Time+in+Geometry

This started off as a few questions on the Postmodern Page, but I think that the concept of time will provide enough discussion to warrant its own page. Feel free to add any information or ponderable questions related to time in geometry to this page, and we can discuss on the discussion page.

The initial questions were: How can we represent time in geometry? Is it simply another dimension to be added or is there something that sets it apart from spatial dimensions, as there clearly is in our experience? Can a point or object be simply represented as a line (curve) moving through space freely but limited to forward movement through time? In geometries used to model social networks and similar phenomena, how can points 'pop' into the geometry when, for example, somebody new joins Facebook?