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Solutions of the linearized Bach‐Einstein equation in the static spherically symmetric case
Author(s) -
Schmidt H.J.
Publication year - 1985
Publication title -
astronomische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.394
H-Index - 63
eISSN - 1521-3994
pISSN - 0004-6337
DOI - 10.1002/asna.2113060411
Subject(s) - physics , minkowski space , mathematical physics , bounded function , schwarzschild metric , einstein , space (punctuation) , mathematical analysis , gravitation , function (biology) , schwarzschild radius , general relativity , classical mechanics , mathematics , linguistics , philosophy , evolutionary biology , biology
Abstract The Bach‐Einstein equation linearized around Minkowski space‐time is completely solved. The set of solutions depends on three parameters; a two‐parameter subset of it becomes asymptotically flat. In that region the gravitational potential is of the type φ = ‐ m/r + ε exp (‐ r/l ) m and ε being small parameters. Because of the different asymptotic behaviour of both terms, it became necessary to linearize also around the Schwarzschild solution φ = ‐ m/r . The linearized equation resulting in this case is discussed using qualitative methods. The result is that for m ≤ 2 l the asymptotic behaviour is φ = ‐ m/r + ε r − m/l exp (‐ r/l ) u, and for m ≧ 2 l φ = ‐ m/r + ε r −2 exp (‐ r/l ) u , where u is some bounded function; m is arbitray and ε again small. Further, the relation between the solution of the linearized and the full equation is discussed.