AMOP Lecture 5 Tue 2.11.2014 Lecture 25. Relativity of ...

Lecture 25. Relativity of lightwaves and Lorentz-Minkowski coordinates V.

(Ch. 0-4 of Unit 8)

Review of space-time (x,ct) and per-space-time (ω,ck) geometrySpace-time (x,ct) and per-space-time (ω,ck) geometry and its physics

All of those contraction and expansion coefficients Detailed views Einstein time dilation

The old “smoke and mirrors” trickDetailed views Lorentz contraction

Heighway’s paradox 1 and 2Phase invariance used to derive (x,ct)↔(x′,ct′) Einstein Lorentz Transformations (ELT)Introducing the stellar aberration angle σ vs. rapidity ρEpstein’s† space-proper-time (x,cτ) plots (“c-tau” plots)

Trigonometry: From circular to hyperbolic and backGroup vs. phase velocity and tangent contacts

AMOP Lecture 5 Tue 2.11.2014

†Lewis Carroll Epstein, Relativity Visualized Insight Press, San Francisco, CA 94107

See also: L. C. Epstein, Thinking Physics Press, Insight Press, San Francisco, CA 94107

1Tuesday, February 11, 2014




Heighway’s paradox 1 and 2


0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectorsυ′,cκ′c·Time-Period c·τ′=λ′(units: λ

A= cτ

A= micron)12

Space-Wavelength x′(units : λ

A= micron)12

Frequency: υ′=2π ·ω′(units : υ

A=600THz)

c·Wave Number c·κ′=c·k′/2π(units : υ

A=c·κ

A=600THz)

cτ′,x′ φ

R′

L′LL

“waves per meter”

“waves per second”

“meters per wave”

“seconds per wave”

1.0

e+ρ = 2

e−ρ = 12

Fig. 7 SRQMbyR&C3Tuesday, February 11, 2014

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

R′

L′LL





1.0

e+ρ = 2

e−ρ = 12

e−ρ = 12

e+ρ = 2

Fig. 7 SRQMbyR&C

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟


0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

R′

L′LL





1.0

G′

P′

e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

Fig. 7 SRQMbyR&C

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slope: Vgroup/c =

tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3


0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

cτgroup

λgroup

=

λphase

(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectorsc·Time-Period c·τ′=λ′(units: λ

A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





cκgroup

1.0 υphase

G′

e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

Fig. 7 SRQMbyR&C

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5

Slope: V phase/c=cothρ

=5/3 Slope: Vgroup/c =

tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3

Einstein"Tick" c ′ttick


0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτA4/5

cτgroup

=cτA4/3

λgroup

= λA4/5

λphase

= λA4/3

(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors

(x′,ct′)G=

=cτA(3/4,5/4)

(x′,ct′)P=

=cτA(5/4,3/4)

c·Time-Period c·τ′=λ′(units: λ

A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





cκgroup

=υA5/4

1.0 υphase

=υA5/4

G′

e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

Fig. 7 SRQMbyR&C

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3

Einstein"Tick"c ′ttick=cτ A5/4


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=

cτA(sinhρ, coshρ)

=cτA(3/4,5/4)

(x′,ct′)P=

cτA(coshρ, sinhρ)

=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

Fig. 7 SRQMbyR&C

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3

Einstein"Tick"c ′ttick=cτ Acoshρ =cτ A5/4


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

Fig. 7 SRQMbyR&C

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3


time rDopp υgroup τ phase υ phase τ group bDopp u/c c/uspace κ phase λgroup κ group λphase Vgroup/c Vphase/c

e−ρ sinhρ sechρ coshρ cschρ e+ρ tanhρ cothρ12

34

45

54

43

21

35

53


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

Fig. 7 SRQMbyR&C

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3




34

45

54

43

21

35

53

c ′ttick

EinsteinTimeDilation


Space-time (x,ct) and per-space-time (ω,ck) geometry and its physicsAll of those contraction and expansion coefficients

Detailed views Einstein time dilationThe old “smoke and mirrors” trick

Detailed views Lorentz contractionHeighway’s paradox 1 and 2


The old “smoke and mirrors” trick http://galileoandeinstein.physics.virginia.edu/lectures/srelwhat.html


http://galileoandeinstein.physics.virginia.edu/lectures/srelwhat.html


The old “smoke and mirrors” trick http://galileoandeinstein.physics.virginia.edu/lectures/srelwhat.html




1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

Fig. 7 SRQMbyR&C

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3




34

45

54

43

21

35

53

c ′ttick



1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

Fig. 7 SRQMbyR&C

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3




34

45

54

43

21

35

53

c ′ttick


LorentzContraction


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3




34

45

54

43

21

35

53

c ′ttick


LorentzContraction


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3




34

45

54

43

21

35

53

c ′ttick


LorentzContraction

Einstein?TimeAsimultaneity

Effects not publicized

(sinhρ)


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3




34

45

54

43

21

35

53

c ′ttick


LorentzContraction

?Lorentz?Expansion


Einstein?TimeAsimultaneity(sinhρ)

(coshρ)


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3




34

45

54

43

21

35

53

c ′ttick


LorentzContraction

?Lorentz?Expansion



(coshρ)


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12



34

45

54

43

21

35

53

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3


c ′ttick


LorentzContraction

?Lorentz?Expansion



(coshρ)


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12


ρrapidity e−ρ sinhρ sechρ coshρ cschρ e+ρ tanhρ cothρ

ρ = 12

34

45

54

43

21

35

53

σstellar∀ tanσ cosσ secσ cotσ sinσ cscσ

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3



LorentzContraction

c ′ttick?Lorentz?Expansion



(coshρ)


1.0

0.5

1.5

2.0

-1.0 -0.5

0

+2.0

1.0

0.5

1.5

2.0

-1.0 -0.5

0

+0.5 +1.0 +1.5 +2.0

cτphase

=cτAsechρ

=cτA4/5

cτgroup

=cτAcschρ

=cτA4/3

λgroup

= λAsechρ

= λA4/5

λphase

=λAcschρ

=λA4/3


(x′,ct′)G=


=cτA(3/4,5/4)

(x′,ct′)P=


=cτA(5/4,3/4)


A= cτ

A= micron)12


A= micron)12


A=600THz)


A=c·κ

A=600THz)

cτ′,x′ φ

P′+G′=R′

P′-G′=L′

P′

LL=GG-PP





e−ρ = 12

e+ρ = 2

e+ρ = 2

e−ρ = 12



ρ = 12

34

45

54

43

21

35

53


p L H λDeB

′υ phase

c ′κ phase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=υA

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟ =υA

5434

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

′υgroup

c ′κ group

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=υA

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=υA

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟ =υA

3454

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

Slop

e: V gr

oup/c=t

anhρ

=3/5



tanhρ=3/5

Slop

e: V ph

ase/c=

cothρ=5

/3



LorentzContraction

c ′ttick?Lorentz?Expansion



(coshρ)


SOURCESOURCE

b=2SOURCESOURCE

b=1/2

b=1=rSOURCESOURCE

Alice: “ No Bob, you’rethe one with short

group-λ ! ”Carla: “I agree with Alice!”

Bob: “ Alice! Your group-λ is short! ”

SOURCESOURCE

λ′=0.4µmλ=0.5µm

(a) Heighway

paradox-1

(b) Paradox-2

SOURCESOURCE

b=2

SOURCESOURCE

Alice: “ No Bob, you’rethe one with short laser-λ ! ”

Bob: “ Alice! Your laser-λ is short! ”λ′=0.25µmλ=0.5µm

Carla: “I’m outa here.They have really lost it!”

(Doppler blue-shifts

to 0.25µm for Alice)(currently part of)

Fig. 8 SRQMbyR&C


SOURCESOURCE

b=2SOURCESOURCE

b=1/2

b=1=rSOURCESOURCE

Alice: “ No Bob, you’rethe one with short

group-λ ! ”Carla: “I agree with Alice!”

Bob: “ Alice! Your group-λ is short! ”

SOURCESOURCE

λ′=0.4µmλ=0.5µm

(a) Heighway

paradox-1

(b) Paradox-2

SOURCESOURCE

b=2

SOURCESOURCE

Alice: “ No Bob, you’rethe one with short laser-λ ! ”

Bob: “ Alice! Your laser-λ is short! ”λ′=0.25µmλ=0.5µm

Carla: “I’m outa here.They have really lost it!”

(Doppler blue-shifts

to 0.25µm for Alice)(currently)

Fig. 8 SRQMbyR&C


Phase invariance used to derive (x,ct)↔(x′,ct′) Einstein Lorentz Transformations (ELT)


Key points inSRQMbyR&C


Very key point inSRQMbyR&C



′φphase = ′kphase ′x − ′ω phase ′t = kphasex −ω phase t ≡φphase


Key point holdsfor Any phase





′φgroup = ′kgroup ′x − ′ω group ′t = kgroupx −ω group t ≡φgroup



′ω phase

c ′kphase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=ω A

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=ω A

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟

′ω group

c ′kgroup

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=ω A

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=ω A

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟


′φphase = ′x ω A

csinhρ − ′t ω A coshρ = 0 ⋅ x −ω A t ⇒ ct = c ′t coshρ − ′x sinhρ

′φgroup = ′x ω A

ccoshρ − ′t ω A sinhρ = ω A

cx − 0 ⋅t ⇒ x = −c ′t sinhρ + ′x coshρ


′φgroup = ′kgroup ′x − ′ω group ′t = kgroupx −ω group t ≡φgroup



′ω phase

c ′kphase

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′P = ′R + ′L

2=ω A

e+ρ + e−ρ

2e+ρ − e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=ω A

coshρsinhρ

⎛

⎝⎜

⎞

⎠⎟

′ω group

c ′kgroup

⎛

⎝⎜⎜

⎞

⎠⎟⎟= ′G = ′R − ′L

2=ω A

e+ρ − e−ρ

2e+ρ + e−ρ

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=ω A

sinhρcoshρ

⎛

⎝⎜

⎞

⎠⎟


′φgroup = ′kgroup ′x − ′ω group ′t = kgroupx −ω group t ≡φgroup Key point holdsfor Any phase







Introducing the stellar aberration angle σ vs. rapidity ρEpstein’s space-proper-time (x,cτ) plots (“c-tau” plots)



u=c sin σ

c

δδ

S S′(a) Fixed Observer (b) Moving Observer

σ

k(↓↓) k′′(↓↓)

c√1-u2/c2=c/cosh υ

ω0

c

z

x

Fig. 5.6 Epstein’s cosmic speedometer with aberration angle σ and transverse Doppler shift coshυZ.

Introducing the stellar aberration angle σ vs. rapidity ρ Together, rapidity ρ=ln b and stellar aberration angle σ are parameters of relative velocity

The rapidity ρ=ln b is based on The stellar aberration angle σ is based on the longitudinal wave Doppler shift b=eρ transverse wave rotation R=eiσ defined by u/c=tanh(ρ). defined by u/c=sin(σ).At low speed: u/c~ρ. At low speed: u/c~ σ.

ρ


www.uark.edu/ua/pirelli/php/lighthouse_scenarios.php

Happening 1: Ship 1 is hit by Blink 1Happening 2: Lighthouse emits Blink 2

(Here: ρ=Atanh(1/2)=0.55, and: σ=Asin(1/2)=0.52 or 30°)

σ=30°stellar ab-angle

Lighthouse ship example of stellar aberration(Here: ρ=atanh(1/2)=0.549)


http://www.uark.edu/ua/pirelli/php/lighthouse_scenarios.php


www.uark.edu/ua/pirelli/php/lighthouse_scenarios.php

Lighthouse ship example of stellar aberration(Here: ρ=atanh(1/2)=0.549)

Happening 1: Ship 1 is hit by Blink 1Happening 2: Lighthouse emits Blink 2

(Here: ρ=Atanh(1/2)=0.55, and: σ=Asin(1/2)=0.52 or 30°)

σ=30°

σ=30°stellar ab-angle




Introducing the stellar aberration angle σ vs. rapidity ρEpstein’s†space-proper-time (x,cτ) plots (“c-tau” plots)





σ

Proper timecτ=√(ct)2-x2

Coordinate timect=√(cτ)2+x2

Coordinatex=(u/c) ct

cτ

xLight (never ages)x=ct

ο

(Age)

(Distance)

PP

γ

Particle going u in (x,ct)is going speed c in (x, cτ)

Epstein’s space-proper-time (x,cτ) plots (“c-tau” plots)

Fig. 5.8 Space-proper-time plot makes all objects move at speed c along their cosmic speedometer.

Time contraction-dilation revisited

This is also proportionalto Lagrangian

L= -Mc2√(1-u2/c2)


A cute Epstein feature is that Lorentz-Fitzgerald contraction of a proper length L to L′=L√1-u2/c2 is simply rotational projection onto the x-axis of a length L rotated by σ.

σ

Proper timecτ=√(ct)2-x2

Coordinatex=(u/c) ct

cτ

xο

PPParticle P going u in (x,ct)is going speed c in (x, cτ)

PP′′

ctσProper length L

Contracted lengthL′=L√1-u2/c2

L

ctComoving particle P′

Epstein’s space-proper-time (x,cτ) plots (“c-tau” plots)Length contraction-dilation revisited


Epstein’s space-proper-time (x,cτ) plots (“c-tau” plots)Twin-paradox revisited

τ

xB xB

τ

A outboundA outbound

A inbound

Bolder than

A

Bstationary

Bstationary

Bway older than

A


τ

xB xAxB

τ τ

A outboundA outbound

A inbound

Bolder than

A

Bstationary

Bstationary

Bway older than

A

Byounger than

A

Bway older than

A

Epstein’s space-proper-time (x,cτ) plots (“c-tau” plots)Twin-paradox revisited

B stationaryviewpoint

A outbound appears stationaryviewpoint

A outbound to right49Tuesday, February 11, 2014

Fig. 5.10 CW cosmic speedometer.

Geometry of Lorentz boost of counter-propagating waves.


Bb =Be+!

Blue shift =4 Br =Be-!

Red shift =1

C

Geometric MeanB= !(4·1)=2

Difference MeanBsinh! = (4-1)/2 = 3/2

Arithmetic MeanBcosh! = (1+4)/2 = 5/2

Transformed Per-Time "# - axis

Per-Time " - axis

Per-Spaceck- axis

2

2-2 3

3

4

4

1

1-1


=4=1

C





2 Transformed Per-Space

ck# - axis


=4=1

C






ck# - axis


=4=1

C






ck# - axis

Stellar AberrationAxis


=4=1

C






ck# - axis



=4=1

C






ck# - axis


Lorentz Contraction circle




ρ = 12

34

45

54

43

21

35

53


p L H λDeB

Trigonometry: From circular to hyperbolic




ρ = 12

34

45

54

43

21

35

53


p L H λDeB



Fig. 5.10 CW cosmic speedometer.

Geometry of Lorentz boost of counter-propagating waves.


AMOP Lecture 5 Tue 2.11.2014 Lecture 25. Relativity of ...

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