Relativistic Momentum Classical physics: Definition of momentum: p = mv Conservation of momentum:p 1 + p 2 = p 3 + p 4 Coordinate transformation (Galilei; velocity of object: u; frame: v): p' = m u' p 1 ' + p 2 ' p1p1 p2p2 m M p3p3 p4p4 Relativistic physics: Same definition of momentum does not work due to complicated velocity transformation Use four-velociy (has simpler transformation law!): define four-momentum: P = m U Assume four-momentum conservation in one frame: P 1 + P 2 = P 3 + P 4 Coordinate transformation (Lorentz): P ' = m U ' P 1 ' + P 2 ' U = u cucu ( ) P = m u cucu ( ) Space component (velocity of object: v): p = mv This is the relativistic momentum (conserved in all inertial frames!) Define dynamic mass: m d = m p = m d v (equivalent to classical case) To distinguish between the masses we call m the rest mass = m(u + v)= p + mv = p 1 + p 2 + mv + Mv = p 3 + p 4 + mv + Mv= p 3 ' + p 4 ' = m L v -1 U = L v -1 m U = L v -1 P = L v -1 P 1 + L v -1 P 2 = L v -1 ( P 1 + P 2 )= L v -1 ( P 3 + P 4 )= P 3 ' + P 4 '
Relativistic Momentum and Energy Coordinate transformation: P = m U = m L v U ' = L v P ' Time component : mc = ––––––– mc 1 – v 2 /c 2 ≈ mc (1 + – –– ) 1 v 2 2 c 2 = – (mc 2 + – mv 2 ) 1 c 2 Time component of four-momentum conservation (same approximation, v << c ): (mc 2 + ½ mv 1 2 ) + (Mc 2 + ½ Mv 2 2 ) = (mc 2 + ½ mv 3 2 ) + (Mc 2 + ½ Mv 4 2 ) E 1 + E 2 = E 3 + E 4 Conservation of Energy! Total energy: E = mc 2 Rest energy: E 0 = mc 2 (Energy for v = 0) Kinetic energy: E kin = E – E 0 = mc 2 – mc 2 = ( – 1) mc 2 Some important relations Analog to the spacetime interval we calculate the ‘magnitude’ of the four-momentum: P 2 = (E/c) 2 – p 2 = For m 0 (p 2 c 2 >> m 2 c 4 ): Velocity in terms of E and p : = ––––––– energy c ( mc) 2 – ( mv) 2 = 2 m 2 c 2 (1 – v 2 /c 2 ) = m 2 c 2 E 2 = p 2 c 2 + m 2 c 4 E = pc v = p/( m) = pc 2 /E ; for m 0 : v = c = m d c 2 E' = (E – v p x ); p x ' = (p x – v/c 2 E); p y ' = p y ; p z ' = p z
Relativistic Momentum and Energy v p Energy transformation and Doppler shift Energy transformation E' = (E – v p x ) Identical to Doppler shift formulas Energy proportional to frequency (for m = 0) Can be used to easily deduce general formula for Doppler shift: E' = (E – v p x ) f E = f R (1 – v/c cos ) pxpx p x = p cos = (E – v E/c) for massless objects ( p = E/c; assume p = p x ) = (E – v p cos ) = E ––––––– 1 – v/c 1 – v 2 /c 2 = E –––– 1 – v/c 1 + v/c = E (1 – v/c cos ) (for m = 0) If p x = 0: E' = E
Relativistic Momentum and Energy v p Energy transformation and Doppler shift Energy transformation E' = (E – v p x ) Identical to Doppler shift formulas Energy proportional to frequency (for m = 0) Can be used to easily deduce general formula for Doppler shift: E' = (E – v p x ) f E = f R (1 – v/c cos ) pxpx p x = p cos L M L M Equivalence of Mass and Energy: Einstein’s box Centre of mass cannot shift in isolated system light pulse has a mass equivalent: Conservation of momentum: p = E/c = Mv Total time of process: t = L/c CM shift of box: M x = Mvt = p L/c = E/c 2 L CM shift of light: m light x light = m light L m light = E/c 2 xx = (E – v E/c) for massless objects ( p = E/c; assume p = p x ) = (E – v p cos ) = E ––––––– 1 – v/c 1 – v 2 /c 2 = E –––– 1 – v/c 1 + v/c = E (1 – v/c cos ) (for m = 0) If p x = 0: E' = E
Headlight Effect v = 0.9 c E-M Radiation emitted isotropically Rest frame of emitter Laboratory frame v = 0.99 c v = 0.9 c Galilean transf.
uncharged wire S S' charged wire at rest in S at rest in S' I v – Q S I S' Lorentz force (attractive) Electrostatic force (attractive) Relativistic Wire B = µI/2 d F B = – Qv×B B = µI/2 d F B = – Qv×B = 0
General Relativity: Spacetime curvature
General Relativity: Gravitational Lensing
General Relativity: Spacetime Diagram, Light Cones x Past Future ct y Elsewhere x'x' ct' Object …. Black hole falling into Special Relativity General Relativity