Strings

We have learned a great deal about black holes by considering the behavior of quantum fields near horizons. But ultimately local quantum field theory fails in a number of ways. In general the failures can be attributed to a common cause-quantum field theory has too many degrees of freedom.

The earliest evidence that QFT is too rich in degrees of freedom was the uncontrollable short distance divergences in gravitational perturbation theory. As a quantum field theory, Einstein's general relativity is very badly behaved in the ultraviolet.

Even more relevant for our purposes is the divergence in the entropy per unit area of horizons. Entropy is a direct measure of the number of active degrees of freedom of a system. Evidently there are far too many degrees of freedom very close to a horizon in QFT. We quantified just how over-rich QFT is.

The remaining portion of this book deals, in an elementary way, with a theory that seems to have just the right number of degrees of freedom: string theory. The problems posed by black holes for a fundamental theory of quantum gravity are non-perturbative. Until relatively recently, string theory was mostly defined by a set of perturbation rules. Nevertheless, even in perturbation theory, we will see certain trends that are more consistent with black hole complementarity than the corresponding trends in QFT.

We explained that the key to understanding black hole complementarity lies in the ultrahigh frequency oscillations of fluctuations of matter in its own rest frame. The extreme red shift between the freely falling frame and the Schwarzschild frame may take phenomena which are of too high frequency to be visible ordinarily and make them visible to the outside observer. As an example, imagine a freely falling whistle that emits a sound of such high frequency that it cannot be heard by the human ear. As the whistle approaches the horizon, the observer outside the black hole hears the frequency red shifted. Eventually it becomes audible, no matter how high the frequency in the whistle's rest frame.

On the other hand, the freely falling observer who accompanies the whistle never gets the benefit of the increasing red shift. She never hears the whistle.

This suggests that the consistency of black hole complementarity is a deep constraint on how matter behaves at very short times or high frequencies. Quantum field theory gets it wrong, but string theory seems to do better. The qualitative behavior of strings is the subject of this lecture.

In order to compare string theory and quantum field theory near a horizon, we will first study the case of a free particle falling through a Rindler horizon. As we will see, it is natural to use light cone coordinates for this problem. The process and conventions are illustrated in Figure 1.1. The coordinates X^± are defined by

 (1.1)

and the metric is given by

 (1.2)

where Xⁱ run over the coordinates in the plane of the horizon. We will refer to Xⁱ as the transverse coordinates, because they are transverse to the direction of motion of the point particle. The trajectory of the particle is taken to be

    (1.3)

As the particle falls closer and closer to the horizon, the constant τ surfaces become more and more light-like in the particle's rest frame. In other words, the particle and the Rindler observer are boosted relative to one another by an ever-increasing boost angle.

Near the particle trajectory X⁺ and τ are related by

 (1.4)

for large τ. This suggests that the description of mechanics in terms of the Rindler (or Schwarzschild) time be replaced by a description in light cone

Fig. 1.1. Free particle falling through a Rindler horizon.

coordinates with X⁺ playing the role of the independent time coordinate. We will therefore briefly review particle mechanics in the light cone frame.