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Special Theory of Relativity

Special Theory of Relativity. In ~1895, used simple Galilean Transformations x’ = x - vt t’ = t But observed that the speed of light, c, is always measured to travel at the same speed even if seen from different, moving frames

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Special Theory of Relativity

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  1. Special Theory of Relativity • In ~1895, used simple Galilean Transformations x’ = x - vt t’ = t • But observed that the speed of light, c, is always measured to travel at the same speed even if seen from different, moving frames • c = 3 x 108 m/s is finite and is the fastest speed at which information/energy/particles can travel • Einstein postulated that the laws of physics are the same in all inertial frames. With c=constant he “derived” Lorentz Transformations light u moving frame also sees v_l =c.NOT v_l = c-u sees v_l =c P460 - Relativity

  2. “Derive” Lorentz Transform Bounce light off a mirror. Observe in 2 frames: A vel=0 respect to light source A’ velocity = v observe speed of light = c in both frames -> c = distance/time = 2L / t A c2 = 4(L2 + (x’/2)2) / t’2 A’ Assume linear transform (guess) x’ = G(x + vt) let x = 0 t’ = G(t + Bx) so x’=Gvt t’=Gt some algebra (ct’)2 = 4L2 + (Gvt)2 and L = ct /2 and t’=Gt gives (Gt)2 = t2(1 + (Gv/c)2) or G2 = 1/(1 - v2/c2) mirror L A A’ x’ P460 - Relativity

  3. Lorentz Transformations Define b = u/c and g = 1/sqrt(1 - b2) x’ = g (x + ut) u = velocity of transform y’ = y between frames is in z’ = z x-direction. If do x’ ->x then t’ = g (t + bx/c) + -> -. Use common sense can differentiate these to get velocity transforms vx’ = (vx - u) / (1 -u vx/ c2) vy’ = vy / g / (1 - u vx / c2) vz’ = vz / g / (1 - u vx / c2) usually for v < 0.1c non-relativistic (non-Newtonian) expressions are OK. Note that 3D space point is now 4D space-time point (x,y,z,t) P460 - Relativity

  4. Time Dilation • Saw that t’ = g t. The “clock” runs slower for an observer not in the “rest” frame • muons in atmosphere. Lifetime = t =2.2 x 10-6 sec ct = 0.66 km decay path = bgtc b g average in lab lifetime decay path .1 1.005 2.2 ms 0.07 km .5 1.15 2.5 ms 0.4 km .9 2.29 5.0 ms 1.4 km .99 7.09 16 ms 4.6 km .999 22.4 49 ms 15 km P460 - Relativity

  5. Time Dilation • Short-lived particles like tau and B. Lifetime = 10-12 sec ct = 0.03 mm • time dilation gives longer path lengths • measure “second” vertex, determine “proper time” in rest frame If measure L=1.25 mm and v = .995c t(proper)=L/vg = .4 ps L Twin Paradox. If travel to distant planet at v~c then age less on spaceship then in “lab” frame P460 - Relativity

  6. Adding velocities • Rocket A has v = 0.8c with respect to DS9. Rocket B have u = 0.9c with respect to Rocket A. What is velocity of B with respect to DS9? DS9 A B V’ =(v-u)/(1-vu/c2) v’ = (.9+.8)/(1+.9*.8)= .988c Notes: use common sense on +/- if v = c and u = c v’ = (c+c)/2= c P460 - Relativity

  7. Adding velocities • Rocket A has v = 0.826c with respect to DS9. Rocket B have u = 0.635c with respect to DS9. What is velocity of A as observed from B? DS9 A Think of O as DSP and O’ as rocket B B V’ =(v-u)/(1-vu/c2) v’ = (.826-.635)/(1-.826*.635)= .4c If did B from A get -.4c (.4+.635)/(1+.4*.635)=1.04/1.25 = .826 P460 - Relativity

  8. Relativistic Kinematics • E2 = (pc)2 + (mc2)2 E = total energy m= mass and p=momentum • natural units E in eV, p in eV/c, m in eV/c2 -> c = 1 effectively. E2=p2+m2 • kinetic energy K = T = E - m @ 1/2 mv2 if v << c • Can show: b = p/E and g = E/m --> p = bgm if m.ne.0 or p=E if m=0 (many massless particles, photon, gluon and (almost massless) neutrinos) • relativistic mass m = gm0 a BAD concept P460 - Relativity

  9. “Derive” Kinematics • dE = Fdx = dp/dt*dx = vdp = vd(gmv) • assume p=gmv (need relativistic) • d(gmv) = mgdv + mvdg = mdv/g dv = gmc2 - mc2 =Total Energy - ‘rest” energy = Kinetic Energy P460 - Relativity

  10. What are the momentum, kinetic, and total energies of a proton with v=.86c? • V = .86c g = 1/(1-.86*.86)0.5 = 1.96 • E = g m = 1.96*938 MeV/c2 = 1840 MeV • T = E - mc2 = 1840 - 938 = 900 MeV • p = bE = gbm = .86*1840 MeV = 1580 MeV/c or p = (E2 - m2)0.5 Note units: MeV, MeV/c and MeV/c2. Usually never have to use c = 300,000 km/s P460 - Relativity

  11. Accelerate electron to 0.99c and then to 0.999c. How much energy is added at each step? • V = .99c g = 7.1 v = .999c g = 22.4 • E = g m = 7.1*0.511 MeV = 3.6 Mev • = 22.4*0.511 MeV = 11.4 MeV • step 1 adds 3.1 MeV and step 2 adds 7.8 MeV even though velocity change in step 2 is only 0.9% P460 - Relativity

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