Lecture 4 As the ﬁrst of our two generalizations, we look at the Minkowski plane, which is R2 with the indeﬁnite bilinear form X, Y M in = X1 Y1 − X2 Y2 . The role of J is played by the matrix 0 1 K= . 1 0 In particular X, K X M in = 0, which is the analogue of det(X, X) = 0 in the Minkowski context. Take two vectors X, Y where Y, Y M in = 1. One can then write X = Y, X M in Y − KY, X M in KY. A regular curve c : I → R2 is called spacelike if c (t), c (t)�M in > 0 for all t. We deﬁne the curvature of c to be c (t), Kc (t) M in κ= . c (t) 3 The equation of motion is then d c = −κ(t)Kc . dt c , c 1/2 M in The curvature is reparametrization invariant. Every spacelike curve admits a reparametrization c̃ = c(ψ) such that (t), c̃ (t)M in = 1 (for the opposite case of timelike curves, this would be called proper time parametrization). For curves with this property, the equation of motion simpliﬁes to (t) = −κ(t)Kc (t). Example 4.1. c(t) = (cosh(t), sinh(t)) is the analogue of a circle. It is parametrized with unit speed, and its curvature is constant equal to −1.