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CHAPTER 1 Introduction
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at spaces. Instead we generalize to include spaces that are curved, like spheres or saddles. Now, since we are in a relativistic context, we need to include not just curved spaces but time as well, so we work with curved space-time. A general way to write Eq. (1.2) that will do this for us is ds 2 = g ( x )dx dx (1.3)
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The metric tensor is the object g ( x ), which has components that depend on space-time. Now we have a dynamical geometry that varies from place to place and from time to time, and it turns out that g ( x ) is directly related to the gravitational eld. Hence we arrive at the central truth of general relativity: gravity geometry
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Gravitational elds are essentially the geometry of space-time. The form of the metric tensor g ( x ) actually stems from the matter energy that is present in a given region of space-time which is the way that matter is the source of the gravitational eld. The presence of matter alters the geometry, which changes the paths of free-falling particles giving the appearance of a gravitational eld. The equation that relates matter and geometry (i.e., the gravitational eld) is called Einstein s equation. It has the form 1 R + g R = 8 GT 2 (1.4)
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where G = 6.673 10 4 m 3 / kg s 2 is Newton s constant of gravitation and T is the energy-momentum tensor. R and R are objects that depend on the derivatives of the metric tensor g ( x ) and hence represent the dynamic nature of geometry in relativity. The energy-momentum tensor T tells us how much energy and matter is present in the space-time region being considered. The details of the equation are not important for our purposes, just keep in mind that matter (and energy) change the geometry of space-time giving rise to what we call a gravitational eld, by changing the paths followed by free particles.
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A Quick Primer on Quantum Theory
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So matter enters the theory of relativity through the energy-momentum tensor T . The rub is that we know that matter behaves according to the laws of quantum theory, which are at odds with the general theory of relativity. Without going into
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String Theory Demysti ed
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detail, we will review the basic ideas of quantum mechanics in this section (see Quantum Mechanics Demysti ed for a detailed description). In quantum mechanics, everything we could possibly nd out about a particle is contained in the state of the particle or system described by a wave function:
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( x, t )
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The wave function is a solution of Schr dinger s equation:
2 + V = i
t
(1.5)
The wave function itself is not a real physical wave, rather it is a probability 2 amplitude whose modulus squared ( x, t ) (note that the wave function can be complex) gives the probability that the particle or system is found in a given state. Measurable observables like position and momentum are promoted to mathematical operators in quantum mechanics. They act on states (i.e., on wave functions) and must satisfy certain commutation rules. For example, position and momentum satisfy [ x, p ] = i (1.6)
Furthermore, there exists an uncertainty principle that puts constraints on the precision with which certain quantities can be known. Two important examples are x p / 2 E t / 2 (1.7)
So the more precisely we know the momentum of a particle, the less certain we are of its position and vice versa. The smaller the interval of time over which we examine a physical process, the greater the uctuations in energy. When considering a system with multiple particles, we have a wave function ( x1 , x 2 , , x n ) say where there are n particles with coordinates xi . It turns out that there are two basic types of particles depending on how the wave function behaves under particle interchange xi x j . Considering the two-particle case for simplicity, if the sign of the wave function is unchanged under
( x1 , x2 ) = ( x2 , x1 )