for light-rays impinging on a center of mass M with impact parameter b.

It is interesting to give a numerical evaluation of this effect in the case of light rays (actually any electromagnetic wave signal) passing close to the sun. In this case the maximal possible effect occurs when the impact parameter is the minimal conceivable one, namely when b = R6 equals the radius of our star. The astronomical

Although tiny this effect is observable and experiments are in full agreement with the theory. The first measure was performed during a solar eclipse in 1919. In the seventies it was confirmed with much higher precision by means of radar signals reflected from a satellite orbiting near the Sun.

References

1.Schwarzschild, K.: Über Das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Reimer, Berlin (1916), S. 189 ff. (Sitzungsberichte der Kniglich-Preussischen Akademie der Wissenschaften; 1916)

Chapter 5

Einstein Versus Yang-Mills Field Equations:

The Spin Two Graviton and the Spin One Gauge

Bosons

Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there. . .

Richard Feynman

5.1 Introduction

Enough evidence has been provided in previous chapters that the law of Newtonian mechanics:

relating the acceleration a of a point-particle of mass m to the total force F applied to it can be successfully replaced by the statement that the world-line of that particle should be a time-like (or null-like if m = 0) geodesic of the pseudo-Riemannian manifold (M , g) which describes space-time. In particular we saw that, by choosing g to be the Schwarzschild metric, we can retrieve the entire corpus of Newtonian Astronomy as brought to perfection by the monumental work of Laplace in his

Exposition du Système du Monde.1

The next question is: what fixes the choice of the metric? Newton theory is composed of two parts. Law (5.1.1) establishes how a particle reacts to the presence of a force-field, while the celebrated attraction law:

mM

F = −G (5.1.2) r2

determines the force on the particle of mass m generated by the presence of another mass M. In General Relativity, Einstein field equations, that constitute the topics of the present chapter, replace Newton’s law (5.1.2). They are 2nd-order differential equations to be satisfied by the metric tensor gμν (x), that are obtained by equating a certain two-index symmetric tensor Gμν , constructed with the components of the

1See Chap. 2 of Volume 2 for more information on Laplace and its contributions to gravitational theories.

Fig. 5.1 A graphical representation of Eddington’s sheet metaphor which summarizes the basic idea of Einstein Gravity Theory

four-index Riemann curvature tensor Rαβγ δ , encoding the geometry of space-time, to the stress-energy Tμν encoding the mass-energy content of the latter.

The fundamental geometric idea is summarized by the metaphor of Eddington’s sheet, shown in Fig. 5.1. Space-time is like a flat elastic sheet which is curved and bend by any heavy mass we might place on it. Light particles, who would just go along straight lines if the sheet were flat, twist instead their paths if the sheet is curved by its mass-energy content.

The path historically followed by Einstein to derive his own field equations was long and complex. Indeed it took about ten years, from the first conception of a generally covariant theory of gravity, to arrive at the publication of the 1915 article containing the final form of the field equations. Considering the matter a posteriori, from the vantage point of contemporary Theoretical Physics, we can present a few different but equally well knit chains of arguments that all lead to the same unique result found by Einstein. These different derivations emerge from the interplay of the two complementary aspects of any field theory, namely

1.the microscopic particle-theory aspect,

2.the macroscopic geometrical aspect,

with the two available formalisms one can utilize to discuss (pseudo-)Riemannian geometry namely:

1.the original metric formalism of Riemann, codified in the tensor calculus of Ricci and Levi Civita,

2.the vielbein or repère mobile formalism invented by Cartan.

The choice between the two formalisms is not only a matter of taste or convenience, but it is intimately related to two fundamental conceptual issues. The first is the question whether gravity is described by a gauge theory, as it happens to be true of all the other fundamental non-gravitational interactions. The second issue is even more fundamental and concerns the gravitational interaction of fermionic particles that, by definition, happen to be the quanta of spinorial fields. In the classical tensor