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

CHAPTER 7. A FIRST EXAMPLE 88From this, one computes that up to second order− ( 1 + r 2) ( ) √1det g αβ = ( )[1 + r 2 1 − C2k 1 − C2r 2 (∂ r δτ) 2 − 2C 1 − C2r 2 (∂ rδτ)r 2 ( ) ]−kr21 + r 2 1 − C2r 2 (∂ t δτ) 2 + O (3). (7.18)This finally allows us to compute the Lagrangian density:√L <strong>DBI</strong> = −T 1 e −Φ 0− (1 + r 2 ) detg αβ[)≈ −T 1 e −Φ 10√ ( ) 1 +(1 r2− C22 r 2 (∂ r δτ) 2k 1 − C2r 2kr 2 ) √−(12 (1 + r 2 − C2) r 2 (∂ t δτ) 2 − C 1 − C2r 2 (∂ rδτ)− 1 ) ) ](4C(1 2 − C28 r 2 (∂ r δτ) 2 + O (3)[L <strong>DBI</strong> = − T 1e −Φ 01√ √ − C (∂ r δτ)k 1 − C2r√2kr 2) 3/2]−2 (1 + r 2 1 − C2) r 2 (∂ tδτ) 2 +(1 r2− C22 r 2 (∂ r δτ) 2 . (7.19)1This result agrees with Eq. (5) in the original paper, except for an overal factor <strong>of</strong> √kwhich has been forgotten in the article, and a minus sign in front <strong>of</strong> the first order termwhich is due to the fact that we used τ = arcsin ( ) (Cr +δτ instead <strong>of</strong> τ = arccos C)r +δτwhich was used in the article. This sign is <strong>of</strong> no importance, as the linear term doesnot contribute to the equations <strong>of</strong> motion. Also note a typo in the article in the term∼ (∂ r δτ) 2 , where the author wrote ( √ ) 3/2 . . . instead <strong>of</strong> (. . .) 3/2 .