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[In all areas of physics and mathematics it is common to search for insight into a theory by finding exact solutions of its fundamental equations and by studying these solutions in detail. This goal is particularly difficult in non-linear theories and the usual approach consists of assuming particular symmetries and searching for solutions with these symmetries. Stripped of inessential features and simplified in this way, the search for exact solutions becomes easier. In a sense, this approach betrays a reductionist point of view but, pragmatically, it is often crucial to gain an understanding of the theory that cannot be obtained otherwise and that no physicist or mathematician would want to renounce to. In this chapter we discuss exact solutions of ETGs with spherical symmetry. In addition to gaining insight into the theory, spherically symmetric solutions are particularly important in astrophysics as models for stars and compact objects, including black holes, which are important theoretical laboratories for theories of quantum gravity. The next section discusses spherical symmetry in GR and in metric f(R) gravity and presents static spherically symmetric solutions and a Noether symmetry approach. Then, the more difficult is- sue of non-static and non-asymptotically flat solutions is discussed. The second part of the chapter is devoted to the study of spherically symmetric solutions in general scalar-tensor theories and of the Jebsen-Birkhoff theorem. The chapter ends with a discussion of black holes in ETGs and of a map from spherical to axially symmetric solutions. An example is given. A spherically symmetric solution in Palatini f(R) gravity has already been given in Sect. 3.4.2.]
Published: Sep 30, 2010
Keywords: Black Hole; Spherical Symmetry; Symmetric Solution; Apparent Horizon; Ricci Scalar
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