Lecture 3 8 251 Spring 2007 Lecture 3 Topics Relativistic electrodynamics Gauss law Gravitation and Planck s length Reading Zwiebach Sections 3 1 3 6 Electromagnetism and Relativity Maxwell s Equations Source Free Equations E 1 B c t 1 0 B 2 With Sources Charge Current E 3 1 E 1 J B c c t 4 Notes 1 E and B have same units 3 is charge density charge volume Here no get messy in higher dimensions 4 J is current density current area 0 or 4 those constants would B are dynamical variables E d p 1 q E v B dt c 1 Lecture 3 8 251 Spring 2007 Solve the source free equations 0 solved by B A Used to have E 0 E B True equation 1 1 A E A E c t c t 1 A 0 E c t So 1 A E c t scalar Thus 1 A E c t B encoded as A E A are the fundamental quantities we ll use Gauge Transformations A A A A A B B function of x t function vector 1 c t 1 1 A E E c t c t and B elds unchanged So under gauge transformations E g t A A Physically equivalent s and B s Not guaranteed to be Suppose 2 sets of potentials give the same E gauge related then A A Suppose we have 4 vector A A 2 Lecture 3 8 251 Spring 2007 Take x Have indices from x and from A so will get a 4x4 matrix Have two important quantities E and B with 3 components each 6 important quantities Hint that we should get a symmetric matrix F A A A A x x F F 1 Ai Ei c t xi F12 x Ay y Ax Bz Foi F 0 Ex Ey Ez Ex 0 Bz By Ey Bz 0 Bx Ez By Bx 0 What happens under gauge transformation A A A Then get F A A A A F F De ne T x F F F Note indices are cyclic Some interesting symmetries T T T T So T is totally antisymmetric A totally symmetric object in 4D has only 4 nontrivial components so T 0 gives you 4 equations 3 Lecture 3 8 251 Spring 2007 T 0 A A A A Charge Q is a Lorentz invar Not everything that is conserved is a Lorentz invar eg energy Since Q is both conserved and a Lorentz invar c J form a 4 vector J Now let s do what a typical theoretical physicist does for a living guess the equation F J No derivatives not right F x J F x 1 J c No constants not right Correct amazingly even sign 0 F 0 x F 0i xi F0i Ei F 0i Ei veri ed So E Electromagnetism in a nutshell F A A F J x c Consider electromagnetism in 2D xy plane Get rid of Ez component 0 Ex Ey 0 Bz F Ex Ey Bz 0 But what about Bz Doesn t push particle out of the plane v Bz with v in the xy plane remains in xy plane but rename Bz as B a scalar 4 Lecture 3 8 251 Spring 2007 How about in 4D spatial dimensions Ex Ey 0 F 0 Ez 0 EN 0 So get tensor B It s a coincidence that in our 3D spacial world E and B are both vectors in all dimensions Let s look at E Notation Circle S is a 1D manifold the boundary of a ball B 2 Sphere S 2 R x21 x22 x23 R2 Ball B 3 R x21 x22 x23 R2 5 Lecture 3 8 251 Spring 2007 When talking about S 2 R call it S 2 R 1 implied Vol S 1 2 Vol S 2 4 Vol S 3 2 2 d Vol S d 1 2 2 d2 All you need to know about the Gamma function 1 2 1 1 x 1 x x n n 1 for n Z x dte t tx 1 for x 0 0 in d 3 and general d dimensions Calculating E d 3 Ed vol B 3 r d vol q B 3 r through S 2 r This represents the ux of E E r vol S 2 r q 6 Lecture 3 8 251 Spring 2007 E r 4 r2 q E r 1 q 4 r2 This falls o much faster at large r and increases much faster as small r General d Ed vol B d r d vol q B d r through S d 1 r This represents the ux of E E r vol S d 1 r q E r d 2 q 2 d 2 rd 1 Electric eld of a point charge in d dimensions If there are extra dimensions then would see larger E at very small distances 7