'Imaginary' Time and the Dreaded Singularity

taken from the book Schrodingers Kittens and the Search for Reality by John Gribbin

...cosmologists have a standard model of how the Universe works on the large scale, involving matter, gravity and the general theory of relativity. One of the big problems - perhaps the big problem - with the cosmologists' standard model, the Big Bang theory, is the presence of a singularity at the birth of the Universe. Astronomers know that the Universe is expanding because their telescopes show that galaxies are moving apart from one another. Einstein's general theory of relativity predicted this expansion, because the theory says that the space between the galaxies must be stretching as time passes. Both theory and observation suggest that if you imagine winding this expansion backwards in time to find out what the Universe was like long ago, you must reach a moment in time when all of the matter and all of the spacetime in the Universe was concentrated in a single point, the singularity.

A singularity is a place where the laws of physics as we know them break down. Taking the equations literally, it is a point of zero volume and infinite density, which seems absurd. Yet in the 1960s Stephen Hawking and Roger Penrose showed that if the general theory of relativity is an accurate description of the way the Universe works (which it certainly seems to be in the light of all the evidence, including that of the binary pulsar), then there is no escape from the requirement of a singularity at the beginning of time. The kind of expansion that we observe around us today, coupled with Einstein's equations, proves that there must have been a singularity in the beginning.

But is this disturbing conclusion simply the result of making the wrong kind of analogy? In the 1980s, Hawking returned to the puzzle of the origin of the Universe, and, with others, attempted to find a way to describe the Universe in a model which incorporates the ideas of quantum mechanics, as well as those of the general theory of relativity. This is the work which leads many cosmologists to feel that some variation on the 'many-worlds' or 'many-histories' idea is required, because there is no way an observer can be 'outside' the Universe to collapse its wave function from a superposition of states into a unique history. But there is another intriguing feature of Hawking's approach, a new analogy which gives a different perspective on the Big Bang.

I said before that there is an important difference in the way time and space are treated by the equations of relativity (both the special and the general theories). Time, I mentioned, appears in the equations with a minus sign in front of it. But this is not quite the whole story, because those equations also deal, like Pythagoras' famous theorem about right-angled triangles, in squares. So the parameters that represent spatial displacements in Einstein's equations are squares: x2, y2 and z2. The parameter that represents temporal displacement, however, is represented by a negative square: - t2. This is what prevents time from being treated in exactly the same way as space, because as we all learned in school you cannot take the square root of a negative number. If you know x2 then x has an easily understood meaning; the square root of 4, for example, is 2. But if you know - t2, what does that tell us about t? What is the square root of; say, minus 9?

Hawking pointed out that the problem of the singularity at the beginning of the Universe - the 'edge' of time - can be resolved by taking on board an almost trivial mathematical device. Mathematicians know all about the square roots of negative numbers. They have been a standard feature of mathematics for more than 200 years, and mathematicians are able to manipulate them in their calculations with the aid of a single simple trick. They invented a 'number' called i, which is defined as 'the square root of minus one'. So i x i is equal to - 1. If, now, you want to know the square root of -9, you say that -9 is equal to (-1) x 9, and that the square root is equal to the square root of - 1 multiplied by the square root of 9, which is simply i x 3. Such 'imaginary numbers' can be manipulated in the same way as ordinary numbers - addition, multiplication, division and all the rest - and are an important part of many mathematical calculations. They provide a model for mathematicians to use in describing the undescribable, the world of square roots of negative numbers; and they operate by analogy with the way 'real' numbers operate.

Hawking's stroke of chutzpah was to suggest that our everyday understanding of time is wrong, and that a better model of the way the Universe works is obtained by changing over to using measurements in what he calls imaginary time, i t. As far as the mathematics is concerned, this is a trivial change to make. It has about as much significance as a change in the choice of projection a map-maker uses in providing us with a picture of the Earth. For example, the traditional Mercator projection gives continents their correct shapes, by and large, but distorts their relative areas; while the Peters' projection, developed in the 1970s, shows continents in their correct relative proportions, but distorts their shapes. Both projections (and others) show the entire surface of the globe mapped on to a flat sheet of paper, and because it is impossible to represent the surface of a sphere perfectly on a flat sheet of paper no single projection can be said to be 'correct' while the others are 'incorrect'. They are just different.

In a similar way, mathematicians are free to choose many aspects of the coordinate systems they use in describing the positions of events in space and time. To take another geographical example, it is a historical accident that we choose to measure longitude relative to the meridian that passes through Greenwich, in London. Navigators could just as well use any of the other meridians, the imaginary lines joining the North and South poles of our planet, as 'longitude zero.

Hawking's switch to 'imaginary time' is not quite that simple, but it involves only a change in choice of mathematical coordinates, and it has the dramatic effect of putting the time parameter in Einstein's equations on exactly the same footing as the space parameters. If time is measured in units of i t, then when time measurements are squared we get units of i2 x t2 which is simply (-1) x t2, or - t2. Now, we have to multiply this negative number by the minus sign that comes into Einstein's equations themselves, which cancels out the (-1) we got from i2 and leaves us with just t2 (remember the old adage 'two negatives make a positive').

This change of model, or choice of a different mathematical analogy, has in effect made time, as far as Einstein's equations are concerned, exactly the same as space. And it turns out that this modest mathematical change removes the singularity from the equations.

The way we now have to think of the expanding Universe, says Hawking, is not in terms of a bubble of spacetime that appears out of a mathematical point (the singularity) and grows, but in terms of lines of latitude drawn on the surface of a sphere which stays a constant size. A tiny circle drawn around the north pole of the sphere represents the Universe when young - all of space is represented by the line that makes up the circle. As the Universe expands, it is represented by lines drawn further from the pole and closer to the equator, each one bigger than the previous circle. Moving from the pole towards the equator represents the 'flow' of time. Once past the equator, the 'universe' starts to shrink again as successive latitude circles get smaller, until they disappear at the south pole.

But what happens at the poles themselves - the beginning and end of time? There is no 'edge' to the sphere at these points, even though time is said to 'begin' at the north pole. Because time has been put on the same mathematical footing as space, the analogy with the geography of the Earth is perfect. At the North Pole of our planet, all directions are 'south', and there is no direction 'north' - but there is no edge to the planet there. At the north pole of Hawking's model of the Universe, all time directions are 'the future' and there is no direction of time corresponding to 'the past' - but there is no edge of time there. The singularity problem does not arise.

If you could travel backwards in time to the Big Bang itself, you would not disappear into a singularity, but would pass through the point (moment) of 'zero time' and find that you were heading off into the future again, in just the way that a person a little to the south of the North Pole of the Earth can walk due north, go across the pole and keep on walking in the same direction, but now finds that this direction is due south. The Universe is seen, on this picture, as a completely self-contained package of spacetime and mass-energy, expanding out of nothing and contracting back into nothing.

All this has been achieved by a simple coordinate transformation, putting time on an equal footing with space. It's unfortunate that in mathematical jargon numbers involving i are traditionally called 'imaginary' numbers, because that means that Hawking's alternative time coordinate goes by the name imaginary time', which makes it sound like something out of a science fiction story, or Alice in Wonderland. [...^] But this is, in fact, a mathematically respectable way of looking at things, which seems to be more physically reasonable than the traditional way of looking at things, since it does not contain the dreaded singularity.

There are other ways to explore the possibilities this raises. Hawking 'spatialized' time; Ilya Prigogine has said that his approach to a description of how things work is equivalent to 'temporalizing' space, treating creation as something that goes on everywhere in spacetime in some sense simultaneously.