fields of Minkowski space are the 3 space translations, time translation, three generators of rotations (the little group) and the three generators of boosts. These are Time and space translations Vector fields generating three rotations, often called the J generators, Vector fields generating three boosts, the K generators, The boosts and rotations generate the Lorentz group. Together with space-time translations, this forms the Lie algebra for the Poincaré group. Killing fields in general relativity Killing fields are used to discuss isometries in general relativity (in which the geometry of spacetime as distorted by gravitational fields is viewed as a 4-dimensional pseudo-Riemannian manifold). In a static configuration, in which nothing changes with time, the time vector will be a Killing vector, and thus the Killing field will point in the direction of forward motion in time. For example, the Schwarzschild metric has four Killing fields: the metric is independent of , hence is a time-like Killing field. The other three are the three generators of rotations discussed above. The Kerr metric for a rotating black hole has only two Killing fields: the time-like field, and a field generating rotations about the axis of rotation of the black hole. de Sitter space and anti-de Sitter space are maximally symmetric spaces, with the -dimensional versions of each possessing Killing fields. Killing field of a constant coordinate If the metric coefficients in some coordinate basis are independent of one of the coordinates , then is a Killing vector, where is the Kronecker delta. To prove this, let us assume . Then and Now let us look at the Killing condition and from . The Killing condition becomes that is , which is true. The physical meaning is, for example, that, if none of the metric coefficients is a function of time, the manifold must automatically have a time-like Killing vector. In layman's terms, if an object doesn't transform or "evolve" in time (when time passes), time passing won't change the measures of the object. Formulated like this, the result sounds like a tautology, but one has to understand that the example is very much contrived: Killing fields apply also to much more complex and interesting cases. Conversely, if the metric admits a Killing field , then one can construct coordinates for which . These coordinates are constructed by taking a hypersurface such that is nowhere tangent to . Take coordinates on , then define