A FEYNMAN LECTURE AND OUR
1961 PH.D. THESIS ON AN
ENERGY THEORY OF GRAVITY
The main features of this new force were:
- The force had a classical limit.
- The classical limit was a 1/r² force.
- Like objects attracted each other.
The question meson physicists wanted to answer was “What kind of elementary particle(meson) was responsible for the new forces?” These three features come close to answering that question.
- From the existence of a classical limit, the gravitational meson had to have an integer spin. Half integer spin particles like the electron and neutrino cannot give rise to a classical field. (We explain why in the thesis.)
- The 1/r² nature of the force implies that the gravitational meson, now called a graviton, has zero rest mass, like the photon which is responsible for the 1/r² electric force.
- Odd integer spin forces, like the spin 1 photon, give rise to a repulsive force between particles. Even integer spins give rise to an attractive force between like particles. Since gravitation is attractive, the graviton had to be a spin 0, spin 2, or a higher even spin particle.
The choice of spin 0 was eliminated by further experiments showing that the source of gravitational fields was proportional to inertial mass, the m in E = mc². The next simplest candidate was spin 2 for the graviton.
Feynman then showed us how to construct a spin 2 theory of gravity. He started by showing how to derive Maxwell's theory for a spin one field coupled to a conserved electric charge. Then he did the same kind of steps to derive the theory of a spin 2 field coupled to its conserved source, namely, energy. The result turned out to be equivalent to what is known as the linearized approximation to General Relativity.
The meson physicists would have noticed a problem with this linear theory. The problem was that gravitons themselves have energy, which was not included in the original source of gravity. Gravitons have energy which also act at a source of gravity.
To fix his linear theory, Feynman looked for the most general Lagrangian that would handle the problem. In doing so, he found a symmetry in the theory that had a geometrical interpretation. Thus he concluded that meson physicists would have finally ended up with Einstein's geometrical approach to the classical theory of gravity, namely, General Relativity.
I went home and tried to repeat the seminar. The main test of the nonlinear theory of gravity was that it correctly predicted the 43 seconds of arc per century shift of the perihelion of Mercury's orbit. The linear theory gives the wrong answer.
I tried constructing a nonlinear theory as follows. First, I treated the sun as a point mass and used the linear theory to calculate its spin 2 gravitational field. Then I read textbooks on how to calculate the energy tensor for a spin 2 field. Next I calculated the energy of the gravitational field, and used this energy as a source for an additional gravitational field. Finally I calculated the motion of the planet Mercury in the combined sum of the two gravitational fields.
My work was sufficiently accurate to give the right answer for the perihelion shift of Mercury's orbit, but it did not. After carefully checking my work, I went to Feynman and said something was wrong with an energy theory of gravity. Feynman was surprised. I had constructed a relativistically invarient theory that took into account that gravity creates gravity, yet I got the wrong answer. But in the seminar Feynman had gotten a unique theory that gave the right answer. What was the difference between the two approaches?
We decided that Feynman got a unique theory because he derived it from an action principle. Thus my approach could not have come from an action principle. Something must have gone wrong with the energy tensors I used. I went back and essentially subtracted my theory from Einstein's to discover that the so-called energy momentum tensors are not unique. If I changed the energy (did not follow the textbook formulas) I could get the correct perihelion shift of Mercury. Thus I could show, not only were energy momentum tensors were not unique, but the lack of uniqueness had physical consequences. The first half of my thesis was a study of the lack of uniqueness of energy momentum tensors.
I wrote up the thesis more or less as a textbook to explain to other graduate students the contents of Feynman's seminar, and to explain how to construct energy momentum tensors. It took over 100 pages to do this clearly.
After completing my thesis I wrote a short summary paper for the American Journal of Physics. It seems that no one was interested in another approach to gravity or the lack of uniqueness of energy tensors, and I could not get the paper published. I just made sure that the thesis was available through a Thesis Service run by Xerox.
A decade later, Murry Gell-Man found that he had to use one of my energy momentum tensor terms to get rid of an infinity in his scale invarient theory of elementary particles. He named the terms he used the "Huggins Term". Since then there have been a number of papers using the Huggins term to get rid of divergent parts of their theory. All these papers refer to my unpublished thesis. But now, thanks to the Web, and Adobe pdf files, it is easy to make the thesis available. Here it is:
Quantum Mechanics of the Interaction
of Gravity with Electrons: Theory of
a Spin-Two Field Coupled to Energy