DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 7. A FIRST EXAMPLE 877.1.3 Adding perturbationsThe next step is to add perturbations. For this, we need to introduce time. We do thisby modifying the metric,ds 2 = − 1 k dt2 + dρ 2 + th 2 ρ dτ 2 , (7.9)in which k is the number <strong>of</strong> NS5-branes spread out on the circle. Instead <strong>of</strong> scaling thetimelike dimension with k −1 we could also have scaled (all) the spacelike dimensionswith k as was done in Eq. 6.6, and one founds both methods in the literature. In terms<strong>of</strong> r this becomes (we also include the restatement <strong>of</strong> e −Φ )⎧⎨ds 2 = − 1 k dt2 + dr21 + r⎩2 + r21 + r 2dτ2 ,(7.10)e −Φ = e 0√ −Φ 1 + r 2 .Now we consider perturbations <strong>of</strong> τ by stating that( ) Cτ = arcsin + δτ (r, t). (7.11)rHence, we see that the components <strong>of</strong> the induced metric becomeUsing∂ r τ =(∂ r τ) 2 =g rt = g tr =g tt = −1k +r21 + r 2 (∂ tδτ) 2 , (7.12)g rr = 11 + r 2 + r21 + r 2 (∂ rτ) 2 , (7.13)r21 + r 2 (∂ tδτ)(∂ r τ). (7.14)−Cr 2 √1 − C2r 2 + (∂ r δτ), (7.15)C 2( ) − 2C √(∂ rδτ)+ (∂ r δτ) 2 , (7.16)r 4 1 − C2r 2 r 2 1 − C2r 2one finds that⎡g αβ =⎢⎣⎡r 2⎣1 + r 2−1k +⎡r21 + r 2 (∂ tδτ) 2 r 2⎣1 + r 2⎤−C√ + (∂ r δτ)r 2 1 − C2r 2⎦(∂ t δτ)⎡1⎣ 11 + r 2 1 − C2−C√ + (∂ r δτ)r 2 1 − C2r 2⎤⎦(∂ t δτ)+ r 2 (∂ r δτ) 2 − 2C √(∂ rδτ)⎦r 2 1 − C2(7.17)r 2 ⎤⎤.⎥⎦