Lecture 27. Relativity of transverse waves and 4-vectors

(Ch. 2-5 of Unit 2 4.05.12)

Introducing per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformationReviewing the stellar aberration angle σ vs. rapidity ρ in Epstein’s cosmic speedometer†

Reviewing “Sin-Tan Rosetta” geometry Reviewing relativistic quantum Lagrangian-Hamiltonian contact relations

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

Time contraction-dilation revisitedLength contraction-dilation revisitedTwin-paradox revisitedVelocity addition revisited

Lorentz symmetry effectsHow it makes momentum and energy be conserved

†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

1Thursday, April 5, 2012

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

Reviewing 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.

u=c sin σ

c

δδ

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

σ

k(↓) k′(↓)

c√1-u2/c2=c/cosh ρ

ω0

c

z

x

2Thursday, April 5, 2012

Reviewing the “Sin-Tan Rosetta” geometry

https://www.uark.edu/ua/pirelli/php/hyper_constrct.php

NOTE: Angle φ is now called stellar aberration angle σ

Fig. C.2-3and

Fig. 5.4in Unit 2

3Thursday, April 5, 2012

ρ

ρHHaammiillttoonniiaann ==HH ==BB ccoosshh ρ

--LLaaggrraannggiiaann

--LL ==BB sseecchh ρRReesstt

EEnneerrggyy

BB ==MMcc22

HH

BB

--LL

VVeelloocciittyy aabbeerrrraattiioonn

aannggllee φ

33−

433221100--11

4

--22

BB ccoosshh ρ

BB ee--ρcp

H=E

BB

BB ee++ρ

BB sseecchh ρ

22

11

BBlluuee sshhiiffttRReedd sshhiifftt

(a) Geometry of relativistic transformationand wave based mechanics

(b) Tangent geometry (u/c=3/5)

(c) Basic construction given u/c=45/53

u/c =3/5u/c =1

cp =3/4

H =5/4

-L =4/5

cp =45/28

H =53/28

-L=28/45

e-ρ=1/2e-ρ=2/7

(d) u/c=3/5

11

1 1

bb--cciirrccllee

g-circle

p-circle

4Thursday, April 5, 2012

-1

-1

+2

0 +1

+1

p-slope=v

dH/dp=+1.0

-1 0 +1

-1

+1

+2

v-slope=p

dL/dv =+1.6

1.6

1

1.0

1

Lagrangian plot

L(v)= v•p - H(p)(a) Hamiltonian plot

H(p)=p•v-L(v)(b)

v-slope

intercept

-H = -0.6 p-slope

intercept

-L = -1.0

Lagrangian

L(v) = 1.0 Hamiltonian

H(p) = 0.6

momentum

pvelocity

v

velocity

v = 1.0

momentum

p = 1.6

L(v) H(p)

v-slope is momentum p

velocity v is p-slope

Lagrangian L(v) is -p-slope intercept

-v-slope intercept is Hamiltonian H(p)

Unit 1Fig. 12.3

Reviewing classical quantum Lagrangian-Hamiltonian relations

5Thursday, April 5, 2012

(a) Hamiltonian

Momentum p

PP′

P′′

-L-L′-L′′

L(q,q)

Velocity u=q

QQ′Q′′

-H

-H′

-H′′

HH′

H′′LL′L′′

H

H′

H′′

slope:

slope:

∂H∂p

= q= u

∂L∂q

= p

(b) LagrangianH(q,p)

radius = Mc2

O

O

Light cone u=1=chas infiniteH

and zero L

Fig. 5.1. Geometry of contact transformation between relativistic (a) Hamiltonian (b) Lagrangian

Reviewing relativistic quantum Lagrangian-Hamiltonian relations

dΦ = kdx − ω dt= −µ dτ = -(Mc2/) dτ. dτ = dt √(1-u2/c2)=dt sech ρ

Differential action: is Planck scale times differential phase: dS = dΦ

For constant u the Lagrangian is: L =−µτ = -Mc2√(1-u2/c2)= -Mc2sech ρ = -Mc2

Start with phase and set k=0 to get product of proper frequency and proper timeΦ µ = Mc2/ τ

dS = Ldt = p·dx − H·dt = k·dx − ω ·dt = dΦ

cosσ

L= p· x − H = p·u − H...with Poincare invariant:

6Thursday, April 5, 2012

Suppose starlight in lighthouse frame is straight down x-axis : ω↓ ,ckx↓ ,cky↓ ,cky↓( ) = ω0,−ω0,0,0( )

k(→→)k(↑↑)

k(↓↓)

k′′(→→)k′′(↑↑)

k′′(←←)

k′′(↓↓)

(a) Laser frame ω0 (b) z-(→Moving) shipω0 coshρω0

ω0ω0ω0 ω0 eρ

k(←←)

CW Laser-pairwavevectors

Apparentpositionof k′(↑)source

Apparentpositionof k′(↓)source

ω0 e−ρz-axis

x-axis

Per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation

−ω0 sinhρz

ω0 coshρz

−ω0

“South”

(Lighthouse frame)

7Thursday, April 5, 2012

Suppose starlight in lighthouse frame is straight down x-axis : ω↓ ,ckx↓ ,cky↓ ,ckz↓( ) = ω0,−ω0,0,0( )

k(→→)k(↑↑)

k(↓↓)

k′′(→→)k′′(↑↑)

k′′(←←)

k′′(↓↓)

(a) Laser frame ω0 (b) z-(→Moving) shipω0 coshρω0

ω0ω0ω0 ω0 eρ

k(←←)

CW Laser-pairwavevectors

Apparentpositionof k′(↑)source

Apparentpositionof k′(↓)source

ω0 e−ρz-axis

x-axis

Per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation

−ω0 sinhρz

ω0 coshρz

−ω0

along -z-axis : ω←,ckx←,cky←,ckz←( ) = ω0,0,0,−ω0( )

“South”

“West”

(Lighthouse frame)

8Thursday, April 5, 2012

Suppose starlight in lighthouse frame is straight down x-axis : ω↓ ,ckx↓ ,cky↓ ,ckz↓( ) = ω0,−ω0,0,0( )

k(→→)k(↑↑)

k(↓↓)

k′′(→→)k′′(↑↑)

k′′(←←)

k′′(↓↓)

(a) Laser frame ω0 (b) z-(→Moving) shipω0 coshρω0

ω0ω0ω0 ω0 eρ

k(←←)

CW Laser-pairwavevectors

Apparentpositionof k′(↑)source

Apparentpositionof k′(↓)source

ω0 e−ρz-axis

x-axis

Per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation

−ω0 sinhρz

ω0 coshρz

−ω0

along -z-axis : ω←,ckx←,cky←,ckz←( ) = ω0,0,0,−ω0( )

up +x-axis : ω↑ ,ckx↑ ,cky↑ ,ckz↑( ) = ω0,+ω0,0,0( )

“South”

“West”

“North”

(Lighthouse frame)

9Thursday, April 5, 2012

Suppose starlight in lighthouse frame is straight down x-axis : ω↓ ,ckx↓ ,cky↓ ,ckz↓( ) = ω0,−ω0,0,0( )

k(→→)k(↑↑)

k(↓↓)

k′′(→→)k′′(↑↑)

k′′(←←)

k′′(↓↓)

(a) Laser frame ω0 (b) z-(→Moving) shipω0 coshρω0

ω0ω0ω0 ω0 eρ

k(←←)

CW Laser-pairwavevectors

Apparentpositionof k′(↑)source

Apparentpositionof k′(↓)source

ω0 e−ρz-axis

x-axis

Per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation

−ω0 sinhρz

ω0 coshρz

−ω0

along +z-axis : ω→ ,ckx→ ,cky→ ,ckz→( ) = ω0,0,0,+ω0( )

along -z-axis : ω←,ckx←,cky←,ckz←( ) = ω0,0,0,−ω0( )

up +x-axis : ω↑ ,ckx↑ ,cky↑ ,ckz↑( ) = ω0,+ω0,0,0( )

“South”

“West”

“North”

“East”

(Lighthouse frame)

10Thursday, April 5, 2012

Suppose starlight in lighthouse frame is straight down x-axis : ω↓ ,ckx↓ ,cky↓ ,ckz↓( ) = ω0,−ω0,0,0( )+ρz -rapidity ship frame sees starlight Lorentz transformed to : ′ω↓ ,c ′kx↓ ,c ′ky↓ ,c ′kz↓( ) = ω0 coshρz ,−ω0,0,−ω0 sinhρz( )

k(→→)k(↑↑)

k(↓↓)

k′′(→→)k′′(↑↑)

k′′(←←)

k′′(↓↓)

(a) Laser frame ω0 (b) z-(→Moving) shipω0 coshρω0

ω0ω0ω0 ω0 eρ

k(←←)

CW Laser-pairwavevectors

Apparentpositionof k′(↑)source

Apparentpositionof k′(↓)source

ω0 e−ρz-axis

x-axis

Per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation

−ω0 sinhρz

ω0 coshρz

−ω0

(Lighthouse frame)

11Thursday, April 5, 2012

′ω↓c ′kx↓c ′ky↓

c ′kz↓

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω↓ckx↓cky↓

ckz↓

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω0−ω0

00

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ω0 coshρz−ω0

0−ω0 sinhρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Suppose starlight in lighthouse frame is straight down x-axis : ω↓ ,ckx↓ ,cky↓ ,ckz↓( ) = ω0,−ω0,0,0( )+ρz -rapidity ship frame sees starlight Lorentz transformed to : ′ω↓ ,c ′kx↓ ,c ′ky↓ ,c ′kz↓( ) = ω0 coshρz ,−ω0,0,−ω0 sinhρz( )

k(→→)k(↑↑)

k(↓↓)

k′′(→→)k′′(↑↑)

k′′(←←)

k′′(↓↓)

(a) Laser frame ω0 (b) z-(→Moving) shipω0 coshρω0

ω0ω0ω0 ω0 eρ

k(←←)

CW Laser-pairwavevectors

Apparentpositionof k′(↑)source

Apparentpositionof k′(↓)source

ω0 e−ρz-axis

x-axis

Per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation

−ω0 sinhρz

ω0 coshρz

−ω0

(Lighthouse frame)

12Thursday, April 5, 2012

After the 4-vector transformation, ω0=ω↓ is transverse Doppler shifted to ω0cosh ρz, while ckz=0 becomes ckz' = -ω0 sinh ρz .(The x-component is unchanged: ckx' = -ω0 = ckx and so is y-component: cky' = -ω0 = cky.)

′ω↓c ′kx↓c ′ky↓

c ′kz↓

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω↓ckx↓cky↓

ckz↓

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω0−ω0

00

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ω0 coshρz−ω0

0−ω0 sinhρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Suppose starlight in lighthouse frame is straight down x-axis : ω↓ ,ckx↓ ,cky↓ ,ckz↓( ) = ω0,−ω0,0,0( )+ρz -rapidity ship frame sees starlight Lorentz transformed to : ′ω↓ ,c ′kx↓ ,c ′ky↓ ,c ′kz↓( ) = ω0 coshρz ,−ω0,0,−ω0 sinhρz( )

k(→→)k(↑↑)

k(↓↓)

k′′(→→)k′′(↑↑)

k′′(←←)

k′′(↓↓)

(a) Laser frame ω0 (b) z-(→Moving) shipω0 coshρω0

ω0ω0ω0 ω0 eρ

k(←←)

CW Laser-pairwavevectors

Apparentpositionof k′(↑)source

Apparentpositionof k′(↓)source

ω0 e−ρz-axis

x-axis

Per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation

−ω0 sinhρz

ω0 coshρz

−ω0

−ω0 sinhρz

−ω0 ω0 coshρz

(Lighthouse frame)

13Thursday, April 5, 2012

After the 4-vector transformation, ω0=ω↓ is transverse Doppler shifted to ω0cosh ρz, while ckz=0 becomes ckz' = -ω0 sinh ρz .(The x-component is unchanged: ckx' = -ω0 = ckx and so is y-component: cky' = -ω0 = cky.)

′ω↓c ′kx↓c ′ky↓

c ′kz↓

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω↓ckx↓cky↓

ckz↓

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω0−ω0

00

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ω0 coshρz−ω0

0−ω0 sinhρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ω0 secσ

−ω00

−ω0 tanσ

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Suppose starlight in lighthouse frame is straight down x-axis : ω↓ ,ckx↓ ,cky↓ ,ckz↓( ) = ω0,−ω0,0,0( )+ρz -rapidity ship frame sees starlight Lorentz transformed to : ′ω↓ ,c ′kx↓ ,c ′ky↓ ,c ′kz↓( ) = ω0 coshρz ,−ω0,0,−ω0 sinhρz( )

k(→→)k(↑↑)

k(↓↓)

k′′(→→)k′′(↑↑)

k′′(←←)

k′′(↓↓)

(a) Laser frame ω0 (b) z-(→Moving) shipω0 coshρω0

ω0ω0ω0 ω0 eρ

k(←←)

CW Laser-pairwavevectors

Apparentpositionof k′(↑)source

Apparentpositionof k′(↓)source

ω0 e−ρz-axis

x-axis

Per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation

−ω0 sinhρz

ω0 coshρz

−ω0

−ω0 sinhρz=ω0 tanσ

−ω0

ω0 coshρz=ω0 secσ

ω0 sinσ

−ω0 cosσ ω0

σ

σ

σ

(Lighthouse frame)

14Thursday, April 5, 2012

After the 4-vector transformation, ω0=ω↓ is transverse Doppler shifted to ω0cosh ρz, while ckz=0 becomes ckz' = -ω0 sinh ρz .(The x-component is unchanged: ckx' = -ω0 = ckx and so is y-component: cky' = -ω0 = cky.)

′ω↓c ′kx↓c ′ky↓

c ′kz↓

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω↓ckx↓cky↓

ckz↓

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω0−ω0

00

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ω0 coshρz−ω0

0−ω0 sinhρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ω0 secσ

−ω00

−ω0 tanσ

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Suppose starlight in lighthouse frame is straight down x-axis : ω↓ ,ckx↓ ,cky↓ ,ckz↓( ) = ω0,−ω0,0,0( )+ρz -rapidity ship frame sees starlight Lorentz transformed to : ′ω↓ ,c ′kx↓ ,c ′ky↓ ,c ′kz↓( ) = ω0 coshρz ,−ω0,0,−ω0 sinhρz( )

k(→→)k(↑↑)

k(↓↓)

k′′(→→)k′′(↑↑)

k′′(←←)

k′′(↓↓)

(a) Laser frame ω0 (b) z-(→Moving) shipω0 coshρω0

ω0ω0ω0 ω0 eρ

k(←←)

CW Laser-pairwavevectors

Apparentpositionof k′(↑)source

Apparentpositionof k′(↓)source

ω0 e−ρz-axis

x-axis

Per-spacetime 4-vector (ω0,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation

Recall hyperbolic invariant to Lorentz transform: ω2-c2k2=ω′2-c2k′2 (=0 for 1-CW light) The 4-vector form of this is: ω2-c2k•k=ω′2-c2k′•k′ (=0 ″ ″)

−ω0 sinhρz

ω0 coshρz

−ω0

−ω0 sinhρz=ω0 tanσ

−ω0

ω0 coshρz=ω0 secσ

ω0 sinσ

−ω0 cosσ ω0

σ

σ

σ

(Lighthouse frame)

15Thursday, April 5, 2012

Fig. 5.10 CW cosmic speedometer.

Geometry of Lorentz boost of counter-propagating waves.

16Thursday, April 5, 2012

Fig. 5.10 CW cosmic speedometer.

Geometry of Lorentz boost of counter-propagating waves.

′ω→c ′kx→c ′ky→

c ′kz→

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω000

+ω0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=ω0

coshρz − sinhρz00

−sinhρz + coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ω0e−ρz

00

−ω0e−ρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

If starlight is horizontal right-moving k→ wave then ship going u along z-axis sees :

The usual longitudinal Doppler blue shifts e+ρz or Doppler red shifts e−ρz appear on both k-vector and frequency ω0.

If starlight is horizontal left-moving k← wave then ship going u along z-axis sees :

′ω←c ′kx←c ′ky←

c ′kz←

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

=

coshρz ⋅ ⋅ −sinhρz⋅ 1 ⋅ ⋅⋅ ⋅ 1 ⋅

−sinhρz ⋅ ⋅ coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ω000

−ω0

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=ω0

coshρz + sinhρz00

−sinhρz − coshρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ω0e+ρz

00

−ω0e+ρz

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

17Thursday, April 5, 2012

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)

18Thursday, April 5, 2012

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

19Thursday, April 5, 2012

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

20Thursday, April 5, 2012

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

21Thursday, April 5, 2012

u/c=1/2=tanhρu=sinσu

u/c=1/2=tanhρu=sinσu

v/c=2/3=tanhρv=sinσv

σu σv

unitcircle

x-axis

cτ-axis Geometry of relativistic velocity additiontanh(ρu+ρv )= tanhρu+tanhρv

1+tanhρu·tanhρvwhere:tanhρ=sinσ

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

22Thursday, April 5, 2012

u/c=1/2=tanhρu=sinσu

u/c=1/2=tanhρu=sinσu

v/c=2/3=tanhρv=sinσv

σu σv1/√3=sinhρu=tanσu

√3/2=sechρu=1/coshρu

unitcircle

1

0 1 x-axis

cτ-axis

2/√5=sinhρv=tanσv

3/√5=coshρv=secσv

2/√3=coshρu=secσu

In frame where ships go v=2c/3:Bullet going Vu(+)v= ??from ship1 passes ship2at ship2-time τ=1 andat bullet-time τ=√3/2=sechρu

In ships’ frame:Bullet going u= c/2from ship1 passes ship2at ship2-time τ=1 andat bullet-time τ=√3/2=sechρu

ship1ship2

ship1

ship2

√32

Geometry of relativistic velocity additiontanh(ρu+ρv )= tanhρu+tanhρv

1+tanhρu·tanhρvwhere:tanhρ=sinσ

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

23Thursday, April 5, 2012

u/c=1/2=tanhρu=sinσu

u/c=1/2=tanhρu=sinσu

v/c=2/3=tanhρv=sinσv

V/c=7/8=tanh(ρuv)=sinσuv=tanh(ρu+ρv)

tanhρu=sinσu

tanhρusinhρu=sinσutanσu

tanhρucoshρu=sinσusecσu

σu σv σuv1/√3=sinhρu=tanσu

sechρu=1/coshρu

unitcircle

1

0 1 x-axis

cτ-axis

2/√5=sinhρv=tanσv

3/√5=coshρv=secσv

2/√3=coshρu=secσu

In frame where ships go v=2c/3:Bullet going Vu(+)v=7c/8from ship1 passes ship2at ship2-time τ=1 andat bullet-time τ=√3/2=sechρu

In ships’ frame:Bullet going u= c/2from ship1 passes ship2at ship2-time τ=1 andat bullet-time τ=√3/2=sechρu

√32

Geometry of relativistic velocity additiontanh(ρu+ρv )= tanhρu+tanhρv

1+tanhρu·tanhρvwhere:tanhρ=sinσ

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

24Thursday, April 5, 2012

u/c=1/2=tanhρu=sinσu

u/c=1/2=tanhρu=sinσu

v/c=2/3=tanhρv=sinσv

V/c=7/8=tanh(ρuv)=sinσuv=tanh(ρu+ρv)

tanhρu=sinσu

tanhρusinhρu=sinσutanσu

tanhρucoshρu=sinσusecσu

σu σv σuv1/√3=sinhρu=tanφu

sechρu=1/coshρu

unitcircle

1

0 1 x-axis

cτ-axis

2/√5=sinhρv=tanσv

3/√5=coshρv=secσv

2/√3=coshρu=secφu

cosh(ρu+ρv)

sinh(ρu+ρv)

(coshρv+tanhρusinhρv)(coshρv+tanhρusinhρv)coshρu=coshρucoshρv+sinhρvsinhρv=cosh(ρu+ρv)=8/√15

coshρusinhρv+sinhρvcoshρv=sinh(ρu+ρv)=7/√15

In frame where ships go v=2c/3:Bullet going Vu(+)v=7c/8from ship1 passes ship2at ship2-time τ=1 andat bullet-time τ=√3/2=sechρu

In ships’ frame:Bullet going u= c/2from ship1 passes ship2at ship2-time τ=1 andat bullet-time τ=√3/2=sechρu

1/2+2/31+(1/2)(2/3) =7/8

√32

Geometry of relativistic velocity additiontanh(ρu+ρv )= tanhρu+tanhρv

1+tanhρu·tanhρvwhere:tanhρ=sinσ

Velocity addition revisited

25Thursday, April 5, 2012

u/c=1/2=tanhρu=sinφu

u/c=1/2=tanhρu=sinφu

v/c=2/3=tanhρv=sinφv

V/c=7/8=tanh(ρuv)=sinφuv=tanh(ρu+ρv)

tanhρu=sinφu

tanhρusinhρu=sinφutanφu

tanhρucoshρu=sinφusecφu

φu φv φuv1/√3=sinhρu=tanφu

√3/2=sechρu=1/coshρu

unitcircle

1

0 1 x-axis

cτ-axis

2/√5=sinhρv=tanφv

3/√5=coshρv=secφ v

2/√3=coshρu=secφu

(coshρv+tanhρusinhρv)(coshρv+tanhρusinhρv)coshρu=coshρucoshρv+sinhρvsinhρv=cosh(ρu+ρv)=8/√15

cosh(ρu+ρv)

sinh(ρu+ρv)

coshρusinhρv+sinhρvcoshρv=sinh(ρu+ρv)=7/√15

In frame where ships go v=2c/3:Bullet going Vu(+)v=7c/8from ship1 passes ship2at ship2-time τ=1 andat bullet-time τ=√3/2=sechρu

In ships’ frame:Bullet going u= c/2from ship1 passes ship2at ship2-time τ=1 andat bullet-time τ=√3/2=sechρu

√32

ρu=tanh-1(1/2)=0.54930

ρv=tanh(2/3)= 0.80472ρuv=tanh(7/8)= 1.35402=ρu+ρv

tanhρu=u/c=1/2=sinφutanhρv=v/c=2/3=sinφutanhρuv=u/c=7/8=sinφuv

φu=sin-1(tanhρu)=0.5236=30°

φv=sin-1(tanhρ)=0.7297=41.8°

φuv=sin-1(tanhρuv)=1.06543=61.04°

10°20°

30°

40°

50°

60°

70°

80°

90°

√53

26Thursday, April 5, 2012

A strength (and also, weakness) of CW axioms (1.1-2) is that they are symmetry principles

due to the Lorentz-Poincare isotropy of space-time (invariance to space-time translation in the vacuum).

Operator has plane wave eigenfunctions with roots-of-unity eigenvalues . (5.18a) (5.18b)

This also applies to 2-part or “2-particle” product states where exponents add (k,ω)-values of

each constituent to K=k1+k2 and Ω=ω1+ω2, and -eigenvalues also have that form .

Matrix of T-symmetric evolution U is zero unless and .

T(δ ,τ )

T ψ k ,ω = Aei(kx−ω t) ei(k⋅δ −ω⋅τ )

T ψ k ,ω = ei(k ⋅δ −ω ⋅τ ) ψ k ,ωψ k ,ω T† = ψ k ,ω e−i(k ⋅δ −ω ⋅τ )

ΨK ,Ω =ψ k1,ω1ψ k2 ,ω2

T(δ ,τ ) ei(K ⋅δ −Ω⋅τ )

′Ψ ′K , ′Ω U ΨK ,Ω ′K = ′k1 + ′k2 = K ′Ω = ′ω1 + ′ω2 = Ω

′Ψ ′K , ′Ω U ΨK ,Ω = ′Ψ ′K , ′Ω T†(δ ,τ )UT(δ ,τ ) ΨK ,Ω (if UT = TU for all δ and τ )

= e− i( ′K ⋅δ − ′Ω ⋅τ )ei(K⋅δ −Ω⋅τ ) ′Ψ ′K , ′Ω U ΨK ,Ω = 0 unless: ′K = K and: ′Ω =Ω

Lorentz symmetry effectsHow it makes momentum and energy be conserved

That’s momentum (P=hK) and energy (E=hW) conservation!

27Thursday, April 5, 2012