Lecture 11 8 251 Spring 2007 Lecture 11 Topics Static gauge transverse velocity and string action Motion of free open string endpoints Reading Section 6 6 6 9 P P 0 f 1 P P 1 S d x P 0 d d x i 0 f 1 X 1 0 P1 d X 0 1 P0 1 X 0 0 P0 1 X 1 1 P1 i L x 0 and 1 are the endpoints of the string i f is the time interval that the string is evolving over String described at end by x f varies from 0 to 1 P How do we make this variation S vanish Let 0 1 so whatever we say about applies to both 0 and 1 x Impose a Dirichlet BC Some x is a constant oas a function of x 0 Then x 0 But can t impose Dirichelt BC on x0 Time always ows Never a constant 1 Lecture 11 8 251 Spring 2007 Impose Free BCs x arbitrary P 0 Must include P 0 0 again since time ows D Branes D2 Brane 2 number of spatial dimensions of the object DP Brane P number of spatial dimensions with free BCs where endpoints can move freely BCs for motion of an open string on a D2 brane x3 0 P0 0 P1 0 P2 0 0 or 1 D0 Brane All strings forced to start and end at the point string looks like a closed string but has di erent equations of motion because it can t move away freely 2 Lecture 11 8 251 Spring 2007 D1 Brane Looks like a string but not actually one Can have up to Dd Branes Have cartesian coordinates for 1 String swejpg out spacetime surface by moving through time 2 Actually matters whether 3 Lecture 11 8 251 Spring 2007 Orientation matters Will see this later Static Gauge Will enable us to draw lines on the surface 2 from lines on 1 Consider line 0 or 1 Draw on worldsheet Q t Q t 4 Lecture 11 8 251 Spring 2007 x0 ct c Description of Coordinates x c x t c x Remember is not the length of the string it s a parameter so 1 is constant BUT the string can elongate or shorten of course Some useful quantities x c x t x 0 x x x x t x x 2 c2 x t 2 x 2 x 2 Consider static string stretched between x1 0 and x1 a x1 x1 independent of 1 Plot x vs Since Nambu Gotta action reparam invar doesn t matter what path chosen as long as not e g 5 Lecture 11 8 251 Spring 2007 So x x c 0 0 x 2 c2 0 dx1 d 0 x 2 dx1 d 2 Nambu Gotta action T0 S c d T0 d 0 1 1 d x x1 2 c2 dx1 2 d d dx1 d 0 T0 d x1 1 x1 0 tf dt T0 a ti Recall S K V dt String not moving so K 0 For stretched string V tension distance So physically the tension of the string in constant T0 a potential energy of static string stretched to length a Where did point P go from t to t dt No physical answer So hard to talk about velocites 6 Lecture 11 8 251 Spring 2007 But let s construct a velocity we can all agree on Construct plane perpendicular to P Say P moves to P where P is the intersection of the plane with the string at t dt This is called the perpendicular velocity String doesn t actu ally neccesarily move perpendicularly but this well de ned quantity pretends it does ds d x d x ds 1 d x ds is a unit vector tangent to the string x v t x x x t x s 2 v x t 2 x t x s 2 Let s simplify Nambu Gotta action 7 Lecture 11 8 251 Spring 2007 2 2 2 x x x x 2 c x x x x t t 2 2 2 ds x x x c2 d t s t ds 2 c2 v d 2 2 2 Nambu Gotta action knew nothing mattered except perpendicular velocity No way to tell how a point moves 2 ds So c d 1 vc2 S T0 1 dt 0 ds d d 1 2 v T0 c2 dt 2 c2 ds 1 v Recall L m0 c2 1 v 2 c2 T0 ds rest energy of small section of string Consider a totally free open string no D branes P 0 T0 x x x x 2 x c 2 x x x 2 s c x t t s T0 2 2 c 1 v c2 P Note magic with the ds d 0 P Numerator 0 x s x t x x T0 s t endpoint 0 2 c2 c 1 v 0 So endpoint So at endpoint either 1 x t string or 2 x t 0 But 2 can t be because 8 x x s t Lecture 11 8 251 Spring 2007 P T0 1 v 2 c2 x s 0 v 2 c2 since x s 0 Motion perpendicular to string always if free 9