Presentation on theme: "Lorentz transformation"— Presentation transcript:
1 Lorentz transformation Transformation of space and timet = g (t' + x' v/c²) t' = g (t – x v/c²)x = g (x' + vt') x' = g (x – vt)y = y' y' = yz = z' z' = zFour-vectorctxyzg g v/c 0 0gv/c g 0 0ct'x 'y 'z 'X = = = X'LvX = Lv X'Spacetime interval:For two eventse0 = (t0, x0, y0, z0) = (t0', x0', y0', z0') ande1 = (t1, x1, y1, z1) = (t1', x1', y1', z1')we find that the spacetime intervals2 = (cDt)2 – (Dx2 + Dy2 + Dz2)= (cDt')2 – (Dx'2 + Dy'2 + Dz'2)is Lorentz-invariant(Dt = t1 – t0 ; Dx = x1 – x0 ; …)Relativistic spacetime diagramxctct'x'Light1Unit hyperbola:
2 Velocity transformation ux = dx/dtdx = g (dx' + vdt')dt = g (dt' + dx' v/c²) dx/dt = (dx'+ vdt')/ (dt'+ dx' v/c²) = (dx'/dt'+ v)/ (1 + dx'/dt' v/c²)dy/dt = dy' / g (dt'+ dx' v/c²) = dy'/dt' / ( 1 + dx'/dt' v/c²) ux = (ux'+ v)/(1 + ux'v/c²) reverse transformation: ux' = (ux – v)/(1 – uxv/c²)uy = uy'/ g (1 + ux'v/c²) replace v by –v: uy' = uy/ g (1 – uxv/c²)uz = uz'/ g (1 + ux'v/c²) uz' = uz/ g (1 – uxv/c²)special case u = ux: (that is: u is parallel to the relative velocity v of the reference frames)u = (u'+ v)/(1 + uv/c²) u' = (u – v)/(1 – uv/c²) dt't = g (t' + x' v/c²) t' = g (t – x v/c²)x = g (x' + vt') x' = g (x – vt)y = y' y' = yz = z' z' = zctxyzg g v/c 0 0gv/c g 0 0ct'x'y'z'X = = = X'Lv
3 Four-Velocity Velocity transformation: ux = (ux'+ v)/(1 + ux'v/c²) reverse transformation: ux' = (ux – v)/(1 – uxv/c²)uy = uy'/ g (1 + ux'v/c²) replace v by –v: uy' = uy/ g (1 – uxv/c²)uz = uz'/ g (1 + ux'v/c²) uz' = uz/ g (1 – uxv/c²)U (τ is proper time of the object)X X XU U'Coordinate transformation:X = Lv X' d X = Lv d X'τ is Lorentz-invariant U = Lv U'
5 The Twin ParadoxA and B are twins of age 20. B decides to join an expedition to a nearby planet. The distance between Earth and the planet is d = 20 light years and the space ship used for the expedition travels at a speed of v = 4/5 c.- What time is it on Earth and on the spaceshiprespectively when B arrives at the planet?After a short stop at the planet the spaceshipreturns with the same speed.- How old are A and B at B’s return?t = 0t = t1t = 2t1t" = t2't" = 2t2'Time passing on Earth while spaceship turns aroundt'= t1'=t2't1 : x1 = xplanet ;x1' = 0 t1 = 25 yt1' :x1 = xplanet ;x1' = 0 t1' = 15 yEarthplanetspaceshipt' = 0t2' : x2 = xplanet ;x2' = 0 t2' = 15 yt2 : x2 = 0 ; t2' = 15 y t2 = 9 yt = t2t = g (t' + x' v/c²) t' = g (t – x v/c²)x = g (x' + vt' ) x' = g (x – vt)
7 Relativistic Doppler Effect fR = fE [ (1 – v/c) / (1 + v/c) ]½ for source moving away from receiver (or vice versa)fR = fE [ (1 + v/c) / (1 – v/c) ]½ for source moving towards receiver (or vice versa)(fR : frequency of receiver in receiver’s frame; fE frequency of emitter in emitter’s frame)“longitudinal Doppler shift”Transversal effect due to time dilation:fR = fE gx1ct1ct'x'Plain wave emitted source at rest in S'Plain wave emitted source at rest in S
8 Cosmic Microwave Background Planck (2009)WMAP (2001)COBE (1989)Cosmic Microwave Background(Nobel prize in physics 2006: George Smoot, John Mather)COBE (COsmic Background Explorer) satellite2.721 K2.729 K0 K4 KT = 0.02 mK(Dipole subtracted)