It may be empty space but it's full of promise for the future of fundamental physics and cosmology - if only we can crack how much energy it contains, says Steven Weinberg.

The energy of empty space" may seem like an exceptionally unpromising subject. Isn't empty space just, well, empty? Not quite. In fact, our failure to understand the energy of empty space is the most important roadblock today to further progress in both fundamental physics and cosmology.

Heisenberg's uncertainty principle tells us that space cannot be truly empty. In its most familiar form, it states that it is impossible for the position of a particle and its rate of change - the particle's velocity - both to have definitive values at the same time. A similar uncertainty principle applies to fields, such as electric, magnetic or gravitational fields. It is impossible for a field and its rate of change to have any definite value at the same time. In particular, it is not possible for any of these fields to remain zero in supposedly empty space; if a field happened to be zero at one instant, then its rate of change could have any possible value, so the field could have any possible value at any later instant. These continually fluctuating fields give an energy to any volume of space itself, proportional to the volume, whether or not the volume contains ordinary matter.

Who cares? Normally, it is not energy that matters, but differences in energy. To learn how much energy we can extract from a ball rolling down a hill, we have to ask what is the difference in the gravitational energy of the ball on top of the hill and at its bottom. But the energy in space is there all the time, whatever else is present, and therefore has no effect on the differences in the energies of any two states.

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There is just one thing in nature that does respond to energy itself, and not just to differences in energies: gravitation. The gravitational field of the Sun receives contributions not only from the Sun's mass but also from its heat energy, so that the Earth goes around the Sun slightly faster than it would if the Sun were cold. There is a lot of space in the universe, and its energy contributes to the cosmic gravitational field, which in turn governs the way that the universe is expanding. This has provided an experimental upper limit on the energy of empty space. Depending on its sign, too much vacuum energy would have either stopped the expansion before now or speeded it up so much that there would have been no time for galaxies or stars to form.

So how much energy is there in a cubic foot of empty space? At first sight, it seems that the quantum fluctuations in the electric, magnetic and gravitational fields would give a cubic foot of vacuum an infinite energy, but this absurd result applies only if one includes the contributions of fluctuations of any possible wavelength, from 1ft down to zero. This is not justified, for we really do not know much about the behaviour of fluctuations of very small wavelength. If in this calculation we include only wavelengths larger than the shortest distance that has been probed in elementary particle accelerators, about 10 -16 cm, then the calculation gives a finite energy per cubic foot, but an energy that is larger than allowed by observations of how the universe is expanding - too large, in fact, by a factor of 10 56 ! If the vacuum energy were this large, then the universe would have evolved much too fast to allow time for the appearance of galaxies or stars or life.

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This is a puzzle, but not yet a paradox because there are other possible contributions to the vacuum energy. Einstein in 1917 proposed introducing a modification to the field equations of general relativity, known as the cosmological constant, that would be equivalent to giving empty space a constant energy per volume. So if quantum field fluctuations give empty space some huge positive energy, larger by a factor by 10 56 than is allowed by observation, we can just assume that Einstein's cosmological constant gives an energy that is equally huge, but negative, so that the two energies cancel. But to avoid a conflict with observation, the cancellation would have to be accurate to 56 decimal places, which seems pretty miraculous.

This puzzle provides a good illustration of the style of modern physics. It is not hard to invent a theory of vacuum energy that would agree with all the data. We can just follow Einstein and introduce a cosmological constant, adjusting its value so that the net vacuum energy is as small as needed to avoid conflict with observation. But our aim is not just to develop theories that agree with observation; our theories also have to explain why nature is the way it is. So far we have failed to reach this sort of understanding of the energy of empty space.

This mystery has been on physicists' minds for decades. Most of us assumed that there was some fundamental principle, not yet discovered, that required an exact cancellation of the vacuum energy. We thought that the largest failure of our most ambitious unified theories of gravitation and other forces was that they had given no rationale for such a cancellation. It was a great surprise then to learn from astronomical observations in 1999 that the energy of empty space, though vastly less than might have been expected, is apparently not zero. It seems to equal about twice the energy contained (according to the relation E=mc 2 ) in the masses of the constituents of the universe. (This presents us with another puzzle: the energy in each volume of space itself is presumably constant, while the energy per volume in particle masses decreases as the universe expands, so why should they have similar values at this particular moment in the history of the universe?) This value of the vacuum energy was inferred from a study of the expansion of the universe, as indicated by the way that distant galaxies are rushing away from us. If the galaxies we observe have been travelling at constant speed since the beginning, then the distance of any galaxy would be proportional to its speed. In the absence of avacuum energy, we would expect the galaxies to be slowing down under the influence of their mutual gravitational attraction, so that the speed they had in the time of emission of the light we see now would have been greater than the speed they have had since the light was emitted, and their distances would therefore be smaller than they would if their speeds were constant. In fact, it seems that the distances are larger than they would be if the speeds were constant, indicating that they are not slowing down but speeding up. This is just the effect that would be expected from a positive vacuum energy. Incidentally, if the expansion of the universe continues to accelerate, then there are galaxies now whose light will never reach us, and eventually all galaxies beyond our Local Group will be invisible to us.

Whether the vacuum energy is actually zero or has the value indicated by these recent observations, there is no question that it is incredibly tiny compared with what we would have expected from our estimate of the energy in quantum field fluctuations. Many explanations have been proposed. Perhaps some field automatically adjusts its value to cancel almost all of the vacuum energy. Perhaps some unknown physical principle dictates that the universe is evolving towards a state with zero vacuum energy, so the vacuum energy is small now because the universe is already pretty old. Perhaps our big bang is just one episode in a universe in which countless big bangs have occurred and will occur, each with different values for fundamental physical constants like the vacuum energy. In that case, any creatures that ask about the vacuum energy must be in a big bang where the vacuum energy happens purely by chance to be small enough to allow time for them to evolve to the point of asking the question. Whatever the true explanation, it is bound to be interesting.

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Steven Weinberg was awarded the Nobel prize for physics in 1979.

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