Special relativity Squashing of the E field line associated to a moving charge is suggestive of Lorentz contraction e m law and eqn of motion should be invariant with respect to Lorentz transformations Let s refresh our memory with some basic concepts of special relativity SR in short P Piot PHYS 571 Fall 2007 Proper time Consider two spherical waves So for photons we can write This holds true for any inertial frames and one generally define the proper time which is an invariant it is a scalar P Piot PHYS 571 Fall 2007 Proper time In SR proper time is an invariant Note that P Piot PHYS 571 Fall 2007 Proper time Proper time defined as This holds true for any inertial frames and one generally define the proper time which is an invariant it is a scalar P Piot PHYS 571 Fall 2007 3 1 dimension space Minkowski s metric Let Then we can write with contravariant and is the Minkowski s metric P Piot PHYS 571 Fall 2007 3 1 dimension space some properties Some useful properties contravariant x g x The scalar product is defined as covariant x x g x x g x x Contravariant and covariant form of the metric are equal g g g g mixed form is the Kroenecker delta function g g g P Piot PHYS 571 Fall 2007 Lorentz transformation LT I Derived to in sure Laws of Physics have the same form in inertial frames Many proofs The transformation must let d invariant A possible transformation is x x cosh ct sinh ct x sinh ct cosh Consider the coordinate in O corresponding to the origin x 0 in O x ct sinh ct ct sinh x tanh Vt sinh cosh Rapidity additive for LT compositions Which gives in 1 1 dim the usual Lorentz transform LT If e m only is considered other transformation can let Maxwell s equation invariant e g just dilations but LT are universal P Piot PHYS 571 Fall 2007 Lorentz transformation II The Lorentz transform from O to O two aligned inertial frames is given by the boost matrix see JDJ eqn 11 98 Note that The Lorentz transformation is Formally If O and O not aligned Lorentz transformation would be multiplied by a rotation matrix P Piot PHYS 571 Fall 2007 Particle dynamics in SR The principle of SR is All laws of physics must be invariant under Lorentz transformations Invariant Physics laws retain the same mathematical forms and numerical constants scalars keep the same value P Piot PHYS 571 Fall 2007 Particle dynamics in SR 4 velocity define Then An invariant can be form via the scalar product Moreover since d is an invariant and then u conforms to Lorentz transformation i e satisfies the principle of SR P Piot PHYS 571 Fall 2007 Particle dynamics in SR 4 momentum Define Then and The fundamental dynamical law for particle interactions in SR is that 4 momentum is conserved in any Lorentz frame So Kinetic energy P Piot PHYS 571 Fall 2007 Particle dynamics in SR example Consider w one incident n at rest Question Question Minimum required energy for the incoming n to enable the reaction Answer Answer At threshold the 4 final neutrons are at rest in the lab frame with we finally get the threshold energy P Piot PHYS 571 Fall 2007