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RP Kerr and A. Schild, a New Class of Vacuum Solutions Of Gen Relativ Gravit (2009) 41:2469–2484 DOI 10.1007/s10714-009-0856-0 GOLDEN OLDIE EDITORIAL Editorial note to: R. P. Kerr and A. Schild, A new class of vacuum solutions of the Einstein field equations Andrzej Krasi´nski · Enric Verdaguer · Roy Patrick Kerr Published online: 1 August 2009 © Springer Science+Business Media, LLC 2009 Keywords Einstein equations · Kerr solution · Kerr–Schild mertrics · Black holes · Golden Oldie Part 1: Explanation of some details of derivation By Andrzej Krasi´nski The Kerr solution is today textbook material and a basic element of education in relativ- ity. However, derivations of it are not easy to find in the literature—most textbooks and monographs simply quote it as a given thing. There exist several uniqueness theorems (see part 2 of this note) that refer to physical and geometrical properties of the Kerr solution, but none of them gives a hint on how to derive it starting from the Einstein The republication of the original paper can be found in this issue following the editorial note and online via doi:10.1007/s10714-009-0857-z. A. Krasi´nski (B) N. Copernicus Astronomical Center, Polish Academy of Sciences, ul. Bartycka 18, 00 716 Warsaw, Poland e-mail: [email protected] E. Verdaguer Departament de Física Fonamental, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona, Spain e-mail: [email protected] R. P. Kerr Mathematics and Statistics Department, University of Canterbury, Private Bag 4800, Christchurch 1, New Zealand e-mail: [email protected] 123 2470 A. Krasi´nski et al. equations. Given the great physical and astrophysical significance of this solution,1 this is a rather frustrating situation. Therefore, we (the editors) decided to go back to the historical origins. In Part 3 of this note, Roy Kerr gives a first-hand account on how he arrived at his famous result. Here, some comments on the Kerr–Schild paper. The original publication [3] reported only the final result and some of its basic properties, but gave no hint on how to derive it. A derivation was published 2years later, in a volume of conference proceedings that has never been easily accessible, and was not frequently referred to. This is the text reprinted here. The idea of the derivation is mostly self-explanatory, but not the details. The authors warn the reader at one point that the calculations are not simple, but even where they say that something follows by “simple calculation”, the results are not necessarily easy to reproduce. This note is meant to be a guide for those readers who wish to verify all the details.2 We will use the following notation for the tetrad vectors: {e1, e2, e3, e4} = {m, m,,k} . (1) The following auxiliary results are helpful in verifying the “simple direct calcula- tion” at the beginning of Sect. 2: α β γ α γ α The combinations βγ l l , βγ lαl and βα have the same value, no matter whether gαβ or ηαβ is used to calculate the Christoffel symbols. Then: α γ α γ 1 ρ α l = (η)l − l l lβ ;ρ , βγ βγ 2 (2) α α 1 ρ lα = (η)lα + l lβlγ ;ρ , βγ βγ 2 α α (η) η where βγ are calculated from gαβ , and βγ are calculated from αβ (remem- α η (η) = ber: αβ is flat, but expressed in arbitrary coordinates, so in general βγ 0). Using these formulae in (2.1), after a lot of algebra, one is led to (2.2). The other “simple calculation” mentioned before (2.8) is again not really simple. To carry it out one needs to note that, in consequence of (2.6), Eqs. (2) above simplify to: α α γ = (η) γ − ρ , α , βγ k βγ k k H ρ k kβ (3) α α = (η) + ρ , βγ kα βγ kα k H ρ kβ kγ 1 This is attested by the very large number of papers discussing its various astrophysical applications—see Refs. [1]and[2]. 2 According to the information from R. P. Kerr (see part 3 of this note), the Kerr–Schild paper is a text specially adapted to deriving the Kerr metric. The derivation by the original method was published still later, see Ref. [4]. 123 Editorial note to: R. P. Kerr and A. Schild, A new class of vacuum solutions 2471 while the general Christoffel symbols of the two metrics are related by α α = (η) − α , − α − α; − , α βγ βγ k kβ H γ Hk kβ;γ Hk γ kβ H β k kγ α α αρ αρ − Hk kγ ;β − Hk ;β kγ + η H,ρ kβ kγ + η Hkβ;ρkγ αρ ρ α + η Hkβ kγ ;ρ + 2Hk H,ρ k kβ kγ . (4) ρ To derive (3.9) one observes that R12 = 0 is equivalent to k H,ρ (z + z) + ρ H z2 + z2 = 0, while z obeys k z,ρ =−z2. This last equation is not trivial either. To obtain it, one must take the Ricci identities for the vector field kα: ρ kα;βγ − kα;γβ = Rραβγ k (5) and contract them first with kγ gαβ , obtaining an equation analogous to the Raychaudhuri equation: γ 2 2 2 1 ρ γ k θ,γ + σ − ω + θ =− Rργ k k , (6) 2 where 2 def 1 α;β ω = k[α;β]k (7) 2 def α θ = k ;α (8) 2 def 1 α;β 2 σ = k(α;β)k − θ (9) 2 are the rotation, expansion and shear of the null geodesic k-congruence. Then one γ α β β α ρ γ contracts Eq. (5) with k m m − m m and uses the fact that Rρ[αβ]γ k k ≡ 0. The result is: γ 1 2 2 k [ ] ,γ + ( ) − ( ) = 0. (10) 4 12 2 412 421 But θ and ω are the real and the imaginary part of the same Ricci rotation coefficient 412, which is another exercise for the reader (Eqs. (2.17) in the paper seem to have misplaced indices). Knowing this, one can rewrite (10)as γ k ω,γ + 2θω = 0, (11) and then observe that (6) and (11) can be written as one complex equation: γ 2 2 1 ρ γ 1 k z,γ + z + σ =− Rργ k k ≡− R . (12) 2 2 44 123 2472 A. Krasi´nski et al. ρ In the shearfree vacuum case this reduces to k z,ρ =−z2, as stated above. Thus, ρ ρ k H,ρ (z + z) + H z2 + z2 = 0 is equivalent to k [H/(z + z)],ρ = 0, whose solution is (3.9). Deriving (3.11) is again tricky. The set (3.10) implies the integrability condition m m P|12 − P|21 = P|m ( 12 − 21) (from (2.21)), which reads explicitly def z ρ z ρ z ρ z ρ S = Y, m z,ρ − m Y, ρ − Y , m z,ρ + m Y , ρ 2 u u 2 u u z z z z z z ρ + − Y, Y , − P,ρ (z − z) = 0. (13) z z u u Other useful formulae are: ρ m z,ρ = (z − z)Y ,u , (14) ρ 2 ρ z,ρ = Y,u Y ,u −Hz + m Y ,uρ , (15) 1 2 and their complex conjugates. Also, it is important to note that R33 = R 313 + R 323 ≡ 0 2R1323 = 2R 323. Although R33 must be real, the equation R1323 = R2313 does not 3 hold identically. Consequently, 2R2313 is not identically equal to R1323 + R2313.In 4 the equation R 132 = 0 one must use the analogue of (2.23) for the full Riemann tensor and all the other equations derived so far. The result is: + ρ 3P z z 3P ρ z e ,ρ (z + z) + 3H Y, Y , + 3ze m Y, ρ z u u u 2 2 z 3P ρ z 3P ρ z 3P ρ − 3 e m Y , ρ + 3 e Y , m z,ρ − 3 e Y, m z,ρ = 0. (16) z u z2 u z u Taking the imaginary part of the above we obtain (z + z)S = 0, where S is the expression defined in (13). Thus, the imaginary part of (16) vanishes by virtue of the ρ integrability condition of (3.10). Using now (13)in(16) to eliminate the m Y,uρ terms ρ and recalling that Y,u = Y,ρ, we obtain (3.11). From here on, until the end of Sect. 4, the derivation is general and covariant, and relatively easy to follow. The non-covariant arbitrary assumption is made at the begin- ning of Sect. 5. What the authors say here is not strictly correct: Eq. (5.1) does not result for any quadratic polynomial (Y ). One must assume = αY 2 + βY − α, where α and β are arbitrary complex constants; only then can (5.1) be achieved by a translation of ξ and u or of ξ and v. 3 The identities obeyed by the Riemann and Weyl tensors are deduced under the assumption that these tensors arise from commutators of second derivatives of tensors. However, if the basic objects in the the- ory are the Ricci rotation coefficients, like in the Newman–Penrose formalism, then the curvature tensors are present in first-order equations, and not all the ‘identities’ will automatically be fulfilled (in fact, only i Rijkl =−Rijlk =−R jikl and C jil = 0). Thus, some of those other ‘identities’ must be imposed as equations to fulfill. The equation R1323 = R2313 holds by virtue of (13).
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