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CHAPTER 15 The Holographic Principle
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5. In AdS/CFT correspondence, the number of degrees of freedom is (a) Proportional to the area of the bounding surface and to Newton s gravitational constant. (b) Proportional to the area of the bounding surface and inversely proportional to Newton s gravitational constant. (c) Proportional to the area of the bounding surface and the fundamental string length.
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(d) Proportional to the area of the bounding surface and the string coupling constant.
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String Theory and Cosmology
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Conventional cosmology, which grew out of general relativity, astrophysics, and quantum eld theory, proposes that the universe began with a big bang at a nite time in the past with an in ationary rush, and will expand forever until the universe dies with a whimper, as a result of increasing entropy eventually sapping the useful life out of it. Proposals which originated in string/M-theory have led to different cosmological models. These models have the unexpected and shocking ability to describe the universe before the big bang. Based on a brane-world-type universe, they involve the collision of two branes which get rid of the singularity of bigbang theory and replace it with an eternal universe, which could be described as cyclic. In this chapter, we give an overview of some of the cosmological models that have arisen from string/M-theory. Unfortunately, the details of these models using string/M-theory are well beyond the scope of this book, so our description will be more of a qualitative nature. The motivated reader is urged to consult the references for details. Cosmology is sure to be an active area of research in the coming years with many new and possibly unexpected developments.
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String Theory Demysti ed
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Einstein s Equations
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In the previous chapter we introduced the Einstein eld equations, which give a classical description of gravity. In this chapter, we discuss the application of the Einstein eld equations to cosmology. For a detailed description of the study of cosmology in the context of general relativity, please see Relativity Demysti ed and any of the references contained therein. Cosmology is the study of the evolution of the universe as a whole. The starting point is the Robertson-Walker metric: ds 2 = dt 2 + a 2 (t )d 2 (16.1)
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Here, d 2 represents the spatial part of the metric. The function a(t ) is called the scale factor. It characterizes the spatial size of the universe and how it changes with time. The Hubble constant is given by H= a a (16.2)
We can characterize the spatial structure of the universe by a curvature constant K. If the space is at, has negative curvature (a saddle) or has positive curvature (a sphere), then K = 0, 1, + 1, respectively. Observational evidence indicates that our universe is at. The behavior of the universe with time is determined by starting with a given metric believed to describe the overall structure of the universe, and then using it to work out the components of the curvature tensor. Then we can solve the Einstein eld equations either with or without matter present. This can also be done with or without a cosmological constant. In standard cosmology treatments, space is assumed to be isotropic, meaning that it is the same in all directions. We may not want to make that assumption in string theory where some spatial dimensions are treated differently. There are two cosmological models that come up rather repeatedly. A de Sitter universe is one without matter (a vacuum solution of Einstein s eld equations), with at space, and a positive cosmological constant. An anti-de Sitter universe (sometimes denoted AdS) is a vacuum solution to the Einstein eld equations with positive cosmological constant and negative scalar curvature.
In ation
The cosmological models studied in relativity theory are only a part of modern cosmology. The second piece which is needed to explain known data is in ation. The standard big-bang model begins the universe with a singularity and it expands
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