All tensors of contravariant rank two which are divergence‐free on one index, concomitants of a spinor field σiAX′ together with its first two partial derivatives, and scalars under spin  transformations are constructed. The Einstein and metric tensors are  the only candidates

Anderson, Ian M.

Lovelock, D.

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A characterization of the Einstein tensor in terms of spinors Ian M. Anderson Department of Mathematics. The University of Arizona. Tucson. Arizona 85721 David Lovelock Department of Mathematics. The University of Arizona. Tucson. Arizona 85721 and Department of Applied Mathematics. University of Waterloo. Waterloo. Ontario. Canada N2L 3G1 (Received 30 December 1975) All tensors of contravariant rank two which are divergence-free on one index, concomitants of a spinor field (Tux together with its first two partial derivatives, and scalars under spin transformations are constructed. The Einstein and metric tensors are the only candidates. 1. INTRODUCTION It is usually assumed that the field equations govern-ing the interaction of the gravitational field with any other field (the latter having associated with it an energy-momentum tensor Tii) are of the form (1. 1) where A/j are the components of a type (2,0) tensor which is constructed only from those field variables characterizing the gravitational field. Furthermore, it is customary to demand that the choice of Ali be re-stricted by the identity AiJ,j=O, (1.2) in order that, as a consequence of (1. 1), we have TIJIj = O. Typically, if it is assumed that (i) the gravitational field is characterized by a sym-metric metric tensor gab' and (ii) the quantities Ali are constructed from gab and its first two partial derivatives, i. e. , then it is known that, 1 in a four-dimensional space, (1. 2) and (1. 3) imply that Ail =aGiJ + bgii , (1. 3) (1.4) where a, b are constants, and Gli is the Einstein tensor. In this case (1.4) and (1. 1) give rise to the usual Einstein field equations, where Tli now satisfy Tij=Tii. (1. 5) However, it has been claimed by some, 2,3 that TiJ should be nonsymmetric, in which case, as has been pointed out by Ehlers, 4 in order to accommodate this possibility, the assumptions (i) and (iil would have to be changed. The purpose of this note is to discuss the consequence of adopting one possible alternative to (i) and (ii). It is known that the gravitational field can be charac-terized in terms of a spinor field ajAX' or a tetrad field h (a=I, ... ,4), these two characterizations being (",)1 1001 Journal of Mathematical Physics, Vol. 17. No.6. June 1976 equivalent by virtue of the relations the a",AX' being the conventional Pauli spin matrices. These quantities are related to the metric by gjl = ajAX,aJBy,EABEX'Y' = ajAX,aJAX' (1.6) Guided by these observations, and (1. 3), we shall therefore seek all tensors Ali satisfying (1. 2) and for which (1. 7) where Aji are also assumed to be invariant under arbi-trary (unimodular) spin transformations. This problem is equivalent to finding all tensors Ali satisfying (1. 2) and A lJ = A ji ( h ; h ; h ) , (""a (a"a.b (""a,bc (1. 8) where Ai} are scalars under arbitrary proper Lorentz transformations. Skew-symmetric tensors satisfying (1. 2) and (1. 8) are known;5 however they are not scalars under arbitrary proper Lorentz transformations. It is clear from (1.6) that every spin-tensor which is a concomitant of gjj and its partial derivatives is always a concomitant of ajAX' and its partial derivatives. If the converse to this were valid, then the above prob-lem, viz. (1.2) and (1. 7) would immediately reduce to (1.2) and (1.3), and so, without further calculation (1.4) would follow. Unfortunately a proof of the converse does not seem to exist in the literature. The spinor notation which we adopt here is essentially that of Pirani. 6 From (1.6) it can be shown that the following is an identity (Schmutzer 7 ): r:;aA Y' ab BY' creBX' = t(g"bacA X ' + gbC(j'AX' _ g"caMx' + iE"bcaa/x'). If we define S"bA B and S"b x' Y'by sabA B = (j' AX' abBX' - ab AX' (j'BXI , then repeated application of (1.9) gives rise to Copyright © 1976 American Institute of Physics (1. 9) (1.10) (1. 11) 1001 Downloaded 27 Sep 2011 to 129.123.125.223. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions(1.12) ,2. CHARACTERIZATION OF Ars Because the Ali are assumed to satisfy (1. 7) and to be the components of a type (2,0) tensor under arbitrary coordinate transformations, certain invariance identi-tiess must be satisfied, one of which is Ars;.AX· .be + Ars;cAX' .ab + Ars;bAX' ,00 = 0, (2.1) where Ars;aAX',be== ilArs! a aaAX' ,be is a spin-tensor sym-metric in be. The invariance of Ali under spinor transformations also gives rise to invariance identities,8 two of which are (2.2) and (2.3) If we define the spin-tensor ATs;.b.ed by (2.4) then (2. 1) gives rise to (2.5) If we multiply (2. 2) by S·b A B, (2. 3) by SOb X' Y' and add the resulting equations we find, by virtue of (1.12), (2.6) From (2.5) and (2.6) it is easily seen that (2.7) Up to the present, no use has been made of (1. 2), which in view of (1.7) can be expressed in the form (2.8) Differentiation of (2. 8) with respect to aaAx' ,cds thus yields ATs;aAX' ,cd + Ard; aAX' ,se + ATe;aAX' ,ds = 0, which, by (2.4), is equivalent to (2.9) If we define ATs;ab,ed;1j .~, by (2.10) and note (2.4), we see that (2.11) Elsewhere! it has been shown that, in a four-dimen-sional space, if Ars;.b.ed;!J ,k' is any quantity which has the properties (2.5), (2.6), (2.7), (2.9), and (2.11) then (2.12) [An alternative proof of (2.12), which readily suggests generalizations to higher dimensional spaces, is presented in the Appendix. ] A comparison of (2.12) and (2.10) establishes that A/i'ab,ed is independent of alllY',kl' which, in turn, im-plies that A/Nab,cd is also independent of a lBy• ,k' 9 i. e. , 1002 J. Math. Phys., Vol. 17, No.6, June 1976 (2.13) Where g'abed is a spin-tensor with the same symmetry properties as Ns;ab,ed, viz. (2.5), (2.6), (2.7), and (2.9). From (2.4) and (2.13) we thus find which, upon integration, yields (2.14) From (1. 6) we see that AX' + AX' + 0 ( ) g'b,ed = a.AX• ,edab aa °bAx' ,cd .bed °IAX' ;oux·.J . When this is taken together with (2.14), it gives rise to which can be reexpressed10 in the form Ars = tgsabedRoobd + gS(oaAX' ;aaAX' .b) where gs is now a spin-tensor, which is therefore independent of aaAx' .b' i. e. , (2.15) All spin-tensors gs:= gS(aaAx') have been constructed9 the result being gs = bffs, (2.16) where b is a real constant. Consequently, the problem of determining ATS has re-duced to the evaluation of the spin-tensor gsabcd where Esabed = ESabcd( alA X') and where Esabcd satisfies (2. 5), (2.6), (2.7), and (2.9). To do this we proceed somewhat indirectly, as follows. It is easily seen that, because of the symmetry prop-erties of Eoabed, gt.lei BSbd _ 2(Fsabed + FOOde. + EdaSeb) to - , from which we obtain g taleJ BSbd + g tbicJ Bsad + Eta/d} o'be + g tbldJ osae - 12gsabed tli IlJ tl} tl} - • If we define (fa I c= - b,EtI}lgtalei, and recall that B~~1:= - ESbdlEtlJ I' we see that (2. 17) reads (2.17) (2.18) gs.bed == Esbdl {fa, e + E'aal {fbi e + ES bel {fa I d + Esael {3"b, d • (2. 19) Consequently a knowledge of ~a,c determines gsabed. From (2.18) we note that and {3"a,e= - j3'ela {3"'11 =0. We now define I.M. Anderson and D. Lovelock (2.20) (2.21) (2.22) 1002 Downloaded 27 Sep 2011 to 129.123.125.223. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions{3ABCDX' y, Z, W' = O'rAX' O'aBY' O'cCZ' al DW,fJ"a Ie, (2.23) in which case (2. 21) and (2. 22) imply f3ABCDX'Y'Z'W' = -f3ACBDX'Z'Y'W" (2.24) and (2.25) respectively. From arguments similar to those presented in Lemma A3 of Ref. 9, we find by virtue of (2.24) and (2.25) f3ABCDX'Y'Z'W' =ic[EBc€AD(Ex'Y'Ez'w' + Ex'z'€y'w') -EY'Z,Ex'w·(EABECD +EACEBD )], where C is a real constant. The latter is substituted in (2.23) which is then solved for fJ"a1c, (1.9) being used repeatedly in the process, this giving rise to (2.26) where a is a real constant. Equations (2.16), (2.19), and (2.26) are now substituted in (2.15) to yield Ars = aG"s + bg"s. We thus have the following theorem. Theorem: If ATs=Ars(O'aAX':O'aAX',b:O'aAX',bc) is a spin-tensor and then ATS = aG"S + bg"s where a, b are constants. Consequently, we see that, even in this case, the symmetry of NS [and hence of rrs by (1.1)] is again an inevitable consequence. APPENDIX The purpose of this appendix is to outline an alterna-tive derivation of Eq. (2.12) which explicitly exhibits the role played by the dimensionality of the space. We begin by introducing quantities !lab and f31J which are both assumed to be symmetric in their indices but are other-wise arbitrary, and consider the expression Arll;ai ,cI 3;!l4,hi sos b d j h tv '" '" (.) 2 11121314i5"b-cal-'ljl-'hk' 1003 J. Math. Phys .• Vol. 17, No.6, June 1976 By expanding the Kronecker delta and by repeatedly invoking the symmetries of Ars;ab.cdHJ,hk, viz. (2.5), (2.6), (2.7), and (2.9), as well as the symmetries in-duced by the expression (lab!lcd f3 IJ!3"k a lengthy, but nevertheless straightforward calculation, shows that = t[4Ars;ab,cd;lJ,hk +ATs;ab,iJ;cd,lk] !lab!lcaf3/Jf3hk' (AI) This equation holds irrespective of the dimension of the underlying space. However, for n=4, the left-hand side of (Al) vanishes identically in which case it is easily seen, on account of the arbitrary nature of the !lab and f3 IJ , that By successively interchanging the pairs (cd), (ij), and (hk) in (A2), it thus follows that as required. ACKNOWLEDGMENTS This work was supported in part by a grant from the National Research Council of Canada. One of us (I. A. ) would also like to thank the National Research Council for the award of a postgraduate scholarship. tD. Lovelock, J. Math. Phys. 13, 874 (1972), 20. Costa de Beauregard, "Translational Inertial Spin Effect," in Perspectives in Geometry and Relativity, Hlavaty Festschrift (University of Indiana Press, Blooming-ton, 1966), p. 44. 3D. W. Sciama, Proc. Cambridge Philos. Soc. 54, 72 (1958). 4J. Ehlers, Survey of General Relativity Theory. Relativity, Astrophysics and Cosmology (Reidel, Holland, 1973), p. 1. 'B.O.J. Tupper, Can. Math. Bull. 16,111 (1972). 6F. A. E. Pirani, in Lectures on General Relativity, Summer Institute in Theoretical Physics, Brandeis (Prentice-Hall, Englewood Cliffs, N.J., 1965), Vol. 1, p. 249. E. Schmutzer, Relativistische Physik (Akademische Verlaggesellschaft, Leipz ig, 1968). BJ. Wainwright, Tensor 19, 217 U968). "I. M. Anderson, Gen. Rel. Grav. 6, 621 U 974). tOU. Lovelock and H. Rund, Tensors, Differential Forms and Variational Principles (Wiley-Interscience, New York, 1975). I.M. Anderson and D. Lovelock 1003 Downloaded 27 Sep 2011 to 129.123.125.223. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions

A Characterization of the Einstein Tensor in Terms of Spinors

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A Characterization of the Einstein Tensor in Terms of Spinors

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