Post on 18-Jan-2022
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
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 2
2Tuesday, 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) 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) 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
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
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
4Tuesday, 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
+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
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
5Tuesday, February 11, 2014
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 2
6Tuesday, February 11, 2014
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
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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
7Tuesday, February 11, 2014
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
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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
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=cτ A5/4
8Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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=cτ Acoshρ =cτ A5/4
9Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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=cτ Acoshρ =cτ A5/4
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
10Tuesday, February 11, 2014
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 2
11Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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=cτ Acoshρ =cτ A5/4
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
c ′ttick
EinsteinTimeDilation
12Tuesday, February 11, 2014
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
13Tuesday, February 11, 2014
The old “smoke and mirrors” trick http://galileoandeinstein.physics.virginia.edu/lectures/srelwhat.html
14Tuesday, February 11, 2014
The old “smoke and mirrors” trick http://galileoandeinstein.physics.virginia.edu/lectures/srelwhat.html
15Tuesday, February 11, 2014
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
16Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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=cτ Acoshρ =cτ A5/4
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
c ′ttick
EinsteinTimeDilation
17Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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=cτ Acoshρ =cτ A5/4
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
c ′ttick
EinsteinTimeDilation
LorentzContraction
18Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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=cτ Acoshρ =cτ A5/4
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
c ′ttick
EinsteinTimeDilation
LorentzContraction
19Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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
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=cτ Acoshρ =cτ A5/4
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
c ′ttick
EinsteinTimeDilation
LorentzContraction
20Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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
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=cτ Acoshρ =cτ A5/4
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
c ′ttick
EinsteinTimeDilation
LorentzContraction
Einstein?TimeAsimultaneity
Effects not publicized
(sinhρ)
21Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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
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=cτ Acoshρ =cτ A5/4
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
c ′ttick
EinsteinTimeDilation
LorentzContraction
?Lorentz?Expansion
Effects not publicized
Einstein?TimeAsimultaneity(sinhρ)
(coshρ)
22Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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
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=cτ Acoshρ =cτ A5/4
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
c ′ttick
EinsteinTimeDilation
LorentzContraction
?Lorentz?Expansion
Effects not publicized
Einstein?TimeAsimultaneity(sinhρ)
(coshρ)
23Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
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
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=cτ Acoshρ =cτ A5/4
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
c ′ttick
EinsteinTimeDilation
LorentzContraction
?Lorentz?Expansion
Effects not publicized
Einstein?TimeAsimultaneity(sinhρ)
(coshρ)
24Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
e−ρ = 12
e+ρ = 2
e+ρ = 2
e−ρ = 12
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
′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=cτ Acoshρ =cτ A5/4
c ′ttick
EinsteinTimeDilation
LorentzContraction
?Lorentz?Expansion
Effects not publicized
Einstein?TimeAsimultaneity(sinhρ)
(coshρ)
25Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
e−ρ = 12
e+ρ = 2
e+ρ = 2
e−ρ = 12
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
′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=cτ Acoshρ =cτ A5/4
c ′ttick
EinsteinTimeDilation
LorentzContraction
?Lorentz?Expansion
Effects not publicized
Einstein?TimeAsimultaneity(sinhρ)
(coshρ)
26Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
e−ρ = 12
e+ρ = 2
e+ρ = 2
e−ρ = 12
time rDopp υgroup τ phase υ phase τ group bDopp u/c c/uspace κ phase λgroup κ group λphase Vgroup/c Vphase/c
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
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=cτ Acoshρ =cτ A5/4
EinsteinTimeDilation
LorentzContraction
c ′ttick?Lorentz?Expansion
Effects not publicized
Einstein?TimeAsimultaneity(sinhρ)
(coshρ)
27Tuesday, February 11, 2014
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
(a) Space-time( ′) geometry of CW -paths (b) Per-space-time ( ) geometry of CW point vectors
(x′,ct′)G=
cτA(sinhρ, coshρ)
=cτA(3/4,5/4)
(x′,ct′)P=
cτA(coshρ, sinhρ)
=cτA(5/4,3/4)
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
P′+G′=R′
P′-G′=L′
P′
LL=GG-PP
“waves per meter”
“waves per second”
“meters per wave”
“seconds per wave”
e−ρ = 12
e+ρ = 2
e+ρ = 2
e−ρ = 12
time rDopp υgroup τ phase υ phase τ group bDopp u/c c/uspace κ phase λgroup κ group λphase Vgroup/c Vphase/c
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
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
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=cτ Acoshρ =cτ A5/4
EinsteinTimeDilation
LorentzContraction
c ′ttick?Lorentz?Expansion
Effects not publicized
Einstein?TimeAsimultaneity(sinhρ)
(coshρ)
28Tuesday, February 11, 2014
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
29Tuesday, February 11, 2014
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
30Tuesday, February 11, 2014
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
31Tuesday, February 11, 2014
Phase invariance used to derive (x,ct)↔(x′,ct′) Einstein Lorentz Transformations (ELT)
32Tuesday, February 11, 2014
Very key point inSRQMbyR&C
′φphase = ′kphase ′x − ′ω phase ′t = kphasex −ω phase t ≡φphase
′φphase = ′kphase ′x − ′ω phase ′t = kphasex −ω phase t ≡φphase
Key point holdsfor Any phase
35Tuesday, February 11, 2014
Very key point inSRQMbyR&C
Key point holdsfor Any phase
′φphase = ′kphase ′x − ′ω phase ′t = kphasex −ω phase t ≡φphase
′φgroup = ′kgroup ′x − ′ω group ′t = kgroupx −ω group t ≡φgroup
36Tuesday, February 11, 2014
Very key point inSRQMbyR&C
′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
⎛
⎝⎜
⎞
⎠⎟
Key point holdsfor Any phase
′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
′φphase = ′kphase ′x − ′ω phase ′t = kphasex −ω phase t ≡φphase
′φgroup = ′kgroup ′x − ′ω group ′t = kgroupx −ω group t ≡φgroup
37Tuesday, February 11, 2014
Very key point inSRQMbyR&C
′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 = ′kphase ′x − ′ω phase ′t = kphasex −ω phase t ≡φphase
′φgroup = ′kgroup ′x − ′ω group ′t = kgroupx −ω group t ≡φgroup Key point holdsfor Any phase
′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
38Tuesday, February 11, 2014
Very key point inSRQMbyR&C
′φphase = ′kphase ′x − ′ω phase ′t = kphasex −ω phase t ≡φphase
′φgroup = ′kgroup ′x − ′ω group ′t = kgroupx −ω group t ≡φgroup Key point holdsfor Any phase
′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
39Tuesday, February 11, 2014
Very key point inSRQMbyR&C
′φphase = ′kphase ′x − ′ω phase ′t = kphasex −ω phase t ≡φphase
′φgroup = ′kgroup ′x − ′ω group ′t = kgroupx −ω group t ≡φgroup Key point holdsfor Any phase
40Tuesday, February 11, 2014
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
41Tuesday, February 11, 2014
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
42Tuesday, February 11, 2014
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)
43Tuesday, February 11, 2014
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
44Tuesday, February 11, 2014
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
†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
45Tuesday, February 11, 2014
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)
46Tuesday, February 11, 2014
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
47Tuesday, February 11, 2014
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
48Tuesday, February 11, 2014
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.
50Tuesday, February 11, 2014
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
†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
51Tuesday, February 11, 2014
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
52Tuesday, February 11, 2014
=4=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
2 Transformed Per-Space
ck# - axis
53Tuesday, February 11, 2014
=4=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
2 Transformed Per-Space
ck# - axis
54Tuesday, February 11, 2014
=4=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
2 Transformed Per-Space
ck# - axis
Stellar AberrationAxis
55Tuesday, February 11, 2014
=4=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
2 Transformed Per-Space
ck# - axis
Stellar AberrationAxis
56Tuesday, February 11, 2014
=4=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
2 Transformed Per-Space
ck# - axis
Stellar AberrationAxis
57Tuesday, February 11, 2014
=4=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
2 Transformed Per-Space
ck# - axis
Stellar AberrationAxis
Lorentz Contraction circle
58Tuesday, February 11, 2014
time rDopp υgroup τ phase υ phase τ group bDopp u/c c/uspace κ phase λgroup κ group λphase Vgroup/c Vphase/c
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
p L H λDeB
Trigonometry: From circular to hyperbolic
59Tuesday, February 11, 2014
time rDopp υgroup τ phase υ phase τ group bDopp u/c c/uspace κ phase λgroup κ group λphase Vgroup/c Vphase/c
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
p L H λDeB
Trigonometry: From circular to hyperbolic
60Tuesday, February 11, 2014
time rDopp υgroup τ phase υ phase τ group bDopp u/c c/uspace κ phase λgroup κ group λphase Vgroup/c Vphase/c
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
p L H λDeB
Trigonometry: From circular to hyperbolic
61Tuesday, February 11, 2014
Fig. 5.10 CW cosmic speedometer.
Geometry of Lorentz boost of counter-propagating waves.
62Tuesday, February 11, 2014
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
†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
63Tuesday, February 11, 2014