Imaginary time is a mathematical representation of time that appears in some approaches to special relativity and quantum mechanics.
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.
In certain physical theories, periods of time are multiplied by
Mathematically, an imaginary time period
[1]: 769 Stephen Hawking popularized the concept of imaginary time in his book The Universe in a Nutshell.
"One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world.
From the viewpoint of positivist philosophy, however, one cannot determine what is real.
All one can do is find which mathematical models describe the universe we live in.
It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons.
"In fact, the terms "real" and "imaginary" for numbers are just a historical accident, much like the terms "rational" and "irrational": "...the words real and imaginary are picturesque relics of an age when the nature of complex numbers was not properly understood.
"In the Minkowski spacetime model adopted by the theory of relativity, spacetime is represented as a four-dimensional surface or manifold.
Its four-dimensional equivalent of a distance in three-dimensional space is called an interval.
Assuming that a specific time period is represented as a real number in the same way as a distance in space, an interval
in relativistic spacetime is given by the usual formula but with time negated:
are distances along each spatial axis and
is the speed of light, however we conventionally choose units such that
Mathematically this is equivalent to writing
may be either accepted as a feature of the relationship between space and real time, as above, or it may alternatively be incorporated into time itself, such that the value of time is itself an imaginary number, denoted by
The equation may then be rewritten in normalised form:
is the speed of light and time is imaginary.
Hawking noted the utility of rotating time intervals into an imaginary metric in certain situations, in 1971.
[4] 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.
[1]: 769–772 With all such singularities removed from the Universe, it thus can have no boundary and Stephen Hawking speculated that "the boundary condition to the Universe is that it has no boundary".
[2]: 85 However, the unproven nature of the relationship between actual physical time and imaginary time incorporated into such models has raised criticisms.
[5] 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.