the final affine field laws. iii

The Final Affine Field Laws. IIIAuthor(s): Erwin SchrödingerSource: Proceedings of the Royal Irish Academy. Section A: Mathematical and PhysicalSciences, Vol. 52 (1948 - 1950), pp. 1-9Published by: Royal Irish AcademyStable URL: http://www.jstor.org/stable/20488487 .

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PROCEEDING S OF

THE ROYAL IRISH ACADEMY PAPERS READ BEFORE THE ACADEMY.

THE FINAL AFFINE FIELD LAWS. 111.1

(From the Dublin Institute for Advanced Studies.)

By ERWIN SCHRODINGER.

[Read 23 JuNE, 1947. Published 23 FEBRuARY, 1948.4

THIS Part is concerned with generalizinig a set of altogether four and twenty familiar identities. The first four express the conservation laws in the way of an invariant divergence, the next four in nonl-invariant form.

namely by plain divergences; the last sixteen give every component of the

total stress-energy-momentum as a plain divergence. The theorems hold and are of equal relevance in the -Einstein-Strauss

theory and in any affine version whose Lagrangian ? is a function of the ltik

only. It is not astonishing that these new theories open a much more direct

and more initeresting way than, to rny knowledge, any previous- conception

of a unified field theory did, of drawing upon the hoard of iinformation which

in pure gravitational theory is vested in the Lagrange function. The reason

is that they are founded on virtually the same Lagrange fuinction, to which

nothing is added, only certain symmetry postulates are dropped; as has beeni

explained in Part II. It seems worth while to work out the close kindred,

notwithstanding the inordinate obstacles met with in the search for exact

solutions of the field equations. Even in the miuch simpler case of pure

gravitation only few are known. Very welcome they wouild be. Yet for

giviing the desired clue to the coveted coalescenice with quantuim theory,

general theorems may well prove to be more effectual, or at any rate they

will be needed. Since in the present case the functions that express the affinity explicitly

by the fundamental tensor are well-nigh inaccessible by tensor calculus,

1 Proc. Roy. I. A. 51 (A), pp. 163, 295, 1947.

PROC. R.I.A.1 VOL. 52, SECT. A. LII



2 Proceedings of the Royal Irish Academy.

singularly suited is the method due to Felix Klein, Gbtt. Nachr. math. plhys

1KI. 1917, p. 469. W. Pauli, at the age of 20, has expounded it masterfully

in sect. 23 of his famous article, Enzyklopiadie der Math. Wissensch. Vol. V

19 (1920). I anm indebted to Dr. Papapetrou, Scholar of the Institute, for

re-directing, my attention to this elegant ilethod and to Pauli's exposition.

lt is Inot widely popular. Text-books as a rule still use non-illunminating

unsurveyable computations instead.

1. The invariant integral.

We start from considering the invariant integral

14 = jgik Rjk dr , (1 1)

whose intergrand, with the factor j, is according to Part I, eqn. (24), a part

of the Lagrangian 2 on which the affine theory is based. However, this is

now less relevant than that the integrand is here regarded as a function of

the 9ik and their derivatives only. For we shall now from the outset define

the *4'kl, of which '*ik is the Einstein tensor, as those-rational but

extremely intricate-functions of the aforesaid quantities as which the equations

8klI ++ qkul r gkopi

g/kl (OJ7 T,f r + T = 0 (1,2)

or, equivalently, the equations

gk4, i -ga ro-ki k irr f = 0 (1, 3)

give them uniquely. In the symmetrical case (qik =- ki) these functions

become time Christoffel-brackets and 1 becomes that Hamiltonian integral on which the analytical developments we wish to generalize are based in the

theory of pure gravitation. We contemplate the variation of (1, 1) for arbitrary gik:

I = J*BRX 8 3k d T + gik Z*Rik dT. (1,4)

We wish to get rid of the second part, in the event that the g gik vanish

at the bounidary. By carefully applying to it the consideration set forth

in the Appendix to Part I, we find, it does not vanish on the strength of

(1, 2), unless we join the demand that for the non-varied 9jik

O pk? k FfO . (1, 5)

This is virtually equivalent to

9V,k 0. (1, 6)



SCHRODINGER The Final A/fine Field Laws. 3

For, by contracting (1, 2) both ways and subtracting the results, one obtains,

as Einstein and Straus have first observed,

ik = l (* Prko

- J 1'fk) (l, 7)

So we can state, that

an = J *Rikfgkdr (1,8)

holds for any a q ik that vanish at the boundary, provided the unvaried g i*

comply with (1, 6). From

=- S - ju/y gg,agMv, (1 9) one easily finds

gik V/_ g(8 g - igikgA 8 g"v)* (1,10)

Hence X tk pq - ak 8 g ik, (1, 1)

where we have put

0i= 4- /g(? s-k 9 ikgv*R/AP) . (1, 12)

An equivalent form of (1, 8), valid under the same restrictions, is therefore

= 1* j - | ik Sgikdr . (1, 13)

By specializing in a variation induced by a suitable change of frame, the

relations (4, 1) below can be obtained forthwith. But we aim at more, viz.

not only at these four, but at twenty-four relations altogether. They are

not-or at any rate I shall not prove them to be-strict identities, but

depend on (1, 6). In principle we shall always adjoin this condition, which the field-laws demand just as they demand (1, 2) and (1, 3). We shall

frequently have to refer to it. It will impose increased carefulness and

compel us to modify some consideratioins, because as a rule also the variations

must be subjected to it and are then no longer iindependent.

How the twenty-four relations change, when this condition is discarded,

and whether the full identities are of the same general type, I cannot tell.

2. The non-invariant integral.

The extended set of relations is obtained, just as in the symmetrical case,

by studying a certain non-invariant integral, whose integrand is onily a part

of that of 1l and contains only first derivatives of the gik. Put

Aik = Fra. rIak - *pa *rik (2, 1)

and A = s ik Aik, (2, 2)




and fix your attention on

fA dr. (2,3)

Let" auean any variation that conserves (1, 6) and thus (1, 5). The

following relation

( roik - 8jak GJi) a (Sik,a) = -A Aikagik _ giik A (2,4)

may then be confirmed by- going to the trouble of replacing 9ik a in the second factor of the first member according to (1, 2) and carefully examininc

the twelve terms that result from executinjg first the variation and then the product. Let us put for abbreviation

.Ia=k ra,k _

aak kr . (2, 5)

From (2,4) we obtain, with regard to (2, 1), (2, 2) and (2, 5)

A= - A kirk - Aa (gik,), (2, 6)

valid if (1, 6) holds for both gik and 9ik ? 33ik, On account- of this restric

tion the A-factors in (2, 6) have not, as they have. in the symmetric case, the

dignity of partial derivatives of A, nay, the Alik are not even the values

the corresponding partial derivatives take for such 3ik's as comply with

(1, 6). But that does not matter.

Using (2, 6) we work out the variation of the initegral (2, 3) for any

set -of gik that complies with (1, 6) but lnot necessarily vanishes at the

boundary:

|J Adr'Ad dr l (A ika Aik)8&4kdr- j ik a dr. (27)

A glance at (2, 5) and (2, 1) shows that

A4ik,a - /ik - Rik (2, 8)

From this anid from the idenitity (1, 11) we obtain, using the notation

(1, 12):

d JNdr = J dAdr = J ik 8ag0 dr - J(Aaik gik), a dr, (2,9)

valid, to repeat that, if (1, 6) holds and is preserved by the variation. If

the latter vanishes at the boundary, then from (1, 13)

8 iAdr = 8AAd = -d. (2,10)

Notice that (2, 9) was obtained without any reference to the integral P.

So for us (2, 10) is not just a special case of (2, 9), but a distinctive state

ment. Both are of crucial relevance in the considerations to follow.



SCHR6DINOER-The Final Affine Field Laws. 5

3. Change of the frame.

We shall now be concerned with the change the non-inlvarialnt integral

(2, 3)-taken over an invariantly fixed region G-suffers under an

infilitesimal change of-the fra-me:

A= X$-k + k (3,1)

where the hk and their derivatives are to be arbitrary inifinitesimal functions of the xSk. I'his is a thing entirely different from a variation. The change

of the integral is conveniently decomposed iito two parts of which the first is

the vo6lumne integral

[8A 8T, (3, 2)

coming. from the local change of A, which we indicate by 8* (following a

widespread custom, with due apology for making thereby of the asterisk a

second, entirely different use). By local c-hange one means the one due to the local increments of the vector-, tensor-. etc. components on which A

depends. E.g.

Bi^= yr/C j~k t(3 3)

where the first two terms come from the transformation formula and wouild, they alone, produce the changed value at XCA- = -XCk + &, while -the third

term serves to haul it homne to r.

The secon7d part, to be added to (3, 2), is a contribution, as it were from

the boundary. It comes from the slight change in the limits of the integral. In computing this contribution we may obviously disregard the local chanige of A, since we are, of course, concerned with the first order only. So this second contribution is simply

J-Adr |fAdr, (3,4)

where G' and C inidicate the respective domains of integration. Now from a mere change of the integration variables (as distinguished from a

change of frame) we have for any function A of the coordiniates -

Js ( a) T= J0A&(Xa + a) si|dr

= |g?i /)+f 1+ a I d = (3,5)

-~~~~~~ rx

xaxe d

F~~a S~~ 3(A&) a) r 'i Xa)r +d




Since the ntame of the integration variables is irrelevant, the last integral represents the difference (3, 4). By adding it to (3, 2) we get

8* AdTr a*AdT +(A Q,a dr, (3,6)

being the total change of our integral under the change of frame (3, 1).

(The asterisk on the left is to distinguish it in writing from the variations

envisaged in the previous sections.) The first contribution may be computed by taking in the second (!)

equality (2, 9) 3 - 3t9ik This is allowed. For, since 3?9k is due to a mere change of frame, it most certainly preserves the condition (1, 6), whlich

is invariant. So we get:

S JASr = ftik&7gekdr + J(Acsf - Ak3a1k)Gdr (3,7

In the first integral on the right we use (3, 3) and integrate by parts:

8* Adr J [(gsk jrc + yks kr),)s + iik 9ik,r] er dr +

+[ACa - Aai + (gak k + gka Ak,) ar],c dr. (3, 8)

Except for the one factor 8*gik, which we let stand for the moment to

save prinlt, this is the final form of the precious expression from which all

the identities spring.

4. The identities.

We specialize the change of frame successively in three manners, all three

leaving our integral unchanged, but for different reasons.

(a) If the eA vanish at the boundary, the last term in (3, 6) vanishes

and the change of frame amounts for our purpose to a variation. If also the

first derivatives of the Ek vanish at the boundary, then from (3, 3) the

variation vanishes at the boundary. In this case the variation of our non

invariant integral if from (2, 10) equal to the variation of the invariant

integral 1P and must vanish for this reason. Since now the last integral in (3, 8) is zero as well, the first must also be

and because t, is arbitrary we conclude

j (9k% rA + Y1 Zkr),t 4 itk gik, = 0 (4 1)

rihis equation-an identity under (1, 6)-must, of course, be invariant. In

fact its first member is a tensor also by form, I mean apart from its



SCHRODINGER- The Final Affine Field Laws. 7

vanishing. In order not to interrupt the proceedings we defer the discussion. For shortness we introduce the mixed tensor

Zar _ i (/k trk + gk %kr). (4, 2)

(b) On account of (4, 1) we get from (3, 8) for any change of frame

S A dr - J (A4g - A aik 8*gik + 2Zar,r),

a dr

. (4, 3)

From the transformation formula of g ik:

i = rk a+ iir k ik Ar - ik,rr (4 4) axr a xr axr

r

Working this into (4, 3) with a convenient change of the dummies and

collection of terms we get

* JAdr = J[(8rat + A"a8ik,r + 2Z ar) ( --

(AGrkgSk + A kr gAcs - Zs rA Aikg) d] (4, 5)

Now it is well known that for linear transformations of the coordinates, the

components of an affinity behave as those of a tensor. Hence for linear

transformations A has, from (2, 1) and (2, 2), the character of a scalar density

and our integral is invariant.

We first take the er to be arbitrary constants. Then the terms with aXr a er~ ~ ~ ~ ~~~~~~~~~~x

and those with aXa drop out. An d since the integral that remains must

vanish, however we may have chosen the bouzndary, its integrand must vanish,

and that is the first round bracket, comma a. With the notation

tA = i (ar A + Aa0A;k kr) (4, 6) we get

(tar + ar)a = O. (4,7)

This again is an invariant identity under (1, 6), though its first member

has not invariant form (it is a tensor only inasmuch as it is zero in every

frame).

(e) On account of this, (4, 5) reduces for any frame to

Z* A dr = f2 (t'r + 1sr) ax,

| (Ak r7k + Aakrflsk - 3s A.k9Th) ?] 4 . (4, 8)




If now we taked the derivatives En,s to be arbitrary tonstants, this leaVes

the transformation of frame (3, 1) stilli linear, and by the same reasoning as

before we get the last set of sixteen invariant identities under (1, 6)

tr + 4r (Ar rk + Aakr ,' 8t, Aa- k gik) (4 9)

5. Discussion.

If you multiply and contract (1, 3) by gkm gn' and use the relations

km mgk= gk gkl= y glk - ;"I (5, 1)

as well as their derivations with respect to xi, you get the third equivalent

form of (1, 2) and (1, 3), namely

9ik ,r + gsk Pr sr + giS rkrg = 0 . (5, 2)

If you substitute the last factor on the left in (4, 1) accordingly, this relation

may be written

1 [(ysk Zr.k)

, - gsak Zik rir + (gks Zkr), - yks Zki '

r8] = 0 (5, 3)

Now, if you compute the following two invariant divergences

(a) that of the mixed'tensor gk %kr with respect to the affinity rirg

as it stands,

(b) that of g?k Zrk with respect to the mirror image of r

you find their arithmetical mean equal to the first member of (5, 3), apart

from terms which vanish by (1, 5), but are tensors anyhow. The peculiar

kind of divergence of 5ik met with here is singularly in keeping with the

peculiar invariant derivatives in (1, 4), (1, 5), (5, 2); briefly, if not quiite

meticulously expressed the rule is: use for the first subscript the Jr-affinity,

for the second its mirror image.

This results ins a strange kind of " symmetry " in the mixed tensor defined

by (4, 2). There is nothing to suiggest the order of the indices in, Zik, even

though Zik be non-symmetric. (If one adopts the attitude of 'Einstein and

Straus, who take non-symmetric tensors as hermitian, Zik is real). Another

consequence of this symmetric form is that the inver se connection-expressing

the Zik by the ZAk-is not at all simple.

The conservation law (4, 1) is more interesting than in the symmetrical

case, not only on account of the extremely complicated functions it involves,

if everything were really expressed by the gik andl their derivatives (to

check the validity by direct compuitation is practically impossible), but also



SCHRODINGER-The Ftnal Affine Pield Laws. 9

because Ztk does not itself vanish as a consequence of the field equations.

If one adopts the affine Lagrangian I have proposed and thus (see Part I):

=Ri Agik - Fik (5, 4) one gets first

g,u)'*R*= 4A g'vFlA,

then from (1, 12)

-ik Fik/g - Aik (ift FIAv - A), (5, 5)

and finally from (4, 2)

k i Fik 8k - gp FV + A g8.k (5,6)

This was the expression given in Part I, eqn. (26). Thiere are other ways of

writing it, for instance, since

A Vg = j(g4v Ruv 4- +

Fwv), one has also

t k = gli F[? - I gV Fv + _ Zik 9C gLM . (5, 7)

Returning to the general discussion, we have at the moment not very

much to say about (4, 6), (4, 7), (4, 9), except that no striking simplifications

occur, when the various A-factors are replaced according to (2, 1), (2, 2), and (2, 5), and gik,r in (4, 6) according to (1, 2). So there would be no

point in writing that out. The only thing worth mentioning is, that

A Aaik tr ' - 2 , (5, 8)

not quite identically, only with (1, 5) auid (1, 6). This relation together

with - Aik gik = A (2, 2)

is connected with the fact, that A is homogeneous of the second degree in

the gik', and of the minius first degree in the gik alone. From this and

from (2, 6) both relations follow directly, even though the A-factors are nlot

true derivatives. From (5, 8) and (4, 6) follows

t = A, (5, 9)

showing that the pseudo-density on which our deductions were based has

the comparatively simnple meaningr: trace of the pseudo-energy-tensor.

PROO. R.I.A., VOL. 52, SECT. A. [2]



the final affine field laws. iii

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