Lecture 15 8 251 Spring 2007 Lecture 15 Topics Solution of the open string motion in the light cone gauge Reading Sections 9 2 9 4 x0 0 c 0 0 is a line goes to intersection of the worldsheet with the x c 0 hyper plane n x 0 n x 1 0 n x 2 0 n x 1 x 2 0 n 1 0 c recover static gauge If n to be timelike n x 0 Same for a 0 x 0 v 1 Lecture 15 8 251 Spring 2007 Set usefully aim at dimensionless n p n x const P 1 P d 0 dP P d d d n Ask for n P 0 at endpoints so that 1 0 p 0 Reminder of Units J Angular momentum of rotating string 1 E 2 since J h J E 2 h is dimensionless Let s use natural units as opposed to Planck units set c 1 h 1 Thus L 1 T M L2 1 M L 1 Thus everything can be written in terms of units of length sometimes people use mass instead 1 1 2 So in natural units E 2 M 2 L So string length ls units to get actual numbers must replace the c s and h s ls h c In natural units T0 1 c 2 To remember ls T0 1 c 2 L2 Back to I L L1 As it turns out n x 2 n p the 2 will be convenient 2 in natural Lecture 15 8 251 Spring 2007 parameterization Static gauge P o T0 x 2 x o c T0 x 2 2 c2 c ds d 1 v x P 0 2 ds d 2 T0 ds d 2 c2 c 1 v 1 Try to make n P constant along the parameterized string 2 Get a range 0 If change parameter how does it Imagine had some parameter P transform Claim transformation law d P d Makes sense that P d is reparam invar P Multiply by n d n P d Can set to be A is constant with respect to might be a function of 1 n P 1 A n P 0 Also n P momentum So A n p A not dependent n P n p 1 n P n p 0 Recall eq of motion of string 3 Lecture 15 8 251 Spring 2007 P P 0 Dot with n n P n P 0 n P 0 We had n P 0 at string boundaries 0 1 and since n P 0 n P 0 and times Closed Strings n x 2 n p For closed strings more convenient to remove 2 n x n p n P n P 0 Do have n P n p 2 0 but don t have endpoints having n P 0 Open strings give rise to E M Closed strings give rise to gravity harder more subtle For a closed string know how to put param on strings at di erent times but we don t know how to correlate these ticks No special points on closed string like endpoints on open string Compute n P 1 x x x 2 n x 2 n x so n x 0 so to get n P 0 make x x 0 So in spacetime sense want x x x tangent to string x line of constant If given space vector x and timelike vector then unique vector orthogonal to x prop to x So we lock the params on the string but still remains the ambiguity of translation of string where do we set 0 No one knows 4 Lecture 15 8 251 Spring 2007 Summary 1 n x n p where 2 if open string or 1 if closed string 2 n P np 2 3 0 2 4 n P 0 everywhere x x 0 P 1 x 2 x 2 x 2 x 2 Dot by n n P 1 x 2 n p n p 2 2 2 x x 2 1 x 2 x 2 x 2 x 2 2 x 2 x 2 x 2 0 x 2 x 2 x 2 x 2 0 Using 4 get x x 2 0 In static gauge got x 1 x 1 c t P 1 x 2 5 Lecture 15 8 251 Spring 2007 P 1 x 2 Eq of Motion x x 0 Wave equation for everyone 6