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3.8 Discontinuities and junction conditions 47If [H ab ] = 0 the possible (step function) discontinuities in the Riemanntensor satisfy n a [G ab ]=0=n a [C a bcd]P c eP d f and can be characterizedby specific Riemann tensor components (Mars and Senovilla 1993b).In the non-null case, the required conditions for a matching withouta δ-function part in terms of the equality of first and second fundamentalforms were given first by Darmois (1927) (a formulation in terms ofthe connection in C 1 coordinates had earlier been given by Sen (1924)).They guarantee the existence of coordinates, e.g. Gaussian normal coordinateson both sides of Σ, in which the metric and its first derivativeare continuous, which is the form of junction condition given byLichnerowicz (1955), and similarly the existence of coordinates in whichthe conditions of O’Brien and Synge (1952) are true. The coordinate forms(Lichnerowicz or O’Brien and Synge) imply the Darmois form. In thatsense the formulations are equivalent (Bonnor and Vickers 1981). However,if other coordinates are used the O’Brien and Synge conditions, forexample, give additional, and physically unnecessary, restrictions.For the null case, the corresponding restriction of the above results hasbeen developed in Taub (1980), Clarke and Dray (1987), Barrabés (1989)and Barrabés and Israel (1991).Junction conditions are hard to use in exact solutions except whenthe hypersurface Σ shares a symmetry with the space-time. Most of theapplications have been to cases with spherical, cylindrical or plane symmetry.The best known example in the non-null case is the Einstein andStraus (1945) or ‘Swiss cheese’ model, in which the Schwarzschild solution(15.19) is matched to a Friedmann Robertson–Walker solution (14.6). Forsome examples in the null case see Chapter 25 on colliding plane waves.Note that if two space-times M 1 and M 2 are each divided into tworegions, giving V 1 + , V 1 − , V 2 + and V 2 −+−, and if V 1 is matched with V2 , thenthe same conditions will match V 2 + with V1 − .The conditions stated above concern the gravitational field, and thus,indirectly, the total energy-momentum (see Chapter 5). However, in nonvacuumspace-times, the matter content will have its own field equationsleading to additional boundary conditions which also have to be imposed.