Contents

28.2 Robinson–Trautman Einstein–Maxwell fields 431If Q ,ζ ̸= 0, then the typeD solutions turn out to beds 2 = r 2 e −x (dx 2 +dy 2 ) − 2du dr−κ 0[ e 3x (au + b) 2 /2r 2 +e 2x a(au + b)/r ] du 2 ,Φ 1 =e 3(x−i y)/2 (au + b)/2r 2 ,(28.46)Φ 2 = −e 2x−3i y/2 [3(au + b)/2r + ae −x ] /r √ 2,a and b being real constants.28.2.4 Type II solutionsOnly some special cases of Petrov type II non-twisting Einstein–Maxwellfields have been treated in detail. They refer to special properties of theMaxwell field or to simple functional structures of the metric functions. Inparticular, the cases h =0,Q = 0 and ∆ ln P = const have been studied(but the complete solution was not always found).If h vanishes, then the field equations (28.37c)–(28.37e) give (with realq and P 0 )m = m(u), Q = q 2 (u)f(ζ), (28.47)∆∆ ln P 0 +12m(ln q) ,u − 4m ,u =0, P = q(u)P 0 (ζ,ζ). (28.48)Comparing these equations with the vacuum Robinson–Trautman fieldequations (28.8), we see that the following theorem holds:Theorem 28.5 Ifds 2 0 =2r2 dζ dζ/P 2 − 2du dr − [∆ ln P − 2rP ,u /P − 2m(u)/r]du 2 (28.49)is a vacuum solution (flat or non-flat) such that P satisfies (28.48), thends 2 =ds 2 0 − κ 0q 4 (u)f(ζ)f(ζ)du 2 /2r 2 ,Φ 1 = q 2 (u)f(ζ)/2r 2 , Φ 2 = q 3 (u)f ,ζ/2r 2 (28.50)is an Einstein–Maxwell field (a charged vacuum solution). q(u) is real,f(ζ) analytic, and by a coordinate transformation q =1can be achieved.With this choice of coordinates, the field equations (28.48) give∆∆ ln P 0 = k = const, 4m = ku. (28.51)