following

POSTULATE OF THE UNIFIED FIELD

The equations

U1 + 1A + 1A IA | (5.1), 6X2 by2 bZ2 a)t2 1

are satisfied at the origin of the local system.According to the above postulate, the field equations of the combined

gravitational and electromagnetic field are of the nature of a system ofwave equations-a type of equation which has already shown itself to be offundamental importance for the study of gravitational or electromagneticphenomena. The relationship of the above system of field equations tothe field equations of the earlier theory of gravitation and to the Maxwellequations of the electromagnetic field is discussed in the following'section.When (5.1) is expressed in general invariative form, we have

4EehJ kk=O (5.2)

k = 1

as the proposed system of field equations for the unified field theory. Thesystem (5.2) is obviously invariant with respect to transformations of the xscoordinates, since the factors which compose the system are themselvesdirectly invariant; it is also invariant with respect to orthogonal trans-formations (1.4) of the fundamental vectors. To prove this we observe that

*h_,,klat = hPrsajqdka'from (4.13). Putting I equal to k in these equations and multiplyingthrough by ek, we obtain

(k~1 ek hjkk) as = ( E erhq rr)

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772 MATHEMATICS: T. Y. THOMAS PROC. N. A. S.

on making use of (1.7). This proves the assertion of invariance of (5.2)with respect to orthogonal transformations of the fundamental vectors.

6. If we denote the components of electric force by X, Y, Z and thecomponents of magnetic force by a, P, y then Maxwell's equations for theelectromagnetic field in free space, are

6z __by bz atax bz a:

6z ox ataY~~~ax_az~~~~(6.1)

Ox by ata a + az=0

ax by Ozand

z _ja_ 6x

by 6z ataac y _ bY

1 Z ax at (6.2)

0 _ 6a _> JZbx by at

tjax by bz

where x, y, z are rectangular Cartesian coordinates and t is the time. Itis possible to write the above equations in a more contracted form. Forthis purpose we define a set of skew-symmetric quantities Fjk as the ele-ments of the matrix

o x Y z

-x 0 -Y j

-Y a 0 -a

-z -, a 0and then construct the following two sets of equations

Fjk,l + Fklj + Fljk = 0 (6.3)and

4E ekFk,k = 0, (6.4)

k = I


where Fjk,l is now used to denote the ordinary partial derivative of Fikwith respect to z'. The expanded forms of (6.3) and (6.4) give equations(6.1) and (6.2), respectively. Functions soy of the coordinates x, y, z andthe time t are called electromagnetic potentials if they are such that

Fjk =.bfk _ b'k (6.5)aJZk aJZ

If the Fjk have the form (6.5), equations (6.3) are satisfied identically.Now consider the covariant components h' referred to the local system,

i.e., the components Al'1, and put

= A1i1 (6.6)

for a fixed value of the index i; also define a set of quantities Fjk by (6.5),using the so given by (6.6) for that purpose. Then

Fjk a k_ az~kl(6.7)

Differentiating and evaluating at the origin of the local system, we have

Fjk, hjkl -hkjlvthe absolute derivatives Fjkl given by these equations satisfy the first setof Maxwell's equations (6.3) in consequence of the identities (4.11). More-over,

4 4 3 4E ekFjkf* E (ekh>k - ekhkik) = E ek hJkk.

Hence, the second set of Maxwell's equations (6.4) is satisfied in conse-quence of the field equations (5.2). In other words, the covariant corn-ponents of the fundamental vectors when considered as electromagnetic po-tential vectors, satisfy, in the local coordinate system, the universally recognizedlaws of Maxwell for the electromagnetic field in free space, as a consequenceof thefield equations (5.2). This fact strongly suggests that the componentshA will play the r6le of electromagnetic potentials in the present theory.Taking account of the field equations (5.2) we easily deduce the system

of equations4 4

E ekgijkk = 2 E ek el hik hi.*. (6.8)k = 1 k,l =1

Or, denoting the components gag by Blij, when referred to the local co-ordinate system, we have

a-2 + a2B + a-2B 1- 2B_4__(-)X2 ~ + at2 2 E ekelh,khj'k (6.9)by2OZ2 ~~~~~k,1 = 1

VOL. 16, 1930 773


at the origin of the local system. Now in the earlier theory of gravitation,the field equations could be given a form analogous to (6.9) in which theright members were equal to zero;3 since the nature of the solution of (6.8)is determined by the form of the left members of (6.9) it follows that thefunctions gaff possess the same general character as in the earlier theory ofgravitation. Moreover, the interpretation of the quantities hl as electro-magnetic potentials would lead us to expect that in a purely gravitationalfield, i.e., more precisely, in a comparatively inappreciable electromagneticfield, the square of the hlk would be negligibly small quantities. In thiscase the right members of (6.9) vanish approximately and we are left withthe system

ij1 + 62BI j + -Blij___Blij = o

ax2 ay2 2 bt2

as a first approximation. It is, therefore, to be expected that the quan-tities ga, will successfully assume the r6le of gravitational potentials as inthe previous theory of gravitation.

1 A. Einstein, Berliner Berichte, 1928-30.2 A. Einstein, "Auf die Riemann-Metrik und den Fern-Parallelismus gegriindete ein-

heitliche Feldtheorie," Math. Ann., 102 (1930), pp. 685-697.' G. D. Birkhoff, Relativity and Modern Physics, Harvard University Press (1923), pp.

124 and 228.4 T. Y. Thomas, "The Principle of Equivalence in the Theory of Relativity," Phil.

Mag., 48 (1924), pp. 1056-1068.6 L. P. Eisenhart, Riemannian Geometry, Princeton University Press (1926), Chap. I.6This has reference to the path in the sense inwhich that word is used in "The Geometry

of Paths," Trans. Am. Math. Soc., 25 (1923), pp. 551-608.7 It is obvious that the system of differential equations (2.6) possesses a unique formal

solution1x pa" +paZ + -paz"zj + (a)

S 2! ~

such that the conditions (2.7) and (2.8) are satisfied. A proof of convergence of theseries (a) can be given in the following manner. Consider a system of equations

azixzk _F = 0, (b)

where the ly are analytic functions of the variables Xc. The conditions of integrabil-ity of (b) are that

_AP,- 115 + Fe, - Fa Fa5 = 0

identically. These conditions are satisfied by taking all the functions FPP equal to one

another in accordance with the equations

774 PROc. N. A. S.


Mrat- ~X1 + *-+ XI

p

ere M and p are constants; the system (b) then possesses a unique solution

X" = pa, + Pia Zs + - pai Z" ? + (C)

isfying the conditionsXa = Pa (zi =),axC1

ai=c, (ki = 0).

)w choose the above constants Mand p so that each function Fib, dominates the corre-)nding functions Ac',,, and, furthermore, choose the pa and Pa such that

pa2lpaI lp,P>Ica

view of the fact that (c) is likewise a solution of the system

(a~x_ F aX ~k )zjzk = O.

s then easily seen that each expansion (c) dominates the corresponding expansion (a).e convergence of the expansions (a) within a sufficiently small neighborhood of theues zi = 0, is therefore established.I The proof is analogous to that given in my paper "The Identities of Affinely Con-Ated Manifo'ds," Math. Zeit., 25 (1926), pp. 714-722.'A method of covariant differentiation based on the non-symmetric componentsDffine connection Lily is used by Einstein.2 The point of view adopted in the presentrestigation in which covariant differentiation, or more precisely absolute differentia-n, is brought into relationship with the local coordinate system, necessarily depends onsymmetric connection; for this reason I have rejected the method of Einstein. Ituld be possible also to define covariant or absolute derivatives on the basis of theristoffel symbols P'm, derived from the components gae and, in fact, this has been ad-ated by T. Levi-Civita, "Vereinfachte Herstellung der Einsteinszhen einheit!ichenIdgleiz-hungen," Berliner Berichte, 1929. This method for the construction of invari-ts can be brought into relationship with the local coordinates; in fact, it is only neces-y to replace Postulate D of the Postulates of the Local System by a similar postulate ongeodesics of the space of distant parallelism. My primary objection to this is that wea good deal of the simplification inherent in the above theory. For example, in

ce of the quantities hik given by (4.5) we would have the more complicated set ofariants

(6h r shah-~facih hlhk

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MATHEMATICS: H. S. VANDIVER

investigation. Now there is a certain psychological influence exerted by the methoditself upon the investigator. What I mean is that one is naturally led to the construc-tion of special invariants which, on the basis adopted for the formation of invariants, arerepresented by simple analytical expressions. So, for example, the field equations pro-posed by Einstein2 have a very simple analytical form in terms of the covariant deriva-tives used by him, but these same equations would be considerably more complicated inform if expressed in terms of the covariant derivatives used by Levi-Civita; also thesimple form (5.1) or (5.2) of the field equations assumed in the above investigation ispeculiar to the method of absolute differentiation which I have adopted. Thus thedifferent methods of construction of invariants will lead us, in practice, to the assumptionof different systems of field equations; these systems of field equations must, roughlyspeaking, be of the same general character but will nevertheless, not be precisely equiva-lent, considered as systems of partial differential equations.

It is possible to develop a process by which the above methods are brought into re-lationship with one another and which will, moreover, permit the ready construction ofinvariants depending on combinations of these methods. This process has its geo-metrical foundation in the study of those surfaces xa = f' (u, v) which are defined assolutions of the system of equations

62Xa a axe axy+ Aza - = 0,

12Xa + r IXa- = 0 (V =o), (a)

i ?oy X .ai = 0 (U = 0),

and so constitutes a generalization of the process of covariant differentiation or exten-sion, as developed in The Geometry of Paths. In this way we are led to a set of relationsXa = g" (y, z) which, for Zt = const., denote a transformation to a system of coordinates9, and which for y9 = const. denote a transformation to a system of coordinates z.If t(y, z) represents the components of a tensor either in the (y) or (z) coordinate system,then

6___y__Z_ a = O, Z = 0

defines, when considered as a function of the xi coordinates, the components of a tensorin the (x) coordinate system. As an alternative method of procedure the system (a)can be replaced by the system

a__a + 0, axe aX*)yJzk 0O62Xa a, ?6X8 bX'Y

(. + ray I yjykA= 0 (z = 0), (b)ayjbyk byl W)k(2C+ xb-y+ a ) Zjzk = 0 (y = 0).

The consideration of system (b) enables us, moreover, by imposing initial conditions corre-sponding to (2.9), to develop a theory of absolute invariants which is a generalizationof that of the above investigation. The details of this process will not be developed hereas it is not necessary for our work on the theory of relativity.

776 PROC. N. A. S.


* NATIONAL RESEARCH FELLOW.1 : Borel, "Sur quelques points de la Th~orie des fonctions," Ann. Acole Norm. sup.,

Ser. 3, 12, 1895.2 H. Poincar6, "Sur les determinants d'ordre infini," Bull. Soc. math. France. 14,

1886.3 t. Borel, "Legons sur les fonctions monog~nes," Paris, 1917.4 Cf. Carleman, "Les fonctions quasi analytiques," Paris, 1926.5 C. de la VaWI~e Poussin, The Rice Inst. Pamph., 12 (1925), No. 2.

ON THE UNIFIED FIELD THEORY. II

BY TRAcY YERKES THOMAS


Communicated November 6, 1930

In our previous note' we constructed a system of field equations for thecombined gravitational and electromagnetic field which, on the basis of theinterpretation of the quantities h, as electromagnetic potentials, led toMaxwell's equations in their exact form at the origin of the local coordi-nates. To secure the exact form of Maxwell's equations in the local co-ordinate system was, in fact, the principal motive for the introduction ofthis system of field equations. Actually, however, a system of field equa-tions constructed with primary regard to a law of conservation appearsto be of deeper physical significance. This latter point of view is made thebasis for the construction of a system of field equations in the present note-and the equations so obtained differ from those of Note I only by the ap-pearance of terms quadratic in the quantities hok. It would thus appearthat we can carry over the interpretation of the h, as electromagnetic po-tentials; doing this, we can say that Maxwell's equations hold approxi-mately in the local coordinate system in the presence of weak electromag-netic fields.

Equations of Note I will be referred to by prefixing the numeral I be-fore the number of the equation in question.

1. In the Einstein theory of gravitation the operation of forming thedivergence of a tensor constituted an almost unavoidable generalizationfrom the analogous operation of pre-relativity physics. When dealingwith quantities of the nature of absolute invariants, however, as we shall doin the development of the present theory, it is by no means obvious howthis operation can best be defined. However, actual investigation of theidentities of the field theory points to a definition of the operation of di-vergence which is well adapted to a statement of the laws of conservation;the reasons for this particular definition will be apparent from the identitiesof the following sections.

830 PRoc. N. A. SE


DIVERGENCE RULE: The formuka4

VmTj. .. m=Ad em [ Tj ...mm + 2hmrT...m]

defines the divergence of the invariant TjI... m with respect to the index m.Any covariant index, i.e., covariant under transformations of the funda-mental vectors, can of course appear in place of the particular index m inthe definition of the divergence operation. It should be observed that theterm involving the absolute electromagnetic forces 2 hk in the aboveformula, corresponds to those which contained the components of affineconnection in the divergence formula of the earlier theory of relativity.A generalization of the above formula to include invariants T with anynumber of contravariant indices could obviously be made but this is notnecessary for the requirements of the present note. On the basis of thedivergence rule we obtain the field equations from the followingPOSTULATE OF THE UNIFIED FIELD. The divergence of the absolute elec-

tromagnetic forces is equal to zero, i.e.,

Vk hk = . (1.1)

There are 16 equations in the system (1.1) for the determination of the16 electromagnetic potentials h,. By reference to the field equations inwhat follows we shall mean equations (1.1) rather than the correspondingequations of our previous note.

2. Before proceeding further we must derive certain special identitieswhich we shall need in the following work; this will be done in as concisea manner as possible. We have

hjt== ha'. (2.1)

Henceh.,k = 2 [h',k - k j] + hmjh'kl + hmkhZ'j (2.2)

when use in made I (4.5) and I (4.10). Interchanging k,l in (2.2) andadding, we obtain a set of identities which can be reduced to the form

h5,kz = 3 [hj,kj, + h.i,k + hm,k h7l + hmlh k]. (2.3)

We observe that these latter identities constitute an inverse form of theidentities (2.2) in that (2.2) expresses the hij,kj in terms of the hjakj while(2.3) gives h',kz in terms of the hk Il invariants.The invariants h',k,l satisfy a set of identities

VOL. 16, 1930 831


hjk,l + h1,Ij + j= 2 [I4, h*, + hm,k14i j + hnst h7'* (2.4)

which is easily deduced from (2.2).Let us denote the quantities A', byX,,l with reference to a system of

local coordinates and define the invariant Ajm by the formula

Ajkim VS

(See Sect. 2 in Note I.) Now transform equations I (4.6) to local co-ordinates, differentiate, and evaluate at the origin of the local system;we thus obtain a set of identities which can be written

hj,kl,m = hj,kl - Ajkim + 1 hit t2 ,kl + 2 h1, hl ,ji + 1 h1,1 hrnjk

+ hM hi,+ ,+ ht k hj t+ h' . (2.5)

Interchanging the indices j,k in (2.5) and subtracting the resulting iden-tities from (2.5) we obtain

hjklm - hkjl,m = h,klm- '4jim +h,,1I4,k + hsk hjj + [h',1(h'k -hk,l) + hrnk (h h11-I4sj) + htmt, (h;,k- hti1)1. (2.6)

To the identities (2.6) we now add those two sets of identities which resultfrom (2.6) by permutating the indices k,l,m cyclically. By a suitablearrangement of terms the set of identities so obtained can be given theform

8h;,klm = 3 (h5,kl,m + hj lmk + hs,mk,1 + hjm htl + h;,mk +

hJ,k h'imI + hik h5,Im + hti hM.km + h;,m hk, (2.7)

These identities express the invariants M,k1m in terms of the invariants14,1,km plus invariants of lower order in the derivatives of the electromag-netic potentials.To obtain the inverse form of the set of identities (2.7) we deduce the

formula

A;Mm = -31 h',jkl + 6 (h; 1h ,ki + hikk h j1 + hlh4 ,jk)which we use to eliminate the invariants Alum from (2.5). The result ofthis elimination is a set of identities which can be put into the form

3hk,k,m = 3h1,klm + hm,,jk + 2 hI, h, kl + hik h' j, + h', h',Jk+ 3 (ha hikl + h",k h,11 + h ,, h;,kl). (2.8)

Finally the set of identities

82 PROC. N. A. S.

VOL. 16, 1930 MATHEMATICS: T. Y. THOMAS 833

kjklm = 3 (hlmk + hjmkjl) + hmkj + hmjjk + hmjkl + hil(3 h1mjk + hi.km) +,tk (3 h4njl + hj!,1) (2.9)

+ 5 hi,1 Inki + hM~m hjkj + 6 h14j hMkj + hrnk (8 kia + htj)

+ h (8 hj,kt + hljk) + 3 (hjk h;,mk + kjk h1,im)can be obtained by eliminating the invaants jl between the identities(2.7) and (2.8).

3. The identities of Sect. 2 can be used to deduce identities of especialinterest for the field theory. By use of (2.2) it readily follows that

4

Vk hj,k = 4 ek (,kk + 4 hrak7,,) (3.1)

identically. Hence

4

| ek (hj,kk + 4 hk h,) = (3.2)k=

constitutes a system of equations completely equivalent to the field equa-tions (1.1).Now put I = k and m = j in (2.9), then multiply these equations through

by ejek and sum on the two repeated indices. This gives4 4

E Ej ej ek (h,kkj + hj hj, + 2 h ,kr) = 0 (3.3)j=1 k=1

identically. We next consider the set of identities4 4

2 E3 E ej ek hkj hij=l k-l

4 4 4 4

= 3 =2 , ej ek hjm (hikk + 8 hrm h j) + 4 X E ej ek (ki,k Wkj)jj=l k=1 j=l k=l

which we use to eliminate the last set of terms from (3.3). As a resultwe obtain the set of four identities

. ei{[ ek (hjkk + 4 hr,k hk,i)] + 2 hj,m

ek (k7~kk + 4 k,.k Wk,j) I = 0.By (3.1) this last set of identities can be given the form

Vj Vk Mk = ° (3.4)


i.e., the divergence of the left member of (1.1) vanishes identically. The iden-tities (3.4) are the mathematical counterpart of the laws of conservationof the physical world.2

4. If we fail to take into account the conditions imposed by the fieldequations on the structure of space, then the number of independentinvariants hM,kh....1 of order r + 1 in the derivatives of the electromag-netic potentials h$, i.e., the number of arbitrary values (hjad ...00which these invariants can assume at a point Q, is

16 K(4, r + 1)- 4K(4, r + 2),

where K(p,q) denotes the number of combinations with repetitions of pthings taken q at a time; this is an immediate consequence of the fact thatI (4.7) and I (4.9) constitute a complete set of identities. We shall denotethe above number by N(r + 1) and observe that we can write

N(r + 1) = 12K(4,r) + 8K(3,r) + 4K(2,r)

It can easily be shown that the number of independent quantities(hJk,11. .ir)Q is likewise given by N(r + 1). To do this we transform theexpression for hk given by 1 (4.5), to a system of local coordinates, dif-ferentiate repeatedly, and evaluate at the origin of the system. Wethereby obtain a system of equations of the form

h'kl..l = h2 ,j[^a...lr hkjl.l]+ *(4.1)

where the * is used to denote terms of lower order than those which havebeen written down explicitly. Now let P denote the operations of holdingjfixed, permuting the indices k.i.... 1 cyclically, and adding the resultingterms. Then

2 P(hk,1. ,.) = P (hMkz1....) - P (h~j1 ... 1) + *

from (4.1) or

(r + 2) hkl1.. . = 2 P (h,k,li ... I) + *. (4.2)In view of (4.1) and (4.2) it follows that if we consider the quantities

W.; his*; .k. §el..M _

to have fixed values at Q, the number of independent quantities (hj'k,l1...00is equal to the number of independent quantities (atz.. .r)Q, i.e., thenumber N(r + 1).By making use of the above result it can be shown that N(r + 1) gives

the number of arbitrary partial derivatives of the rth order of the quanti-ties h5,k at the point Q. The reader can easily work out the details of thisdemonstration.

5. When account is taken of the conditions imposed by the field equa-

834 PRoc. N. A. S.


tions, the number of arbitrary partial derivatives of the quantities hj1,kat the point Q is decreased. A lower bound to this number can, however,be deduced in the following manner. Taking the absolute derivative of theleft member of (1.1) we obtain

(Vk h;,k),l = 0. (5.1)By (3.4) four of the equations of the set (5.1) are linearly dependent onthe remaining equations (5.1) and the field equations (1.1). In fact,

(Vk hik), =

where the dots have been used to denote a linear expression in certain ofthe left members of (5.1) plus a linear expression in the quantities in theleft members of (1.1) with coefficients equal to the absolute electromagneticforces. Assuming that the h5,k and al their partial derivatives to those oforder r (. 1) inclusive have fixed values at the point Q, it follows that thenumber of partial derivatives of the quantities h.,k of order (r + 1) atQ, whose values are determined as a consequence of equations (1.1), isat most equal to

16K(4,r) - 4K(4,r - 1). (5.2)

Hence the arbitrary derivatives of the hj,k of order r + 1 at Q cannot beless than the difference of N (r + 2) and (5.2), i.e.,

16K(3,r + 1) + 8K(2,r + 1);

for r = 0 this expression likewise gives the number of arbitrary derivativesof the first order of the hk*. In our next note we shall apply the abovelower bound to the problem of constructing the general existence theoremfor the field equations.

1 These PROCEEDINGS, 16, 761-776 (1930).2 If we replace (1.1) by

Vk h,k =D

where D is the analogue of the vector of charge and current density of the classicaltheory, then (3.4) yields

Vj Dj = 0.These latter equations are the analogue of the equation of the classical theory whichshows that electric charge is conserved. We have purposely failed to introduce theinvariants D of unknown structure into our field theory as it is our wish to investigatethe extent to which the physical world can be described by equations of the type (1.1).

VOL. 16, 1930 835-

MA THEMA TICS: T. Y. THOMAS

ON THE UNIFIED FIELD THEORY. III



Communicated December 9, 1930

It is the object of the present note1 to deduce the general existencetheorem of the Cauchy-Kowalewsky type for the system of field equationsof the unified field theory. The existence theorem at which we arrivediffers, however, from the usual form of such theorems in that certainlinear combinations of partial derivatives of the h', namely, the absoluteelectromagnetic forces g1,k, occur among the arbitrary functions, ratherthan these derivatives themselves. The theory is carried sufficientlyfar to determine the characteristic surfaces of the four-dimensional world,i.e., those three-dimensional surfaces which are characterized by the factthat if taken to "bear" the data of the problem our existence theoremceases to apply, inasmuch as certain of the coefficients of the power seriesexpansions of the h. become indeterminant. These surfaces are analogousto the characteristic surfaces which arise in connection with the well-knownequation

1 2 -S-c2 bt2

which represents the propagation of a disturbance with constant velocityc; we have not, however, undertaken a discussion of the wave surfaces inthe present note.

Before proceeding to the investigation of the mathematical problemat hand I would like to make a remark concerning the general nature ofthe field theory previously presented. Einstein2 has pointed out thatthe vanishing of the invariant hj, is the condition for the four-dimensionalworld to be Euclidean, or more properly pseudo-Euclidean. From thepoint of view of our previous notes this fact has its interpretation in thestatement that the world will be pseudo-Eucidean only in the absence ofelectric and magnetic forces. This means that gravitational and electro-magnetic phenomena must be intimately related since the existence ofgravitation becomes dependent on the electromagnetic field. Thus wesecure a real physical unification of gravitation and electricity in the sensethat these concepts become but different manifestations of the same funda-mental entity- provided, of course, that the theory shows itself to betenable as a theory in agreement with experience.

Equations of preceding notes will be referred to by prefixing the serialnumber of the note to the number of the equation in question.

1. In the statement of the existence theorem which we shall deduce

48 PROC. N. A. S.


the components ht, are divided into groups which exhibit different degreesof arbitrariness; for the purpose of exposition we shall presuppose thisfact by dividing these components into suitable groups in accordancewith the followingRULE: The group G, (a = 0, 1, 2, 3) for the components h. is composed

of all components that can be formed from ha by taking a = a + 1 andi = 1, 2, 3, 4. There are four components h, in G, and we shall denotethese components by Pi, when reference is had to their division into groups.In an analogous manner the following rule will be selected to divide thecomponents h,k into groups.RULE: The group ~5m(m = 0, 1, 2) for the components hi,k is composed

of all components that can be formedfrom h,,k by taking k = m + 1 and i,j-1, 2, 3, 4 subject to the inequality j > m + 1. If there are Km componentshj,A in group 5m, then

Km = 12 - 4m.

We shall denote the components hM in group km by Kim, where I = 1,Km. It should be observed that the groups G and 05 are mutually ex-clusive; also that the elements Kim in any group @$m are independent,i.e., unrelated by equations of the type I (4.10).

2. In terms of the notation introduced in Sect. 1 equations I (4.5)can be written

-)i, 6pim+ZP2K, (2.1)

(?= 1, 2,3,4ff= 1,2,3\r =1, ..., a/

where the E denotes a homogeneous polynomial, quadratic in the Pi, andlinear in the Kim; also the inequalities ,u < , ,u < v are satisfied by theindices in these equations.Now consider equations II (2.4), i.e.,

hj,k,i + hk,j + hl,j,k = 2 [h', j hk, + ht,k h1'MJ + hm,l h7,k]. (2.2)

Putting k = 2 and I = 1 in (2.2), these equations show that hj,2,1 forj > 2 can be expressed in terms of derivatives of the K10 plus a quadraticpolynomial in the Kim. If the contravariant vector component h' $ 0and if R(P) denotes a rational function of the potentials Pi,, these latterequations can be given the form

-xl R (P) ax l+ E K2,

where I = 1.... 8 and the inequalities q . 1, v > q are satisfied. More

VOL. 17, 1931 49

50 MATHEMATICS: T. Y. THOMAS PROC. N. A. S.

generally, it can be observed that the system of equations (2.2) can bewritten

dKxm =ZR (P) K R (P) K (2.3)

a< 1, ..., |M

provided that the quantities

h and |h h2l

constructed from the hi components do not vanish; the inequalitiesq _ m, v > q are also satisfied by the indices in (2.3).

3. When account is taken of the field equations

Akhj,k = 0 (3.1)the form of (2.3) will remain unchanged, provided that an arrangement ofthe components h.9,k into new groups 6) is effected. To see how this is

bh' Wi b, bhi Whi Whbi hi'_ .k3,1 4,1 3,2 4,2 2,1 2,1 4,34,3_axl aX (,XI ax, aix, aX2 ax I aX2

123 h/ ° -hl 0 -hl -h2 0 0

124 0 hl 0 -hl -h4 -h4 0 0

134 -h4 h 0 0 0 0 -hl -h2

234 0 0 -h4 h/ 0 0 -hl -h2

j=1 hI 4 0 0 h/ h2 0 0

.j= 2 0 0 4h h4 h/ h2 0 0

j=3 4h 0 -hl 0 0 0 h4 h4j=4 0 /4 0 -hl 0 0 -hl -h2

TABLE 1

brought about we consider the four sets of equations (2.2) correspondingto j k I equal 123, 124, 134 and 234, respectively; likewise the four setsof equations obtained by letting j have the values 1, 2, 3 and 4 in (3.1).The equations so obtained can be solved for the following derivatives

dh3,l h4, d3,2 h4,2 h2, h2, h4, h4,3a)X11 axly axlp aXl - aXl' 6X2' a-Xl' a-X2"


We have, in fact, indicated the determinant formed by the coefficientsof the above derivatives in Table 1 which is so constructed that (1) eachrow corresponds to an equation of the system, namely, an equation (2.2)or (3.1) determined by the indicated values of j k I or j, respectively, and(2) the elements h, in any column are the coefficients of the derivativesat the top of this column.By an ingenious method J. M. H. Wedderburn3 has shown that the

determinant in Table 1 is equal to the product of two fourth-order con-jugate imaginary determinants. This leads to the easy reduction of thisdeterminant to the form

(ekhi'I~2h' 2} (3.2)( khk h'){|h h2 + ahlh2 1 32

I shall take this opportunity to express my thanks to Professor Wedder-burn for this contribution.

Let us now denote the components h1,,k in the four sets of derivativesin the first four columns of Table 1 by KI(7 and the components h/,k in thetwo sets of derivatives in the last four columns of the table by K72; weindicate this readily by the following schematic arrangement

K1*1 h$3,1, h4,l,h3,2h4,K1*2 h2,1, hi4,3

All independent h/4k are included in the components K*. There areK* = 16 components K7l, and K2 = 8 components K72, to introduce anotation corresponding to that employed in Sect. 2. In terms of thesedesignations it is therefore clear that if the expression (3.2) is differentfrom zero, equations (2.2) and (3.1) can be put into the form

a </ = R (P) a tq + ER (P)K*2 (3.3)

/m=1, 2 \1,=l...,K*|

a\a 1w ... M/

where the inequality v > q is satisfied by the indices in these equations.4. The system composed of (2.1) and (3.3) is harmonic in the sense

of Riquier.4 Moreover, this system is completely integrable. To seethis let us observe that if we form the conditions of integrability we obtaina system R involving the following quantities

piIj (4.la)

VOL. 17, 1931 51


j l1 2, 3, 4; A = O, 1,p 2, 3Vv, =1, 2, 3, 4; v > t;v, t >,u

K;q ~2K;qand Kgq ax a Kv. (4.1b)

1q=12; p= 1; ...,KgVv, 2, 3,4; v >; v, {> q

Now it can be shown that the components hA and also that, subject to thecondition of symmetry in their lower indices, the components A' and A'8can be assigned arbitrary values at a point Q of the space;' moreover,arbitrary values can be assigned to these quantities at Q irrespective ofthe values at Q of the quantities in (4.lb). The number of arbitraryquantities hA,A'7 and A' a is therefore equal to 136, and since these quan-tities are determined by the quantities in (4. la), it follows that there cannot be less than 136 of the quantities in (4. la) to which arbitrary valuescan be assigned at the point Q. There are also 24 arbitrary quantitiesK;p in (4. lb). Now in Note II we established the expression

16K(3, r + 1) + 8K(2, r + 1)

as a lower bound to the number of arbitrary derivatives of the (r + 1)storder of the quantities h,k. We deduce from this that no less than 184of the derivatives of the Kpq in (4.1b) can have arbitrary values at Q.Hence 136 + 24 + 184, or 344, is a lower bound to the number of arbi-trary quantities in (4.1). Now 344 is also the number of arbitrary quan-tities in (4.1) as deduced by actual calculation based on the admissibleranges of the indices of these quantities. It follows, therefore, that allquantities in (4.1) can be given arbitrary values at a point Q of spaceand hence that the above system R, giving the conditions of integrabilityof (2.1) and (3.3), must be satisfied identically. This gives the followingEXISTENcE THEoREM: Let

v (xU+1 X4) 4lm (Xm+1, X4)[i = 1, 2, 3, 4; a = ,1, 2, 3] m = 1, 2; 1 = 1, ...,Km]

denote sets of functions of the variables x'+1, .. ., x4 and xm+l, ..., x4, re-spectively, analytic in the neighborhood of the values xa = qa of their argu-ments; furthermore the via are subject to the condition that at ea = qa thecorresponding determinant Ih and the expression (3.2) do not vanish.Then there exists one, and only one, set of potentials ha1, in a system of co-ordinates xa, each function ha(x) being analytic in the neighborhood of thevalues xa = qa, which constitutes a set of integrals of the field equations (3.1)such that

52 PROC. N. A. S.


P= $i (X1,x2,X3X4) Pia = q3ia (X+1 .4..X4)K1 = * .x4)[i = 1, 2,3, 4] i= 1 2,314; = 1,2,31Im = 1 2; = 1, . ..,K1

xi = qlp ..Xl = go,lL x1 = q1, . m = qm J

5. Let us denote by Sk a k dimensional surface in the four dimensionalspace $4, and consider the three surfaces Si, S2, S3 defined by

S3: xI = p(X2, x3, x4)5x = c(x2, x3, x4)x2 = (X3, X4) (5.1)

(xI = '(x2, x3, x4)S1: = +1(x3, x4)

(3= W (X4),

where the functions so, w are analytic in the neighborhood of some setof values xs = qs of the coordinates; as so defined any surface S, is con-tained within the surface Sk provided that the inequality i< k is satisfied.We shall take the above surfaces Si, S2, S3 and the surface S4, or fourdimensional space itself, as those surfaces which "bear" the data in ourproblem. In other words, we now assign the Pi, over the surface S4-, andthe Ki. over the surface 54-m as shown by the scheme

Over S4-: Over S4-m:Pic = 1i0 (Xc+l . ..., X4) Klm = R* .(X'+ls, . .*, X4), (5.2)

[i= 1, 2, 3, 4;= 0,1, 2, 3] [m = 1, 2; 1 = 1, ..., K"]

where the v and 2* are analytic functions of the indicated surface co-ordinates.Now let qs denote the co6rdinates of a point on the surface Si and make

the transformation

yl = xi _ (, y2 = X2-, y3-= X3- ,y4 = X4-q4 (5.3)

so that the above surfaces Si, S2, S3 become

iS3: Yl 0

{ y:: (5.4)

SI: y2 =0ty3 0.

In order that the existence theorem of Sect. 4 may apply it is necessarythat the data given for the x coordinate system be determined with respectto the y coordinate system over the surfaces S. It is obvious that the

VOL. 17, 1931 53


absolute electromagnetic forces will be so determined since these quan-tities, i. e., Klm, are absolute invariants under coordinate transformations.If we denote the components h' by l with respect to the y coordinate sys-tem, then the components 1 in group G, are likewise determined over thesurface S4-,. In connection with the demonstration of this fact we laydown the following:

DEFINITION: A group of components G which is related by a transforma-tion T to the corresponding components in groups G of the same and lowerserial order, will be said to be closed with respect to the transformation T.Since the groups Go are closed with respect to the transformation (5.3)in this sense, the above statement, namely, that the components 1, ingroup G,, are determined over the surface S4-,, is immediately evident.

Let us now put

xi= _p(X2, XI, X4), P = -{J(X3, X4)= X- (X4), '' - -q4

and consider the equation

|ga~ a = ° (5.5)

in which the ge denote the ordinary contravariant components of thefundamental metric tensor, i.e.,

4

g = E ek hk hk-k=i

Under the transformation (5.3) the left member of (5.5) becomes anexpression in the l having the form of the first factor of (3.2); similarlythe left members of the two equations

laEl @E1 -_h2 h I ar aff = ° I (5.6a)Of=a=1 r2h h"2 ( = 0

and

E ha h | a = 0 (5.6b)assume a form in the components P. corresponding to the first and seconddeterminants in the last factor of the expression (3.2). Suppose that theequation (5.5) and at least one of the equations (5.6), i.e. (5.6a) or (5.6b),are not satisfied over the surface Si; then take the above-mentionedpoint on Si with coordinate q4 occurring in the transformation (5.3) tobe a point on SI at which the left member of (5.5) and the left memberof either (5.6a) or (5.6b) do not vanish. The existence theorem of Sect. 4will then apply with respect to the y coordinate system and on the basisof this application the following existence theorem can be stated.

54 PROC. N. A. S.


EXISTENCE THEOREM: Consider a set of three surfaces Si, S2, S3 definedby (5.1) and a set of analytic functions $5 and M* defined over these surfacesand the four dimensional space S4 as represented by (5.2), the functions 13being subject to the condition that (1) the corresponding determinant ha |does not vanish over S1, and (2) the equation (5.5) and either equation (5.6a)or (5.6b) are not satisfied over the surface Si. Then there exists one, andonly one, set of potentials h, in a system of co6rdinates x', the ha, (x) beinganalytic functions of the xa coordinates, which constitutes a set of integrals ofthe field equations (3.1) such that the quantities Pi, and K7m assume the as-signed values $ and k * over the surfaces S.

6. Equations (5.5) and (5.6a) satisfied over a curve S, are invariantunder coordinate transformations. It is possible, however, to make anorthogonal transformation of the fundamental vectors, i.e., a transforma-tion of the form

*h'. = a' ha, (6.1)

so that with respect to the *h potentials an equation of the type (5.6a)will not be satisfied over the curve S1.6 Equations (5.5), on the otherhand, remain invariant in form under an orthogonal transformation (6.1)of the fundamental vectors. A curve S, over which the equation (5.5)is satisfied, will be called a characteristic line with respect to the surface S3.Those surfaces S3 over which the equation (5.5) is satisfied will be calledcharacteristic surfaces; they are analogous to the characteristic surfacesdetermined by the wave equation as mentioned in the introduction tothis note, and must therefore be expected to represent the wave surfacesin the present theory.By a formal process based on the general theory of characteristic surfaces

as developed by Volterra and Hadamard, it was recently shown by Levi-Civita7 that an equation of the form (5.5) determines the characteristicsurfaces for the Einstein gravitational equations in regions free of matter;the existence theorem for the equations of Einstein, however, was notestablished by Levi-Civita. An investigation of the characteristic sur-faces determined by the field equations (3.1) will be undertaken in alater communication.

1 Previous notes on the Unified Field Theory have appeared in these PROCEEDINGS,16, 761-776 and 830-835 (1930).

2 A. Einstein, "Die Kompatibilitat der Feldgleichungen in der einheitlichen Feld-theorie," Sitzungsberichte preussischen Akad. Wissenschaften, 1930, 18-23.

3 Wedderburn's method is as follows: For simplicity the revised notation

a = h', 8 = h', = h', a =ha = h, b = h2, c = h, d =h

is introduced. Then (1) if we rearrange the rows and columns, and (2) if we multiply by-] certain of the rows and columns in Table 1, this table assumes the form represented

VOL. 17, 1931 55


by Table 2. Denoting V/E1 by i as usual, we now add i times each even column to thepreceding odd column in Table 2 and then subtract i times each odd row from the pre-

ceding even row. Table 3 shows the result obtained. By an obvious rearrangementof rows and columns Table 3 assumes the form given in Table 4, which shows that thedeterminant of the elements in Table 1 is equal, apart from algebraic sign which we havedisregarded, to the product of two-fourth order conjugate imaginary determinants.Expansion of the fourth-order determinant in the lower right hand corner of Table 4shows that this determinant has the value

(a2 _ 2 _ 72 52)[(a/ - ba) + i(c -d)].

Taking the absolute value of this expression, the expression (3.2) is obtained. Thefact that (3.2) and the determinant formed from the elements in Table 1 have the samealgebraic sign is easily observed by replacing the h? by Kronecker's 5? in (3.2) and Table 1.

0 -c 0 -y a 0 -/ 0

c 0 'y 0 0 a 0 -,Bd 0 a 0 -0 0 a 0

0 d 0 5 0- 0 aa 0 a 00 Y 0O

0 a 0 a -y 0 0 5b 0 ,B O/ O0 0 Y

0 b 0 /305O -y 0

TABLE 2

-ie -e -iy-y -y a 0 -,8 0

0 ic 0 iy 0 a 0 -,8d 0 5 0 -/3 0 a 0

O dO 5 0 -# 0 aa 0 a O iy y 5 0

0 a 0 a O -i-y O ab 0 / 0 0 i-yO y

0 b 0 3 0 5 0 -iy

TABLE 3

-ic -iy a -/ -c --Y 0 0

d 5 -/ a 0 0 0 0

a a iy 5 0 0 y 0

b ,B iy 0 0 0 y0 00 0 ic iy a -,B

0 0 0 0 d 5 -/3 a

0 0 0 0 a a -iy5

0 0 0 0 b /3 5 -ib

TABLE 4

4 C. Riquier, "De l'existence des integrales dans un systame differential quelconque,"Annales de l'ecole normale superieure, serie 3, 10 (1893), pp. 65-86; Ibid., 123-50.Two of the three conditions in Riquier's definition of the harmonic system are obviously

56 PROC. N. A. S.


satisfied for the system composed of (2.1) and (3.3). To show that the third conditionis likewise satisfied we assign to each of the quantities xa, Pi., and K7,, a single "cote"in the following manner:

x" has the "cote" -a,

Pi, has the "cote" a,

K7m has the "cote" m.

Then (a - r) and (m - a) are the "cotes" of the derivatives in the left members of theequations (2.1) and (3.3), respectively; similarly (m - a) and (q - v) are the "cotes"of derivatives in the right members of these equations. Hence

(a -T) - (,u -v) >O, (m - a) - (q -v) >0,

since 7-T 0, m - a > 0, ,-v < 0, and q-p < 0. The system composed of(2.1) and (3.3) is therefore harmonic in the sense of Riquier. See also T. Y. Thomas,"Invariantive Systems of Partial Differential Equations," Ann. Math., 31 (1930),687-713; and Ibid., 714-726.

5 For an analogous proof, see T. Y. Thomas, "A Theorem Concerning the AffineConnection," Am. J. Math., 50 (1928), 518-520.

6 We have

(*hal *hB - *h2 *) [(*h h.4 -ha hO) ak al (a)

in which the expression in brackets in the right member is assumed to be taken at apoint P on Si. Now assume that the left member of (a) is linearly dependent on theindependent polynomials of the set

4E ek ap a. ep. (b)k=1

Then we can write4

kk4

[(hkhh -hl hk) ak] al (APq ek 1) ap a' ek Akk,I k) Yx; ax a2 - -k1 k-i

where the APq are constants. Equating coefficients, we obtain

(hk h1 - ha= Aek ,

where A is a constant. Putting k = I in this last equation we see that A must vanish.Hence we can deduce that

x-a ax-P ixf5Xdxa -

In other words the rank of the matrix

aXI TX2 TX3 6X4ax' ax2 ?x3 a-X4

is less than two at P. However, this is not possible since the above matrix is seen tohave the form

VOL. 17, 1931 57

PHYSICS: E. H. KENNARD

01°1 * * 1when account is taken of the functions 4 and ,6 in Sect. 5. Hence at a point P on S,the right member of (a) is linearly independent of the independent polynomials of theset (b). It is therefore possible to choose the values of the constants a4 so that at apoint P on SI the right member of (a) and the independent polynomials (b) can haveindependent values; in particular, the constants a4 can be so chosen that the rightmember of (a) will be different from zero while all expressions (b) vanish. This provesthe statement in Sect. 6 that by making a suitable orthogonal transformation (6.1) ofthe fundamental vectors, the expression in the left member of (a) can be given a valuedifferent from zero over the curve Si.

7 T. Levi-Civita, "Caratteristiche e bicaratteristiche delle equazioni gravitazionalidi Einstein," Rendiconti Accad. Lincei, 11, 1930, pp. 1-11; Ibid., pp. 113-121. Inthis connection, see also T. Y. Thomas, "On the Existence of Integrals of Einstein'sGravitational Equations for Free Space and Their Extension to n Variables," thesePROCEEDINGS, 15 (1929), pp. 906-913.

QUANTUM-MECHANICAL MOTION OF FREE ELECTRONS INELECTROMAGNETIC FIELDS

BY E. H. KENNARD

DEPARTMENT OF PHYSICS, CORNELL UNIVERSITY

Communicated November 25, 1930

The motion of an electron in an electric or magnetic field was treatedquantum-mechanically by the author and by Darwin for the case of acertain type of wave-packet in a uniform field.1 2 Later writers3 4'5 haveconcerned themselves for the most part with the energy and the chatacter-istic functions, which are needed for radiation problems. In this paperformulas are obtained by the Ehrenfest method for the motion of thecentroid of any packet in an electromagnetic field of general type, thefield being treated in the usual way as a perturbing term in the Hamiltonianfunction for the electron. Only the non-relativistic case is considered,based upon Schrodinger's equation; for the Dirac electron the usualmethod of approximation leads to this same equation plus terms expressingthe spin effect,4 but the attempt to justify the approximation for suchpurposes encounters an extraordinary difficulty that seems to be connectedwith the "negative energy" problem and has not yet been overcome.The general result obtained below is that in a uniform electric and

magnetic field the packet-centroid moves as the electron should moveaccording to classical theory, while in non-uniform fields its accelerationis a kind of average of the classical value.

58 PROC. N. A. S-

V ,MATHEMA TICS: T. Y. THOMAS

3. Since sodium thiocyanate peptizes proteins, it should alleviate andperhaps counteract all disturbances due to coagulation of the nerve proteins.

4. It has been shown experimentally that intravenous injections ofsodium thiocyanate solutions bring rabbits out of the unconsciousness dueto ether, amytal or morphine more rapidly than is normal. The rabbitsbreathe approximately twice as rapidly under these conditions.

5. It has been shown experimentally that intravenous injections ofsodium thiocyanate solutions into rabbits can prevent death from strych-nine or histamine; and can prevent anaphylactic shock in rabbits pre-viously sensitized by subcutaneous injection of an egg-white sol.

6. Potassium thiocyanate cannot be substituted safely for sodium thio-cyanate because of the greater toxic action of potassium salts. This hasbeen known for over half a century and yet many medical men give po-tassium and sodium salts interchangeably. The only justification forthis is the as yet unproven assumption that potassium salts do not affecthuman beings as they do dogs and rabbits.

1 This work is part of the programme now being carried out at Cornell Universityunder a grant from the Heckscher Foundation for the Advancement of Research estab-lished by August Heckscher at Cornell University.

2 Eli Lilly Research Fellow.3 Bancroft and Richter, Proc. Nat. Acad. Sci., 16, 573 (1930).4 Bayliss, Principles of General Physiology, 399 (1915).5 Claude Bernard, Leqons sur les effets des substances toxiques et medicamenteuses, 5,

(1857.)6 Keith, Canadian Med. Assn. J., 16, 1171 (1926).7 Rabuteau, .8le'ments de toxicologie, 541 (1887).

ON TIHE UNIFIED FIELD THEORY. IV

By TRACY YERKES THOMAS


Communicated December 22, 1930

In this note' we shall consider the characteristic lines or bicharacteristics(Hadamard) determined by the differential equation of the characteristicsurfaces. The bicharacteristics are the so-called geodesics of zero lengthwhich give the light tracks in the Einstein theory of gravitation.2 It isshown that a characteristic surface is generated by the totality of bi-characteristics issuing from a point P of the continuum. In a system ofmetric local coordinates (see Sect. 3) this characteristic surface has the form

4E ekww = 0k=1

VOL. 17, 1931 ill

MATHEMATICS: T. Y. THOMAS P

and so appears to an observer in the local system as a spherical wavepropagated with unit velocity.While local coordinates were first mentioned in connection with the

Unified Theory in Note I, my attention has been called since writing thatnote to the fact that such coordinates are not new in mathematical litera-ture. In 1901 Poincare published a paper ("(juelques remarques sur lesgroup continus," Rend. Circ. Matem. Palermo, 15, p. 321) in which heintroduced a set of quantities of the character of affine local coordinates(see Sect. 3). More recently a similar system of variables was used byJ. A. Schouten ("Kontinuierliche Transformationsgruppen," Math. Ann.,102 (1929), p. 244). Also A. D. Michal ("Scalar Extensions of an Ortho-gonal Ennuple of Vectors," Am. Math. Monthly, 37 (1930), p. 529) hasdefined a system of geodesic coordinates Sk which become identical withthe metric local coordinates when k is allowed to become indefinitely large.

1. In Note III a three-dimensional surface

T _ Xi - p(X2, X3 X4) (1.1)

was called characteristic provided that the equation

ge -x- -cX = 0 (1.2)

was satisfied over this surface. If we substitute (1.1) into (1.2) thisequation becomes

4 4 4

F=g'1 -2 E g1P, + E Egapp = o (1.3)PJ=2 a=2 =2

where Pa =Xa ( 2, 3, 4).

The characteristic surfaces (1.1) are defined by the differential equation(1.3).

If we put

ID" =ap (a = 2, 3, 4)6Pa

dX2 dx3 dx4 xthen = =d

p2 p3 p4 4E (1.4)a=2

defines a system of curves on the characteristic surface (1.1). Throughany point P of the surface (1.1) there passes in general one and only one

112 PROC. N. A. S.


curve C of the system (1.4); these curves C are the bicharacteristics de-termined by the field equations of this theory.3

2. We have supposed the equation of the characteristic surfaces tobe solved for the coordinate xl in the previous work. Let us now takethis equation in the more general form

7r(xl, . . ., X4) = 0 (2.1)

subject to the condition that the function 7r depends explicitly on the xlcoordinate. Then from (2.1) we have

Pa (=2 3, 4) (2.2)ql

where qa (a =1,2, 3, 4).

Henceq2F = geGq qe (2.3)

Also qPa = 2gaeq (ai = 2, 3, 4);

and ql E papa) 2g'pqwhen the condition

ge'qaq = 0 (2.4)

is taken into account. The set of equations (1.4) can therefore be giventhe more symmetrical form

dx= 2g"qo. (2.5)dv

On solving equations (2.5) for the quantities qc, we obtain

ga = 1/2 ga dx. (2.6)

Now putdqa = qa dx- (2.7)

so that qa, is the second derivative of the function wr(x), and then differ-entiate (2.4) with respect to x7 where y = 2, 3, 4. This gives

- a3 + 2gaqay2 y(-a l qaql, + 2g aqlqo,)

(= 2,3,4)Making use of (2.5) and (2.7) this last system of equations becomes

VOL. 17, 1931 113


dq7+

gaD ld1 eogI qlf

dv + X qaq0 =dv(qad+ .q (2.8)

(ey = 2, 3, 4)Now differentiate the equations (2.5), which are satisfied along a bi-characteristic C on the characteristic surface (2.1), with respect to theparameter v; from the equations so obtained eliminate the q" and thederivatives of the q, by means of (2.6) and (2.8), respectively. We thusobtain a set of equations which takes the form

d2xa+ di dxy 1 ( ql dxi 1 bg, dx dxy\ dxadv2 + ~ dv dv q, aX? dv 4 axI dv dv JTdv (2.9)

where the Ip, are the Christoffel symbols derived from the componentsgap At a point P with coordinates xo on a bicharacteristic C, equations(2.5) determine the values of the derivatives of the xa with respect to theparameter v, i.e.,

/dxa\ (2.10)' dv )o 2

If for v = 0, we suppose that xa = xo and the derivatives of the xa withrespect to the parameter v have the above values (2.10) then equations(2.9) determine a solution

x = E (v) (2.11)

representing the bicharacteristic C. By a change of the parameter vequations (2.9) can be given a simpler form. In fact, under a transforma-tion of parameter (2.9) becomes

d2xa a di dx-~~+ ro-fdu2 du du

d2u+11 ag,, di dxv aql dx8 du_ Ldv2 ql 4 ax 1 dv dv bx" dv JdvJ dea (22

jdu\2 du 2.12)

kdv!

Choosing the new parameter u to be such a function of v that the bracketexpression in the right member of (2.12) vanishes along C, we have

d2e iwdxi dxy

- U2 + dud = 0 (2.13)dj2+ du du

114 PROC. N. A. S.


along the bicharacteristic C. Since the condition

dxa dx8g1du d = 0 (2.14)

is satisfied on account of (2.4) and (2.6) along C, this curve is of the typecommonly described as a geodesic of zero length. Moreover, the factthat the equation (2.1) of the characteristic surface depends explicitlyon the xl coordinate is in no way involved in (2.13) and (2.14) so thatthese latter equations must be satisfied along the bicharacteristics on anycharacteristic surface regardless of the particular form 7r(x) = 0 of theequation by which the characteristic surface is defined.

3. In Note I we introduced a system of local coordinates zi' whichpossessed the property that the equations of a path through the originof the local system had the form of the parametric equations of a Euclideanstraight line through the origin of a system of rectangular cartesian co-ordinates. As the paths are determined by the affine connection of thecontinuum it seems natural to refer to the above coordinates as affinelocal co6rdinates. The equations (2.13) to which we are led by the fore-going work suggest that* we likewise construct a system of metric localco6rdinates. defined by the metric properties of the continuum in an analo-gous manner. More precisely, a system of metric local coordinates w'will be characterized geometrically by the postulates in Sect. 2 of Note Iexcept for the fact that postulate D regarding the paths of the continuumwill be replaced by a similar postulate involving the geodesics (2.13). Bya method similar to that employed in Note I it can be shown (1) that themetric local co6rdinates wi remain unchanged when the underlying co6rdinatesxa undergo an arbitrary analytic transformation T, i.e.,

and (2) that when the fundamental vectors hi undergo an orthogonal trans-formation

* = ak Ii (3.1)

the metric local coordinates associated uith any point P likeuise undergoan orthogonal transformation, i.e., a linear homogeneous transformation,

W l ak W*

which leaves the form

VOL. 17, 1931 115


4

E ekw w (3.2)k=l

intariant. The local coordinates wi are related to the underlying co-ordinates ea by a transformation of the form

x = p - h(Pw' --s I (Pwwi - I! ajPw'w -I...2! 11-.

in which the coefficients I are determined by successive differentiation ofthe equations

62ea axflae 2VjWk)ww(WIC-kk + Zx%;) w=O

and evaluation at the origin of the local system.If we transform the components of a tensor 7i"' (x) to a system of

metric local coordinates thereby obtaining tkr..41j (w) and evaluate thecomponents t at the origin of this system, we obtain a set of absoluteinvariants Tk '. under transformations of the Xa coordinates. Moregenerally, if we differentiate the components t any number of times andevaluate at the origin of the (w) system, we obtain a set of.quantities,namely,

Tk .j..p.q = ( (-kVq w (3.3)

each of which is an absolute invariant with respect to transformations ofthe ea co6rdinates. Let us denote the components h, and gtp by B1l andCl,, respectively, in a system of metric local coordinates. Then theequations

h.m = ( 1AB ) (3.4)

and gijlk.. .m= (ak. 1 (3.5)

define sets of absolute invariants h and g under arbitrary analytic trans-formations of the xe coordinates. When the fundamental vectors undergoan orthogonal transformation (3.1) the absolute invariants (3.3), (3.4)and (3.5) transform by the tensor transformation implied by the indicesin the symbol of the invariant.The explicit formulae for any of the invariants (3.3), (3.4) and (3.5)

116 PROC. N. A. S.


as well as the identities satisfied by the invariants (3.4) and (3.5) caneasily be deduced.

4. It is a known fact that the totality of characteristics through apoint P of the continuum of a partial differenti.al equation of the firstorder, generates a characteristic surface.3 Hence an integral surface of(1.3) is generated by the totality of characteristics through P of thisequation. When the equation (1.3) is taken with reference to a systemof metric local co6rdinates and the point P is at the origin of this system,the discussion of this integral surface is of especial interest.Denoting the components gel by Clill when taken with respect to a

system of metric local coordinates the eqtiation (1.3) becomes

4 4 4Cllll- 2 E Cililri + E E C1ii- rirj = 0, (4.1)

i=2 i=2 -;=2

where~w

rj W (i = 2, 3, 4). (4.2)

The geodesics (2.13) through the origin have equations

wt= rtu (4.3)

in which the vi are arbitrary constants; hence the condition that (2.13)should be a bicharacteristic of the field equations, i.e., the condition that(2.14) be satisfied along (2.13), is

4

Z ei7i = 0. (4.4)i=1

It is seen that q 1 0O since if this were not true it would follow from (4.4)that the remaining quantities n7' would likewise vanish. The normalizingcondition n' = 1 can therefore be imposed. Equation (4.4) then becomes

(,q2)2 + (Xq3)2 + (X74)2 = 1. (4.5)

TakingX2 = sinO cos so,f3 sinO sin so, X4 = COS (4.6)

the condition (4.5) is satisfied automatically. With these values of the-'s equations (4.3) for i = 2, 3, 4 are the equations of transformationbetween a system of spherical co6rdinates 0, so, u and a system of rectangularcartesian co6rdinates W2, W3, w4.We shall have occasion to use the following identities4

Wi = Clj1w, w = -ClkjiW' (k = 2, 3, 4).

VOL. 17, 1931 117

(4.7)


From these identities we have that

1 = ClU@ Xk = -Clkjl7i (k = 2, 3, 4),if we take the point with coordinates wt to lie on one of the geodesics ofzero length passing through the origin.A set of functions w'(i = 1, 2, 3,.4) and ri(i = 2, 3, 4) depending on

three independent variables is said to constitute an integral of (4.1) inthe general sense of Sophus Lie (1) if the functions wi and r, satisfy (4.1)and (2) if

4

dwl = E rdwt. (4.8)t=2

We can readily construct an integral of (4.1) in this sense. Let us putri= 'i = 2, 3, 4), thereby defining the ri as functions of the variables0, s on account of (4.6). Equations (4.3) define the wi as functions ofthe variables 0, so, u. At the origin, i.e., when u = 0, equations (4.1) aresatisfied since these equations reduce to (4.5). Using (4.7) equations(4.1) can be put into the form

Cjijji77j= 0; (4.9)

in fact, the left members of (4.1) and (4.9) are identically equal. Now theabove functions w' and ri satisfy (4.1). This is seen from the fact (1) that(4.9) is satisfied for u = 0 and (2) that the left member of (4.9) is in-variant along a geodesic of zero length issuing from the origin. To showthat (4.8) is also satisfied we observe that the differential du is identicalwith the left member of this expression; the right member of (4.8) likewisebecomes equal to du when account is taken of (4.3) and (4.5). Hencethe above functions wt and ri of the independent variables 0, so, u con-stitute an integral of (4.1) in the sense of Lie.From (4.3) for i = 2, 3, 4 and (4.5) we have

u = + V(W2)2 + (W3)2 (W4)2.

Hence (4.3) for i = 1 gives

WI= - (W2)2 + (w3)2 + (w4)2 (4.10)

or

W1 = + V(w2)2 + (W3)2 + (W4)2 .

118 PROC. N. A. S.

(4.11)


The parametric equations (4.3) are equivalent to (4.10) for u _ 0 and to(4.11) for u > 0. By elimination of the variables 0, ic, u from the aboveintegral formed by wt and ri it follows immediately that both (4.10) and(4.11) constitute an integral of (4.1) in the ordinary sense where thequantities ri are defined by (4.2).The complete integral surface which is the graph of (4.10) and (4.11) is

a cone with center at the origin. This is illustrated in the accompanyingfigure in which the co6rdinate W4 is suppressed for the purpose of graphicalrepresentation.

If wl has a constant negative value (4.10) repre-sents a sphere with center at the origin of a systemof rectangular cartesian coordinates W2, W3, W4;similarly (4.11) is the equation of a spherical sur-face with center at the origin if wl is taken as apositive constant. Hence if wl increases algebrai-cally starting with some negative value the com-plete integral surface composed of (4.10) and (4.11)appears as a series of moving concentric spheres \which first shrink down to a point, namely, theorigin, and then expand. Introducing the co-ordinate w' as the time coordinate, the abovesurface is seen to shrink or expand with unitvelocity.

In a later communication we shall investigatethe possibility of interpreting the integral surface (4.10) or (4.11) as awave surface in the four-dimensional continuum.

1 Previous notes have appeared in these Proceedings, 16, pp. 761-776 and pp. 830-835,1930; also, 17, pp. 48-58, 1931.

2 T. Levi-Civita, "Caratteristiche e bicharatteristiche dellee quazioni gravitazionalidi Einstein," Rendiconti Accad. Lincei, 11, pp. 1-11, 1930; Ibid., pp. 113-121.

3 E. Goursat, Leqons sur l'integration der e'guations aux derivees partielles dupremier ordre. 2nd ed., 1921, pp. 184-192.

4 These identities can be derived in a manner similar to the analogous identities innormal coordinates (Riemann-Birkhoff). See 0. Veblen, "Invariants of QuadraticDifferential Forms," No. 24, Cambridge (1927), p. 96.

VOL. 17, 1931- 119


the operators which correspond to themselves under this automorphismis that the order of G is odd. This is also a necessary and sufficient condi-tion that G is the direct product ofK and C. When the order of G is of theform 2m the number of the invariants of K is the same as the number ofthe invariants of H and at least equal to the number of the invariants ofC, and the group generated by H and C contains the square of all theoperators of G and is of index 2t under G, where i represents the numberof the invariants of C. Hence the order of the cross-cut of C and H is 2'.When either H or K is cyclic then the three subgroups H, K and C must becyclic and their common cross-cut is of order 2. If this condition issatisfied and G is also decomposable into H and K there are m/2 - 1 or(m - 1)/2 such groups as mn is even or odd.

G. A. Miller, Trans. Amer. Math. Soc. 10, 472, (1909).

ON THE UNIFIED FIELD THEORY. VI



Communicated April 2, 1931

Let us suppose that the functions hI in group Go have the values 6 through-out a finite region R of the four dimensional continuum; these specialvalues of the hI can obviously be imposed, at least throughout a sufficientlysmall region of the continuum, by a suitable coordinate transformation.A coordinate system for which the h' have the above special values willbe referred to as a cannonical co6rdinate system. Interpreting the co-ordinate xl as the time t let us now assume that a disturbance is producedat a time t = 0 throughout a closed region Ro of space.' After a timet = p(X2, X3, X4), the effect of this disturbance will be felt at a point(X2, X3, X4) outside the region X?. For a definite value of t the equationSp(X2, X3, X4) = t will represent the surface of the space (x2, X3, X4) which,at the instant t, separates the region affected from the region unaffectedby the disturbance. If we assume that the values of the h, and their de-rivatives vary in a continuous manner when we pass from one region tothe other, then it follows from the result in sect. 8 of Note V under thehypothesis of canonical coordinates that the surface t -~p(X2, X3, X4) = 0must be a characteristic surface of the field equations. Wave surfacesare thus identified with the characteristic surfaces.Now assume the expression for the element of distance in the region R

in the form

V'OL. 17, 1931 325


4 4

ds2= dt2 - E E ya dxa dx (a)a=2 p= 2

where we have put t = xl and -y,Xo = - g,, in terms of the notation usedin previous notes. We shall refer for the moment to the co6rdinates forwhich the form (a) of the element of distance is valid as Gaussian co-ordinates in conformity with the terminology used by Hilbert.2 It willbe assumed likewise that the co6rdinates of a system of Gaussian co-ordinates have the significance of coordinates of time and space as impliedby the above expression for the element of distance. Use has been madeof Gaussian coordinates in this sense in recent work on cosomology.3Owing to the special form of the element of distance (a) in Gaussian

coordinates the function s must, therefore, satisfy the equation4 4 a

paS aS

a=2 P=2 be Z5

over the surface (p = t where ya0 is the cofactor of the element -ya, in thedeterminant divided by this determinant. The equation

(p(X2, X3, X4) = const. (b)

therefore represents a family of parallel surfaces, and hence the successivepositions of the wave frontform a family of parallel surfaces which are propa-gated with unit velocity.The hypersurface formed from the totality of bicharacteristics issuing

from a point P is a characteristic surface (see Sect. 4 of Note IV). Letus denote the sheet of this hypersurface corresponding to IV (4.11) byS; let us also denote the surface of intersection of S with the hypersurfacet = const. by S*. It can be shown that from the given position Ro of thewave at time t = 0, the position of the wave (b) at any later time t can beobtained by taking the envelope of the surfaces S* corresponding to the differentpoints of Ro.4 In other words Huygen's principle can be applied for theconstruction of the wave fronts. A discussion of wave surfaces in generalfrom this point of view has lately been made on the basis of contact trans-formations by L. P. Eisenhart.5,

In particular if the surface Ro reduces to a point P the correspondingwave surface (b) assumes the form

(w2)2 + (w3)2 + (w4)2 = const. (c)

in a system of metric local coordinates with origin at P. The invarianceof (c) under orthogonal rotations of the fundamental vectors he, i.e. Lorentztransformations of the metric local coordinates, has its interpretation inthe experimental fact that the velocity of light is independent of themotion of the source or the observer.

326 PROC. N. A. S.


The orthogonal trajectories of the family of wave surfaces (b) arelight rays in the usual sense; these curves, however, are geodesics of zerolength6 when considered with respect to the Gaussian ccordinates (t, x2,x3, x4). This establishes the geodesics of zero length or bicharacteristicsof the field equations as the light tracks in the unified field theory.A treatment of the problems of optics can be made on the basis of the

above theory of gravitational and electromagnetic waves. In fact, astep in this direction has recently been taken by Synge and McConnell7who have deduced formulae for the angle of reflection and refraction oflight from a moving surface by a method adaptable to the point of viewof the present theory. Other interesting problems in geometrical andphysical optics await investigation from this standpoint.

1. We shall add to this note a brief discussion of the problem of thedetermination of the structure of the continuum by the assignment ofdata in a system of local coordinates. Let us define a three-dimensionalvariety S3 and a two-dimensional variety S2 in a system of affine localcoordinates zl by

4)(Z1l ... Z4) = 0

S3:4.(Z', ...,Z4) =O,2 J'l(1 .,4*(ZI, . ,z4) =0

where the functions 4) and ' are assumed to be analytic in the neighborhoodof the origin. We shall suppose that 4) and I are independent as functionsof z' and z2; this causes no loss of generality as it involves at most a re-lettering of the coordinate axes. Now put

XI = ¢D(Z), X2 = '(z), X3 = Z3, X4 = Z4. (1.)

This defines an analytic transformation of coordinates which possessesa unique inverse in the neighborhood of the origin of the local system.We shall say that the origin of the local coordinate system is a character-

istic point with respect to the hypersurface S3 if the expression

(aq,)2 --2 ()2 - 2

VaZ1) VYZ2) V5Z3) V5Z4)vanishes at the origin of these co6rdinates. Our discussion will be dividedinto two parts: (A) the case for which the origin is a characteristic point,and (B) the case for which the origin is not a characteristic point withrespect to the hypersurface S3. We shall refer to these two cases ashypothesis (A) and hypothesis (B), respectively.Now suppose that the quantities Kll defined in Note III are given over

S3 and the K72 over S2; this can, in fact, be accomplished by taking theK1, to be analytic functions of the coordinates (x2, X3, X4) and the K72as analytic functions of the coordinates (X3, X4), i.e., analytic in the neighbor-

VOL. 17, 1931 327


hood of the values xo corresponding to the origin of the local system.Now observe that the coefficients H of the power series expansionsI (2.10) are uniquely determined on account of (1.1). Hence, if we knowthe derivatives at x' of the h. and Klm up to the order r inclusive, thederivatives of the h' of order r + 1 at x4 will be determined by II (4.2).Differentiation of III (3.3) under hypothesis (A) leads to the determinationat x' of the derivatives of the Klm of order r + 1, etc. By this processwe obtain uniquely determined convergent power series expansions8 ofthe hX (x) in the neighborhood of the point x4; transformation by meansof (1.1) gives the functions h' in the affine local coordinate system. As amatter of fact the functions h, are likewise uniquely determined in thesystem of metric local coordinates having its origin coincident with theorigin of the affine local coordinates and we, therefore, arrive at thefollowingEXISTENCE THEOREM. The specification of the functions K71 and Ki,

over the varieties S3 and S2, respectively, as analytic functions of the surfacecoordinates determines uniquely a set of analytic integrals h, of the fieldequations in a system of affine (or metric) local coordinates having its originat a point P on S2 not characteristic uwth respect to S3.

In particular we can take 4 = z' and T = Z2. We then arrive at atheorem which on the point of view of our previous notes can be statedroughly as follows: The specification at a particular instant t = 0 of theabsolute electromagnetic forces K7m throughout a small region Ro determinesthe structure of the space-time continuum in the neighborhood of Ro.

2. A modification of the above existence theorem is necessary whenthe hypothesis (B) is adopted. We can suppose that the functions 4 andT satisfy the normal conditions C1 and C2 at a point P as explained inSect. 1 of Note V. Taking the point P as the origin of a system of affinelocal coordinates z' we shall then have

az1 az1 6Z2 OJz2 OZ3 CZ3 6z4 6z4 0Z2/ (2.1)

6ZI 6Z2 OZ2 OZ2 6Z 6Za4 6Z4 OZ3J

at the origin of these coordinates. At the point x' of the (x) coordinatesystem defined by (1.1) the inequalities V (1.7) will, therefore, be satisfied.Call S* the surface defined by T = 0 and consider the functions U and Vdefined in Sect. 4 of Note V. If we specify the functions U over thehypersurfaces S3 and S3 and the functions V over the surface S2 as analyticfunctions of the surface coordinates, we can uniquely determine the suc-cessive coefficients of the power series expansions8 of the components

PROC. N. A. S.328


h, about the point x4 by applying the method of the preceding section tothe equations V (4.1), V (4.2) and V (4.3). This gives the followingEXISTENCE THEOREM. The specification of the functions U over the

varieties S3 and S* and the functions V over the variety S2 as analyticfunctionsof the surface co6rdinates determines uniquely a set of analytic integrals h,of the field equations in a system of affine (or metric) local co6rdinates havingits origin at a point P on S2 characteristic with respect to S3 and such thatat P the inequalities (2.1) are satisfied.As mentioned in Note V the normal conditions (2.1) place no restriction

on the hypersurface S3.1 Cp. E. Goursat, Cours d'analyse mathematique, 3, 94 (1927).2 D. Hilbert, "Die Grundlagen der Physik," Math. Ann., 92, 1-32.(1924).3See, for example, H. P. Robertson, "On the Foundations of Relativistic Cosmology,"

Proc. Nat. Acad. Sci., 15 (1929), 822-829, where other references are also given; inparticular see footnote 4 to this paper.

4 Cp. J. Hadamard, Lecons sur la propagation des ondes, Hermann, 290 (1903).5 L. P. Eisenhart, "Contact Transformations," Ann. of Math., 30, 211-249 (1929).6 Proof of this statement can be based on the discussion in Sect. 2 of Note IV.7J. L. Synge and A. J. McConnell, "Riemannian Null Geometry," Phil. Mag., 5,

7th Series, 241-263 (1928).9 The proof of convergence involved here will possibly be given later in a compre-

hensive exposition of the present theory.

VOL. 17, 1931 329


ON THE UNIFIED FIELD THEORY. V

BY TRAcY YERxEs THOMAS

DEPARTMENT OP MATHEMATICS, PRINCETON UMVERSI

Communicated March 12, 1931

The general existence theorem established in Note III ceases to applywhen the data of the problem is ascribed over a characteristic surface Ss.In treating this important exceptional case we arrive at an existencetheorem which is, roughly speaking, analogous to the existence theoremfor the ordinary wave-equation when the data is given over a characteristicsurface and which also bears certain resemblances to a theorem of Hada-mard.' This theorem is capable of dynamical application. It followsfrom it that there exists 'an infinite number of sets of integrals h' of thefield equations such that each set of integrals and their first derivativesassume the same values over a characteristic surface S3; in fact, the condi-tions under which these integrals exist are such as to preclude the oc-currence of more than a single set of integrals in case the surface ,S3 isnot a characteristic. This result leads to the interpretation of the char-acteristic surfaces as gravitational and electro-magnetic wave surfacesin the four-dimensional continuum. A brief discussion of these wave sur-faces will be given in Note VI.

1. Let us denote for the moment by Xl, ..., X4 the coordinates ofthe continuum; let us also denote by Hi the contravariant componentsof the ftmdamental vectors and by G'O the contravariant components ofthe fundamental metric tensor. A characteristic surface S3 is then athree-dimensional surface cI(X) = 0 such that over it the equation

Gaxe = (1.1)is satisfied. Suppose that P is not a singular point on the hypersurfaceb = 0, i.e., all first derivatives of the function 4) do not vanish at P, andconsider another surface S. defined by L(X) = 0 which passes throughthe point P. This latter surface will be subjected to the following condi-tion, the reason for which will later appear.

C1. The inequality

Gap axaP

X-C'* ;~(1.2)

holds at the point P.

The above condition is independent of the orientation of the fundamentalvectors. It is therefore possible, and we shall later find it expedient, to

VOL. 17, 1931 199


orientate the vector configurations throughout the continuum in thefollowing manner.

C2. The vector configurations are so oriented that the inequalities

z2 as t O (1.3)

0 (1.4)aEl 1 |Ha HI OXa OX"t°(14

H2H 2

4El ;1E1 H4H'H" 6)a ax °(1.5)

are satisfied at the point P2.

We shall refer to the above conditions C1 and C2 as normal conditions.It is evident that C2 might also be regarded as giving conditions on thesurfaces S3 and S*. However, the above point of view has been preferredas it does not involve an apparent restriction on the characteristic surfaceS3.

In consequence of condition C1 it follows that the rank of the matrix

aXl aX2 6X8 6X4(16

6X1 6X2 6X3 6X4must be two at the point P. In fact, if the rank of this matrix were lessthan two, condition C1 would fail to be satisfied. It is therefore possibleto choose functions Q(X) and 2(X) such that the jacobian of the coordinatetransformation

Xi = D(X), X2 = I(X), X3 = Q(X), X4 = :(X)will not vanish at P; this means that the above transformation possessesan inverse throughout the neighborhood of the point P. Under thistransformation the equation of the characteristic surface S3 assumes theform xl = 0, while the equation of the auxiliary surface S3 becomes x2 = 0.Denoting the components of the fundamental vectors and the fundamentalmetric tensor in the (x) coordinate system by the usual designation ofh and g, respectively, we have from (1.1) that the contravariant component

= 0 over S3; also by Cl we have that gl2 00 at P. Moreover, thecondition C2 results in the fact that the inequalities

h1 > 0, hlh _- hlh hl0,h14 -h4h1 t 0 (1.7)are satisfied by the contravariant components h, at the point P. In

PROC. N. Al.|S200


treating the problem of the determination of integrals of the field equationsby the specification of data over a characteristic surface S3 we shall assumein the following work the normalization of this problem afforded by theselection of the (x) coordinate system and the assertion of the conditionsC1 and C2.

2. We shall find it expedient in making certain calculations to introducea slight change in notation. Let us put

ca = 14,= h, -y = h1, 5 = h

a = h2, b = h2, c = h1, d = 12for the contravariant components hK; also

W= aC2-2_ 72 62, W* = aa-Bb--yc-Ad.The two-rowed determinants formed from the matrix

a a 7 1a b c d|

will also enter into our calculations so that we shall use the abbreviations

A = ab-p#a, B = ac- ya, C= ad-baD = Pc-yb E=1d-bb, F= yd-Sc.

The above quantity W is nothing more than the contravariant componentg11 and hence vanishes over the characteristic surface S3; also W* standsfor the contravariant component g12 So that W* t 0 at P. Finally theinequalities P Z 0, A Z 0 and F Z 0 are equivalent to (1.7).

3. Consider the matrix represented by Table 1. Each row in table 1corresponds to an equation III (2.2) or III (3.1) determined by the indi-cated values of j, k, I or j, respectively, and the elements in any columnare the coefficients of the derivatives at the top of this column; moreprecisely each row in Table 1 corresponds to a set of four equations butthe above terminology is convenient and has been used throughout thisnote. It is necessarv to deduce a number of lemmas regarding certainmatrices and determinants formed from the elements in Table 1.

Let us denote by M1 the matrix formed from the elements in the firstsix columns in Table 1. The fourth order determinant in the upper left-hand corner of M1 will be denoted by J. More generally a determinantconstructed from the elements in rows 1, m, . . ., n and columns p, q, .., rin Table 1 will be designated by the symbol

m ...nP ... r

Let us consider the eight determinants of order five in M1 which are formed

VOL. 17, 1931 201


by bordering J in all possible manners. Expansions of these deter-minants show that

12345 12346 -a7W, 1|2345123466 a2aw.12345 12346 12346 12345

12347 |12348 _ 2 W 3477 2348

12345 12346 ' 12346- 12345W

Since W = 0 at a point P on a characteristic surface S3 it follows by a

theorem in algebra3 that if J 0 0 at P the rank of the matrix Ml is fourat P. We can, in fact, easily prove a more specific result, namely, thatevery determinant of order five in Ml contains W as a factor.4 As we shalllater have use of this result we state it as the following

jkl b.hI ah' ?ht a ahsaht ahI ahI ahI ahI ahZ A'2,14,2 4,1 3,2 4,2 3,1 4,1 8,2 4,2 3,2 2,14,l

a~ 1a ax, 6x1a 15alx' ax aX2 ax' aX2 (X2 a C2 aX2

123 -Y 0 0 -a 0 13 0 -a 0 b -c 0

=4 O-_Y a 0O-f0 a 0 -b 0 0 -c

j 3 0 a 0 0 a 0 -b 0 a 0 d

124 -a 0 i 0 -a 0 b 0 -a 0 -d 0

134 0 -a y 0 0 _ c 0 0 -d 0 -a

J= 2 a 0 0 'y - 0 0 c d 0 a 0

234 0 -0 0 -5 'y 0 0 -d c 0 0 -b

j= 1 3 °0 5 0 0 y d 0 0 c b 0

TABLE 1

LEMMA I. Every determinant of order five of the matrix M1 has the formWR(a, 9, -y, 5), where R (a, j,P'y, 5) denotes a polygnomial in the variabksindicated.

Let us denote by M.l the matrix determined by the elements in thefirst four rows and first six columns in Table 1. The following lemmacan then be proved.LEMMA II. The matrix M2 is of rank four at a point P of a characteristic

surface.This lemma shows in particular that the rank of M1 is four at P. To

prove Lemma II we consider the determinant

l 2 3 4 (a 23 4 5 61

PROC. N. A. S.202


contained in M2. The assumption that the above determinant vanishesgives a2 = B2 and since W = OatP it follows that -y = 5 = Oat P. HenceF = 0 at P contrary to condition C2 in Sect. 1. This proves Lemma II.By calculation we have that5

123 4 5 6 7 8 -4A2P + (B2 + C2-D2-E2)2.3 4 5 6 7 8 9 lOj

Since A t 0 and F t 0 we obtain the result stated in the followingLEMMA III. The inequality

11234567 8 >03 4 5 6 7 8 9 10

is satisfied at the point P.4. We shall apply the symbols U and V to represent components

h1k as indicated by the scheme

U h',1, h',3; V hh44,, h3,2, h4,2,l4,' .

Equations III (2.2) and III (3.1) can now be solved for first derivativesof the V with respect to xl and x2 in consequence of the fact that thedeterminant

1234567 83 4 5 6 7 8 9 10

does not vanish at P by Lemma III. This solution gives

av ER(h) au + E R(h) av + * (4.1)

(a = 1, 2, 3, 4; = 3, 4)

a 1 W= WER

-+ EZ R +R E

-+ *B (4.2)

(a = 2, 3, 4; # = 3, 4)

where R denotes a rational function of the h, and the* as usual denotesterms of lower order than those which have been written down explicitly.The occurrence of the quantity W as a factor in the first term of the rightmember of (4.2) follows as a result of Lemma III; this fact is essentialin our later work.By multiplying the four equations corresponding to the first four rows

in Table 1 by suitably chosen quantities p, q, r, s and adding to the equationcorresponding to the seventh row in Table 1 it is possible to obtain anequation in which the coefficients of the derivatives of the componentsU and V with respect to xl will vanish at P. This follows from Lemmas

VOL. 17, 1931 203


I and II. Similarly it is possible to multiply the four equations corre-sponding to the first four rows in Table 1 by quantities p, q, r, s, and addto the equation corresponding to the last row in Table 1 so as to obtainan equation in which the coefficients of the above derivatives will likewisevanish at P. We can, in fact, construct in an obvious manner a systemof linear equations in the unknown quiantities p, q, r, s or p, q, r, s withthe non-vanishing determinant

,,=11 2 3 4|3 4 .5 61

and solve by Cramer's rule. This gives

P = 2 2'et,q2-2r a2C (2' S a2 2

-(3 - a _ -a 6P aa_(2'2 a2_ 2'r a2- #2's a2( 2

In the two equations so obtained the coefficients of the derivatives inin the last six columns of Table 1, i.e., the derivatives of the quantitiesU and V with respect to x2, will be given by the matrix

u v w x y z

-v u -x w - z y

ihA uE-oaC aB- #Din which U = a2 - (32' a2 - (32 ' W = a2 - (2

-6A -aF yD+SE-bWX a2 - 22Y2 _2 Z a2 (32

Let us call the four equations corresponding to the first four rows in Table

0 -a 0 a -ey 0

a 0 -3 00 - Y

0 - 0 0 a 0 a

a 0 -a 0 -6 0

U V W X y Z

-v u -x w -z y

TABLE 2

which does not vanish at

1 the system L1 and the two equations whichwe have constructed by the above process, thesystem L2. Now differentiate L1 with respectto x2 and L2 with respect to xl. On combin-ing the resulting equations we have a systemL3 of six equations which can be solved for thesecond derivatives with respect to xl, x2 ofthe quantities U and V. Table 2 indicatesthe determinant of the coefficients of thesederivatives in L3. Expanding5 this determin-ant and neglecting additive terms containingW we obtain the simple quantity (2(3W*)2P in consequence of the normal conditions

imposed in Sect. 1; this fact permits the unique determination of the

204 PROC. N. A. S.

VOL. 17, 1931 MATHEMATICS: T. Y. THOMAS 205

above second derivatives and leads to a system of equations L3 of theform

O2U6XI-X2( 62U O2U 62V

=WF3R x ER a + R + *x (4 3)62V (xl(- xla-xx2l (a6* l ifI = 1, 2)

It is to be observed that the first term in the right member of these equa-tions contains W as a factor and that terms involving second derivativesof V with respect to xl do not appear.

5. Let us write I (4.5) or III (2.1) in the form

-2 +EZ RU+>ERV (5.1a)

- ax + ERU+RV614 614 (5. lb)

3x2 2 + E RU +ZRV)

4 + E RU + E RV

2 + E RU + E RV (5.1c)

6h4 614

Other equations of this type likewise result from the condition W = 0over the surface S3; in fact we have

Eglh' a' = O(a = 2,3,4) (5.2)~2 6

over S3. Now a * O at P since a =O would give =y = a = O atP and we would have a contradiction with the normal conditions imposedin Sect. 1. Making use of this fact and also the fact that W* 0 at Pequations (5.2) can be solved for the derivatives of the covariant com-ponent h' with respect to X2, X3 and X4. Hence (5.2) can be given theform

621 _62 _ 6 36X2 - ***''X3 - =*...-** (53


where the dots denote terms of the sort occurring in the right membersof (5.1) with the exception of the derivatives in the left members of (5.3).It is to be borne in mind that equations (5.3) hold only over the surfaceS3 and must not, therefore, be differentiated with respect to the coordinatexl in the process of finding derivatives of the component h' which we shalllater employ. This circumstance, however, causes no difficulty sincesuch differentiations can be made on the equation (5.1a) which containsthe derivative of h1 with respect to xl in its left member.

6. Suppose that each component U is defined over the surfaces S3and S3* as an analytic function of.the surface coordinates. We representthis by writing

U = J(X2, X3, X4) for xl = 0 (6.1)U = K(x1,x3,x4) for x2 = o

where it is to be understood that over the two-dimensional surface Sdefined by xl = x2 = 0 the functions J and K are identical, i.e.,

J(0, x3, x4) = K(0, x3, X4).

Over the surface S2 we shall ascribe the components V as indicated bythe equation

V = L(x3, X4) for x1 = = 0. (6.2)

Similarly the components h' will be defined throughout the four-dimen-sional continuum, over the above surfaces S3, S2 and along the curve Sigiven by x1 = =X3 = 0 as shown by the following equations

h- PS(X'1 x2, x3, x4) h = Qi(X2, X3, X4)[i=1, 2, 3, 4]

-XI = °__ . (6.3)

h= R(x3, x4) hi SI(X4)

-XI X2 = O j X1 =X2 = X3 = O0

A value (h')o will be assigned the component h1 at the point P, which isnow taken as the origin of the (x) coordinate system, such that the con-ditioiis W = 0 and ,B 0 are satisfied at P. It will, moreover, be assumedthat the values of the h, in (6.3) are such that the remainder of the normalconditions imposed in Sect. 1, namely, W* t 0,4 t 0 aiid F t 0 holdlikewise at the point P; there is also the condition, which is a consequenceof the underlying postulates of the space of distant parallelism, that thedeterminant I jiI t 0 at P.

206 PROC. N. A. S.


The specification of the above data is sufficient to determine the powerseries expansions of the components hs in the neighborhood of the pointP. We observe, for example, that the quantities in the right membersof (4.1) and (4.2) are known at P so that the left members of these equa-tions are determined. Likewise the left members of (5.1) and (5.3) aredetermined at P since all quantities occurring in the right members ofthese equations are known from the assignment of the above data. Inother words, all first derivatives of the quantities U, V and hs are de-termined at the point P. To assist in the calculation of the higher de-rivatives of these quantities we lay down certain rules of anteriority whichindicate the order in which these derivatives are to be determined. If21 and e3 denote derivatives of U or V we shall say that 2I is anterior toe when one of the following conditions is satisfied:

(1) 21 is of lower order than !;(2) 2 is of the same order as e8 but involves fewer differentiations

with respect to xl;(3) 21 is of the same order as Q, involves the same number of differen-

tiations with respect to x1, but fewer differentiations with respect to X2;(4) 1 is of the same order as !8, involves the same number of differ-

entiations with respect to xl and the same number of differentiationswith respect to X2.2, but 21 is a derivative of a quantity U whereas 58 isa derivative of a quantity V.

The rules of anteriority for the derivatives of the components h1 canbe made on the basis of the idea of the "cote" which we have previouslvused (footnote 4 to Note III). Let us assign "cotes" to the ha and thecoordinates xa as follows:

Xa has "cote" -ahi has "cote" 5ha has "cote" a

(i * 1 ifa= 2)

Then if G and Z denote derivatives of components ht we shall say thatE is anterior to Z if:

(1) G is of lower order than Z;(2) E is of the same order as Z but the "cote" of Q is algebraically

less than that of Z6Finally we shall say that 21 is anterior to S if 21 is of lower order than

: and conversely that z is anterior to 21 if G is of lower order than W.Now differentiate one of the equations (4.1), (4.2) or (4.3) any number

of times with respect to the co3rdinates x" and evaluate at the point P.The resulting equation will then express the derivative in its left member

VOL. 17, 1931 207


as a sum of polynomials in anterior derivatives. In case we are dealingwith an equation (4.2) the vanishing of the factor W at P prevents theoccurrence in the right member of (4.2) of derivatives which are not an-terior to the derivative in the left member of this equation; a similarremark applies to an equation (4.3). If we differentiate one of the equa-tions (5.3) any number of times with respect to coordinates x2, x3, x4 andevaluate at P the derivatives in the right member of the resulting equationwill be anterior to the derivative in the left member. This will likewisebe the case if we differentiate one of the equations (5.1) any number oftimes with respect to the coordinates ea except in the case of an equation(5.1b) which involves in its right member a derivative of the componenthI; in this case, however, the derivative of the component h1 resultingfrom differentiation of the equation (5. ib) can be eliminated by an obvioussubstitution so as to obtain an equation in which the right member con-tains only derivatives anterior to the derivative in its left member. Fromthe above considerations it is obvious that any derivative of U, V or h,can be determined at P when the values are known at P of all derivativesanterior to the derivative in question. It follows, therefore, that thepower series expansions of the components hs about the point P will befully determined by the value (h')o of the component 14 at P and thespecification of data given by (6.1), (6.2) and (6.3) in accordance with thecondition 14 t 0 at P and the normal conditions in Sect. 1.

7. It was shown in Note II that the expression

16K(3, r + 1) + 8K(2, r + 1) (7.1)

constituted a lower bound to the number of arbitrary derivatives of thequantities U and V of the (r + I)st order; also the method explained inSect. 4 of Note III enables us to say that there cannot be less than4K(4, r + 2) derivatives of the h1, of order (r + 1) to which arbitrary valuescan be assigned at the point P when the condition W = 0 over the surface83 is disregarded. When this latter condition is taken into considerationit follows, therefore, that the difference between 4K(4, r + 2) andK(3, r + 1), or

4K(4, r + 1) + 3K(3, r + 1) + 4K(2, r + 1) + 4 (7.2)

is a lower bound to the number of derivatives of the h1, of the (r + 1)storder to which arbitrary values can be assigned at the point P.

If we differentiate r times the equations (4.1), (4.2), (5.1), (5.3) andr - 1 times the equations (4.3), and then form the conditions of integra-bility of the resulting equations we obtain a system M involving the U,V and h1 and the derivatives of these quantities to the order (r + 1) atmost. Now the data specified by (6.3) shows that the number of arbitrary

208 PROC. N. A. S.


derivatives of ht, of the (r + 1)st order at P is at most equal to (7.2);the number of arbitrary derivatives of U and V of the (r + 1)st order iseasily seen to be equal to (7.1) in consequence of (6.1) and (6.2). If,therefore, one of the equations of M is not satisfied identically and if,moreover, this equation involves at least one of the derivatives of thecomponents U, V or h. of order (r + 1) we shall be led to a contradictionwith the fact that (7.1) or (7.2) constitutes a lower bound to the numberof these derivatives which are arbitrary at the point P; in case an equationof M does not involve a derivative of U, V, or h, of order (r + 1) and yetis not satisfied identically we shall have a similar contradiction with regardto the lower bound for derivatives of order less than (r + 1). It follows,therefore, that all conditions of integrability involved in the determina-tion of the power series expansions of the h, must be satisfied and, hence,that these expansions are unique. As the power series expansions of theh, can be shown to converge within a sufficiently small neighborhood atthe point P we arrive at the following7EXISTENCE THEOREM. Let us specify the functions U over the surfaces

S3 and S. and also the functions V over the surface S2 as arbitrary analyticfunctions of the surface co6rdinates represented by equations (6.1) and (6.2),,respectively. Moreover, let us specify the value of the component h' at P,i.e., xa = 0, and also the values of the remaining components ha as arbitraryanalytic functions in accordance with (6.3), the above values being takensubject to the following conditions:

W = 0, IhI t O, , t OW* 0, A t 0, F O0

at the point P. Then there exists one, and only one, set offunctions h, givenby convergent power series expansions about the point P, which constitutesa set of integrals of the field equations III (3.1) and for which the surfaceS3 is a characteristic.

It should be observed that the normal conditions incorporated in thestatement of the above existence theorem have been imposed merely toavoid the ambiguity which would otherwise occur in the selection ofcertain non-vanishing determinants formed from the coefficients of certainof the previous equations; these conditions in no way restrict the char-acteristic surface S3.

8. Suppose that we confine our differentiations of the equations(4.1), (4.2), (4.3), (5.1) and (5.3) in the process of determining the powerseries expansions of the functions h, about the point P to differentiationsinvolving the indices X2, X3, X4 alone; this is evidently possible from therules of anteriority in Sect. 6. We then arrive at a determination of thequantities h, and their first derivatives over the characteristic surface

VOL. 17, 1931 209


S3. Now if we consider the functions K in (6.1) which we can supposeto be of the form

(P3(X1)3 + (4(X1)4 +

we see that this determination of the h, and their first derivatives overS3 is independent of the (Pk. Different selections of the functions Pk(X3, X4)will give different sets of integrals ht of the field equations and we arriveat the fact that there exists an infinite number of sets of integrals ha of thefield equations such that each set of integrals and their first derivatives assumethe same values over a characteristic surface S3.The above result can be extended in the following manner: Let us

supplement the above process by allowing a single differentiation withrespect to the coordinate xl. Then the quantities hs as well as theirfirst and second derivatives will be determined over S3 independently ofthe functions O4, V5, . . . and there is an analogous result when we restrictourselves to p(>l) differentiations with respect to the coordinate xl.Let us say that two sets of integrals ht of the field equations have contactof order C over a characteristic surface S3 if the two sets of functions h ,

and all their derivatives to those of order C itclusive, but not derivativesof the (C + 1)st order, assume the same values over S3. We can thensay that there exists an infinite number of sets of integrals h, of the fieldequations having contact of order C(2. 1) with one another over a char-acteristic surface S3. It is important to notice that if the surface S3 werenot a characteristic the functional data common to each of these sets ofintegrals would be sufficient to give a unique determination of a set ofintegrals h, of the field equations.

I J. Hadamard, Lecons sur la Propagation des Ondes, Hermann (1903), pp. 296-310.2 See footnote 6 to Note III.3 Bocher, Introduction to Higher Algebra, MacMillan, p. 54 (1929).4 Let Q denote any determinant of order five in Ml. Assuming J * 0, we have

Q = 0 whenever W = 0 by the result in the text. Hence JQ = 0 for W = 0 withoutrestricting J to non-vanishing values. Hence JQ WT. Since the polynomial Wis irreducible it follows from the last equation that either J or Q must involve W asa factor. But J is equal to a252 + 62yl2 so that W is not a factor.of J; hence W is afactor of Q. While the above discussion is sufficiently general to admit the possibilityof J having zero values, the vanishing of J at the point P would, as a matter of fact,be in contradiction to the normal conditions in Sect. 1.

I See footnote 3 to Note III.6 A "cote" of a derivative is the integer obtained by adding to the "cote" of the

function which is differentiated, the "cotes" of all the variables of differentiation,distinct or not.

7 The proof of convergence involved here will be given later in a comprehensiveexposition of the present theory.

210 PROC. N. A. S.

unified field theory - 4

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