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The Kasner Metric
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The Kasner metric is a solution to the Einstein eld equations that has an interesting property that makes it useful from a string theory perspective. We can characterize the Kasner metric by considering the notion of isotropy. If space is isotropic, then it is the same in all directions. This is a reasonable assumption that is used routinely in cosmology when considering the 3 + 1 dimensional space-time we appear to
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CHAPTER 16 String Theory and Cosmology
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live in. On large scales, it doesn t matter which direction you look the universe looks the same. In contrast, the Kasner metric is anisotropic, meaning that not all spatial dimensions evolve in the same way. As time increases, the universe expands in n of the spatial directions but contracts in the other D n directions. So this metric could describe a universe in which some of the dimensions become small (compacti ed) as the universe evolves. As you might imagine, this makes the metric appealing within the context of string theory. The Kasner metric can be written in the following way: ds 2 = dt 2 + t
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j =1 2p D 2 pj
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(dx j )
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(16.6)
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The presence of the term t j multiplying each spatial direction dx j makes the behavior of each dimension dependent on the passage of time. We call the pj Kasner exponents and they must satisfy two conditions aptly named the Kasner conditions:
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j =1
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D 1
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=1 =1
(first Kasner condition) (16.7) (second Kasner condition)
( p )
j =1 j
D 1
The Kasner conditions enforce a constraint on the pj. What these tell us is that the pj cannot all have the same sign. Since the metric term related to each spatial dimension depends on t 2 p j, this tells us that some dimensions will expand as time increases and some will contract as time increases. That is If pj is positive, then t
2 pj 2 pj
> 1 and the direction x j is increasing with time. < 1 and the direction x j is shrinking with time.
If pj is negative, then t
To see this, note a simple illustration. Let pj = 0.2. Then at t1 = 5, we have t = 2p 50.2 = 1.38. At a later time t2 = 15, we have t j = 150.2 = 1.72 , so the dimension has increased by a factor of 1.72/1.38 1.25. Now suppose that instead pj = 0.2. At t1 = 5, 2p 2p we have t j = 5 0.2 = 0.72 . At a later time t2 = 15, we have t j = 15 0.2 = 0.58, so clearly the dimension is shrinking when the Kasner exponent is negative. When the Kasner metric is studied in string theory, it must be supplemented by equations for the dilaton eld f. The dilaton eld is related to the metric through the Kasner exponents pj. In particular, it is possible to take
D = 1 p j ln t j =1
(16.8)
String Theory Demysti ed
Interestingly, the dilaton eld introduces a type of duality into the model. In fact, this duality is related to T-duality, because it relates large and small distances. Given a set of Kasner exponents pj and a dilaton eld f, there exists a dual solution with p = p j j
= 2 p j ln t
j =1
(16.9)
Notice that since p = p j, expanding dimensions in the theory are the contracting j dimensions in the dual theory and vice versa. Pre-big-bang cosmology can be described in terms of this duality. It allows for the universe to go through the following stages of evolution: It starts out in a large, at, and cold state. It contracts to a self-dual point. The universe enters a state where it is small, highly curved, and very hot. This is the big bang. It enters an expansion phase which is the universe we live in. This was the rst attempt at a cosmological model using string theory. However, it has since been discarded in favor of brane-based cosmological models. This is because several problems with the model could not be resolved, and brane models of the universe are compelling because of how the elds of the standard model and gravity are described. Before going on to brane-world cosmology though, let s see how the Kasner metric can describe an accelerating universe. An interesting effect that can arise when considering some spatial dimensions contracting and others expanding is that the contracting dimensions actually cause the expanding dimensions to accelerate.1 Suppose that we have n > 1 contracting dimensions with three expanding spatial dimensions. It can be shown that they cause the three spatial dimensions not only to expand, but to do so in an in ationary manner without a cosmological constant. We write the number of space-time dimensions as D = n + 4, where we understand that the n dimensions which contract are all spatial and the remaining dimensions are 3 + 1 dimensional space-time. The metric can be written in a general form which is split between time, the expanding dimensions, and the contracting dimensions as D 1 2 3 ds 2 = dt 2 + a 2 (t ) dxi2 + b 2 (t ) dx m m=4 i =1 (16.10)
Levin, Janna, In ation from Extra Dimensions, Phys. Lett. vol. B343, 1995, 69 75.
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