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Course notes for General Relativity Instructor: Bernard Whiting Sept. 11 2008 Notes TEXd by: B.T. Hall, T.N. Kypeos 1 Tangent Vectors / spaces In Euclidian space, in cartesian cordinates, all vectors, derivatives etc. lie in the space of points Rn / This does not hold for more general spaces. Insert picture The set {p ∈ M : Tp } is called the tangent bundle T (M ); if dim(M ) = n, dim(T (M )) = 2n (n to label p, n to give components of ~v ). 2 Manifolds Definition: A mapping (or set of mappings) between a set of point (and their local open regions corresponding to these points and Rn / This mapping must be continuous in the open ball surrounding each point. In general, there will not be a single map that spans the manifold. To check whether something is a manifold, check differential nature of the (local) map. A set of maps that span the manifold is called an atlas; to each (unique) atlas there corresponds a manifold provided the maps are smoothly connected at the overlaps. The presence of a manifold allows us to use behavior in cartesian Rn and infer (for tensors) the behaivor in the corresponding part of M. 2.1 Informal Def of Curvature Insert picture φ δ Sum of Interior Angles π excess 0 π 2 3π 2 π 2 π 2π π 3π 2 5π 2 3π 2 3π 2π 2π − δ Excess scales linearly with φ, and with the area of the triangle. To investiate curvature in M , draw triangles, take A → 0 and watch lima→0 (Σφi − π) 3 Symmetries (killing vectors) 0 For a vector k µ to be a symmetry of gab , we require that gab (xc ) = gab (xc ); c doing the transformation for arbitrary (∈ k ) gives 1 0 gab (xc ) = gab (xc ) − [gcb ∂a k c + gac ∂b k c + (∂c gab ) k c ] (1) thus, for k c to be a symmetry, the term in brackets must vanish. 3.1 Examples ds2 = −dt2 + dx2 + dy 2 + dz 2 (2) 3.1.1 kc = ∂ c x = (a, b, c, d) (a,b,c,d constant is obvious by inspection)) ∂x (3) 3.1.2 kc = ∂ = Lz = (0, −y, x, 0) ∂φ (4) 3.1.3 others: ∂ ∂θ ∂ sin θ ∂θ cos θ − cot θsin φ∂φ (5) + cot θcos φ∂φ (6) insert picture If k c corresponds to a symmetry, it is called a killing vector. 2