by John Gribbin
In an introductory commentary to Gravitation, Brian Hatfield emphasizes that this need to develop his own understanding of any problem he worked on was typical of Feynman, who for many years had the slogan, ‘What I cannot create, I do not understand’ written on the corner of one of the blackboards in his office. If Feynman wanted to study gravity, the only way he could do it was by creating his own theory of gravity, not by looking for ways to improve Einstein’s theory. Hatfield describes Feynman’s approach to gravity theory as being ‘from the bottom up, instead of from the top down’, in contrast to the top-down approach of Einstein himself, based on a geometrical description of spacetime in four dimensions, which is the way students are usually introduced to the subject.3
Hatfield also comments on Feynman’s sometimes cavalier way with conventions such as the way indices are written in mathematical equations: ‘Feynman once told me that getting minus signs, and factors of i, 2 and pi down correctly was something to be bothered with only when it came time to publish the result.’ In the first six of the gravity lectures, Feynman wrote almost every index down (for example, xi) where the usual convention is to have them up (in this case, xi). This doesn’t matter at all as long as the use is consistent, but Hatfield has restored the more familiar convention in the book, where he mentions the first time he saw Feynman’s van, in 1981 in a parking lot at Caltech. This was the famous van covered in Feynman diagrams, and he knew who it belonged to because ‘the diagram on the back, the only diagram with labels, had all indices in the down position … After looking in one of the windows of the van and seeing a bale of hay in the back, my suspicion that the van was Feynman’s was confirmed.’ (There was a perfectly logical explanation for the bale of hay, since Michelle was a keen horserider; but to Hatfield only Feynman would be driving around campus with a bale of hay.)
We should stress that there is still no completely satisfactory quantum theory of gravity. The approach pioneered by Feynman works very well at reproducing the successes of Einstein’s approach in describing the Universe at large, the orbits of the planets around the Sun and so on. But, also like Einstein’s version, it is less successful in providing a description of what goes on in the true quantum realm, at very high energies and over very short distances. Nevertheless, the successes are striking, not least because gravity is so weak. The electrical force between two electrons, for example, is a little over 4 × 1042 times as strong as the gravitational force between the same two electrons. Because of this, you need to put a lot of particles together in one lump before their combined gravitational influence on any one particle in the lump is as strong as the influence of neighbouring particles on each other produced by electromagnetic forces. In the first of his lectures on gravity, in which he provides a broad overview of the subject, this leads Feynman to consider, in a spirit of open-mindedness, some extreme possibilities. ‘I would like to suggest’, he says, ‘that it is possible that quantum mechanics fails at large distances and for large objects. Now, mind you, I do not say that I think quantum mechanics does fail at large distances, I only say that it is not inconsistent with what we do know.’ And he explains that in this context a ‘large’ object would be one with a mass of about one hundred-thousandth of a gram, containing about a billion billion particles. In an aside to his main theme, talking in 1962, he says that we must ‘not neglect to consider’ that it is possible for quantum mechanics to fail on this scale, because of some process involving gravity, and that this could resolve such puzzles as the ‘Schrödinger’s cat paradox’.
The ‘paradox’ is actually a kind of reductio ad absurdum which Erwin Schrödinger put forward in 1935 to show how ridiculous the standard interpretation of quantum mechanics was (this, remember, was after Schrödinger had said he didn’t like quantum mechanics and wished he’d never had anything to do with it). The puzzle concerns an (imaginary!) cat locked up in a room with a quantum device that has a 50:50 chance of triggering a cat-killing mechanism. Because the so-called Copenhagen Interpretation (developed by Niels Bohr and others at the end of the 1920s) says that it is the act of looking to see whether the quantum device has been triggered or not which ‘collapses the wavefunction’ and makes it decide what state it is in, it can be argued that the cat itself is neither dead nor alive, but exists in a ‘superposition of states’ until somebody opens the door of the room and takes a look.
Although it looks ridiculous when pushed to such extremes, nevertheless (and in spite of Schrödinger’s best efforts) this Copenhagen Interpretation, involving a role for the observer in determining the behaviour of the quantum world just by looking at it, is the standard picture that has been taught since the 1920s. So the idea of a gravitational (or any other) explanation for the distinction between the everyday world and the quantum world, removing the role of the observer, has obvious appeal; it has recently been revived and is widely discussed today (although seldom, if ever, with credit to Feynman’s insight).4
As for the cat puzzle itself, Feynman clearly expresses his objections to the conventional explanation of how the quantum world works, the Copenhagen Interpretation which holds that it is the act of observation which forces the quantum world to choose one reality out of the array of probabilities described by the wave function. To Feynman,
This is a horrible viewpoint. Do you seriously entertain the thought that without an observer there is no reality? Which observer? Any observer? Is a fly an observer? Is a star an observer? Was there no reality in the universe before 109 BC when life began?5
He also deliberates on the ‘many worlds’ idea, that the Universe constantly splits into slightly different versions of reality every time it is faced with a ‘choice’ at the quantum level, and points out that according to the conventional understanding of quantum mechanics this is the only way to describe the entire Universe, in terms of ‘a complete Monster Wavefunction’, because there is no outside observer to ‘collapse the wavefunction’ and bring one of the possible quantum realities into a unique existence. This is precisely the line that has been taken up by a leading school of cosmologists in recent years, leading to a quantum description of the Universe which is based on a combination of the many worlds idea and the sum over histories approach; one of the leading lights of this school has been James Hartle, one of those students from Feynman’s gravity course. What Feynman himself described in 1963 as ‘very wild speculations’ are part of mainstream discussions today.
Feynman’s own discussions of the cosmological and astronomical implications of his work look, with hindsight, even more prescient. He stresses the importance of the fact that everywhere we look in the Universe we see objects that are far from equilibrium, with hot stars pouring energy out into a cold Universe. Studies of non-equilibrium states are also at the forefront of physics today, where researchers are trying to find out how complexity (including life) can arise out of chaos.6
But perhaps the most staggering insight into Feynman’s feel for physics comes from the way he pre-empted, twenty years in advance, the theory of the origin of the Universe that now goes by the name ‘inflation’. The key to this picture of how something appeared out of nothing at all some 15 billion years ago is the realization that the energy in a gravitational field associated with an object of mass m is not only negative, but exactly balances the rest mass energy of the particle, mc2. The way to picture this is to imagine taking all the constituents of the mass m and spreading them out until they are infinitely far apart. Because the gravitational force between the particles goes as one over the separation squared, when the separation is infinite the force goes as one divided by infinity, which is zero. So the constituents can do no work on each other – they can-not tug each other about – when they are infinitely far apart, which means that the energy of the gravitational field is zero in that situation.
Now, imagine the constituents falling together to make the mass m.* Because gravity is an attractive force, the constituents release energy as they fall together. This is why collapsing clouds of
gas in space get hot in the first place, as they shrink down to form protostars; energy comes out of the gravitational field as the cloud collapses, and this heats the cloud up. But if you start with zero energy, and take energy out of the field as the object collapses, that means that for everyday objects the energy in the associated gravitational field is negative! Indeed, if you were to collapse the object all the way down to a mathematical point (a singularity), the energy of the associated gravitational field would indeed be –mc2. Interestingly, although the exact balance between rest mass energy and gravitational energy comes naturally out of the General Theory of Relativity (either Einstein’s version or Feynman’s version), in Newtonian theory the gravitational field ends up with infinite negative energy, which would be even harder to comprehend.
This curious fact – the balance between mass energy and gravitational energy – had been known (as a mere curiosity) for about twenty years by the time Feynman gave his lectures on gravitation. Back in the 1940s, on a visit to Einstein in Princeton, the pioneering cosmologist George Gamow casually mentioned, while they were out walking, that a colleague, Pascual Jordan, had realized that a star might be made out of nothing, since at the point zero its negative gravitational energy is numerically equal to its positive rest mass energy.
Einstein stopped in his tracks, and, since we were crossing a street, several cars had to stop to avoid running us down.7
In spite of its impact on Einstein, Jordan’s idea was regarded as no more than a curiosity, and probably Feynman had never heard of it. Certainly nobody had thought of applying it to the Universe as a whole. In 1962, the idea that the Universe had a definite beginning – the Big Bang – was still very much in doubt, and the famous ‘three degrees Kelvin’ background radiation which is regarded as the echo of the Big Bang had yet to be discovered. The rival Steady State hypothesis, which holds that the Universe has existed in more or less its present form for ever, was still very much a viable alternative, and was, indeed, discussed at length by Feynman in his gravity lectures. But he was also deeply impressed by the possibility ‘that the total energy of the universe is zero’. He pointed out that ‘it is exciting to think that it costs nothing to create a new particle’, and went on to say that:
We get the exciting result that the total energy of the universe is zero. Why this should be so is one of the great mysteries – and therefore one of the most important questions of physics. After all, what would be the use of studying physics if the mysteries were not the most important things to investigate?8
All of this requires that the amount of matter in the Universe should be just enough to match the so-called ‘critical’ density, for which spacetime is described as being flat and the Universe is just poised on the knife edge between expanding for ever or one day recollapsing into a big crunch. For the critical density (and only the critical density) the Universe does indeed tend to disperse itself to infinity, like our imaginary mass m, and end up hovering in a stationary, infinitely spread-out state.
This requires the presence of large amounts of ‘dark matter’, not yet directly detected but now very fashionable in cosmology, not least because improved observations of the way galaxies move have shown that they are indeed being tugged on by the gravitational influence of large amounts of dark stuff. But this view was distinctly unfashionable in 1962. That didn’t worry Feynman, who said that ‘the critical density is just about the best density to use in cosmological problems’, largely because it is the density for which the creation of matter ‘costs nothing’. But he still cautioned against accepting the idea just because it was so attractive:
It is exciting to speculate that it indeed is the ‘true’ density – yet we must not fool ourselves into thinking that a beautiful result is more reliable simply because of its ‘beauty,’ which is in part an artificial result of our assumptions.9
The idea that the Universe might have appeared in this way out of nothing at all passed completely unnoticed by the cosmologists, and was reinvented, independently, by Edward Tryon, of City University in New York, in 1973. Nobody took much notice even then (although the idea was published in the journal Nature), because it seemed that a tiny seed of the Universe, created out of nothing but containing as much mass as our entire Universe, would immediately collapse back into a singularity because of its own intense gravitational pull. But at the end of the 1970s and in the early 1980s, several researchers (most notably, Alan Guth in the United States and Andrei Linde in the Soviet Union) developed the idea of inflation, a kind of antigravity that would operate in the first split-second of the existence of the Universe, whooshing it up in size from something much smaller than a proton to something roughly the size of a grapefruit, and giving it so much outward thrust that even after inflation switched off and gravity began its work of pulling things back together the expansion would continue, slowing all the time, for tens of billions of years, allowing a Universe like the one we see around us to develop.10 None of these pioneers seems to have been aware that a central plank in their platform, the possibility of creating a Universe out of nothing at all because of the balance between mass energy and gravitational energy, had first been suggested by Richard Feynman in 1962. To someone (JG) who studied cosmology in the 1960s, and followed and reported on the developments in the 1970s and 1980s that led to the inflationary scenario becoming accepted as the standard paradigm, it was a breathtaking revelation to open up Gravitation in the summer of 1995 and find the extent of the insights provided by Feynman so long ago.
Perhaps this should not have come as quite such a surprise, though, because thanks to Willy Fowler I already knew about one of Feynman’s astrophysical insights, which is highlighted in a foreword to Gravitation, by John Preskill and Kip Thorne.
As we have already mentioned, it was early in 1963, shortly after the objects now known as quasars were discovered, that Fred Hoyle gave a seminar at Caltech in which he suggested that these objects might be superstars, and both Hoyle and Fowler were astonished when Feynman immediately pointed out that effects described by the General Theory of Relativity would make such supermassive stars unstable. The background to this ‘bolt from the blue’, as it seemed to Fowler and Hoyle, has now been pieced together by Preskill and Thorne, and part of the story is presented by Feynman in Lecture 14 of the gravity series. It seems that early in January 1963 Feynman visited the astrophysicist Icko Iben, then working at the Kellogg Radiation Laboratory at Caltech, and showed Iben the basic set of equations required to describe the structure of a star, taking full account of the General Theory of Relativity. Feynman had worked these out himself, from first principles. He asked Iben how astrophysicists used the equivalent, much simpler, Newtonian equations to make theoretical models of the behaviour of ordinary stars, where general relativistic effects are not important. Iben showed him. Those classical calculations of stellar structure represented the culmination of about 30 years’ work by astrophysicists. A few days later, Feynman came to see Iben again. ‘Feynman flabbergasted me,’ Iben recalls, ‘by coming in and telling me that he had [already] solved the … equations. He told me that he was doing some consulting for a computer firm and solved the equations in real time on what must have been that generation’s version of a workstation.’11 A couple of days after that, on 28 January, Feynman delivered Lecture 14 in the gravity series. It describes a fully general relativistic model of supermassive stars, still valid today, although Feynman’s interpretation of his calculations is not quite correct. It was a few weeks after this lecture that Hoyle gave the now-famous talk at Caltech.
Impressive though all this is, it was to some extent peripheral to the main object of Feynman’s investigation of gravity, which was to develop a complete quantum theory. Without ever completing the work, he pointed the way very clearly for the next generations of researchers. Just as in QED, in quantum gravity Feynman diagrams without any ‘loops’ describe interactions that follow the rules of the classical theory. In QED, you can add one loop to the diagram, and ca
lculate the resulting quantum correction, then add two loops, then three, and so on, developing an ever more accurate calculation (for example, of the magnetic moment of the electron) as long as you have sufficient patience and enough computer power to carry out the calculations (this is the perturbation approach mentioned by Hartle). For gravity, largely because of the way in which gravitons can interact with each other, even setting up the right equations to solve is more difficult, and when you do set them up they are plagued by infinities. Feynman only ever got as far as making the one-loop correction – itself a considerable achievement, accomplished in the summer of 1962 (it was probably this success which encouraged him to give the lectures on gravitation a few months later). Importantly, Feynman found that in order for this approach to work at all, he had to include the influence of ‘ghost’ fields, responsible for the presence of particles which exist only as self-contained loops in the Feynman diagrams and have no ‘real’ existence at all. It was this insight which enabled others to take the approach further, developing the techniques to describe how effects involving larger numbers of loops (‘higher-order’ calculations) should be included in the calculations, using path integral techniques. According to University of Texas researcher Bryce DeWitt, one of the leading investigators of quantum gravity today, ‘his work on quantum gravity ultimately had great impact on the standard model and on the quantization of gauge fields in general … people are well aware of his contribution’.12 Modern quantum gravity theory is one of the most exciting developments in theoretical physics, and it has Feynman’s fingerprints all over it. Feynman himself was happy with his achievement:
I feel I have solved the [problem of the] quantum theory of gravity in the sense that I figured out how to get the quantum principles into gravity. The result is a nonrenormalizable theory, showing it to be an incomplete theory in the sense that you cannot compute anything. But I am not dissatisfied with my attempt to put gravity and quantum mechanics together. I accept whatever consequences that this putting together produces, mainly that it can’t be renormalized. I was slightly disappointed that I did it only to lowest order. I could not figure out what to do with arbitrary numbers of loops, which was later solved by others, but I was not dissatisfied with that. The fact that the theory has infinities never bothered me quite so much as it bothers others, because I always thought that it just meant that we’ve gone too far: that when we go to very short distances the world is very different; geometry, or whatever it is, is different, and it’s all very subtle.13