Toying with Spacetime Consider the two dimensional vector space R2, endowed with an inner product given by (v.w) = -20W + vw1 fo
r all vectors v = (10.01) and w = (wo, w.). This is called a pseudo-inner product on R2 and, when gen- eralized to R, is of fundamental importance in Einstein's special theory of relativity. More specifically, two-dimensional Minkowski spacetime, often denoted by Ri, has this metric structure with the zeroth- component denoting time), and it is called a toy spacetime since it resembles our actual four-dimensional spacetime with fewer spatial dimensions. Though not realistic, it can make some calculations easier to do, whence the descriptor "toy" in its name. (a) Show this inner product satisfies all the criteria of an inner product except the positive-definite property, (uu) > 0. (b) Determine all nonzero vectors satisfying (v, v) = 0. Such vectors are called null vectors (c) Determine all vectors satisfying (v.v) <0. Such vectors are called timelike vectors. (d) Determine all vectors satisfying (v, v) > 0. Such vectors are called spacelike vectors. (e) Make a sketch of Rſ and indicate the position of the null, timelike, and spacelike vectors.
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