Understanding Admit & Adapt: Timelike Killing Vector Field

[GR191511]

"A space-time is said to be stationary if and only if it admits a timelike Killing vector field"
"...given a a timelike Killing vector field,then there always exists a coordinate system which is adapted to the Killing vector field$X^a$，that is,in which$X^a=\delta^a_0$holds..."
How to understand "admit"and"adapt"?Does it mean that $X^a=\delta^a_0$ may not hold even if there exists a timelike Killing vector field?Thanks!

 

[Ibix]

Admits, in this context, is a fancy way of saying "has". Stationary spacetimes are those that have a timelike Killing vector field.

"Adapted to" means "well matched to". You can always pick a coordinate system with a timelike basis parallel to the Killing field. It's often a good idea because then your definition of space can be independent of time, and Schwarzschild coordinates are an example of this. But as always with coordinates you are free to make any choice you like. In Kruskal-Szekeres coordinates, for example, the integral curves of the Killing vector field form hyperbolas. You don't even have to have a timelike basis vector if you don't want to.