Lecture 12 8 251 Spring 2007 Lecture 12 Topics The parameterization Equations of motion and Virasoro constraints General motion for open strings Rotating open string Reading Chapter 7 So far X 0 ct c x X X X 2 v t t s s v2 ds x x 2 x 2 x 2 c 1 d c2 x x t s 2 2 x ds x 2 d 2 x x 2 c2 t x x d x ds x x 0 t x x 0 x v t T0 x 2 x T0 ds d x 2 2 c c 1 v c 2 x T0 1 v 2 c2 x T0 x 0 P ds d c PT P P P 0 1 Lecture 12 8 251 Spring 2007 0 P 0 0 P 0 0 t T0 ds d 0 c 1 v 2 c2 Consider a constant xed d d T0 ds 0 dt 1 v 2 c2 P P 0 t 1 v 2 c2 x T0 ds d 2 x T0 0 c ds d 1 v 2 c2 t2 Wouldn t it be nice if the 1 v 2 c2 disappeared Then we would have a nice wave equation Let s x magnitude of s t ds d 1 v 2 c2 1 ds d 1 d v2 c2 1 T0 ds 1 dEnergy T0 T0 2 Lecture 12 8 251 Spring 2007 E 0 1 T0 Note not equal to the length more convenient this way proportional to en ergy Our cleverness so far Static gauge Time on world time on worldsheet Keep lines orthogonal Set proportional to energy ds d 2 v2 1 2 c 2 x 1 2 c x t 2 1 Recall x x 0 t These two boxed equations are the parameterization conditions Now wave equation is simple 2 x 1 2 x 2 2 0 2 c t For normal non rel string get wave equation For new rel string get wave equation and 2 parameterization conditions Combine the equations x 1 x c t 2 1 Now P T0 x c2 t P T0 Nice and simple 3 x Lecture 12 8 251 Spring 2007 Open String Motion Totally Free 1 ct F ct G 2 This is all the wave equation tell you x position of string X t Now BCs x 0 0 1 x 1 ct F ct G 2 Primes indicate derivative with respect to BC 1 x 0 0 ct 0 F u G u G u a0 F ct G 1 F ct F ct a0 Absorb a0 into F Back to X 2 1 F ct F ct X 2 x t 0 F ct F tells you the motion of one endpoint BC 2 x 0 1 F ct 1 F ct 1 F u 2 1 F u F periodic 2 1 F u 2 1 F u v0 c Let t t 2 1 c x c t F ct F ct t 2 then velocity doesn t change since F u 2 1 F u 4 Lecture 12 X t 8 251 Spring 2007 2 1 1 2 1 F ct 2 1 F ct 2 1 x t v0 c 2 c So v0 average velocity of any xed point on the string This explanation is a bit di erent than the book s x 1 x F ct c t x 1 x F ct c t These yield x 1 x F ct c t F u 1 dF u 1 du u length parameter on curve dF du Example Most famous example Open string doing circular motion 5 Lecture 12 8 251 Spring 2007 l length of string t 0 l cos t sin t X 2 Recall t 0 F ct F u l cos u c sin u c X 2 l F u sin u c cos u c 2c Unit vector l c 2 1 l c 2 String endpoints move at speed of light Periodicity of F 2 1 c m 2 m 1 x 0 1 l F F cos m 1 0 2 2 2 1 2 c l c 2 1 String has 2 E E lT0 T0 2 more energy since rotating 6