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1 Show that the rest-velocity at an event is unique. A rest frame of the fluid at an event is a frame in which (W a ) = (1, 0, 0, 0) and in which the components of the matrix (T ab ) can be written in block form (T ab ) = ρ 0 0 σ , where σ is a 3 × 3 matrix. In general, the σ has three distinct eigenvectors and these pick out three special directions in the fluid. A perfect fluid is one for which there are no special directions and therefore one for which σ is a multiple of the identity. Such a fluid is isotropic: it looks the same in every direction at the event.

So the definition of a manifold captures the idea that the coordinate systems are local, and ties down the permitted transformations between local coordinates. 6 Manifolds 55 to say infinitely differentiable, transformations. Our manifolds are therefore of class C ∞ . 7 An n-dimensional manifold is (a) A connected Hausdorff topological space M , together with (b) A collection of charts or coordinate patches (U, xa ), where U ⊂ M is an open set and the xa s are n functions xa : U → R, such that the map x : U → Rn : m → x0 (m), x1 (m), .

2) The local inertial coordinates set up by an observer in free-fall at an event A are the coordinates xa such that xa = 0 at A and ⎞ ⎛ 1 0 0 0 ⎜ 0 −1 0 0 ⎟ ⎟ (gab ) = ⎜ ⎝ 0 0 −1 0 ⎠ + O(2) 0 0 0 −1 as xa → 0. In local inertial coordinates, special relativity holds over small times and distances. 3 Particle Motion In a local inertial coordinate system at an event A, ∂c gab = 0 at A. 7) at A, where the dot is differentiation with respect to proper time τ . This equation determines their motion, but not in a very practical way because we have to 50 4.