Killing Vector Fields
The Lie algebra of Killing vector fields of a spherically symmetric static spacetime takes the same form in the isotropic chart as in the Schwarzschild chart. Namely, this algebra is generated by the timelike irrotational Killing vector field
and three spacelike Killing vector fields
Here, saying that is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that the spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime. One immediate consequence is that the constant time coordinate surfaces form a family of (isometric) spatial hyperslices (spacelike hypersurfaces).
Unlike the Schwarzschild chart, the isotropic chart is not well suited for constructing embedding diagrams of these hyperslices.
Read more about this topic: Isotropic Coordinates
Famous quotes containing the words killing and/or fields:
“It’s a very delicate surgical operation—to cut out the heart without killing the patient. The history of our country, however, is a very tough old patient, and we’ll do the best we can.”
—Dudley Nichols, U.S. screenwriter. Jean Renoir. Sorel (Philip Merivale)
“On fields all drenched with blood he made his record in war, abstained from lawless violence when left on the plantation, and received his freedom in peace with moderation. But he holds in this Republic the position of an alien race among a people impatient of a rival. And in the eyes of some it seems that no valor redeems him, no social advancement nor individual development wipes off the ban which clings to him.”
—Frances Ellen Watkins Harper (1825–1911)