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9.5 Limits of families of space-times 127investigated first by Geroch (1969) and more recently by Paiva et al.(1993). There is also the possibility of singular limits which can be interpretedas shocks or impulsive waves (cp. Chapters 24 and 25), cp. §3.8.The idea is that one has a family of metrics sufficiently smoothly dependenton some parameter λ, and wishes to study the limit as λ → 0.Geroch (1969) defined hereditary properties to be those which the limitingsolution must inherit from the general family. For example, if thegeneral family all have an isometry group G r , the limit will have an isometrygroup G t where t ≥ r. From the methods of Petrov classificationdescribed in Chapter 4 and §9.3, and the fact that invariants will be continuousfunctions of λ, one can see that ‘being of a Petrov type at leastas special as . . . ’ is a hereditary property. The possible limiting processesare as implied by Fig. 4.1. The corresponding specialization diagram forSegre type has been given by Paiva et al. (1998); essentially the sameinformation arises from considering perturbing the Weyl or Ricci tensorof the limiting type, in the sense of Arnol’d (Ellis and McCarthy 1987,Guzman S. et al. 1991).Since the coordinate transformations giving the form of the line elementfrom which the limit is to be derived may themselves be λ-dependent, andin a way that may be singular as λ → 0, most of the examples have beenderived in an ad hoc manner. Examples of limits found in this way aregiven in §§20.6 and 21.1.2, but such methods do not provide a systematicway of finding all possibilities. Paiva et al. (1993) pointed out thatthe possibilities can be enumerated without directly seeking the coordinatetransformations, by using the Cartan invariants defined in §9.2. Thisapproach arises because the Cartan invariants give a unique local characterizationof the geometry, and will be continuous functions of λ which, forregular limits, will have finite values as λ → 0 if the limiting space-time isnon-singular. Thus one can enumerate the limits by studying the possiblelimiting values of the Cartan invariants: in particular any λ-independentequation relating Cartan invariants must be a hereditary property. Oncethe limiting spaces are known one can then find appropriate coordinatetransformations.This can lead to cases overlooked in applications of other methods.For example, Paiva et al. (1993) rediscussed the limits of the family ofSchwarzschild metrics (15.19) as 1/m → 0 and found five possible limits,rather than the two (flat space and the axisymmetric Kasner metric(13.53) with p 1 = p 2 =2/3, p 3 = −1/3) found by Geroch (1969). The extracases are one inhomogeneous and two homogeneous forms of vacuumplane wave solutions. For m → 0 one gets only flat space. The methodhas also been used to show that the Levi-Civita metric (22.7) appears asa limit of the Zipoy–Voorhees metrics (20.11) (Herrera et al. 1999).