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

CHAPTER 7. A FIRST EXAMPLE 897.1.4 Perturbed equations <strong>of</strong> motionWe proceed to computing the equations <strong>of</strong> motion for the perturbation δτ:∂L0 = ∂ µ∂ (∂ µ δτ) = ∂ ∂Lt∂ (∂ t δτ) + ∂ ∂Lr∂ (∂ r δτ)⎡ √ ⎤ [= ∂ t⎣ −kr2 1 − C2( ) 3/2]r 21 + r 2 (∂ t δτ) ⎦ + ∂ r r 2 1 − C2r 2 (∂ r δτ)= − kr21 + r 2 √1 − C2+(∂2r 2 t δτ ) + 2r( ) 3/2) (r 2 1 − C2 (∂2r 2 r δτ )0 = − kr (∂21 + r 2 t δτ ) )+(2 + C2r 2 (∂ r δτ) + r[) 3/2 (1 − C2r 2 + 3C2r√1 − C2r 2 ](∂ r δτ)(1 − C2r 2 ) (∂2r δτ ) . (7.20)In order to eliminate the time derivative, we fill in δτ = X (r)e iEt , giving us)0 = r(1 − C2 (∂2r 2 r X ) )+(2 + C2r 2 (∂ r X) + krE21 + r2X. (7.21)7.1.5 Solutions <strong>of</strong> the PDETo find the solutions to the differential equation expressed in Eq. 7.21, we will transformit into a hypergeometric equation, which is a second order partial differential equationobeying the general form0 = z (1 − z) ∂ 2 zX + ((c − (a + b + 1) z)∂ z X − abX, (7.22)with a, b and c some arbitrary coefficients. The solutions to the hypergeometric equationcan be expressed in terms <strong>of</strong> the hypergeometric functions. These can be found e.g. in[1] Ch. 2 and [8] Ch. 9.To be able to rewrite Eq. 7.21 into a hypergeometric equation, we introduce thechange <strong>of</strong> variables:r 2 = w, (7.23)w = ( 1 + C 2) z − 1. (7.24)The first substition makes for the following change in derivatives:∂ 2∂r2= ∂w∂r∂∂r = ∂w ∂∂r ∂w = 2√ w∂ w ,[ ]∂ ∂w ∂= 4w∂w 2 + 2∂ w . (7.25)∂w ∂r ∂w