A Strange Detail Concerning the Variational Principle of General Relativity Theory

A Strange Detail

Concerning the Variational Principle

of General Relativity Theory

Peter Ostermann*

A mathematical mistake due to an unjustified as-sumption in the variational principle of generalrelativity theory, which obviously confused Ein-stein and Hilbert, is shown and cleared up here.

There is a strange mathematical detail hidden in the varia-tional principle of general relativity theory (GRT) as used byEinstein [1], Pauli [2] to Landau-Lifshitz [3] e.g., who claimthe tensor density

Eik≡gEik≡Rik−12Rgik

(1)

with Rik Ricci’s tensor density, R its trace, to result from

∂G

−F

∂G∂gik∂lGHI∂gikJK

.(2)

l

Here gik is the contravariant fundamental tensor (i, k ..= 0 .. 3),g ≡ – 1/ |

gik

| is a function of the gik only (therefore∂g/∂gik=– ½ggik is used in the following) and ∂l ≡ ∂/∂xl

the partialderivation with respect to the coordinate xl , glik ≡ ∂lgik aswell as glik ≡ ∂l

gik are mere abbreviations. – The LagrangianG of the gravitational field may be written in the form

G≡

ggumgsvgrwnΓv,urΓw,ms−Γv,umΓw,srs

,(3)as is well known, where Christoffel’s symbols bg

Γl,ik are

Γl,ik≡12gkli

+gilk−glik.

(4)

That something might be wrong with expression (2) above,is indicated by the peculiarity, that – as far as I know – thereactually does not exist any straightforward calculation ofEinstein’s tensor density Eik from (2). I will show here thatthis expression at least suffers from a misleading notation.The obstacle is avoided by other authors from Weyl [4] toWeinberg [5] e.g., who treat the variational principle alter-natively, in that they calculate Eik without using any partialderivatives like ∂/∂gik, ∂/∂glik, thus properly converting:

δzGdx=z

Eikδgikdx.

(5)

The simple reason why (2) does not work, however, is thatG cannot be written as a function ofg,gik, glik (without gik)solely, as seemed necessary so far (though it is an unjusti-fied requirement, that the Lagrangian should always becomposed of quantities together with their direct partial de-rivatives only). In particular, from (3), (4) it is clearly seenthat G ≡ G(g,gik, glik), thus (5) is immediately leading to:

Eik=

∂G

+F∂G

∂gikgiagkb∂lGHI∂g.labJK(6)

P. Ostermann – 10 October 2005

One may easily verify that contrary to (2) this relation reallyworks, calculating Einstein’s tensor from (6) straightfor-wardly. Since ∂

G(gwrs, ..)/∂

glik = – giagkb ∂ G(gwrs, ..) /∂ glab,Eik cannot result from (2) because of ∂G/∂glab×∂l (giagkb) ≠ 0.Relation (6) is not the only valid representation accordingto (5), however. If one insisted on handling G as a functionG(g,gik, glik:=–gia gkb glab) instead, one may also find

Eik=

∂G∂g

ik−∂FlGH∂GI∂gikJ−giagkb

∂G

lK∂g(7)

ab

e.g. (where δgab = –giagkb δgik was used again). Since (7) isright – as I verified, too – expression (2) apparently must bedifferent from Eik (given one is keeping in mind what partialderivatives like ∂/∂gik, ∂/∂glik, ∂/∂gik, ∂/∂glik usually mean).In 1915, discussing (2) and (5) – s. Doc.s 60,62,,66,67,69,71,74,75,77,78,80 of [6] v. 8 – Levi-Civita showed,though with an earlier G, that (2) would not be a tensor den-sity (Doc. 67). Einstein, however, erroneously insisted thatthe tensor character of (2) definitely arises from (5) [“I havenot yet experienced such an interesting correspondence”,Doc. 69, translated by author]. This amicable controversynow dissolves by the result Eik ≠ (2).

The incomplete expression (2) does not affect the validityof Einstein's wonderful equations, of course. That it, how-ever, seems to have confused Einstein even while writinghis basic 1916 paper [7] on GRT, might be seen from amanuscript published posthumously (s. Doc. 31 of [6] v. 6),which “was originally intended as §14 of ... and later as anappendix to ...”, before it was left out in the end. In thatmanuscript he tried to handle the variational principle with-out the restrictiong= 1 (otherwise mostly used), but obvi-ously was not satisfied (s. also Doc.s 183/84 of [6] v. 8).It took about seven months, before Einstein came backwith “Hamilton’s Principle and the General Theory of Rela-tivity” [1, in German], where he refers to the achievementsof Hilbert [8] (and Lorentz) explicitly. Thus, he insists on(2), apparently without further trying to verify it by directcalculation. The ironic background, however, is that Hilbert– though using R instead of G – had claimed an expressionanalogous to (2), which unfortunately is misleading for Eik ,too (at least by its notation), “ ... readily without calcula-tion” [8, translated by author].

* Electronic address:

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[1]Einstein A., Sitz.ber.Preuß.Akad.Wiss. (1916) 1111-1116(reprint in Doc. 41 of [6] v. 6)

[2]Pauli W.: Theory of Relativity, 1958 (1st Ed. in German 1921)

[3]Landau L.D., Lifschitz E.M., Lehrbuch d. theor. Physik, Bd. II, Klas-sische Feldtheorie, 12. Aufl., 1992

[4]Weyl H., Raum - Zeit - Materie, 1./3. Aufl., 1918/20[5]Weinberg S., Gravitation and Cosmology, 1972

[6]The Collected Papers of Albert Einstein, v. 6 ed. by Kox A.J., KleinM.J., Schulmann R., 1996; v. 8 ed. by Schulmann R., et al., 1998[7]Einstein A., Ann.d.Phys. 49 (1916), 769-822(reprint in Doc. 30 of [6] v. 6)

[8]

Hilbert D., Nachr.Ges.Wiss.Gött. (1915) 395-407

A Detail of GRT – v22

P. Ostermann – 10 October 2005

Appendix

In a most instructive paper Corry, Renn, Stachel [A1]have already pointed to the delicate fact, too, that Hilbert [8]presented the tensor density (1A) ≡ (1) without calculation.They conclude from different well justified reasons that hedid not anticipate Einstein’s explicit form of the field equa-tions, thereby rightly dismissing his false argument, that thetensor density (1) had been the only thinkable result becauseof its constituents. Hilbert might tacitly have used Einstein’sfinal insight, that in addition the covariant derivative mustvanish identically, which insight – in my view – is necessaryto complete his argumentation, but is not mentioned there.Now those authors’ conclusion seems strongly supportedby the fact that Hilbert would not have found the correct re-sult calculating straightforwardly from his basic relations.The reason is the wrong presupposition that Riemann’s cur-vature scalar could be handled as if algebraically compos-able of gik, glik, glmik alone (without gik). Although this impos-sibility is clear from the version v1 of this note [an unneces-sary aside in the line below (7) is corrected in v2], it will beshown in more detail because of the implications supportingEinstein’s priority stated by the authors above.

As an extension of relation (2), Hilbert explicitly claimedEinstein’s tensor density∂R

F Eik to result from

∂RIF∂RI∂gik−∂lGH∂gikJ+∂lmGikJ,

(2A)

lKH∂glmKwhere he used the full Riemannian curvature scalar density

R≡

ggum{∂vΓumv−∂mΓuv

v}

−G

(3A)

in his variational principle instead of G only (his K ≡ –R).To show that analogously to the argumentation above, R canbe understood to be a function ofik g,g, glik, gmlik ≡ ∂m

glikindeed, the term in curled brackets may be written as{∂v

Γumv

−∂mΓuv

v

}=gsv

(Xums

v

+Yumsv),

Xewhere:

Γ(4A)

umsv=∂vs,um−∂mΓs,uvj,Yumsv

=grwdgmv

wΓs,ur−grv

wΓs,umi.

It is commonly known, that δ∫Rdx can be reduced by partialintegration to δ∫Gdx, presupposing as usual that all varia-tions vanish at the integration limits (see e.g. [3]). Never-theless, taking Hilbert’s full action integral without this re-duction to (5) exactly as it is, then

δzRdx=zδRdx=z

Eikδgikdx

(5A)

results in the working relation

Eik=

∂R∂g

ik

+giagkb∂l

LMMN∂R−∂FGH∂RIO∂gmlab∂glmabJKPPQ

(6A)

analogously to relation (6) above (verified by direct calcula-tion, too). And again, converting R of (3A), (4A) into afunction R(g,gik, glik:=–gia gkb glab, ...), one may also find

Eik=

∂R∂gik

−∂l

FGH∂RI∂gl

ikJK+∂lmFGH∂RI∂glm

ikJK−giag∂Rkb∂g

,(7A)ab

thus necessarily completing Hilbert’s (2A). In some detail,the derivation of (6A) is done as follows. At first

δR=∂Rik

∂R∂R∂gik

δg+∂gδg+δglablab∂glmablmab(8A)

results from R ≡ R (g,gik, glik, gmlik) given by (3A), (4A).

Then a repeated partial integration yieldsδzRdx=

zl...qdx+∂lQldxwhere:

z

l...q≡RST∂R

∂g

ik

δgik−∂lLMN∂R∂g−∂Flab

mH

∂R

∂gIlmab

KO(9A)

PQδgabU

VW.

Evidently, substituting here δgab = –giagkb δgik proves (6A),

since according to the usual understanding the variation

Ql≡

LMN

∂R

∂g−∂m

FRIR

δglab

H∂∂glmab

KOPQ

δgab+∂∂glmab

mab(10A)

vanishes at the boundary. It seems an interesting aspect,

however, that using R instead of G is extending Einstein’soriginal energy-momentum pseudo-tensor t

ik of the gravita-tional field by

∆tik≡12{bR−Ggδki+∂igglkl+2gkl∂lg}, (11A)

yielding another form t

ik(R) ≡ t

ik +∆t

ik according to (10A)immediately. Though, ∂k (∆t

ik ) ≡ 0 means that in fact there isno additional exchange of energy or momentum.

In this context, the same lack is obviously questioning anyadditional exchange of angular momentum in case of thewell-known symmetric alternative pseudo-tensors [3], [5],too, thus reducing their attraction significantly.

Coming back to the question of priority last, it may bepointed to the simple fact, that there are two sides of Ein-stein’s equations. As shown above, Hilbert would not havefound the left hand side calculating from (2A) straightfor-wardly. For the right hand side, in addition, he offered somespeculation insufficient to reach physical applicability.There, in contrast, Einstein had set the phenomenologicalenergy-momentum tensor of matter and pressure alreadyyears ago, which including his ‘geodesic’ equations of mo-tion has successfully proven applicable since that time invarious situations, again and again. Therefore – in spite ofsome impressive mathematical achievements, in my viewnobody can actually claim Hilbert to have derived Einstein’sequations in his first note [8] or its draft at all.

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[A1] Corry L., Renn J., Stachel J., Science 278 (1997) 1270-1273