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A NEW GEOMETRICAL INTERPRETATION OF THE LORENTZ TRANSFORM AND THE SPECIAL THEORY OF RELATIVITY

A NEW GEOMETRICAL INTERPRETATION OF THE LORENTZ TRANSFORM AND THE SPECIAL THEORY OF RELATIVITY. Lewis F. McIntyre, MS GRD, Inc. 6303 Little River Turnpike, Ste 320 Alexandria, VA 22312. AGENDA . PURPOSE BACKGROUND THE NEW GRAPHICAL APPROACH Lorentz Transform Relativistic Doppler

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A NEW GEOMETRICAL INTERPRETATION OF THE LORENTZ TRANSFORM AND THE SPECIAL THEORY OF RELATIVITY

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  1. A NEW GEOMETRICAL INTERPRETATION OF THE LORENTZ TRANSFORM AND THE SPECIAL THEORY OF RELATIVITY Lewis F. McIntyre, MS GRD, Inc. 6303 Little River Turnpike, Ste 320 Alexandria, VA 22312

  2. AGENDA • PURPOSE • BACKGROUND • THE NEW GRAPHICAL APPROACH • Lorentz Transform • Relativistic Doppler • Four-Vector Solutions • MASS, MOMENTUM & ENERGY

  3. PURPOSE • DEVELOP A GRAPHICAL SOLUTION WHICH • Preserves Equal Units of Measure and Orthogonality in All Reference Frames • Can Accommodate Multiple Reference Frames • TO ASSIST STUDENTS IN GRASPING FUNDAMENTALS OF SPECIAL RELATIVITY

  4. BACKGROUND • TRANSFORMATIONS • Galilean • Lorentz • THE MEASUREMENT & THE EVENT • REVIEW OF OTHER GRAPHICAL TECHNIQUES

  5. TRANSFORMATIONS RELATE AMEASUREMENT (x,y,z,t) OF AN EVENT IN ONE REFERENCE FRAME TO A MEASUREMENT (x*,y*, z*, t*) OF THAT SAME EVENT IN ANOTHER REFERENCE FRAME

  6. The Galilean Transform • Parallel t and t* • Measurement of the Event e , and the Event, are Identical • c>>v t t* x*=x-vt y*=y t*=t x t=0 @ x=0 y* y vt x* e

  7. The Lorentz Transform x t=T Y v= x/t t=T- t t x t=0 @ x=0

  8. THE MEASUREMENT & THE EVENT • RADIAL DISTANCE IS INDETERMINATE! • INFERRING THE DISTANCE & TIME • Parallax • Active Interrogation • Simultaneous Solution of Lightline and Worldline

  9. Radial Distance Indeterminate To A Passive Observer

  10. Determining Radial DistancePassive Measurement of Parallax PARALLAX AT ORIGINATOR PARALLAX AT OBSERVER OR RELATIVE BRIGHTNESS

  11. Determining Radial DistanceActive Interrogation ctreturn x, ct cttransmission

  12. Determining Radial DistanceSimultaneous Solution Between Lightline and Worldline ctreception cttransmission x=vt

  13. REVIEW OF OTHER GRAPHICAL TECHNIQUES • THE MINKOWSKI SPACETIME DIAGRAM • TECHNIQUE • ADVANTAGES & DISADVANTAGES

  14. Minkowski Spacetime Diagram 5 4.5 x* 4 t* 3.5 x* 3 2.5 Time t 2  1.5  1 x*  0.5 0 0 1 2 3 Distance x The Minkowski Space-time Diagram t* t* t*

  15. Minkowski Spacetime Diagram 5 4.5 The point indicated x=2.0, t=3.0 is read as x*=0.577, t*=2.308 4 t* 3.5 3 2.5 Time t 2 1.5 1 x* 0.5 0 0 1 2 3 Distance x The Minkowski Space-time Diagram One Event, Different Measurements

  16. The Minkowski Space-time Diagram Advantages & Disadvantages • ADVANTAGES • Events And Measurements Are Identical • DISADVANTAGES • Only One Pair of Reference Frames • Unique Construction for Each Velocity • One Reference Frame Distorted • Units of Measure “Stretched” • Not Orthogonal

  17. THE NEW GRAPHICAL APPROACH • LORENTZ TRANSFORM • Events on the Worldline • Doppler • The Generalized Lorentz Transform • FOUR-VECTOR SOLUTIONS • MASS, MOMENTUM & ENERGY

  18. A2 A3 O The Velocity TriangleDetermine the Proper Time  timeline of S worldline of S* in S x3 ct3 A A4 A1 (c)2= (ct3) 2 -x3 2 ct3*=c

  19. A2 A3 A4 A O The Velocity TriangleThe Lorentz Angle  timeline of S worldline of S* in S timeline of S* x3 ct3 

  20. A3 B3 O Hyperbolic and Radial TauProper vs. Inferred Time & Distance C C3 B x3 ct3 A A4 B4 C4 radius c

  21. A2 A3 O Relativistic DopplerTime of Receipt from Proper Time of Event The Time of Receipt is Relativistically Doppler-Shifted from the Time of Transmission: Equal Units of Distance in the Plane of Origination to Equal Units of Time in the Plane of Receipt timeline of S worldline of S* in S ct2 x3 ct3 A ct*3 A4 A1 ct1

  22. Relativistic Doppler Up and Down Doppler-Moving Source timeline of S worldline of S* in S DOWN-DOPPLER: Leaving Collocation, t>0 ct2 ct*3 UP-DOPPLER: Approaching Collocation, t<0 -ct*3

  23. Relativistic Doppler Up and Down Doppler-Fixed Source timeline of S worldline of S* in S ct*3 ct2 -ct2

  24. A2 A3 O Relativistic Doppler Time of Event from Time of Interrogation timeline of S worldline of S* in S ct2 The Product of Two Relativistic Doppler Shifts Yields a Classical Doppler Shift x3 ct3 A ct*3 A4 A1 ct1

  25. A2 A3 A A4 O Relativistic Doppler Proper Time vs. Inferred Time timeline of S worldline of S* in S ct2 x3 ct3 A1 ct1 

  26. A2 A3 O Relativistic Doppler Proper Time of Transmission and Proper Time of Receipt timeline of S worldline of S* in S ct2 x3 ct3 A ct*3 A4 A1 ct1

  27. Generalized Lorentz Transform

  28. A2 x3= (ct2-ct1)/2 ct3=(ct2+ct1)/2 A A3 A1 O Generalized Lorentz Transform A Measurement Not on the Worldline • S*’s Measurement Simultaneous with S’s at x3, t3 • Arrives at x3, t3 simultaneously with S*’s • Must start at ct’1 • Must end at ct’2 Timeline of S Worldline of S* in S timeline of S* ct2 ct’2 ct’1 ct1

  29. A2 x3= (ct2-ct1)/2 ct3=(ct2+ct1)/2 A A3 A1 O Generalized Lorentz Transform S*’s Measurement, Using the Same c Timeline of S Worldline of S* in S • S*’s Measurement Simultaneous with S’s at x3, t3 • Arrives at x3, t3 simultaneously with S*’s • Must start at ct’1 • Must end at ct’2 timeline of S* ct2 ct*2 A3* ct*1 ct1

  30. Generalized Lorentz Transform Solving for x*, t* in Terms of x, t

  31. Eddington’s Cigar • A SPACECRAFT FLIES BY EARTH AT 0.866C • AT FLY-BY, OBSERVERS ON EARTH AND IN THE SPACECRAFT BOTH LIGHT 30 MINUTE CIGARS • AT THE END OF THE SMOKE, EACH INFORMS THE OTHER OF THE EVENT • BOTH DETERMINE THAT THE OTHER’S CIGAR LASTED TWICE AS LONG

  32. A2 A3 A1 B1 O Eddington’s Cigar A’s Receipt and Inference worldline of S* in S ct3, x3

  33. B2 B3 A1 B1 O Eddington’s Cigar B’s Receipt and Inference worldline of S in S* ct*3, x*3

  34. FOUR VECTOR SOLUTIONS • A SIMPLIFIED GRAPHICAL SOLUTION • THE DISPLACEMENT FOUR-VECTOR • THE VELOCITY FOUR-VECTOR

  35. ct -0.707 -0.500 -0.266 0.0 0.266 0.500 0.707 0O -15O 15O 30O -30O 0.8 45O -45O 0.866 -0.866 0.6 -60O 60O 0.4 -0.966 0.966 -75O 75O 0.2 90O -90O -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 x APPROACHING VELOCITIES RECEDING VELOCITIES APPROACHING VELOCITIES RECEDING VELOCITIES -0.2 105O -105O 0.966 -0.966 -0.4 -120O 120O -0.6 0.866 -0.866 -135O 135O -0.8 -150O 150O -165O 165O 180O 0.707 0.500 0.266 0.0 -0.500 -0.707 -0.266 THE WORKSHEET Velocity Preferred Frame Time Axis Lorentz Angle  Preferred Frame x- Axis

  36. ct -0.707 -0.500 -0.266 0.0 0.266 0.500 0.707 0O -15O 15O 30O -30O 0.8 A 45O -45O 0.866 -0.866 0.6 A’ -60O 60O 0.4 A* -0.966 0.966 -75O 75O 0.2 90O -90O -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 x APPROACHING VELOCITIES RECEDING VELOCITIES APPROACHING VELOCITIES RECEDING VELOCITIES -0.2 105O -105O 0.966 -0.966 Proper Time -0.4 S’s Measurement of S* -120O 120O S*’s Measurement of S -0.6 0.866 -0.866 -135O Intermediate Point 135O -0.8 -150O 150O -165O 165O 180O 0.707 0.500 0.266 0.0 -0.500 -0.707 -0.266 x=0.6 ct=0.8 THE SIMPLIFIED SOLUTION Retard/Advance the Intercept Read S*’s Coordinates x x* ct Intermediate Plane: t*=t ct* x*=0.2309 ct*=0.577 O

  37. ct -0.707 -0.500 -0.266 0.0 0.266 0.500 0.707 0O -15O 15O 30O -30O 0.8 D A B C 45O -45O 0.866 -0.866 0.6 B’ A’ D’ -60O 60O 0.4 D* A* B* -0.966 0.966 -75O 75O 0.2 90O -90O -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 x APPROACHING VELOCITIES RECEDING VELOCITIES APPROACHING VELOCITIES RECEDING VELOCITIES -0.2 105O -105O 0.966 -0.966 Proper Time -0.4 S’s Measurement of S* -120O 120O S*’s Measurement of S -0.6 0.866 -0.866 -135O Intermediate Point 135O -0.8 -150O 150O -165O 165O 180O 0.707 0.500 0.266 0.0 -0.500 -0.707 -0.266 SIMULTANEOUS EVENTS IN S Not Simultaneous in S* x* ct Intermediate Plane: t*=t ct* O

  38. ct -0.707 -0.500 -0.266 0.0 0.266 0.500 0.707 0O -15O 15O 30O -30O 0.8 A B 45O -45O 0.866 -0.866 0.6 B’ A’ -60O 60O 0.4 A* -0.966 0.966 -75O 75O 0.2 B’A’= 90O -90O -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 x APPROACHING VELOCITIES RECEDING VELOCITIES APPROACHING VELOCITIES RECEDING VELOCITIES -0.2 105O -105O 0.966 -0.966 Proper Time -0.4 S’s Measurement of S* -120O 120O S*’s Measurement of S -0.6 0.866 -0.866 -135O Intermediate Point 135O -0.8 -150O 150O -165O 165O 180O 0.707 0.500 0.266 0.0 -0.500 -0.707 -0.266 PROOF For x into x* BA=x-vt x* ct Intermediate Plane: t*=t ct* O

  39. ct E’A’=x tan() E’A= -0.707 -0.500 -0.266 0.0 0.266 0.500 0.707 0O -15O 15O 30O -30O OD’= 0.8 A 45O -45O 0.866 -0.866 0.6 D’ E’ A’ -60O 60O 0.4 A* -0.966 0.966 -75O 75O ct*=ct-A’A* = 0.2 90O -90O -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 x APPROACHING VELOCITIES RECEDING VELOCITIES APPROACHING VELOCITIES RECEDING VELOCITIES -0.2 105O -105O 0.966 -0.966 Proper Time -0.4 S’s Measurement of S* -120O 120O S*’s Measurement of S -0.6 0.866 -0.866 -135O Intermediate Point 135O -0.8 -150O 150O -165O 165O 180O 0.707 0.500 0.266 0.0 -0.500 -0.707 -0.266 PROOF For t into t* D’E’=x AA’= x* ct Intermediate Plane: t*=t ct* O

  40. ct -0.707 -0.500 -0.266 0.0 0.266 0.500 0.707 0O -15O 15O 30O -30O 0.8 A 45O -45O 0.866 -0.866 0.6 A’ -60O 60O 0.4 A* A** -0.966 0.966 -75O 75O 0.2 90O -90O -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 x APPROACHING VELOCITIES RECEDING VELOCITIES APPROACHING VELOCITIES RECEDING VELOCITIES -0.2 105O -105O 0.966 -0.966 Proper Time -0.4 S’s Measurement of S* -120O 120O S*’s Measurement of S -0.6 0.866 -0.866 -135O Intermediate Point 135O -0.8 -150O 150O -165O 165O 180O 0.707 0.500 0.266 0.0 -0.500 -0.707 -0.266 x=0.6 ct=0.8 w=0.75 ADDITIVE VELOCITIES The Product of Two Lorentz Transforms x Intermediate Plane: t*=t x* ct ct* x*=0.2309 ct*=0.577 u=0.4c ct**=0.529 O

  41. Four-Vector Velocity • FOUR-VECTOR VELOCITY COMPONENTS • u0= dt/d • u1= dx/d • DISPLACEMENTS x EVALUATED AT UNIT  • MULTIPLYING u BY ARBITRARY  YIELDS DISPLACEMENT VECTOR X()

  42. ct -0.707 -0.500 -0.266 0.0 0.266 0.500 0.707 0O -15O 15O 30O -30O 1.6 45O -45O 0.866 -0.866 1.2 -60O 60O 0.8 -0.966 0.966 -75O 75O 0.4 90O -90O -1.6 -1.2 -0.8 -0.4 0.4 0.8 1.2 1.6 x APPROACHING VELOCITIES RECEDING VELOCITIES APPROACHING VELOCITIES RECEDING VELOCITIES -0.4 105O -105O 0.966 -0.966 -0.8 -120O 120O -1.2 0.866 -0.866 Proper Time -135O 135O S’s Measurement of S* -1.6 -150O 150O S*’s Measurement of S -165O 165O Intermediate Point 180O 0.707 0.500 0.266 0.0 -0.500 -0.707 -0.266 FOUR-VECTOR VELOCITY Displacement Vectors at Unit Time Expressed in Terms of Unit Vectors A A* ct* Proper Velocity uo= ct* u1=0x O

  43. Four Vector Summary • ALL MATERIAL OBJECTS TRAVEL AT PROPER VELOCITY c • DIRECTION OF TRAVEL DEFINES LOCAL t • RELATIVE VELOCITY IS COMPONENT OF PROPER VELOCITY PROJECTED ONTO ANOTHER REFERENCE FRAME

  44. MASS, MOMENTUM AND ENERGY • DEVELOPED FROM THE VELOCITY FOUR-VECTOR • Multiply the Proper Velocity by mo • MOMENTUM & ENERGY ARE FOUR-VECTOR COMPONENTS

  45. ct -0.707 -0.500 -0.266 0.0 0.266 0.500 0.707 0O -15O 15O 30O Momentum -30O 1.6 45O -45O 0.866 -0.866 1.2 -60O 60O 0.8 -0.966 0.966 -75O 75O 0.4 90O -90O -1.6 -1.2 -0.8 -0.4 0.4 0.8 1.2 1.6 x APPROACHING VELOCITIES RECEDING VELOCITIES APPROACHING VELOCITIES RECEDING VELOCITIES -0.4 105O -105O 0.966 -0.966 Proper Time -0.8 S’s Measurement of S* -120O 120O S*’s Measurement of S -1.2 0.866 -0.866 -135O Intermediate Point 135O -1.6 -150O 150O -165O 165O 180O 0.707 0.500 0.266 0.0 -0.500 -0.707 -0.266 MOMENTUM ENERGY FOUR-VECTOR Multiply Proper Velocity by Rest Mass Energy is in the t Direction Momentum is in the x Direction A A* Energy moct* Rest Mass O

  46. ct -0.707 -0.500 -0.266 0.0 0.266 0.500 0.707 0O -15O 15O 30O Momentum -30O 1.6 45O -45O 0.866 -0.866 1.2 Kinetic Energy -60O 60O 0.8 -0.966 0.966 Rest Mass -75O 75O 0.4 90O -90O -1.6 -1.2 -0.8 -0.4 0.4 0.8 1.2 1.6 x APPROACHING VELOCITIES RECEDING VELOCITIES APPROACHING VELOCITIES RECEDING VELOCITIES -0.4 105O -105O 0.966 -0.966 Proper Time -0.8 S’s Measurement of S* -120O 120O S*’s Measurement of S -1.2 0.866 -0.866 -135O Intermediate Point 135O -1.6 -150O 150O -165O 165O 180O 0.707 0.500 0.266 0.0 -0.500 -0.707 -0.266 MOMENTUM ENERGY FOUR-VECTOR Relationship of Components A A* x* Rest Mass moct* O

  47. SUMMARY • THE MEASUREMENT IS NOT THE EVENT! • ORTHOGONAL, EQUAL UNITS • FAST, SIMPLE TO USE • EXPLICIT FOUR-VECTOR SOLUTION

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