Imaginary time

Concept in special relativity
Imaginary time is a mathematical representation of time that appears in some approaches to special relativity and quantum mechanics. It finds uses in certain cosmological theories. Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary unit i. Imaginary time is not imaginary in the sense that it is unreal or made up, it is simply expressed in terms of imaginary numbers.

Hawking noted the utility of rotating time intervals into an imaginary metric in certain situations, in 1971. In physical cosmology, imaginary time may be incorporated into certain models of the universe which are solutions to the equations of general relativity. In particular, imaginary time can help to smooth out gravitational singularities, where known physical laws break down, to remove the singularity and avoid such breakdowns (see Hartle–Hawking state). The Big Bang, for example, appears as a singularity in ordinary time but, when modelled with imaginary time, the singularity can be removed and the Big Bang functions like any other point in four dimensional spacetime. Any boundary to spacetime is a form of singularity, where the smooth nature of spacetime breaks down. Roger Penrose has noted that there needs to be a transition from the Riemannian metric (often referred to as "Euclidean" in this context) with imaginary time at the Big Bang to a Lorentzian metric with real time for the evolving Universe. Also, modern observations suggest that the Universe is open and will never shrink back to a Big Crunch. If this proves true, then the end of time boundary still remains.