HOME

TheInfoList



OR:

Imaginary time is a mathematical representation of
time Time is the continuous progression of existence that occurs in an apparently irreversible process, irreversible succession from the past, through the present, and into the future. It is a component quantity of various measurements used to sequ ...
that appears in some approaches to
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
and
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. It finds uses in certain
cosmological Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
theories. Mathematically, imaginary time is real time which has undergone a
Wick rotation In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that sub ...
so that its coordinates are multiplied by the
imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
''i''. Imaginary time is ''not'' imaginary in the sense that it is unreal or made-up; it is simply expressed in terms of
imaginary number An imaginary number is the product of a real number and the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square (algebra), square of an im ...
s.


Origins

In mathematics, the imaginary unit i is \sqrt, such that i^2 is defined to be -1. A number which is a direct multiple of i is known as an
imaginary number An imaginary number is the product of a real number and the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square (algebra), square of an im ...
. A number that is the sum of an imaginary number and a real number is known as a complex number. In certain physical theories, periods of time are multiplied by i in this way. Mathematically, an imaginary time period \tau may be obtained from real time t via a
Wick rotation In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that sub ...
by \pi/2 in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
: \tau = it.
Stephen Hawking Stephen William Hawking (8January 194214March 2018) was an English theoretical physics, theoretical physicist, cosmologist, and author who was director of research at the Centre for Theoretical Cosmology at the University of Cambridge. Between ...
popularized the concept of imaginary time in his book '' The Universe in a Nutshell''. In fact, the terms " real" and " imaginary" for numbers are just a historical accident, much like the terms "
rational Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
" and "
irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
":


In cosmology


Derivation

In the
Minkowski spacetime In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a s ...
model adopted by the
theory of relativity The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical ph ...
,
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
is represented as a four-dimensional surface or
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
. 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 mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
in the same way as a distance in space, an interval d in relativistic spacetime is given by the usual formula but with time negated: d^2 = x^2 + y^2 + z^2 - t^2 where x, y and z are distances along each spatial axis and t is a period of time or "distance" along the time axis (Strictly, the time coordinate is (ct)^2 where c is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
, however we conventionally choose units such that c=1). Mathematically this is equivalent to writing d^2 = x^2 + y^2 + z^2 + (it)^2 In this context, i 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 An imaginary number is the product of a real number and the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square (algebra), square of an im ...
, denoted by \tau. The equation may then be rewritten in normalised form: d^2 = x^2 + y^2 + z^2 + \tau^2 Similarly its four vector may then be written as ( x_0, x_1, x_2, x_3 ) where distances are represented as x_n, and x_0 = ict where c is the speed of light and time is imaginary.


Application to cosmology

Hawking noted the utility of rotating time intervals into an imaginary metric in certain situations, in 1971. In
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fu ...
, imaginary time may be incorporated into certain models of the
universe The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
which are solutions to the equations of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. 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 Hartle–Hawking state, also known as the no-boundary wave function is a proposal in theoretical physics concerning the state of the universe prior to the Planck epoch. It is named after James Hartle and Stephen Hawking. History According ...
). The
Big Bang The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including th ...
, 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 In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. Any boundary to spacetime is a form of singularity, where the smooth nature of spacetime breaks down. With all such singularities removed from the Universe, it thus can have no boundary and Stephen Hawking speculated that "the
boundary condition In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
to the Universe is that it has no boundary". However, the unproven nature of the relationship between actual physical time and imaginary time incorporated into such models has raised criticisms.
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
has noted that there needs to be a transition from the
Riemannian metric In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
(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 The Big Crunch is a hypothetical scenario for the ultimate fate of the universe, in which the expansion of the universe eventually reverses and the universe recollapses, ultimately causing the cosmic scale factor to reach absolute zero, an eve ...
. If this proves true, then the end-of-time boundary still remains.


See also

*
Euclidean quantum gravity In theoretical physics, Euclidean quantum gravity is a version of quantum gravity. It seeks to use the Wick rotation to describe the force of gravity according to the principles of quantum mechanics. Introduction in layperson's terms The Wick ...
*
Multiple time dimensions The possibility that there might be more than one dimension of time has occasionally been discussed in physics and philosophy. Similar ideas appear in folklore and fantasy literature. Physics Speculative theories with more than one time dimen ...


References


Further reading

* * *


External links


The Beginning of Time
— Lecture by Stephen Hawking which discusses imaginary time.

{{Webarchive, url=https://web.archive.org/web/20160303174253/http://www.pbs.org/wnet/hawking/strange/html/imaginary.html , date=2016-03-03 — PBS site on imaginary time. Quantum mechanics Philosophy of time