Contents

528 34 Application of generation techniques to general relativityL ′′ admits more than one symmetry. Moreover, it turns out in each casethat L ′′ has exactly the same form as L. The functions, however, are notthe same; they are related by equations of the form (34.32). At each step afinite invariance transformation of the form (34.8) can be used to generatea new solution of the field equations (Hoenselaers 1978b). None of thesesolutions, however, is asymptotically flat.To summarize: The metric associated with the Lagrangian (34.29) admitsthree non-trivial (in the sense that L ′ admits more than one Killingvector) linear combinations of its Killing vectors. Each of them can beused to derive an L ′ . Each of the three Lagrangians L ′ has the same formas (34.33) and the metrics associated with them admit three non-triviallinear combinations of the Killing vectors, one of which leads back to L.There are thus six inequivalent (in the sense that they are related to theoriginal Lagrangian by equations of the form (34.32) with different righthandsides) Lagrangians L ′′ all of which are of the form (34.29). Thereare thus twelve Lagrangians L ′′′ . The process of Legendre-transformingthe Lagrangians can thus be continued ad infinitum. The group of transformationsyielding new solutions of the stationary vacuum equations isthus infinite-dimensional (the same can be shown by analogous methodsfor the Einstein–Maxwell equations).Asymptotically flat solutions can only be obtained if the infinity ofpotentials ψ is considered. We shall see in §34.3 how a convenient way ofdoing so can be found.That the Lagrangians (34.29) and (34.35) admit the same invariancegroup suggests that they can be cast into the same form. To this end weintroduce new potentials and findE ± = W e −2U ± A, k = k ′ + U − (ln W )/4,L =2∇˜k∇W − 2W (E + + E − ) −2 ∇E + ∇E − .(34.36)Hence we can formulate (cp. Theorem 19.3).Theorem 34.5 From a given stationary axisymmetric vacuum solution(E =e 2U +iψ, k) one gets a new solution (U ′ ,A ′ ,k ′ ) by the substitutionU ′ = −U + (ln W )/2, A ′ =iψ, k ′ = k − U + (ln W )/4. (34.37)The new solution will be real if it is possible to continue the parametersanalytically such that U remains real and ψ becomes purely imaginary(Kramer and Neugebauer 1968b).To derive a set of equations analogous to (10.101) we note that themetrics associated with both Lagrangians (34.35) and (34.36) admitfour Killing vectors and one homothety. The group acting on the E(E)potential is SU(1,1) which has two-dimensional subgroups. We have