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10.2 Lie symmetries of Einstein’s equations 13110.2.2 How to find the Lie point symmetries of a given differentialequationIf a point transformation (10.2) is a symmetry of a system H A = 0, then,as well as u α (x i ), ũ α (˜x i ) is also a solution of that system, i.e.H A (˜x i , ũ α , ũ α ,n, ũ α ,nm,...) = 0 (10.9)has to be a consequence of (10.1). Inserting the infinitesimal transformation(10.4) into this equation, one obtains the necessary conditionXH A ≡ 0(modH A =0). (10.10)Here the identity sign ≡ indicates that this condition on the coefficients ofX has to be satisfied identically in x i ,u α , and all the derivatives, since ithas to be true for every solution, which means that at each point arbitraryvalues can be assigned to all these variables. These conditions are alsosufficient if e.g. the differential equations are written so that they arelinear in the highest derivatives.Written in full, the symmetry conditions are a system of linear partialdifferential equations for the components ξ i ,η α of the generator. It may,or may not, have non-trivial solutions, which may depend on arbitraryparameters or functions. For ordinary differential equations, which areincluded here as a special case, the generators cannot depend on arbitraryfunctions. In most cases occurring in practice, the system (10.10) can besolved and the symmetries can thus be determined. For a review of theavailable computer programs see e.g. Hereman (1996).For Einstein’s vacuum field equations, the symmetries have the form(Ibragimov 1985)X = ξ i (x n )∂/∂x i − (ξ k ,mg kn + ξ k ,ng mk − 2ag nm )∂/∂g nm , (10.11)where the ξ i (x n ) are arbitrary functions, and a is a constant.For a perfect fluid, the four-velocity components u i and pressure p andmass-density µ have to be added to the list of dependent functions. Thesymmetries for this case areX = ξ i (x k )∂/∂x i − (ξ k ,ig kn + ξ k ,ng ik − 2ag in )∂/∂g in+(ξ i ,ku k + au i )∂/∂u i +2ap ∂/∂p +2aµ∂/∂µ. (10.12)Equations (10.11)–(10.12) show that the finite Lie symmetries ofEinstein’s equations are (only) diffeomorphisms (˜x i = x i + εξ i (x n )+···,together with the appropriate change of vectors and tensors), and scalings˜g in =e 2εa g ik . If one is dealing only with a subset of the solutions of