Definition
Imaginary time is a mathematical technique — formally a Wick rotation — that replaces the ordinary time coordinate t with τ = it, where i is the square root of −1.
The substitution turns Lorentzian spacetime (one time and three space dimensions, with a minus sign in the metric) into Euclidean four-space (four space-like dimensions, all signs equal). What looked like time becomes geometrically indistinguishable from a fourth spatial direction.
Why it matters
How it works
In quantum mechanics, the probability amplitude for a particle to go from one point to another is a sum over all possible paths, weighted by the phase e^(iS/ℏ) where S is the action. This integral oscillates wildly and is hard to evaluate. Replacing t with iτ turns the oscillating exponential into a real, exponentially damped one — e^(−S_E/ℏ), where S_E is the "Euclidean action." The integral becomes mathematically well-behaved, and a similar trick is used in lattice gauge theory, condensed-matter physics, and quantum field theory generally.
The deepest application is in quantum gravity. A black-hole spacetime, when continued into imaginary time, becomes a Euclidean manifold with a smooth point at the would-be horizon. Demanding regularity at that point fixes the period of imaginary time — and that period is exactly ℏ/k T_Hawking, recovering Hawking's temperature from pure geometry. This is the cleanest derivation of black-hole thermodynamics.
In cosmology, Hartle and Hawking proposed a "no-boundary" wave function for the universe: integrate over all four-dimensional Euclidean geometries that have a smooth, rounded bottom in imaginary time. The Big Bang singularity disappears — replaced by a closed, edgeless surface, like the south pole of a sphere. In imaginary time the universe has no beginning; the singularity is an artifact of the analytic continuation back to real time.