Lecture 6 | Topics in String Theory

The speaker begins by setting the stage for a discussion on String Theory and its implications for understanding the entropy of black holes. They mention that a system with high entropy has a vast number of microscopic degrees of freedom that are not easily tracked. The entropy of a black hole, as proposed by Bekenstein, is said to be proportional to its surface area, but the nature of the microscopic degrees of freedom that carry this entropy was not clear from general relativity. String Theory offers a potential explanation by describing the microscopic objects that could be responsible for this entropy.

The speaker delves into the constants of String Theory and gravity, highlighting the difference between the Planck length (l_subp) and the string length scale (L_string). They discuss how these lengths are fundamental to the structure of strings and the gravitational force. The Planck length is derived from fundamental constants, while the string length scale is related to the size of a wiggling string. The speaker also introduces the string coupling constant (G), which is a probability amplitude for string interactions, playing a role similar to electric charge in electrodynamics.

The speaker explores how forces are thought of in electrodynamics and gravity, using Feynman diagrams as an analogy. They explain that the force between two charged particles is proportional to the product of their charges, while in gravity, the force is proportional to the square of the string coupling constant and the product of the masses, divided by the distance between them squared. The speaker also discusses the relationship between the small string coupling constant (G) and the large Newtonian constant (Big G), and how they are connected through the string length squared.

The speaker introduces the concept of entropy in the context of black holes and strings. They explain that the entropy of a black hole is proportional to the area of its horizon and can be expressed in terms of the mass of the black hole and the Planck length squared. For a string, when energy is added and it becomes highly vibrated and tangled, the entropy is related to the number of configurations of the string that are indistinguishable due to its complexity. The speaker uses a simple model of a string as a collection of links on a lattice to illustrate this point.

The speaker attempts to estimate the entropy of a vibrating string using a simplified model. They consider a string as a series of links on a lattice and discuss the number of possible configurations for a string starting from one end. They argue that the entropy of a string is proportional to the logarithm of the number of possible configurations, which in turn is proportional to the length of the string. The speaker also touches upon the concept of the mass of a link in string theory and how it relates to the string length.

The speaker describes a thought experiment where the string coupling constant (G) is slowly increased or decreased, allowing a string to morph into a black hole and vice versa. They explain that as G increases, gravity becomes significant and a highly energetic string can collapse into a black hole. Conversely, decreasing G from a black hole scenario would result in the black hole transitioning back into a string. The speaker emphasizes that during this adiabatic change, the entropy remains constant, which is a key point in deriving the entropy of a black hole from string theory.

The speaker identifies the transition point between strings and black holes as when the Schwarzschild radius of the black hole is comparable to the string length scale. They argue that a black hole smaller than the string length scale does not make sense in the context of string theory. The speaker also derives a relationship showing that the Schwarzschild radius equals the string length at the transition point, indicating a critical interplay between mass, the string coupling constant, and the Planck length.

The speaker discusses the principle of entropy conservation in adiabatic processes. They explain that when a system undergoes a slow change in a control parameter, such as the coupling constant G, the entropy remains constant. This principle is crucial for deriving the entropy of a black hole from the perspective of string theory, as it allows the speaker to equate the entropy of a black hole to that of a string after an adiabatic transformation.

The speaker outlines the strategy for deriving the entropy of a black hole using string theory. They propose starting with a black hole at a certain mass and coupling constant, then slowly changing the coupling constant until the black hole morphs into a string. Since the entropy is conserved during this adiabatic process, the entropy of the string can be used to determine the original black hole's entropy. The speaker highlights that this approach has been validated for various types of black holes, including rotating, charged, and higher-dimensional ones.

The speaker concludes the discussion by inviting the audience to further explore the concepts introduced in the talk. They direct interested individuals to visit Stanford University's website for more information, indicating a broader scope of study and resources available for those wanting to delve deeper into String Theory and its implications for understanding the nature of black holes and the fabric of spacetime.