А New Lagrangian for the Vacuum Einstein Equations and Its Tetrad Form

А New Lagrangian for the Vacuum Einstein Equations and Its Tetrad Form Abstract We give а modification of the Palatini Lagrangian for the free gravitational field that yields the vanishing of the torsion as а result of the field equations and requires only the assump­ tion of the symmetry of the metric. We transcribe this Lagrangian into the tetrad formalism and show how the tetrad form of the Einstein field equations follows from it. Some remarks on possiЫe generalization to а theory with nonvanishing torsion in the presence of matter conclude the paper. §(1): Introduction The derivation of а set of field equations from а Lagrangian is often very helpful in discussing the symmetry properties of the equations, conservation laws, invariance properties of the field, and so on. It is useful to have as many properties as possiЫe follow from the variational principle, without the need to impose additional constraints on the variations themselves when deriving the field equations. As is well known (see, for example [9], р. 107), in the Palatini variational principle for the Einstein field equations, the metric and the affme connection both are symmetric, the Einstein field equations as well as the definition of the affine connection as the metrical one, given Ьу the Christoffel symbols, follow from these variations. As is equally well known, without the assumption of the symmetry of the affinity, the correct equations for this definition do not follow from the Palatini variational principle. The usual way out of this difficulty is to impose the condition of symmetry on the coordinate components of the affine connection. When а tetrad formalism is used, however, the symmetry of the affine connection is not reflected in а similar symmetry of the anholonomic (tetrad) components of the affine con­ nection. This difficulty has been sidestepped in the past Ьу implicitly imposing the condition that the affinity Ье metric (though not symmetric), when taking variations. (See for example [10] or [5]. This point will Ье discussed in more detail in the concluding section.) This is certainly unsatisfactory, since it violates the spirit of the Palatini approach, which assumes no а priori relation between metric and affinity. In addition, in the tetrad formalism used in [10] and [5] а mixed form of the affine connection is employed, with one coordinate and two tetrad indices. This is also necessary when the theory is rewritten in terms of tensor-valued exterior forms (see [11 ]). But the tetrad analogues of the coordinate compo­ nents of the affinity are the anholonomic components of the connection (а.с.с.); and the affine geometry of а manifold can Ье completely described in terms of а set of tetrad vector fields and these anholonomic components. As for the metri­ cal properties, we need only use the tetrad and the metric tensor in the tangent space, or tetrad metric as we shall call it for short, to describe these. Therefore, it should Ье possiЬle to write а variational principle for the Einstein equations entirely in terms of the tetrad metric, the а.с.с., and the tetrad vectors, without ever introducing coordinate components of the affine connection. In this paper we shall show that both these proЫems can Ье solved. First we shall prove that there is а modified Lagrangian with the following remarkaЫe property. When we apply the Palatini variational technique to it, we obtain the correct affinity and the Einstein free gravitational field equations with only the assumption that the metric is symmetric. Then we shall show that the tetrad transcription of the Lagrangian yields, when we apply the Palatini variational technique to it, the tetrad transcription of the Einstein equations as well as the correct relation between the tetrad metric, the tetrad vectors, and the а.с.с. As а preliminary to this second part we shall review the tetradial formalism in Section 3. In а concluding section, we shall make some remarks about the generalization of our Lagrangian to the case when matter is present. §(2): Тhе Lagrangian We consider а four-dimensional manifold, with coordinatesx µ (µ = О, 1, 2, 3), оп which there is defined а metric tensor field gлµ and an affine connec- tion Гtv • For the moment we shall not assume the symmetry of either field. The usual Lagrangian may now Ье formed: where g -д. µ = ...,Г-i g -д.µ and fo = ·/Л. µR -д.µ (2.1) (2.2) R -д.µ = Г{µ ,к - г:µ ,-д. + г;Р Гfµ - Гfр Г�µ Variation of this Lagrangian in the general nonsymmetric case has been dis­ cussed in detail in connection with the Einstein-Straus unified field theory (see [1]; [9], р. 108). The results of this discussion are as follows. The variation with respect to гtv yields an equation that does not completely determine the г;v in terms of the g-д. µ · Indeed the vector (2.3) Г-л. = ½(Гfµ - Гt-л. ) = Гtл.µ J (which is indeed а vector, since it is the trace of the torsion tensor Г (µ -д.J) is com­ pletely undetermined Ьу this equation, which on the other hand does impose the following condition on g-д.µ : (2.4) If we assume the g-д.µ to Ье symmetric, the last equation is satisfied, but the гtv still remain arЬitrary to the extent of an arЬitrary choice of Г-л. . On the • other hand, variation of (2.1) with respect to g-д.µ yields R -д.µ =О (2.5) without, of course, implying the symmetry of R -д.µ in the general case. Now let us trace the terms in the Lagrangian that are causing the difficulty. Since the proЫem comes from the antisymmetric part of the affine connection, it seems reasonaЫe to separate out the terms in (2.2) that arise from the existence of this antisymmetric part. The first and third terms will automatically Ье sym­ metrized in µ and л when R -д.µ is contracted with g-д.µ (assumed symmetric from now on, except in Section 3) in the Lagrangian (2.1); so we need merely to symmetrize the remaining index pair in the third term and all the index pairs in the second and fourth terms and see what additional terms arise. 1n this way, we find that -д.µ fo = g [Г(-д.µ ),к - Г(к µ ),-д. + Г(кр)Гf-д.µ) - Г(-д.р) Гfкµ)] - g µ Г(-д.рJ Г/кµJ -д. + gлµ [- Г[к µ],л + Гfл.µ) Г[кр]] (2.6) Note that the Lagrangian has been decomposed into three terms, each of which is itself а tensor density. The first term is just the Einstein-Palatini Lagrangian for the symmetrized affine connection, which is itself an affme connection, of course. The second term is algebraic quadratic in the torsion tensor; and as we shall see, its variation with respect to the torsion tensor implies the vanishing of the latter. The third term is the one that is causing the proЫem; therefore its elimination will eliminate the difficulty. Noting that the last term is just g' л" µ contracted with the covariant derivative of Гµ , we see that if we suЬtract gлµ гµ ;л from .С0 , we get а new Lagrangian лµ .С= g л [Rлµ - Гл ; µ] = g µ [Г(лµ ),к - Г(кµ),л + Г(кр)Г(лµ ) - Г(лр)Г(кµ )] ~лµ гк[ гр -g л р] (кµ ] (2.7) Thus, we have а Lagrangian completely separated into two terms, depending on the symmetric and antisymmetric parts of the affme connection respectively. The first term is just the usual Palatini Lagrangian for the symmetrized affine connection. With the assumption that the metric is symmetric, it will yield the usual relation between the metric and the symmetric part of the affine connec­ tion. The second term is an algebraic quadratic in the antisymmetric part of the affine connection and will yield the vanishing of that antisymmetric part upon variation. It is clear that we may vary the symmetric and antisymmetric parts of the affme connection separately. After throwing away а total divergence, in the usual way, we get �J и рл � µ ~µ р 0л )] " J.,Pd4 х - f {[-g~лµ llк + 21 (~ к + g 11Р к оГ(лµ ) g llp U where 11 denotes the covariant derivation with respect to the symmetrized affine connection. Variation of the antisymmetric part thus yields рµ л Рл µ (2.9) g~ Г(рк) - g~ Г(рк] = Q Multiplying this equation Ьу iн•gµµ' = gл л:iµµ', where ~ (i.e.,gд_µ = g-д.µ/n), we find ~л11_�11 gл µg - иµ gµµ'Г(vл'] = Тµ'vл' = Тл'vµ' (2.10) (2.11) Thus, the tensor Тµ 'vл.' is symmetric in µ,', л' but antisymmetric in v, л.'; conse­ quently it must vanish. This means that the torsion tensor vanishes and the affine connection is symmetric. Гtvл.J = О Variation of the symmetric part of the connection gives лµ -g ll pл к + ½(g ll P o� + gµpllP o�) = О (2.12) (2.13) Ву the standard Palatini method, this reduces to the usual connection between the symmetric metric and the symmetrized affine connection. Variation with respect to the contravariant metric tensor density gл.µ yields the field equations (2.14) R(л.µ) - Г(л.;µ) = Rл.µ [Г(ра)] - Г[л. р I ГtкµJ = О where Rл.µ [Г(р а)] means the Ricci tensor formed from the symmetrized affine connection. Since the torsion tensor vanishes, this reduces to the usual Einstein equations. §(3): Tetrad Formalism In this section we shall review the tetrad formalism (see [7]) to show clearly what parts of the formalism require no more than а bare manifold, which parts need an affmely connected manifold, and which parts require а metric on the manifold. We consider again а four-dimensional manifold and introduce four linearly independent contravariant vector fields z/:i (т = О, 1, 2, 3). This contravariant basis induces at each point of the manifold а dual covariant basis z:Г defmed Ьу (3.1) In а bare manifold we can define the curl ofz� and the Lie bracket [zm,zп ] ofz� andz�; and we can form scalars Ьу projection onto the basis nl __ nl - �•т п - �• пт 21 (zlv,µ _ zlµ,v) zµт zvп _ (3.2а) (3.2Ь) If П�п vanishes, each of the covariant basis vectors has vanishing curl; conse­ quently four scalar fields exist whose gradients form the covariant basis. Hence the name anholonomic object. Now we assume the manifold to Ье affinely connected. The basic property of an affine connection гtv is that given any vector field А л. we can construct its absolute or covariant derivative (3.3) Using any one of the tetrad vectorsz1л. and projecting the resulting tensor on the tetrad, we get the following set of scalars: (3.4) There are 64 scalars 'Уiп п , which we shall call the а.с.с. If the affine connec­ tion is metric and symmetric, they reduce to the Ricci rotation coefficients. If we use а holonomic basis, then the 'Уiп п are numerically equal to rtv, in coordi­ nates adapted to that basis. Taking the antisymmetric part of(3.4), and with the help of(3.2), we find 'У/тп l = Sfп п - П�п where S!n n is the projection of the torsion tensor onto the tetrad 1 µ v Sтп l - Г"Л.[µv]Z-л_ZmZn (3.5) (3 6) Equation (3 .5) shows that the symmetry of the affine connection Ьу no means shows up as а symmetry of its anholonomic components. We need to find the breakup of the а.с.с. corresponding to the one used in Section 2, but we have just seen that it does not correspond to а breakup of the latter into symmetric and antisymmetric parts because of the anholonomic object. We can easily find the desired breakup, however, Ьу inserting the breakup of the affine connection into(3.4): (3.7) where t!n n =(z; ,v + Г�µ) z/;) z� z� = z;II v z� z� (3.8) Thus, even though not symmetric, t�n are the а.с.с. corresponding to the sym­ metrized affine connection. 3 Even though it involves а slight abuse of language, we shall refer to the {!п п as the symmetrized а.с.с. to distinguish it from 'Уiтп) , which we shall call the symmetric part of the а.с.с. Using(3.5) and the fact that the 'У� п are the sum of their symmetric and antisymmetric parts, it is easy to show that sl _ 1 _ [],1 (3.9) 'Ymn - 'У(тп) тп Now we shall introduce the metric into our manifold. For the moment we consider an arbltrary metricgµv • With the given tetrad z/:i we can form the tetrad components of the metric: (3.10) with (3.11) 1/lm 1/ln = o::Z We shall call 1/im the tetrad metric of the manifold. 3Тhеу were first introduced Ьу Schouten and Struik in 1935, as far as we can tel1. See [8], р. 84. 1 transform like the anholonomic components of It should Ье emphasized that the {тп an aff"me connection under anholonomfc coordinate transformations, and the sfпп transform like the anholonomic components of а tensor under these transformations. On the other hand, neither 'Уfтп ) nor 'У/ тпJ have such transformation properties and thus cannot Ье considered as the anholonom1c components of an aff"me connection or а tensor, respectively. This is the basic reason that the decomposition (3.7) is so important. Equa­ tion (3.9) shows that this breakup can Ье def"med entirely in terms of the tetradial quantities. In applications one often makes some special choice of the zf:z, leading to а simple form of ТJт п; e.g., an orthonormal tetrad, in which case the numerical values of ТJт п are equal to the components of the usual Мinkowski metric. But in general the ТJт п can Ье arЬitrary functions of the coordinates. There exists а class of affine connections characterized Ьу the condition (3.12) gл_µ; v = О, which preserve the scalar product of any pair of vectors, with respect to the metric, under parallel transport. If gл_µ is symmetric, (3.12) alone does not lead to а symmetric connection. The torsion tensor will vanish provided that the re­ lation between metric and affine connection has been derived from а variational principle of the type discussed in Section 2. When the condition (3.12) is valid, there is а simple relation between the tetrad metric and the а.с.с. Defining the directional derivation of а scalar field ф with respect to the tetrad vectors Ьу the relation Ф,1 = Ф,л.z/ (3.13) ТJzm,n = ТJzq'Y�m + ТJqm 'Yiz lq т lm qт l Т) , п + Т) 'Yn q + Т) 'Ynq - Q (3.14) we find (З.14а ) If the metric is symmetric, we can write (3.14) ТJzm,n - 'Уzпт - 'Ymn l =О (3.14Ь) where we have defined 'Ylnm = ТJ1q'Yim (3.15) Equation (3.14Ь) shows а well-known special property of the а.с.с.: when ТJzm = const, then 'Ylmn is antisymmetric in l, п. §(4): Тhе Lagrangian !Expressed iп Tetrad VariaЫes Since !, as given in (2 .7), is in terms of components with respect to а co­ ordinate basis, our first task is to express it in terms of the three sets of tetrad variaЫes z{, 'У�т , and Т) lт . То simplify this calculation, we can take advantage of the fact that the first term in the Lagrangian is just the Ricci tensor for the symmetrized affmity rtµv) contracted with the contravariant metric tensor density; while the second term is just the covariant derivative of the vector Гµ sirnilarly contracted. Ву expressing the contravariant metric tensor density in terms of the tetrad variaЬles, we can thus reduce the proЬlem to fmding the tetrad components of tensors. The contravariant tensor i'·µ, is expressed in terms of 'Т/ lт and zl Ьу the inverse of equation (3.7а): (4.1) We still need the determinant g = det gл. µ,, which is derived from the inverse of (3.7), (4.l a ) and has the value g = 'Т/Z 2 ; Therefore 'Т/ = det 'Т/1т , Z = det zl (4.2) (4.3) where (4.4) In the following we shall use as а third set of tetrad variaЫes the ii 1m instead of 11 1m . We notice that gл. µ, depends only on ii 1m and z{ [the zf:i are functions of zl according to (3.1)]. Thus, л. .!> g µ, {R л. µ, [Г(р u)] - Г[лр J Гfкµ,J} (4.5) where R1m [t:S] stands for the tetrad components of the Ricci tensor for the symmetrized а.с.с. The tetrad components of the Riemann tensor are well known (see [7], р. 172, for example), and Ьу contraction those for the Ricci tensor follow at once. k k k r k r пr k R lm = 'Y lm,k - 'Ykm,l + 'Ykr'Ylm - 'Ylr'Ykm + 2 �иkl'Yrm Thus, lm sk " = z~ { 'Т/ ,1.., sk sk sr 'Ylm,k - 'Ykm, l + 'Ykr'Ylm sk sr пr sk - 'Ytr'Ykm + 2�иkl'Yrm } - ~ lm k r z'Т/ SlrSkm (4.6) (4.7) Again there is а clean separation in the Lagrangian between а term that is just the tetrad transcription of the Palatini Lagrangian for the symmetrized а.с.с. and а second term that is an algebraic quadratic in the torsion components. We can vary (4.7) directly in terms of the three sets of tetrad variaЫes; but if we express the variations of the components of the metric and the affine connection in (2 .8) in terms of the variations of our tetrad variaЬles, we shall get the same answer. Variation of (4.3) gives us the relation between variations of the tetrad vari- аЫеs and the variation of Similarly Ьу noting that g . к lт т к г -л_µ "Y 1kmzк k zл.zµ -zµ Zт,л л.µ = (4.) 8 we can find the relation between variation of the tetrad variaЬles and variation of Г�µ 5 Variation of the Rotation Coefficients and Tetrad Metric : §() We first vary the а.с. keeping с., the tetrad metric and tetrad vectors fixed. Equation( . 4 shows ) 8 that the variation induced in Г�µ Ьу variation of 1fm is just the tensorial projection of the latter variation: � к = k Zкlm 5.1 () uГ-л_ O"Yzm kZл.Zµ µ We now break up the variation of the 1tп into variations of the 't� and the S1�. Since the tetrad vectors are frxed, the anholonomic object does not vary; and (3. and ) 5 (3. show ) 9 that the variation of the torsion tensor and the symme­ trized а.с. now с. coincide respectively with the variation of the antisymmetric and symmetric parts of the а.с. Thus с.each , may Ье varied independently. The variation of the torsion components then leads to the vanishing of the torsion Ьу а proof exactly analogous to that in Section for 2 the torsion tensor. Now we consider the variation of the symmetrized а.с. Looking с. back at (2. then ) 8 shows that the coefficient of оrfп п is just the tetrad components of So, the first set of field equations is the coefficient of оГ(л.µ). + µp л .2) (5 0 g ll р о�) =] g l"Л.lpo� z;z:z( [-µg IIк+ ½(P The standard Palatini method of solving the tensorial version of(5 .2still )works: we take the trace of(5. which 2), shows that (5.3) Thus,(5 .2reduces ) to к т l~л ZkZµ Z-;,.._g µllк= Q 5 4) ( . Taking two tetrad V!;)ctors into the covariant derivative and noting that Z11к =Zzf z( 11 к we find that this reduces to ~mn _ ~mn s p + ~m p sn + ~пр sm _ О 1/ ,l 1/ "Yl 1/ "Yl 1/ "Ylp p p Multiplying(5. Ьу6) 1/тп , we find ~ qp sр _ 1 ~ "Yl p - 21/qp 1/ ,l (5.5) (5.6) Introducing this into (5.6) we get 1 ~m n ~ ~pq ~m p s n ~пр sm _ ~m n Т/ 'Ylp + Т/ 'Ylp --Т/ ,l + 2 Т/ Т/рq Т/ ,l From the definition (4.4) of ii1m , we find ~m n _ 1 ~m n ~ ~pq _ . г-= тп Т/ ,1 zТ/ Т/рq Т/ ,1-v-Т/Т/ ,1 (5.8) Therefore equation (5.7) reduces to ~mp s n ~пр s m _ . r-= тп Т/ 'Ylp + Т/ 'Ylp - - у-Т/ Т/ ,l (5.9) (5.7) which is identical with (3 .14а) written for the symmetrized а.с.с. Ву lowering the indices т, п in (5.9), we get equation (З.14Ь), written for the symmetrized а.с.с. We have thus proved that (5.2) yields the correct value of t:nп . То find the complete expression for t:nn we write, besides (З.14Ь) for the symmetrized а.с.с., the two equations derived from it Ьу the cyclic permutation l ➔ т ➔ п ➔ l. Comblning these three equations, we find 2{,zпт = Т//т ,п + Т//п ,т - Т/тп,l + 2Птпl + 2Ппт l + 2П1тп (5.10) And when we raise the index l 2-t�m == Т/1р (Т/тр ,п + Т/пр , т - Т/тп,р + 2Т/тq п�р + 2Т/пq П%.z р ) + 2п:nп (5.11) The first five terms give the symmetric part of 21-�т and the last term its anti­ symmetric part. Variation of the tetrad metric directly yields (5.12) Since the tetrad components of the torsion tensor vanish, this reduces to the tetrad components of the Einstein equations. §(6): Variation of the Tetrad Vectors One can see without difficulty that the equation oЬtained Ьу variation of the tetrad vectors z{ cannot Ье independent of the equations that we derived in Section 5 from the variations of the а.с.с. 'Уiп п and the tetrad metric 1m . Indeed, in the tensor description the Lagrangian depends only on gлµ and гtv ; and consequently with the two equations derived from the variations оgлµ and oГtv are sufficient for minimizing the action integral. We saw in Section 5 that in the tetrad description the equations obtained from the variations о 1m and бriпп are equivalent to those oЬtained from бgлµ and oГtv - Consequently the two equations obtained from the variations о 1m and orfnп are sufficient for extremalizing the action integral, and so the equation oЬtained from the varia­ tion oz{ has to Ье satisfied automatically. The exact relation between the left-hand sides of these three equations can ri ri ri Ье determined without difficulty. The equation obtained Ьу variation of z2 is readily shown to Ье o.f "q,:y_ =О UZ (6.1) with o.f = Б.f ar;v oz2 ог;v az2 - ( o.f ar; v ) Б.f ог;v az2,(3 ,(3 + о'iлµ (6.2) The equations derived from the variations й'Уiп п and orj 1m are o.f1 =О, 0'Ут п (6.3) where л o.f л m п {j.f o.f arµv -/- = --л --/- = -л- Z/ Zµ Zv Бrт п БГµ v аrт п БГµv -л. µ o.f Б.f аi Б.f -л. z µ ----= = --Zz µ лµ 1 т 1m lm л g g ri a 811 Б Б (6.4) The inverses of the last equations are (6.5) Introducing (6.5) in (6.2), we obtain the identity connecting the left-hand sides of the three tetrad equations (6.1) and (6.3). The relation (6.2) shows that one can alternatively start Ьу satisfying (6.1) and the first equation of (6.3), in which case the second of (6.3) will also Ье satisfied. This is of no practical interest, however, since (6.1) is more compli­ cated, at least for the Lagrangian (2.7) used in this work. §(7): Conclusion 1t may Ье useful to compare our Lagrangian with that of Hebl et al. (2] used in their development of the Einstein-Cartan theory. In that case as well, in the absence of matter the torsion tensor vanishes. However, in their treatment the metric and the affine connection are not independent fields. In the Hehl et al. Lagrangian, the metric and affine connection are varied subject to the condition that the affme connection Ье metric, so that independent variation of both is not possiЫe; rather, variation is subject to the constraint that g-л_µ;v =О. The basic difference between the two approaches is thus that for our Lagrangian the sym- metrized affine connection is metric as а result of the field equations; while in the Hehl et al. approach the full affine connection is assumed to Ье metric before any field equations are derived. If we add to our gravitational field Lagrangian (2.7) а matter Lagrangian lм(gap , Г[µ v] ,DA) depending on the symmetric metric tensor, the torsion tensor, and some other dynamical variaЬlesDA , then the resulting field equations will still keep the symmetrized affine connection metric, while the torsion tensor will Ье determined Ьу оlм/oГ(µ vJ .4 Thus, the resulting theory will differ from that of Hehl et al. We shall investigate this theory in а later paper. 1n particular, it should Ье noted that our Lagrangian is not unique. As Trautman has indicated [11], the basic reason for the defect we have noted in Lagrangian (2.1) in the case of а nonsymmetric connection is that the Lagrangian is invariant under projective transformations of the connection; and conversely, the basic reason for the success of our new Lagrangian (2.7) is that we have broken this projective invariance. We could accomplish the same purpose, for example, Ьу adding terms of the formgл µГ;1.. Гµ or gлµГ[л ,вJГrаµ J with arbl­ trary coefficients to the Lagrangian (2.7). Perhaps study of the theory with non­ vanishing torsion will yield some criterion for choosing а unique Lagrangian.