A new geometrical interpretation of the lorentz transform and the special theory of relativity
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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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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

    • Four-Vector Solutions

  • MASS, MOMENTUM & ENERGY


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


BACKGROUND

  • TRANSFORMATIONS

    • Galilean

    • Lorentz

  • THE MEASUREMENT & THE EVENT

  • REVIEW OF OTHER GRAPHICAL TECHNIQUES


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


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


The Lorentz Transform

x

t=T

Y

v= x/t

t=T- t

t

x

t=0 @ x=0


THE MEASUREMENT & THE EVENT

  • RADIAL DISTANCE IS INDETERMINATE!

  • INFERRING THE DISTANCE & TIME

    • Parallax

    • Active Interrogation

    • Simultaneous Solution of Lightline and Worldline


Radial Distance Indeterminate To A Passive Observer


Determining Radial DistancePassive Measurement of Parallax

PARALLAX

AT ORIGINATOR

PARALLAX AT OBSERVER

OR RELATIVE BRIGHTNESS


Determining Radial DistanceActive Interrogation

ctreturn

x, ct

cttransmission


Determining Radial DistanceSimultaneous Solution Between Lightline and Worldline

ctreception

cttransmission

x=vt


REVIEW OF OTHER GRAPHICAL TECHNIQUES

  • THE MINKOWSKI SPACETIME DIAGRAM

  • TECHNIQUE

  • ADVANTAGES & DISADVANTAGES


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*


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


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


THE NEW GRAPHICAL APPROACH

  • LORENTZ TRANSFORM

    • Events on the Worldline

    • Doppler

    • The Generalized Lorentz Transform

  • FOUR-VECTOR SOLUTIONS

  • MASS, MOMENTUM & ENERGY


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


A2

A3

A4

A

O

The Velocity TriangleThe Lorentz Angle 

timeline of S

worldline of S* in S

timeline of S*

x3

ct3


A3

B3

O

Hyperbolic and Radial TauProper vs. Inferred Time & Distance

C

C3

B

x3

ct3

A

A4

B4

C4

radius c


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


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


Relativistic Doppler Up and Down Doppler-Fixed Source

timeline of S

worldline of S* in S

ct*3

ct2

-ct2


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


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


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


Generalized Lorentz Transform


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


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


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


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


A2

A3

A1

B1

O

Eddington’s Cigar A’s Receipt and Inference

worldline of S* in S

ct3, x3


B2

B3

A1

B1

O

Eddington’s Cigar B’s Receipt and Inference

worldline of S in S*

ct*3, x*3


FOUR VECTOR SOLUTIONS

  • A SIMPLIFIED GRAPHICAL SOLUTION

  • THE DISPLACEMENT FOUR-VECTOR

  • THE VELOCITY FOUR-VECTOR


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


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


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


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


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


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


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()


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


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


MASS, MOMENTUM AND ENERGY

  • DEVELOPED FROM THE VELOCITY FOUR-VECTOR

    • Multiply the Proper Velocity by mo

  • MOMENTUM & ENERGY ARE FOUR-VECTOR COMPONENTS


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


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


SUMMARY

  • THE MEASUREMENT IS NOT THE EVENT!

  • ORTHOGONAL, EQUAL UNITS

  • FAST, SIMPLE TO USE

  • EXPLICIT FOUR-VECTOR SOLUTION


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