Condition that an Electron Describe a Geodesic

Citation preview

103

MA THEMA TICS: A. BRAMLEY

VoL. 10, 1924

and there is no torsion. Each pair of opposite faces determines a nonbounding 2-circuit, each set of parallel edges, a non-bounding 1-circuit. In place of J22, however; we obtain the matrix 1 5~~~2 32 s2 Si ~~~~~~'3 t

52 ~

(J22)

S2

-0

S5 S3

S2

-Si

0

This matrix is evidently not reducible to J22, since no linear combination of the three rows with integer coefficients not all zero can be made to vanish. A fuller report on the above matrices and on others obtained by studying the intersection of more than two systems of circuits will be made elsewhere. In particular I.shall prove that the matrix J,, may be replaced by a 3dimensional array with integer elements. The possibility of finding some such invariants as the ones obtained in this note was suggested a long time ago by Professor Veblen.

CONDITlON THA T AN ELECTRON DESCRIBE A GEODESIC By ARTHUR BRAmLny DEPARTMENT OF PHYSICS, PRINCETON UNIVERSITY

Communicated January 17, 1924

Consider a Riemann space, whose linear element is written in the form ds2 = gikdxidxk The equations of motion of an electron' are

+ rap d..

lAgij

fX) = (Sk)k

where j, is the density and 1 2'i7kgak + bg3k _ ga\ ik

If Maxwell's equations are

bkJA

kv

and

=", JIA where

j,,

(dx' dx2 dx3 dx4\

(1)

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MA THEMA TICS: A. BRA MLE Y

P N. A. S. PROC.

and p is the charge density, then (2)

Sik

=

-FaiF+F Fa,sFa8-ik

The equation

Sk = T,k + A,k represents the most general law of conservation consistent with relativity mechanics, where T k is the material energy tensor, and where 4k is a tensor which represents the effect of the non-Maxwellian forces operating on the electron.2 The set of equations (Tij + Aij - Phgij)X'h = 0 define an orthogonal ennuple, whose directions are by definition the principle directions of the tensor3 Tij + Aij. Then we have gijXVhX'k = 6hk k= h. {hk = 1 0 k sh In order that these principle directions be completely indeterminate at every point, it is necessary and sufficient that the coefficient of the X'h in the above equation be zero, i.e., (2) Tij + Ai = pgij where p is a scalar function. Eisenhart4 has shown that when the equation (1) of motion of an electron in general relativity are written in the form

lAg5] (d + r,adXadX)a then by a suitable change of the independent variable the vector 4Si can be identified with the gauge vector of Weyl's geometry. Thus we have (Sik)k = i If the principle directions are indeterminate for the tensor Aik + Tik, from then we have (2)

(Tik+ Aik

=a

But

(Sik)k

=

(Tik +

hence

o

apxt

Aik)k2

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MA THEMA TICS: A. BRAMLE Y

VOLP 10, 1924

This equation shows that Weyl's gauge vector is necessarily a gradient; i.e. the forces acting on the electric particle are conservative since we can regard the O's as the pondermotive forces of the field. The conservative nature of the forces gives a value clue to the conclusion that the system emits no radiation of energy. This, however, has been shown by Eisenhart and Veblen5 to be the condition that the Weyl's space be Riemannian, i.e. that the geodesics which the electron describes are defined by the equation 0 d2xi ,i dxadx dS2 r -s Ts where [6 (Xgak) +6 (Xgflk). 6 (Xgaq)] ra 1 a ra,6 -2 L 6x bo+6xa Jt and -p = log X. The laws of mechanics are contained in the expression that the variation of the action is zero for any continuous deformation of the universe. Suppose that the point xi passes into the point x' where Xi = Xi + 3@x(X.... X.) then of Ldx'. .... dx" = OfLdx = 0 where L is such a function that

AL

=

T

2Tkgik

2

=

T'-g

and the g is the determinant-of the gik's.6 Due to the transformation of the coordinate system a gik = gir t + gkr

+ 7r

the gauge system remaining fixed. Now OjLdx = 6 (Lr) dx + SfLdx

and 5L=

If we define

VW

S Tk6X-k a k+ 2 T'bgr xi 2

= T

i

L.s6

the equation may be written

-OfLdx = J

(Wx

dx + f

-a-a + 1

gxrs Tr$ }dx

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MA THEMA TICS: A. BRAMLE Y

PROC. N. A. S.

Since the variation t' is entirely arbitrary, we must have

a W,) =Vk at +

-V

0

s

and

a

1 ?grs Trs = Tik .i= 2 bxi The first equation leads to the conclusion that Vk = 0 whence L. 6s =Ti from which we have L = T. ?xk

The second of the above equations is the equation of mechanics for a continuous distribution of matter, while the first result shows that the principle directions of the tensor Tik are indeterminate. If we impose the additional restriction that the principle directions of the tensor Aik are indeterminate, we have satisfied the conditions of the problem. A solution satisfying this condition is evidently Aik = 0. which corresponds to the ordinary law of conservation of energy. If we take R. Aik = R gik where R = Rilgik Rij being the contracted curvature tensor, then the conditions are satisfied. The choice of Aif corresponds to the case considered by Einstein7. In both cases the space is homogeneous.8 Conversely, if the electron describes a geodesic in a Riemannian space and the laws of mechanics are valid, then the relation between electric. and material energy tensors is necessarily of the form Sik = Tik + Aik (3) where Aik is an arbitrary function whose principle directions are indetermin-ate. In this case

bp but

thus

cki = (Sik)k = (Tik + (Tik + Aik _p.5ik)k

ik) =

0

Voi,. 10, 1924

GENETICS: W. E. CASTLE

107

which proves the statement, since the principle directions of Tik are indeterminate. Thus a necessary and sufficient condition that an electron describe a geodesic is that the principle directions of Aik are indeterminate. 1 Eddington, Mathematical Theory of Relativity, p. 120. 2 Eddington, Ibid., p. 182. 3Ricci, Atti R. Ist. Veneto, 62, 1230. 4 Proc. Nat. Acad. Sci. Washington, 9, 175 (1923). Eisenhart and Veblen, Ibid., 8, p. 19. 6 Weyl, Space, Time and Gravitation, Section 28. 7 Einstein, Sits. Pr. Ak. Wiss., April, 1919. 8 Eisenhart, Proc. Nat. Acad. Sci., 8, p. 28.

LINKAGE OF DUTCH, ENGLISH, AND ANGORA IN RABBITS By W. E. CASTUz BussEY INSTITUTION, HARVARD UNIVIRSITY, BOSTON Communicated January 17, 1924

In 19191 I presented evidence that in rabbits two patterns of white spotting, English and Dutch, are either allelomorphs or closely linked with each other. English pattern is dominant over self in crosses; Dutch may be called recessive, although Dutch cross-breds usually show the Dutch character feebly expressed. A cross between English and Dutch produces English marked rabbits, similar in appearance to homozygous English, although they are really English-Dutch heterozygotes. Such F1 heterozygotes form two (and only two) kinds of gametes, so far as experiments have been carried. They transmit either English or Dutch, but no gametes are formed which transmit neither of these patterns as would be expected if the two are not allelomorphs. This conclusion was based on a study of 192 cases. Possibly a study of a larger number of cases would show the occurrence of cross-overs. For in rats we obtained no cross.overs between red-eyed yellow and albinism until some 400 or 500 cases had been studied. Then linkage was observed so nearly complete that only a fraction of one per cent of crossing-over has ever been found. Subsequently it was found that English pattern is linked with angora coat. The cross-over percentage, as estimated from 1233 young produced by a back-cross between F1 animals and double recessives, is 13.70 i 0.96. For female F1 parents alone, the case with which comparison will presently be made, the cross-over percentage is 11.94 + 1.97. If English and Dutch are linked with each other, and English is linked