On Exact Kerr-Schild Type Vacuum Solutions to Salam's Two-Tensor Theory of Gravity*

Transcription

1 Commun. math. Phys. 6, (197) by Springer-Verlag 197 On Exact Kerr-Schild Type Vacuum Solutions to Salam's Two-Tensor Theory o Gravity* R. MANSOURI and H. K. URBANTKE Institut ur Theoretische Physik der Universitat, Wien Received January 17, 197 Abstract. A Kerr-Schild type ansatz or the / and g tensor ields leads to a tractable orm o the ield equations o Salam's two-tensor theory o gravity in vacuo. While the general solution contains the Schwarzschild and Kerr metrics in the pure Einstein vacuum case, we can show in the g case that all "non-trivial" (/φ#) solutions are restricted to have the orm o "plane-ronted waves". I. Introduction and Conclusion A two-tensor theory has been set up by Salam etal. [1] to describe the gravitational interactions o leptons and hadrons. Its ield equations in regions ree rom matter are (1.1) where μ g + κ» ) J * V (1.) J 9 gp a π & μ * = K} 1 \det μv \*[jg μq σ g στ Γ* - g μρ ρ *(g κλ λκ - 3)], G μv (), G μv (g) are the Einstein tensors constructed rom the symmetric nonsingular tensor ields / μv,g μv when considered as "metrics". m>0 is the mass o the /-ield which in regions containing matter would interact directly only with hadrons, while g would interact directly only with * Supported by,,fonds zur Ford. d. wiss. Forsch. in OsterrΛ Nr. 155.

2 30 R. Mansouri and H. K. Urbantke: leptons. Thus the gravitational interaction between leptons and hadrons goes via the "mixing terms" on the right hand side o Eqs. (1.1). Several attempts have been reported [, 3] to ind exact nontrivial solutions (i.e. / μv + g μv ) o the vacuum equations (1.1), which are rather complicated. One typical approach is to impose symmetry conditions to simpliy (1.1), but so ar there was not too much success. Another approach consists in imposing algebraic assumptions on,g. As an example o this approach, we consider the case where / and g are o the Kerr-Schild [4, 5] orm, i.e. g = η + %k k where η is the lat Minkowski metric and k is a ield o Minkowski null vectors (η μv k μ = O). The reason or doing so is that on the one hand the class o Kerr-Schild metrics in general relativity contains several interesting metrics such as the plane-ronted waves [6, 7], the Vaidya metric [8,9] and the Kerr amily [10, 5], and can be explicitly determined in the Einstein vacuum case [4, 5]. On the other hand, the drastic simpliication o (1.1, ) when (1.3) is inserted allows a complete treatment in the / g case. The result, however, is rather poor when compared with the corresponding pure Einstein vacuum case. Namely, Eqs. (1.1) permit only Kerr-Schild solutions which are plane-ronted [7] waves, i.e. ields containing a geodesic null congruence with vanishing shear, twist and expansion and with a Weyl tensor o the algebraic type N [11,1]. Two cases are to be distinguished. The irst is characterized by vanishing rotation [7,1] o the congruence, i.e. parallel rays ("pp waves"), and has been described earlier [3]. In the second case the rotation does not vanish. Choosing the coordinates in Kundt's [7] canonical way, the solution can be written where μv dx μ dx v = ds + x^du, ds Q = Idudυ + \dz - υx~ x du\ - 3v x~ du (1.5) is the lat metric in some noninertial coordinates (z = x + iy\ <& = &(u,x,y) and #"(M, X, y) are arbitrary unctions o u whose x, y-dependence is given by (A = d /dx + d /dy ) Δ.F=m (& r -&), Δ & = -(κ g /κ})m (&-<&). Equation (1.6) is ormally the same as in the pp case [3] and can easily be decoupled and solved or suitable boundary conditions.

3 Two-Tensor Theory o Gravity 303 In the rest o the paper we want to prove our statements made above. Several obvious generalizations suggest themselves which will be treated elsewhere; it is, or instance, easy to read o type III / g solutions rom Re. [7]. The interest in all the solutions obtainable in this way lies not so much in the particular solutions that might result, but in the clariication o the general structural properties o the underlying theory [although some solutions which appear as a byproduct might be interesting in themselves; see the Kerr example]. One might, e.g., think that not only our very special kind o ansatz (1.3), but also the particular orm o the mixing terms (1.) (stemming rom "covariantization" oϊ μv η μv ) might be a reason why the class considered here is not too rich. It seems worth while to test this by considering the various generalizations. Finally we want to mention that all what has been said about solutions o (1.1) o the orm (1.3) remains valid in the two-tensor theory o massive gravitation [13], a co variant extension o the theory o Freund et aί [14]. Acknowledgment. We wish to thank Dr. P. Aichelburg or many helpul discussions and or drawing our attention to Re. [5]. II. Determination o the Optical Scalars and the Petroy Type Assume that g μv, μv are o the orm (1.3); then μv -g μv = j4?k μ with j = ^-^. (.1) The ield equations (1.1) now simpliy drastically 1, reducing to G μv () = m JίTk μ, G μv (g)= -\m ^k μ k y. (.) κ Obviously, both $F and ^ must not vanish or nontrivial solutions p φθ). By contraction, both curvature scalars vanish, and (.) is equivalent to R μv () = m ^k μ, R μv (g)=~\nι jek μ. (.3) The contracted Bianchi identities imply rom (.) that k μ is geodesic. For urther analysis, we use the tetrad ormalism o Sec. 5 o Re. [5] (henceorth reerred to as KS\ the unction h appearing there replaced 1 This simpliication is purely algebraic, depending only on a relation like (.1) with k a null vector o g, but not on (1.3). Here is a starting point or generalizations. κ

4 304 R. Mansouri and H. K. Urbantke: by ^ or ^ or g μv, μv. The two tetrads adapted to g, dier only in the vector 3 3 = δ u hd 4 ; note, however, that the eect o both d 3 s on unctions F with d 4 F = 0 is the same (this remark will remove most o the apparent ambiguities that might arise in the ollowing, as we will not distinguish the tetrads and indices or notational convenience). The tetrad orm o (.3) is then R ab {) = 0 = R ab (g) or (α, b) φ (3, 3), R 3 3() = m JP 9 R 33 (g)=-\m ^. κ When we set {KS (5.7)) k μ dx μ = e 3 = du+ Ϋdξ + Ydξ- YΫdυ, (.4) the geodesic condition mentioned aboved reads Y 4 = Γ 44 = 0 (and complex conjugate). With this, k μ given by (.4) is automatically ainely parametrized, i.e. k μ;v = O. The contracted Bianchi identity applied to (.) then leaves = 0. (.5) (Here ^ 4 = k μ d μ &? again there is no ambiguity; all optical scalars o k μ are the same with respect to both metrics!) Now we show that the shear o k μ must vanish or nontrivial solutions. Indeed, the complex shear in this tetrad ormalism is given by F, the ield equations R = 0 imply ^ ^ Z)^] = Y [^4 + (Z-Z)^] = 0 (.6) (c. KS (5.5)). Assuming Y + 0, we would have ^4 = 0 = ^,4 Z = Z, (.7) so that the imaginary part o Z would vanish, but according to (.5) also its real part, the divergence or expansion. Since now Z = Y x = 0, the integrability condition KS(J) applied to 7 [14] gives Y Ϋ Λ = Y \ = 0, a contradiction. Thereore we must have vanishing shear: Y = 0. Now the equations R 44 = R 4 = R = 0 (and c.c.) are satisied. Next we show that the complex expansion Z must vanish also. Our procedure is to assume ZΦ0, which enables us to ollow KS(5) urther: we satisy the equations R 1 = JR 34 = 0 by setting h = \M{Z + Z) with M 4 = 0, i.e. with \{ + ), %( Z) (.8) Z,4 = N 4 = 0. (.9)

5 Two-Tensor Theory o Gravity 305 I now #e = 9? - < and J- are ormed and, together with Z + Z = 4 kμ. μ and Z A = -Z (KS(534% inserted into (.5), we obtain (L-N)ZZ = 0. (.10) Thus, the assumption ZφO leads to trivial solutions, L = N, proving our assertion. It is now seen rom the expressions or the relevant components o the Riemann or Weyl tensor that the ield equations together with Z = 0 imply Petrov type N with k μ as a quadruple Debever-Penrose vector. III. Canonical Form o Solutions Kundt [7] has determined all metrics which contain a geodesic null congruence o vanishing shear, twist and expansion and whose Ricci tensor is o the orm R λv =-μk λ, (3.1) where k μ is tangent to the null congruence. In particular, he has given a geometric deinition o the concept o plane-ronted waves, and has shown that the plane-ronted waves are identical with that subclass o his metrics where the Weyl tensor has type N. Two essentially dierent cases have to be distinguished, or each o which he has given a canonical orm o the metric, whereby the task o solving Einstein's equations is reduced to solving Laplace-Poisson-type equations. Our results o Sec. II show that we are dealing with g μv, μv which have exactly these characteristics o planeronted waves. What remains to be done is to write down our ields in Kundt's canonical orm and to solve those ield equations which are not yet identically satisied. The two essentially dierent cases are distinguished by the vanishing or nonvanishing o an additional optical scalar which appears when both, complex shear and expansion, vanish: the rotation Ω (this name is oten used or the imaginary part o Z also, notably by KS, but we should like to use the word "twist" or the latter; Ω is equal to Y 3 in the KS ormalism). The case o vanishing rotation, i.e. planeronted waves with parallel rays ("pp waves"), appears in the Kerr-Schild orm in Kundt's canonical coordinates; its generalization to / g solutions has been described earlier [3]. In the case o non-vanishing rotation, this quantity can be used as invariant coordinate, and the corresponding canonical orm o the metric is ds = η + Adu, (3.) where η = \dz- υχ- ι du\ + Idυdu - 3v x~ du (3.3)

6 306 R. Mansouri and H. K. Urbantke: Two-Tensor Theory o Gravity is the lat metric in some curvilinear coordinates (z = x + iy), as one easily checks by calculating the Rieman tensor; it is even easy to ind a transormation to Minkowski coordinates and discuss its meaning. The null congruence is given by k μ dx μ o: du. The unction A is independent o v, depends arbitrarily on w, and to satisy (3.1) one must have dx dy - ί A) = μ. (3.4) It is almost obvious now that the corresponding / g solution is given by (1.4, 5, 6). Actually, one has to check that,g can be brought to the canonical orm (3., 3) simultaneously, but this ollows rom the tensor relation (.1). The unction A belonging to g, say, is independent o v in the canonical coordinates. The transormation achieving this leaves open a possible v dependence in the corresponding unction or/, and (3.1) is still satisied. But as or our ield equations (.3) μoc#" { S, this υ dependence must actually be absent. Thereore (1.4, 5, 6) is the general solution. Reerences 1. Isham,C.J., Salam,A., StrathdeeJ.: Phys. Rev. D3, 867 (1971).. Pirani,F.A.E.: Contribution to the Topical Meeting on Gravitation and Field Theory, Trieste Aichelburg,P.C, Mansouri,R., Urbantke,H.K.: Phys. Rev. Letters 7, 1533 (1971). 4. Kerr, R. P., Schild, A.: Proceedings o the Meeting on General Relativity (Anniversary Volume, Fourth Centenary o Galileo's Birth) G.Barbera, Ed. (Firenze 1965). 5. Debney,G.C, Kerr,R.P., Schild,A.: J. Math. Phys. 10, 184 (1969). 6. Jordan,P., Ehlers.J., Kundt,W.: Akad. Wiss. Mainz, Abh. Math.-Nat. Kl. No., (1960). 7. Kundt,W.: Z. Physik 163, 77 (1961). 8. Lindquist,R.W., Schwartz,R.A., Misner,C.W.: Phys. Rev. 137B, 1364 (1965). 9. Misra,M.: Proc. Roy. Irish Acad. 69, A3, 39 (1970). 10. Carter,B.: Phys. Rev. 174, 1559 (1968). 11. Pirani,F. A.E.: Brandeis Lectures 1964, Vol. I; S.Deser, K.W.Ford, Ed. New Jersey: Prentice-Hall, Jordan,P., Ehlers,J., Sachs,R.K.: Akad. Wiss. Mainz, Abh. Math.-Nat. Kl. No. 1, (1961). 13. Aichelburg,P.C, Mansouri, R.: Proceedings o the Topical Meeting on Gravitation and Field Theory, Trieste (1971) (IC/71/144). 14. Freund,P.G.O., Maheshwari,A., Schonberg,E.: Astrophys. J. 157, 857 (1969). H. K. Urbantke Institut ur Theoretische Physik der Universitat A-1090 Wien Boltzmanngasse 5 Austria