Zeitvektor

Zeitvektor, literally "time vector" in German, is a term used in some German-language discussions of spacetime and temporal evolution. It denotes a vector associated with the temporal dimension in a spacetime model and is typically a timelike vector indicating the direction of time flow at a point or along a worldline.

Mathematically, a Zeitvektor can be realized as the coordinate basis vector in the time direction, commonly written as ∂/∂t in a coordinate chart, or more generally as a unit timelike vector field u^μ describing an observer's four-velocity: u^μ = dx^μ/dτ, with g_{μν} u^μ u^ν = -1 in the usual (-,+,+,+) signature. Thus the Zeitvektor captures the notion of temporal evolution within the chosen spacetime description and is closely related to the concept of an observer's frame of reference.

In the ADM formalism of general relativity, the flow of time is decomposed into a lapse function N and a shift vector N^i, and the unit normal to the spatial hypersurfaces often serves as a prototypical Zeitvektor. In this context, the Zeitvektor helps describe how spatial slices are threaded through spacetime and how time evolution is parameterized.

Usage and terminology vary: the term Zeitvektor is mainly found in German texts, while English-language literature tends to speak of timelike vectors, time directions, or four-velocity. The underlying idea, however, corresponds to the direction in spacetime that represents the passage of time for a given observer or along a worldline.

See also: four-velocity, timelike vector, spacetime, general relativity, Lorentzian geometry.