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Away from t = 0 2 2q 2 2 2 ds = t [dx + dy + dz ] - dt 2 so that distances expand forever in this universe. Also, at each time t ≠ 0, we can locally 4 change coordinates to get a copy of flat Minkowski space M . We will see later that this implies zero curvature away from the Big Bang, so we call this a flat universe with a singularity. i t Consider a particle moving in this universe: x = x (t) (yes we are using time as a parameter 2 here). If the particle appears stationary or is traveling slowly, then (ds/dt) is negative, and we have a timelike path (we shall see that they correspond to particles traveling at sub-light speeds).

We let gij = j · i , so that ∂x ∂x i Cj = gijV . Cj = We shall see the quantities gij again presently. 9 If V and W are contravariant (or covariant) vector fields on M, and if å is a real number, we can define new fields V+W and åV by (V + W)i = Vi + Wi and (åV)i = åVi. It is easily verified that the resulting quantities are again contravariant (or covariant) fields. (Exercise Set 4). For contravariant fields, these operations coincide with addition and scalar multiplication as we defined them before.

It is easily verified that the resulting quantities are again contravariant (or covariant) fields. (Exercise Set 4). For contravariant fields, these operations coincide with addition and scalar multiplication as we defined them before. These operations turn the set of all smooth contravariant (or covariant) fields on M into a vector space. Note that we cannot expect to obtain a vector field by adding a covariant field to a contravariant field. Exercise Set 4 1. Suppose that X j is a contravariant vector field on the manifold M with the following property: at every point m of M, there exists a local coordinate system xi at m with Xj(x1, x2, .