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

CHAPTER 7. A FIRST EXAMPLE 937.2.3 Adding perturbationsWe again need to introduce time in order to be able to perturb the system. Just as forthe cigar, we do this by modifying the metric tods 2 = − 1 k dt2 + dρ 2 + coth 2 ρ dψ 2 , (7.46)which in terms <strong>of</strong> r becomes⎧⎨ds 2= − 1 k dt2 + dr2r⎩2 − 1 +e −Φ = e 0√ −Φ r 2 − 1.r2r 2 − 1 dψ2 ,(7.47)The perturbations are given byψ = arcsin( Cr)+ δψ (r, t), (7.48)resulting in the induced perturbed metric⎡g αβ =⎢⎣⎡r 2⎣r 2 − 1−1k +r2r 2 − 1 (∂ tδψ) 2 r 2−C√ + (∂ r δψ)r 2 1 − C2r 2Up to second order, one finds thatkr2r 2 − 1⎤⎦(∂ t δψ)− ( r 2 − 1 ) 1det g αβ = ( )k 1 − C2r 2−(1 − C2⎡⎣r 2 − 1⎡1r 2 − 1−C√ + (∂ r δψ)r 2 1 − C2r 2⎣ 11 − C2⎤⎦(∂ t δψ)+ r 2 (∂ r δψ) 2 − 2C √(∂ rδψ)⎦r 2 1 − C2(7.49)( ) √[1 + r 2 1 − C2r 2 (∂ r δψ) 2 − 2C 1 − C2r 2 (∂ rδψ)r 2 )(∂ t δψ) 2 ]+ O (3). (7.50)r 2 ⎤⎤.⎥⎦The Lagrangian density thus becomes√L <strong>DBI</strong> = −T 1 e −Φ 0− (r 2 − 1) detg αβ[= − T 1e −Φ 01√ √ − C (∂ r δψ)k 1 − C2r√2kr 2) 3/2]−2 (r 2 1 − C2− 1) r 2 (∂ tδψ) 2 +(1 r2− C22 r 2 (∂ r δψ) 2 . (7.51)