Bosh muharrir o’rinbosari: Заместитель главного редактора
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№4 | 2021 34 the expressions for these transformations is represented by the functions which can be chosen arbitrarily and which are the similar “coordinates” of the initial solution in the space of solutions. Integral reductions of Einstein and Einstein-Maxwell equations Metric and electromagnetic potential. Integral reductions of vacuum Einstein equations and of electrovacuum Einstein - Maxwell equations arise if the metric and electromagnetic potential components possess the forms: ds2 = gµνdxµdxν + gabdxadxb, Ai = {Aµ, Aa}, Aµ = 0, xi = {xµ, xa}, µ, ν, . . . = 1, 2 a, b, . . . = 3, 4 (1) where the components of metric gµν, gab and of electromagnetic potential Aa are independent of the coordinates xa and may depend on coordinates xµ. Each of these reductions belongs to one of two types depending on whether the space-time isometry group G2 admits a time-like Killing vector field (the “elliptic” case) or not (the “hyperbolic” case), i.e. whether the signature of two-dimensional metric gµν on the space of orbits is respectively Euclidean or Lorentzian. In the expressions below, the sign symbol ǫ and its “square root” j will remind us about a difference between these cases: ǫ = −11 − − hyperbolic case elliptic case j = 1 i , ǫ , ǫ = 1 = −1. (2) The metric components gµν determine two-dimensional metric on the orbit space of the space-time isometry group G2. By an appropriate choice of local coordinates xµ, the metric gµν can be presented in a conformally flat form, where we use the sign symbols ǫ1, ǫ2 for a unified description of all cases: gµν = f ηµν, ηµν = ǫ01 ǫ02, ǫǫ12 == ±±11,, ǫ1ǫ2 = −ǫ. (3) Here, by definition, f > 0. The relation between ǫ1, ǫ2 and ǫ arises here from the condition of the Lorentz signature (− + ++) of four-dimensional metric. For the metric components gab, which determine the metric on the orbits of the space-time isometry group G2 and for components of electromagnetic potential Aa, we introduce the parameterizations where, by definition, H > 0, α > 0 and ǫ0 = ±1. Then the Einstein equations as well as the Einstein - Maxwell equations for the fields (1) imply that the function α(x1, x2), defined in (4), is a “harmonic” function, i.e. it should satisfy the linear equation which is a two-dimensional d’Alembert equation in the hyperbolic case and the two-dimensional Laplace equation in the elliptic case. Therefore, for the function α(x1, x2) one can define its “harmonically conjugated” function β(x1, x2). жүктеу/скачать 3,08 Mb. Достарыңызбен бөлісу:
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