Problem 2: 4-Vectors and one-forms Problem (30) in Chapter 3.10 in B. Schutz: Two vector fields (vectors as functions of coordinates t and x) and D are given by U = 1+t2, U1 = t?, U2 = V2t, U3 = 0 and D = r, D+ = 5tx, D2 = V2t, D3 = 0. Also, a scalar p has the value p= 22 +? y2. Here we set c=1. Find: (a) Find - , - D, D D. Is suitable as a four-velocity field? Is D? (b) Find the spatial velocity v of a particle whose four-velocity is , for arbitrary t. What happens to it in the limits t 0,1 ? (c) Find Ug for all a. (d) Find U",s for all a,B. (e) Show that UQ UC,= 0 for all B. Show that UUQ,= 0 for all p. (f) Find D.B. (g) Find (U D),for all a. (h) Find U,(UD), and compare with (f) above. Why are the two answers similar? (i) Find p,g for all a. Find pill for all a. (Recall that pill := napP.B.) What are the numbers {p.C} the components of? Note the notation in the book: arrows indicate 4-vectors, while commas in front of indices indicate partial derivatives, i.e.: U", 1,8 = 2:UM [1 point each, 9 points in total]