7.9 Domain Walls in Diverse Space-Time Dimensions |
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Obviously the solution (7.9.31) could have been obtained by directly solving the Einstein equations associated with the action (7.9.5) starting from a conformal ansatz of the type:
dsDW2 /conf = exp A(z) ημν dxμ dxν + dz2 |
(7.9.32) |
Yet we preferred to obtain it from the general solution (7.7.8) for supergravity p- branes in order to emphasize its interpretation as a domain wall, namely a (D − 2)- brane. The direct method of solution can be used to find the conformal representation of the domain wall metric in the exceptional case = −2. As shown in [60] one obtains:
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|z| ημν dxμ dxν + dz2 |
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d − 2 |
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where k is now given by |
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(7.9.34) |
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which is real for negative Λ. There is another important point that we should note. Our starting point, prior to all the subsequent manipulations, has been the form (7.9.6), (7.9.9) which is that of an electric p-brane and not that of a solitonic one (see 7.7.10)). This implies that our domain wall solutions are not exactly bona fide solutions of the action (7.9.5) but require also the coupling to a source term that is the world-volume action of the domain wall, localized at z = 0 in the last coordinate frame we have used. Namely the true action is
A = M |
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∂μφ∂μφ − 2Λe−aφ |
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dD−1ξ Lsource |
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(7.9.35) where Lsource is world-volume Lagrangian of the (D − 2)-brane and the parameter T denotes its tension. An important issue is to relate the wall-tension to the parameters appearing in the classical domain wall solution. This was done in [60] following a standard analysis developed in previous papers [58, 59]. The matching conditions across the singular domain wall source imply that the energy density (tension) of the wall is related to the values of the cosmological constant parameters on either side of the wall, namely the authors of [60] found:
σ = T = 2 Az=0− − Az=0+ |
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where the prime denotes a derivative with respect to z. This leads to |
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298 |
7 The Branes: Three Viewpoints |
Fig. 7.3 The volcano potential
Thus positive-tension domain-wall solutions exist for ≤ −2 with k > 0 and for > −2 with k < 0. Conversely, negative-tension domain walls arise for ≤ −2 with k < 0 and for > −2 with k > 0. So for our domain walls with ≤ −2, we assume the lower bound (7.9.11). To avoid naked singularities we also need k > 0. Using the simple conformal gauge (7.9.31) the authors of [60] have analyzed the fluctuations of the metric around such a background and have found that the graviton wave function obeys, as predicted by Randall-Sundrum [61–63] a Schrödinger equation with a potential that is completely fixed by the value of . More precisely one finds that in the conformal gauge the fluctuations of the D-dimensional graviton
satisfy the Klein-Gordon equation of a scalar field in the gravitational background |
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namely ∂M (√−ggMN ∂N Φ) = 0. Parameterizing: |
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Φ = φ(z)eip·x = e−kzψ(z)eip·x |
(7.9.38) |
where p is the (D − 1)-dimensional momentum the Klein-Gordon equation becomes the following Schrödinger-type equation,
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p2ψ |
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where the potential, calculated in [60] is given by |
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Such an equation has a normalizable zero-mode wave function if the following condition is satisfied ≤ −2. Indeed it is evident from these expressions that for ≤ −2, U has a volcano shape as in Fig. 7.3 since the delta function has a negative coefficient, and the “bulk” term is non-negative for all z. Hence the trapping of
gravity occurs for positive tension (D − 2)-branes in the following window:
AdS ≤ ≤ −2 |
(7.9.41) |