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Welcome back to Physics 211

Welcome back to Physics 211. Today’s agenda: summary Lorentz transformation relativistic mechanics. Final. Friday 12 Dec 2:45 pm - 4:45pm here Comprehensive. (NOT Newtonian gravity). One sheet of formulae allowed Practice questions: waves and relativity + solns online

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Welcome back to Physics 211

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  1. Welcome back to Physics 211 Today’s agenda: summary Lorentz transformation relativistic mechanics

  2. Final • Friday 12 Dec 2:45 pm - 4:45pm here • Comprehensive. (NOT Newtonian gravity). One sheet of formulae allowed • Practice questions: waves and relativity + solns online • Review session – Tuesday 9 Dec 10:30-11:30 Stolkin

  3. Review • Constancy of speed of light forces radical revision of ideas of space and time • Absolute space and absolute time replaced by absolute spacetime • spacetime distance same for all observers • Bizarre consequences – time dilation, length contraction and maximum speed.

  4. Demo – MM experiment  v -- speed of Earth mirrors c+v beam splitter c-v screen interference fringes – sensitive to phase difference of two beams – depends on optical path difference

  5. What replaces Galilean rule for relating coordinates ? • Remember 2 FOR moving with relative velocity v x2=x1-vt1 t2=t1  absolute time • Einstein was able to derive the new rule using arguments based on relativity and constant speed c for light. • Lorentz transformation

  6. Lorentz transformation x2=g(x1-vt1) ct2=g(ct1-(v/c)x1) where g=(1-(v/c)2)-1/2 notice: • space and time mix together • as v/c0 these approach Galilean rules • Can show that x22-(ct2)2= x12-(ct1)2

  7. Two events which are separated in space occur simultaneously in one frame of reference. A second frame of reference moves in the positive x direction with respect to the first. What will be the time separation of these events in this second frame ? • zero (i.e simultaneous) • positive • negative • need more information

  8. Simultaneity • There is no absolute (agreed upon by all observers) notion of two events happening simultaneously!

  9. Energy and mass • Radical rethink of spacetime forces us to change Newtonian mechanics. • Both Newtonian velocity, acceleration are not same in different inertial frames • Thus they cannot be used to formulate a new mechanics (would violate relativity principle)

  10. Replace ? • Need to find quantities which • reduce to the usual ones for small v/c • Allow us to construct quantities which are numerically same for all inertial observers

  11. Consider usual momentum 2D momentum p points in direction of space displacement p=m(Dx /Dt,Dy/Dt) y p x

  12. Relativistic momentum By analogy: relativistic momentum P is proportional to displacement in spacetime P=m0(cDt/Dt,Dx/Dt) ct worldline P Must use proper time t in denominator if P vector in spacetime x What is m0 ?

  13. Interpretation • P measures rate of motion through spacetime • For v/c->0, t -> t and spatial component of P is just usual momentum. • Time component -> constant m0c – component of momentum which is non-zero even when particle at rest!

  14. Relativistic Energy • Interpret this time component of P as the total energy E/c • Notice E=m0c2Dt/Dt=m0c2/(1-v2/c2)1/2 • For small v/c  E=m0c2+1/2m0v2+ … • Interpretation as energy justified ! • Time component P  K+rest mass energy as v/c0

  15. Relativistic momentum, energy The Newtonian expressions must be modified  p=movg E=moc2g

  16. Rest mass • Notice that the expression for relativistic energy only makes sense if allow for a non-zero energy for object at rest! • This rest mass energy = m0c2 • Equivalence of mass and energy ! small mass yields large energy since c large

  17. An object of mass 3 kg moves at speed v/c=0.8 in x-direction. What is its energy ? • E=0 • E=1/2x3x(0.8)2c2 • E=5/3x3c2 • E=3c2

  18. Solution • What is g ? • What is expression for relativistic energy E= ?

  19. An object of mass 3 kg moves at speed v/c=0.8 in x-direction. What is its kinetic energy ? • E=1/2x3x(0.8)2c2 • E=2c2 • E=3c2 • E=0

  20. Solution • Relativistic expression for K ?

  21. Rest mass – same for all observers! • Just like all inertial fames agree on spacetime distance they all agree on quantity • (E/c)2-p2 • Furthermore, can evaluate in frame where object has p=0 moc the rest mass!

  22. Bonus • Conservation of P implies conservation of p and E ! • E is conserved now for all types of collisions! • For E to be real need v less than c • Mass and energy equivalent –

  23. That’s all folks !

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