Eddington was a brilliantly gifted man, with the ability to combine mathematical skill with philosophical depth, but his investigation into two-dimensional time was eventually forgotten. But in 2007 Itzhak Bars of the University of Southern California in Los Angeles came up with a situation which two-dimensional time could solve.
He noted that in quantum theory there seems to be a deep link between position and momentum – in fact the two are linked in the uncertainty relationship which tells us that the more precisely we know the one, the less sure we can be about the other.
So he thought that to express quantum theory in the most natural way, we need a mathematics in which the two are on a completely equal footing. He looked for this; and he found that the only way he could get there was to add in two extra dimensions. One of the extra dimensions was of space, and the other one was of time. In a six-dimensional universe, with a total of four dimensions of space and two of time, he could get the deep underlying symmetry between position and momentum that he needed.
And strangely enough, this six-dimensional universe with its two dimensions of time was not so much beyond experience as we might have feared. The various rules of symmetry kept it quite close to ‘normal’. The two dimensions of time did not lead to time travel or any bizarre paradoxes.
He used a beautiful image to depict this six-dimensional world and its relation to us. The situation, he says, is rather like what happens when we hold our hand up by a lamp and see a two-dimensional shadow on the wall. Bars says that the six-dimensional world is the underlying one, and that it throws up a variety of forms of four-dimensional ‘shadows’ – of which our universe is one.
We live then in a Shadowland, a four-dimensional world which is rich and varied – but which is still only a slice of the bigger picture.
So here indeed is a strange new world to explore – first of all mathematically, and then by experiment, if suitable predictions can be made that we can try to test. And we have an image from the world of poetry to help us think about the new possibilities – a picture of time in two different dimensions, the time of the Wood of Hallaig and the time of the deer flitting through it.
And the words of the poet echo in our minds, as in a film made by Neil Kempsell with the voice of Sorley MacLean himself and music by the late and richly talented Martyn Bennett.