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Understanding Lorentz Transformations: A Deep Dive into Relativistic Mechanics

In this comprehensive review session on December 12th, explore the radical shifts in our understanding of space and time through the lens of Lorentz transformations. We'll cover key concepts such as time dilation, length contraction, and the constancy of the speed of light. Join us as we delve into the principles that replace Newtonian mechanics with relativistic formulations. Practice questions will be provided online to reinforce your understanding. Prepare to rethink energy, mass, and spacetime in this engaging session focused on the implications of Einstein's theories.

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Understanding Lorentz Transformations: A Deep Dive into Relativistic Mechanics

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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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