Y1

For the warp-drive spacetime in Example 7.4, show that, at every point along the curve xs(t), the four-velocity of the ship lies inside the forward light cone.In the warp-drive spacetime in Example 7.4, how much ship time elapses on a trip between stations that takes coordinate time T?

Example 7.4. Warp-Drive Spacetime. This example, due to Alcubierre (1994), uses coordinates (t, x, y, z) and a curve x = Xs (t), y : 0, z = 0, lying in the t-x plane passing through the origin. The line element specifying the metric is ds2 = - dt2 + [dx - V, (t) f(rs)dt12 + dy? + dz2, (7.24) where Vs (t) = dx; (t)/dt is the velocity associated with the curve and rs = [(* - *s (t))2 + y2 + z2]. The function f (rs) is any smooth positive function that satisfies f(0): and decreases away from the origin to vanish for rs > R for some R. Evaluating (7.24) on at = constant slice of spacetime gives dS2 dx- + dy- + dz'. The geometry of each spatial slice is flat and r's is just the usual Euclidean distance from the curve xs (t). Spacetime is flat where f(rs) vanishes, but curved where it does not. Figure 7.2 is a spacetime diagram of the t-x plane. The shaded region is where spacetime is curved. The light cones at a point in the t-x plane are the curves emerging from the point with ds- = 0, that is, with ds- = -dt? + [dx - V, (t) f(rs)dt]? = 0, (7.25) or, equivalently, dx = 1 + Vs (t) f (rs). (7.26) dt