Aikafunktion

Aikafunktion, or time function in the context of Lorentzian geometry and general relativity, is a function t: M → R defined on a spacetime (M, g) that is continuous and strictly increasing along every future-directed causal curve. If the function is smooth and its gradient is everywhere timelike, it is called a temporal function. If, in addition, its level sets t = const are Cauchy hypersurfaces, t is a Cauchy time function.

Time functions encode the causal structure of spacetime and allow a foliation by spacelike hypersurfaces. They are central in questions about the global geometry and the formulation of initial value problems for Einstein’s equations. The existence and regularity of time functions depend on causal conditions: strongly causal spacetimes admit time functions, while globally hyperbolic spacetimes admit smooth Cauchy time functions. Geroch’s splitting theorem shows that every globally hyperbolic spacetime is diffeomorphic to R × S, with S a Cauchy surface, providing a natural time parameterization. More refined results, such as those by Bernal and Sánchez, establish that globally hyperbolic spacetimes admit smooth Cauchy time functions.

Examples include Minkowski space, where the standard time coordinate t is a time function, and Friedmann–Lemaître–Robertson–Walker cosmologies, where cosmic time serves as a time function. Time functions are used to analyze causal relations, construct foliations, and set up initial data surfaces for numerical and analytical studies in relativity. The term aikafunktion reflects the word for time in some Nordic languages, highlighting cross-linguistic terminology for the same geometric concept.