Special Relativity and 8 0 Electromagnetism 0 2 y r a u Yannis PAPAPHILIPPOU n a J , S CERN A P S U , m s i t e n g United States Particle Accelerator School, a m o r University of California - Santa Cruz , Santa Rosa, CA t c e l E th th 14 – 18 January 2008 d n a y t i v i t a l e R l a i c e p S 1 Outline Notions of Special Relativity Historical background Lorentz transformations (length contraction 8 0 0 2 and time dilatation) y r a u n a J 4-vectors and Einstein’s relation , S A P S Conservation laws, particle collisions U , m s i t e n Electromagnetic theory g a m o r t c Maxwell’s equations, e l E d n a Magnetic vector and electric scalar potential y t i v i t a l Lorentz force e R l a i c e Electromagnetic waves p S 2 Historical background Maxwell’s equations (1863) attempted to explain electromagnetism and optics through wave theory 8 A few doubtful hypotheses and inconsistencies 0 0 2 y r A medium called “luminiferous ether” exists for the transport a u n of electromagnetic waves a J , S A “Ether” has a small interaction with matter and is carried P S U along with astronomical objects , m 8 s i t Light propagates with speed c = 3×10 m/s in " but not e n g invariant in all frames a m o r t Maxwell’s equations are not invariant under Galilean c e l E transformations d n a In order to make electromagnetism compatible with classical y t i v i t mechanics, assume that light has speed c only in frames where a l e R source is at rest l a i c e p S 3 Experimental evidence Star light aberration: a small shift in apparent positions of distant stars due to the finite speed of light and Fizeau and Foucault’s 8 Fizeau and Foucault 0 0 2 experiments (1850) on the y r a u n velocity of light in air and a J , S liquids A P S U , Michelson-Morley experiment m s i t e (1887) to detect motion of the n g a earth through ether m o r t Michelson c e l Lorentz-FitzGerland E d n a contraction hypothesis (1894 – y t i v i 1904): perhaps bodies get t a l e R compressed in the direction of l a i c e their motion by a factor p S 4 Postulates of Special Relativity 1. First Postulate: Principle of Relativity Every physical law is invariant under inertial co-ordinate transformations. Thus, if an object in space-time obeys the mathematical equations describing a physical law in one inertial frame of reference, it must necessarily obey the same equations when using any other inertial 8 0 0 frame of reference. 2 y r a u 1. Second Postulate: Invariance of the speed of light n a J , There exists an absolute constant with the following S A P S property. If A, B are two events which have co-ordinates U , m and in one inertial frame F, and have co-ordinates s i t e n and in another inertial frame F', then g a m o r t c e l E d n a y t i v if and only if i t a l e R l a i c e p S 5 The Lorentz Transformation From a frame to a frame moving with velocity v along the x-axis the space-time coordinates are transformed as: 8 0 0 2 y r a u n a J , S A P S U , with m s i t e n g a m o r t c e l E d n a The space-time interval is invariant y t i v i t under Lorentz transformations a l e R l a i c e p S 6 Length contraction Frame F Frame F’ z’ z v y y’ A Rod B 8 0 x x’ 0 2 y r a u n a J , S Rod AB of length L' fixed in F' A P S at x'A, x'B. Its length L seen by U , m s the observer is contracted i t e n g a m o r t c e l E d n a y t i v i t a l e R l a i c e p S 7 Time dilatation Clock in frame F at a point with coordinates (x,y,z) and different times tA and tB.In moving frame F' the time difference Δt’ is dilated Μ 8 0 0 2 y r γ a u n 1 a J , S A P S U , m R B s i t e n g a γ v m o r t c For c = 1, distance between e l Red and Blue with identical Blue moves with his mirror at E d clocks, (light beam bouncing velocity v. Blue measures the mirror and Red RM =1 tick. n a y off mirror). When Red and same time between ticks and Red thinks that distance t i v i t Blue at rest, tick and tocks tocks but according to Red, between Blue and mirror is BM a l e are simultaneous the clock of Blue runs slow R = γ ticks. Distance between l a i c e Blue Red and RB = γ v. So, p S 8 Example: Relativistic Train What does B’s clock read when G goes into tunnel? The two events are coincident and What does G’s clock read as he 8 F frame of tunnel with observers 0 0 2 enters? at both ends A, B y r a u n a J , S A Relativistic train on frame F’ with P S U driver D and guard G , Where is the guard G, with m s i t respect to A, when his clock reads e n g a Observers A and B see train 0? Setting the time to zero in the m o r t contracted c previous equation e l E d n a y t i v i t a l Tunnel moves relative to train e R So the guard is still 100m from l and the D,G see tunnel of 50 m! a i c tunnel entrance!!! e p S 9 4-Vectors and Einstein’s relation First note that and . 4-Position: 4-Velocity: 4-Acceleration: 8 0 0 2 y 4-Momentum: r a u n a J 4-Force: , S A P S Invariants: and U , m s i t e Differentiating the momentum invariant and n g a m o r t i.e. which gives But the rate of change of the kinetic c e l E d n a energy is and by integrating y t i v i t a l e At rest and thus the constant is Finally, the energy is R l a i c e p S 10 Velocity and Kinetic Energy Relative velocity Relative velocity and Lorentz factor Momentum Kinetic energy For , and 8 0 0 2 y r a u n a J , S A P S U , Electron Proton m s i t e n g a m o r t c e l E d n a T [MeV] y t i v i t Heavy particles become relativistic at higher energies a l e R l a i c When relativistic, small velocity change provides big change e p S in energy 11 Special Relativity and Electromagnetism, USPAS, January 2008 R e l a t i o n s h i p s f o r s m a l l p a r a m e t e r v a r i a t i o n s 1 2 Momentum conservation laws Equivalent expression for 4-momentum From the momentum invariance we get 8 0 0 2 y As the 4-momentum is conserved r a u n a J , S A P S U , Total energy is conserved m s i t e n g a m Classical momentum is conserved o r t c e l E d n a y t i Norm of the 4-momentum is conserved v i t a l e R l a i c e Momentum is conserved but mass is not (mass is a form of energy)!!! p S 13 Particle collisions Two particles have equal rest mass m0. Laboratory Frame (LF): one particle at rest, total energy is Elab. 8 0 0 2 Centre of Mass Frame (CMF): Velocities are equal and y r a u n opposite, total energy is E . a cm J , S A P S U , m s i t e n g a m The quantity is invariant. o r t c e l E In the CMF, we have d n a y t In general i v i t a l e In the LF, we have and R l a i c e p Finally S 14 Particle on target vs. colliding particles A proton p1 collides with an anti-proton p2 (same rest mass m0), producing two particles W1 and W2 with mass MW W1 1. p1 , p2 with equal and opposite velocities in lab frame p1 p2 8 0 0 2 y r and a u n a J , S W A 2 P W S 1 U , m s i t 2.