This figure displays the two-dimensional surface constructed from the pair of timelike worldlines and . The curve connecting to is a spacelike geodesic with length . Along these geodesics . Note that the tangent vectors and are unit vectors but they need not be mutually orthogonal.Reuse & Permissions
Figure 2
In this figure the lower (straight) curve is the geodesic that joins to . The upper curve (which is not shown in Fig. 1) arises from the intersection of the Cauchy surface with the two-dimensional surface . The unit vectors and are orthogonal. Note that, in general, is not constant on each Cauchy surface.Reuse & Permissions
Figure 3
This is a situation that we explicitly exclude. In this case the points and are so far apart that the geodesic intersects the Cauchy surface at points other than and . In this case and have the same signs, contrary to the assumptions made in the text.Reuse & Permissions