spacelikeseparated

Spacelike separation, or spacelike separated, refers to a relationship between two events in spacetime in which the interval between them is spacelike. In such a case no signal or influence traveling at or below the speed of light can connect the events; their light cones do not intersect, and they lie outside each other's light cone. Spacelike separation is a Lorentz-invariant property: whether two events are spacelike separated does not depend on the observer.

Mathematically, for events with coordinates (t1, x1, y1, z1) and (t2, x2, y2, z2) and c the speed of light, the spacetime interval is s^2 = c^2 (t2 - t1)^2 - [(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]. If s^2 < 0 (in the common (+,-,-,-) sign convention), the separation is spacelike; equivalently, the spatial distance Δr = sqrt((Δx)^2 + (Δy)^2 + (Δz)^2) exceeds c|Δt|. For s^2 > 0 it is timelike; s^2 = 0 is lightlike (null).

Implications: Since they cannot be causally related, events separated spacelike cannot influence each other. There exists a frame in which the two events are simultaneous. In practice, spacelike separation is important in discussions of locality and causality in relativity and in quantum experiments, such as Bell tests, where measurements are arranged to be spacelike separated to prevent any subluminal communication from explaining correlations.

Related concepts include the light cone, timelike separation, and null (lightlike) separation.