on distant affine connection

On Distant Affine ConnectionAuthor(s): Erwin SchrödingerSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 50 (1944/1945), pp. 143-154Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20520639 .

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[ 143

Ix.

ON DISTANT AFFINE CONNECTION.

(From the Dublin Institute for Advanced Studies.)

BY ERWIN SCHRODINGER.

[Read 11 DECEMBER, 1944. Published 23 MARCHI, 1945.]

A. EINSTEIN and V. Bargmann have recently' discovered a new form of

geometrical connection of a continuum, the distant affine connection. They discuss several variants. We deal here only with the reciprocal one

(eqn. (2, 3 b) below). Moreover, for clearness we fix attention to the

"symmetric"2 case and handle the "skew-symmetric"2 one along with it by two remarks at the ends of ?? 2 and 3.

In ? ? 1 and 2 I show how the new geometrical structure emerges, by

generalisation, from the one that was at the basis of Einstein's "Distant

Parallelism" (Fernparallelismus),3 and consisted in the natural union of an integrable (but in general non-symmetric) infinitesimal affine

connection and a (in general not flat) Riemannian metric.

In ? 3 I deduce the necessary and sufficient conditions for

"symmetrisation" and "skew-symmetrization" in an arbitrary frame.. An

interesting by-product is, that the two cases are by no means mutually

exclusive: very much non-trivial fields exist which are both symmetric

and skew-symmetric. The ? 4 contains remarks about "field equations" and about the meaning

the new symmetry postulates have for infinitesimal affine connections.

? 1. INTEGRABLE AFFINE CONNECTION.

In a previously unconnected manyfold of n = 4 dimensions we

establish an infinitesimal affine connection by associating with any

contravariant vector A, at the point x, the parallel-displaced vector

1 A. Einstein and V. Bargmann, Annals of Mathematics, 45, p. 1, 1944 ; A. Einstein,

ibid., p. 15. 2 Not the notion familiar from infinitesimal affinity. The meaning is made quite

clear in the following. ?A. Einstein, Sitz. Ber. d. Preuss. Akad. d. Wissensch. (Phys. Math. CL), pp. 217,

223, 1928; R. Weitzenb?ck, ibid., p. 466; A. Einstein, ibid., pp. 2, 156, 1929; p. 18,

1930; A. Einstein and W. Mayer, ibid., pp. 110, 401, 1930; p. 257, 1931.

PROC. R.I.A., VOL. LI SECT. A, [18]



144 Proceedings of the Royal Irish Ac-ademy.

A v + Av at tle point x, + d x, where

SAY = - AVp APdx . (tP 1)

The array of the n3 (= 64) quantities A;, arbitrary continiuous functions of the x,, constitute the affine connection or the affinity. A

covariant vector B. shall be displaced thus

~BB - 14 dx, , (1t2)

in order to preserve the scalar product B. AY on displacement. P

Let us denote a vector at point (a) by Aa By continuous transfer,

according to (1, 1), along a given path to point (,B) it turns into a

vector at (/3), which we call A A but which will in general depend on the path. The well-known condition on the AP for this not to

par be the case, is the vanishing of their Riemannian-Christoffel-tensor. (The affinity is then called integrable.) We shall express the condition in a

simpler form. Assuming integrability, choose four linearly independent vectors

hA , I lb, AP , hV (in general notation hy" a t 1, 2, 3, 4) at some

point and expand them into four fields by parallel transfer according

to (1, 1); The linear independence will be preserved, because a linear relation would be preserved. Please notice that a is not a tensor-index,

just a label. We always write it as subscript, but we herewith extend to

it the summation convention.

You have everywhere and for every a and for ever-y d x,

a h ".3 = _ A"vp0, b ha d x, a h= d-X,; Axo

hence

- A"p, ItPa ah% (1, 3)

Let the 16 functions hpa be defined by

hph"o = a vp (1,4)

(That is to say, the hp a are the normalized minors of the determinant

of the Pe.) Then from (1, 3)

=t h7~ a h .a (1,5)

From (1, 4) we can write for this also

A 7 = A a hr a (1, 6)



SCHRODiNOEa-On Distant Affine Connection. 145

Conversely it is easy to see, that the affinity (1, 6) is integrable, if the

hv, are four arbitrary linearly independent vector-fields and the hp,a the normalized minors of the determinant hla Those minors form four

covariant vector fields related to the hI, by

ha hvt c abX (1,7)

which is an immediate consequence of (1, 4). From (1, 7) the following equations are true

(h&)ats = (hVa)atp (hxa)av (hAb)ata . (1,8)

Hence the distant connection for any vector AP must read

v v A P= g A , (1, 9)

where V

g A =(h ,)8t' (hka)ata (,t 10)

In the same way the distant connection for a Bk is found:

A Bk = BA g . (1, 11)

p a A

The entity g introduced in (1, 10) behaves on coordinate transformation as the direct product of a contravariant vector at (/) and a covariant vector' 'at (a) since it is the sum -of four such products. It is the first

example of a " bivector," a notion explicitly introduced in the recent

Einstein-Bargmann papers. Put

IrA - hYa hAa (1, 12)

anid define frA by f vAfA, =

,# I ( ,13)

so that from (1, 4), (1, 7) and (1, 12) obviously

fVA = h,Va hAa . (1,14)

Adopt the f-tensor as metrical tensor, of which it fulfils all require ments; in other words, institute an association of co- and contra-variant

vectors by

A = fpT A , BP = fPu B , (1, 15)

with the corresponding association for tensors of higher rank and with the imperative postscript, that for raising or lowering an index in such an



146 Proceedings of the Royal Irish Academy.

entity as g , you have to use the f-tensor taken at the point to which

that index refers. This has the following consequences, which are either obvious or easy

to verify:

(i) The vector hv becomes associated with It ".

(ii) The four vector fields become mutually orthogonal unity fields.

(iii) The association (1, 15) is preserved on parallel transfer.

(iv) The metrical tensor f is everywhere its own parallel displaced

(this is materially the same statement as iii).

(v) The covariant associate of the " distant-affinity-tensor (1, 10)

fulfils the symmetry condition

g, v A , . (1, 16) A8< a ,B

(vi) The f-tensor is obtained from it by letting the two points (a)

and (/3) coincide

(fpA)at = . A (1, 17)

But let there be no mistake: the metric is in, general not flat. For,

the integrable affinity (1, 6) need not be symmetric in the subscripts. And

only if it happens to be so, can it (and actually does it) coincide with the

Riemannian affine connection, the one mediated by the Christoffel brackets

of the metric.

? 2. OUTumNE OF THE E.B.-CONNECTION.

In the recent Einstein-Bargmann-theory, (1, 9) and (1, 11) stipulate

a direct connection between any pair of points (a) and (/3), without

reference to continuous transition. There are no vectorfields hv,, no

relation (1, 10). The mixed g-tensor-mixed in the double meaning, that

it has one upper and one lower index aond that the two refer to different

points-shall depend continuously on the coordinates of the two points, but is otherwise regarded as a primitive datum, just as A ill an pa0

infinitesimal affinity. We prefer now to write Latin tensor-indices for better contrast with the Greek letters below them, which indicate the points

of the continuum; thus the distant connection reads:

k k I I

Af q Pi1Aa ; Bk = B1 gak k (2, 1) a 9 a B



SCHRODINGE-On Distant Affine Connection. 147

The infinitesimal coninection, contained in it (as (1, 6) is in (1. 9)), would

read

Aiim =1 L y&23 (

9l: )1 MP (2, 2)

But it is far from being a full equivalent of the complete distant connection

(as (1, 6) actually is of (1, 9)), and plavs a subordinate rble, if any.

In the case of ? 1 you easily verify from (1, 10), (1, 4) and (1, 7)

ga, - k, (2, 3a)

k na

9 atn g3g 87k1 (2, 3b) /3 a

k in t gk g p gyg = k, (2, 3C)

/3 y a

k in .

ga g flan g, g - (2, 3d) /3 V, a

In the E.B.-theory, at any rate in the variant I wish to discuss here,

the first and the second of these relations are postulated; in other words, the connection of a point with itself is to be the identity, and the

connection of any two points is to be reciprocal. If the third relation were adopted as well, it would entail all the following ones, and lead back to the case of ? 1. It is, therefore, discarded, or rather it is said

to characterize in this theory the special condition of flatness. Of the six statements at the end of ? 1 the first two become void,

sinee there are -no h-vector-fields. The fifth is, in the variant I am

discussing, postulated, and constitutes a third and very substantial k

restriction on the yxa, From (v) the (iii), (iv) and (vi) are

immediate consequences. As to (iv), the symmetry condition (1, 16) can be written

r r

(.ftkr)at I g (fn - (J,lnr)ata yak (1, 16') a /3

in MIultiply this by g aL and you get from (2, 3 b) and the symmetry

of the f-tensor

r m

(fk c)Oat P (frnDat a g ak g , (2, 4)



148 Proceedings of the, Royal Irish Academy.

showing that the symmetrizing metric is carried over into itself by the affine connection. Then (iii) is obvious, while (vi) follows from (2, 3 a).

Remrk on the skew symmetrical case: If in (1, 16) and (1, 16') we had demanded skew symmetry by adding a minus sign on the right, fkA would have to be skew symmetric. Hence (2, 4) is unaltered and says again, that the connection carries fkl over into itself.

Since this tensor is not really used for metrical purpose, but only for raising and lowering indices, the two cases are very much alike and could be treated in one, except for one additional condition. As Einstein observed, in the skew case the number of dimensions of the continuum must be even, because otherwise all skew tensors are singular (i.e. have a vanishing determinant). For the sake of clarity we shall

keep our ideas fixed on the symmetric case, but the following section virtually embraces both. One would only have to generalize the vocabulary, viz. to replace

"symmetrizing metric" by "tensorfield fkl&

"quadratic form 1I k I" by "bilinear form 1 klI

"pseudo-orthogonal transformation" by "'linear transformation that

leaves qki numerically invariant."

? 3. THE MENING OF THE SymwmY PRoPERTY.

In the E.B.-papers the process of " rinuning" is given great prominence. It is a linear transformation of contravariant and covariant vectors (or

more generally speaking: indices)

k k I * I

A = w}alAa B k -BzIt,,,k , (3, 1) a a a a

where the w) and w are arbitrary continuous non-singular matrices, k

functions of the coordinates of one point (a) (the sign w v is void !),

only subject to kI

w L kg 1ts ^t = z te n(3, 2) a

in order that the scalar product be preserved on rimming. In matrix notation, which is sometimes handier, we would write for (3, 2)

w (a).(a) = 1 .

Thus i s the reciprocal of w taken at the same point.



ScHRmtiNGnt-On Distant Affine Connection. 149

Only rimmittg-invariant rekations are recognized. The main purpose

of this is, to make all differential relations void. Indeed, the vector d Xk

or a gradient cannot be "rimmed"; the former would cease to Xk

be the differential of anything, and the latter to be the gradient of anything. All non-differential tensor equations, if invariant to coordinate transformation, are also invariant to rimming, as far as I can see.

By a suitable rimming the "symmetrizing metric" fk I (if there is one) can be made constant throughout the continuum. Call it then likg

k In the new rimming frame eqn. (2, 4) says that the (new) matrices g I

p are pseudo-orthogonal transformations with respect to the quadratic form ' k . Sinbe the converse is equally obvious, our authors formulate the

necessary and sufficient restriction, imposed by the postulate (1, 16) on k

the go as follows: in a suitable rimming frame they must all become A

pseudo-rotations of the same quadratic form J/ k 1, which achieves symmetrization in that frame when adopted as a constant metric in it.

k We wish to know: what does that mean for the g themselves

as they are before rimming? (This question is relevant, whether or not

you adhere to the idea of rimming invariance; if you reject it, that is

clear; if you accept it, the question virtually means: what is the condition

in an arbitrary rimming frame?) To answer this we distinguish one fixed point (a) and rim with

k k r r

we.r -9 r g w I = gP1 , (3, 3) p l ,8 a

which complies with (3, 2) in virtue of (2, 3 b). The mixed g-matrices "issuing from point (a),", are thereby rimmed to unity matrices:

k 7c k r m k r m gPy w. @rm Ual g Pig I= ga7 m ag = 8 c g = . "I (3 4)

a a a B a k

A matrix g9 V, connecting two arbitrary points (fi), (y) is

by the rimming (3, 3) transformed into * k k r a k

fg = g? r y fm /g - xa (3,5)

13 v fi a A

We call the second member a "three-point-expression based on (a)" and k

we have introduced for it also the sign fag X in order to indicate its a

tensorial nature in the original frame, before rimming (our p is Einstein Bargmann's W).




If all the matrices (3, 5) are pseudo-rotations of one and the same

J?k1, the condition is fulfilled and we can take gkG = i il as the

symmetrizing metric in the frame reached by the rimming (3, 3).

If they are not, no further rimming can help. For, in order to turn all

the unity matrices (3, 4) into pseudo-rotations of one and the same q k ,

this second rimming-call it in matrix notation p(3) anid 4(43) = p-1 (3)

must satisfy, in matrix notation, to

p(K) p'p(a) L (tf3)

where L(fi) is a pseudo-rotation of that i kH. Hence

p(43) _ LQg[3) p(a), p"'(j3) p-'(a) L-1(3)

Remember that here (a) is our fixed point, while (fl) can be any

point. Hence the p-rimming transforms the matrix g from (3, 5) into

g L (,y) pQ(a)g p (a) L- (/)

**,

Now, if g is also to be a pseudo-rotation of that Y k , then

p(a)g p -1(a) - L Qy)g L(y)

must also be one, because they form a group. Since this is to hold with

the same p (a) for all the matrices (3, 5), it is easy to see, that they

themselves, or speaking in the original frame, the three-point-matrices 'c

p a: must all be pseudo-rotations of some other4 quadratic forim;

vwhich, however, we shall henceforth call v k a cancelling the useless

p-rimming. We have the notable result:

The necessary and swfficient condition for symmetrization is, that the

three-point-matrices based on the same point (a)-the p's explained by (3, 5)-all be pseudo-rotations of the same quaxratic form sq. If the

condition bolds for one choice of the point (a) it holds for any other

one.

The form v will in general depend on the choice of (a). In the

general case (to which we restrict attention) it will be uniquely determined5

to within an arbitrary common factor in its ten components.

The symmetrizing metric in the original frame, gk , is found by

undoing the rimming (3, 3) by counter-rimming. But (3, 3) is identical

4 Viz. of p v p, where p is the transposed of p. 6 But there are notable exceptions, e.g. the case of flatness, characterised by (2, 3c).



SCHRIDINGER-nOn Distant Af/ine Connection. 151

at /3 a IHence obviously

gkI = nkk . (3, 6)

According to statement (iii) at the end of ? 1, which in (2, 4) was proved

to follow from the symmetry postulate, the symmetrizing metric at any

other point (,B) is found by affine transfer, thus

m n g k I [gm n gaka ga (3, 7)

18 ;1 a at 3 A

(Counter-rimming gives, of course, the same result.) Eqn. (3, 6) must hold for any other choice of the point (a) (which

we have kept fixed hitherto) say for point (/3), provided we understand

by nkl the quadratic form we obtain by basing our whole procedure on point (,B). But the ratio of the gauge-factors, which are undeter

mrlined in I k I must be drawn from (3, 7):

JDet. [ ? P k

D-;-- e-$ = Det. gal * (3, 8) 41)et. g k I

_We formulate:

In any frame the symmetrizing metric is determined-in the general case uniquely to within a gauge factor-by the three-point expressions

k P'z1 those based on (a) determine the metric -at (a) as the one

of which they are pseudo-rotations. The gaugefactor is fixed by, the fact that the square root of the

determinant of the metrical tensor at (/3) bears to the one at (a) the

ratio: determinant of the mixed g-matrix which transfers a covariant

vector from (a) to (/8). k

Remark on skew symmetry6: Are there connections gza which are

both symmetrizable and skew-symmetrizable? For them the three-point k

expressions paz based on one point (a), would have to leave

invariant both a quadratic form n1kh and a skew bilinear form, call it

kl . Then the same must hold for the mixed form

k= .kr Xrl (3, 9)

where nkr is defined by

hI kr = k (3 10)

The rest of this section was worked- out by F. Mautner, Scholar of the Institute. PROC. R.L.A., VOL. L, SECT. A. [19]




Thus the p's must be suchI pseudo-rotations (with respect to n kl) as leave also k invariant. But t constitutes itself (in the formi 8 1 + E k A where is a small constant) an infinitesimal pseudo-rotation of 1 kl . Indeed

k+ El) km (m Ms + m a) - Is + E { sI + EE Is5.

(3, 11)

Hence, if X kA is not degenerate, i.e. if it has four and only four well defined

eigenvectors, it is easy to see that those and only those pseudo-rotations which share these eigenvectors with Ekl will leave the latter invariant. Hence, the condition on the p's is that they must all belong to the two-parameter Abelian group of pseudo-rotations which have the same couple of invariant planes as kl , but arbitrary angles of rotation.

While this case, though not "flat," is still fairly trivial, more interesting cases of double symmetry arise, when E A;I is degenerate, viz. when its two angles of rotation are equal or oppositely equal. Envisage, e.g. the two forms,

hi 00 o9 h 010 o9

0 1 0 0 1 0 0 0 I = 0 0 1 0 0 0 0 1

KG 0 0 K 0 0 -0 9

(Ior k IA we could equally well take any one of its orthogonal transformed.) Now the matrix E k, is known to commute with any one

out of the well-known three-parameter group of orthogonal transformations whose matrices are obtained from the following one

(cos 9 sin 9 0 0

-sin G cos O 0 0

0 0 cosll -sin 0 (3,13)

K 0 0 sinG cosO 9

(with 0 arbitrary) by arbitrary orthogonal transformation. Thus, if all the p's, based on one point (a) are of this fonn, the connection

admits both of symmetrization and of anti-symmetrization. Actually the p's are allowed even a little more. Indeed, any rotation

whose invariant planes are the 1-2-plane and the 3-4-plane leaves the ekA adopted in (3, 12) invariant. The group allowed for the p's is, therefore, the direct product of the three-parameter group mentioned above and the (Abelian) group of rotations in the 1-2-plane alone. The allowed

group has thus four parameters, as against two in the general case.



ScmRO-DIN6 R-On Distant Affine Connection. 153

? 4. THF INVARIANTS OF THE THREE-POINT-MATRICES.

k Neither a gmal itself, nor any chain product of the type

k k m n p

qaI = gaI, g% n g7 r gEI (4 1)

a ,i v c r

that depends also tensorially (as opposed to scalarly) on two different points has any characteristic property. For it has no invariants. Indeed,

it can be transformed into unity or into any other non-degenerate matrix, not only by rimming, but even by a suitable coordinate transformation.

Only by taking (cr) (a) you obtain a matrix qa , which suffers,

both on rimming and on change of coordinates, only a similarity trans

formation

q = wqw'. (4,2)

Hence, such a q has its eigenvalues invariant and nothing else, as is

well known. The simplest q's are our three-point-matrices p, sinee the "one

point-" and "two-point-matrices" are unity, by (2, 3 a) and (2, 3 b),

which we have adopted. Moreover from (2, 3 b) anv qa can be

k P

represented as a produict of Pam s and of a single g-a; thus as a a af

product of p's alone, it (ar) (a). The prominent role of the p's is obvious.

The authors have proposed, though hesitantly, the field equation

A; p aA; 4 (4, 3)

as a weakening of the condition of flatness

pat = _ k (4, 4) a

They observe that, if the symmetrizing metric is positive definite, (4, 3) is

not a weakening of, but equivalent to (4, 4), because the only real orthogonal

transformation with trace 4 is in this case the unity matrix.

Now if we take the signature of special relativity (-, - -, +) and

keep to the tacit assumption, that the g's are to be real, then the

eigenvalues of a p (since it changes continuously to unity when (,8) coincides with (y)) are of the form

eis % e-is a, 1, (4. 5)




with 9 real and a real and positive. 0 is the angle of spatial

rotation and we may take

a - _.

where i is the velocity involved in the Lorentz-transformation to which our p is equivalent. The demand (4, 3) reads

2 cos 0 + + 4_ 4 (4, 6)

Whatever physical initerpretation of the p's might be adopted, this seems a strange coupling betweeni 0 and (3. Inter alia it restricts the. velocity involved to an absolute maximum, reached for a - r

is max = 0 94281 . . . (4 7)

The p 's based on the same point (a) have mutual invariants, describing their relative orientation. Could one put field-equations that refer to them?

Let me finally confess (though I am afraid it is contrary to the basic

ideas of our authors): I feel some regret that the symumetrizing metric,

which is in general uniquely determined by the distant affinity, has to go

to the waste-paper basket, as indeed it does, if the principle of rimming

invariance is accepted.

In this context the following question deserves attention. Since among the "three-point expressions based on (a)" are also the infinitesimal ones, the symmetrizing metric ought to be determined already by the

infinitesimal affinity (2, 2). It is, therefore, interesting to examine what the new symmetry postulate imposes on an ordinary, infinitesimal affine

connection. This question will be dealt with in a joint paper with

F. Mautner, Scholar of the Institute, whose lucid discussion was very helpful in the present work. The answer seems to be that the infinitesimal

affinity must be one that transfers a metric into itself (i.e. it must make

the absolute derivative of some symmetrical covariant tensor of second rank vanish). The Riemannian aftine connection (the one mediated by the Christoffel brackets) is a special case. But there are others.

If symmetry is replaced by skew symmetry, the answer is very similar. Again the two cases are not mutually exclusive.



on distant affine connection

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