Lecture 27. Relativity of transverse waves and 4-vectors

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1 Lecture 27. Relativity of transverse waves and 4-vectors (Ch. 2-5 of Unit ) Introducing per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation Reviewing 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 revisited Length contraction-dilation revisited Twin-paradox revisited Velocity addition revisited Lorentz symmetry effects How it makes momentum and energy be conserved Lewis Carroll Epstein, Relativity Visualized Insight Press, San Francisco, CA 9417 See also: L. C. Epstein, Thinking Physics Press, Insight Press, San Francisco, CA

2 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=e ρ transverse wave rotation R=e iσ defined by u/c=tanh(ρ). defined by u/c=sin(σ). At low speed: u/c~ρ. At low speed: u/c~ σ. (a) Fixed Observer x (b) Moving Observer S S u=c sin σ c δδ c 1-u 2 /c 2 =c/cosh ρ σ c k( ) k( ) z Fig. 5.6 Epstein s cosmic speedometer with aberration angle σ and transverse Doppler shift coshυz. 2

3 Reviewing the Sin-Tan Rosetta geometry NOTE: Angle φ is now called stellar aberration angle σ Fig. C.2-3 and Fig. 5.4 in Unit 2 3

4 (a) Geometry of relativistic transformation and wave based mechanics H=E (b) Tangent geometry (u/c=3/5) Velocity aberration angle φ 4 H ρ 3 -L B p-circle g-circle 2 b-circle 1 B sech ρ B B cosh ρ Rest Energy B =Mc 2 -Lagrangian -L =B sech ρ -2-1 B e -ρ Red shift 1 2 B e +ρ Blue shift (c) Basic construction given u/c=45/53 3 cp 4 Hamiltonian =H =B cosh ρ (d) u/c=3/5 ρ H =53/28 H =5/ L =4/5 u/c =3/5 u/c =1 -L=28/45 e -ρ =2/7 1 cp =45/28 e -ρ =1/2 cp =3/4 1 4

5 Reviewing classical quantum Lagrangian-Hamiltonian relations (a) Lagrangian plot L(v)= v p - H(p) Hamiltonian plot (b) H(p)=p v-l(v) Unit 1 Fig L(v) H(p) velocity v is p-slope velocity v = 1. v-slope=p dl/dv = Lagrangian L(v) = 1. v-slope is momentum p -1 v-slope +1-1 intercept velocity -H = -.6 p-slope v intercept -L = Lagrangian L(v) is -p-slope intercept -v-slope intercept is Hamiltonian H(p) +1 Hamiltonian H(p) =.6 p-slope=v dh/dp= momentum p = momentum p 5

6 Reviewing relativistic quantum Lagrangian-Hamiltonian relations Start with phase Φ and set k= to get product of proper frequency µ = Mc2 / and proper time τ dφ = kdx ω dt= µ dτ = -(Mc 2 / ) dτ. (a) Hamiltonian slope: H p = q = u H H H -L -L -L ds = Ldt = p dx H dt = k dx ω dt = dφ Differential action: is Planck scale times differential phase: ds = dφ O H(q,p) P H P H P H Light cone u=1=c has infinite H and zero L Momentum p dτ = dt (1-u 2 /c 2 )=dt sech ρ For constant u the Lagrangian is: L = µτ = -Mc 2 (1-u 2 /c 2 )= -Mc 2 sech ρ = -Mc 2 cos σ...with Poincare invariant: L= p x H = p u H (b) Lagrangian radius = Mc 2 L L L -H -H -H O L(q,q) Fig Geometry of contact transformation between relativistic (a) Hamiltonian (b) Lagrangian Q Velocity u=q Q slope: Q L q = p 6

7 Per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation k( ) (Lighthouse frame) (a) Laser frame (b) z-( Moving) ship k( ) k( ) k( ) ω coshρ of k( ) x-axis k( ) k( ) k( ) CW Laser-pair wavevectors e ρ k( ) sinh ρ z e ρ =,,, Suppose starlight in lighthouse frame is straight down x-axis : ω,ck x,ck y,ck y ω of k( ) z-axis South 7

8 Per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation k( ) (Lighthouse frame) (a) Laser frame (b) z-( Moving) ship k( ) ω coshρ of k( ) x-axis k( ) k( ) k( ) CW Laser-pair wavevectors k( ) e ρ k( ) West k( ) ( e ρ ) =,,, along -z-axis : ω,ck x,ck y,ck z sinh ρ z =,,, Suppose starlight in lighthouse frame is straight down x-axis : ω,ck x,ck y,ck z ω of k( ) z-axis South 8

9 Per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation k( ) (Lighthouse frame) (a) Laser frame (b) z-( Moving) ship =,+,, k( ) ω coshρ up +x-axis : ω,ck x,ck y,ck z of k( ) x-axis North k( ) k( ) k( ) CW Laser-pair wavevectors k( ) e ρ k( ) West along -z-axis : ω,ck x,ck y,ck z sinh ρ z k( ) ( e ρ ) =,,, =,,, Suppose starlight in lighthouse frame is straight down x-axis : ω,ck x,ck y,ck z ω of k( ) z-axis South 9

10 Per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation (a) Laser frame k( ) k( ) k( ) (Lighthouse frame) k( ) CW Laser-pair wavevectors (b) z-( Moving) ship =,+,, k( ) k( ) e ρ k( ) ω coshρ up +x-axis : ω,ck x,ck y,ck z sinh ρ z k( ) =,,, Suppose starlight in lighthouse frame is straight down x-axis : ω,ck x,ck y,ck z of k( ) =,,,+ along +z-axis : ω,ck x,ck y,ck z West along -z-axis : ω,ck x,ck y,ck z ( e ρ ) =,,, ω of k( ) x-axis z-axis South North East 1

11 Per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation k( ) (Lighthouse frame) (a) Laser frame (b) z-( Moving) ship k( ) k( ) k( ) ω coshρ of k( ) x-axis k( ) k( ) k( ) CW Laser-pair wavevectors e ρ k( ) sinh ρ z Suppose starlight in lighthouse frame is straight down x-axis : ω,ck x,ck y,ck z +ρ z -rapidity ship frame sees starlight Lorentz transformed to : e ρ ω of k( ) z-axis =,,, ( ω,c k,c k,c k ) x y z =,,, sinh ρ z 11

12 Per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation k( ) (Lighthouse frame) (a) Laser frame (b) z-( Moving) ship k( ) k( ) k( ) ω coshρ of k( ) x-axis k( ) k( ) Suppose starlight in lighthouse frame is straight down x-axis : ω,ck x,ck y,ck z k( ) +ρ z -rapidity ship frame sees starlight Lorentz transformed to : ω c k x c k y c k z = CW Laser-pair wavevectors sinh ρ z 1 1 sinh ρ z ω ck x ck y ck z e ρ k( ) = sinh ρ z e ρ ω =,,, ( ω,c k,c k,c k ) x y z =,,, sinh ρ z sinh ρ z 1 1 sinh ρ z of k( ) = z-axis sinh ρ z 12

13 Per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation k( ) (Lighthouse frame) (a) Laser frame (b) z-( Moving) ship k( ) ω coshρ of k( ) x-axis k( ) k( ) Suppose starlight in lighthouse frame is straight down x-axis : ω,ck x,ck y,ck z ω c k x c k y c k z = k( ) CW Laser-pair wavevectors +ρ z -rapidity ship frame sees starlight Lorentz transformed to : sinh ρ z 1 1 sinh ρ z k( ) ω ck x ck y ck z e ρ k( ) = sinh ρ z =,,, ( ω,c k,c k,c k ) x y z =,,, sinh ρ z sinh ρ z 1 1 sinh ρ z k( ) e ρ After the 4-vector transformation, =ω is transverse Doppler shifted to, while ckz= becomes ckz' = - sinh ρ z. (The x-component is unchanged: ckx' = - = ckx and so is y-component: cky' = - = cky.) ω of k( ) = z-axis sinh ρ z sinh ρ z 13

14 Per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation (a) Laser frame k( ) k( ) k( ) ω c k x c k y c k z = (Lighthouse frame) k( ) CW Laser-pair wavevectors sinh ρ z 1 1 sinh ρ z (b) z-( Moving) ship Suppose starlight in lighthouse frame is straight down x-axis : ω,ck x,ck y,ck z ω k( ) k( ) +ρ z -rapidity ship frame sees starlight Lorentz transformed to : ck x ck y ck z e ρ k( ) = ω coshρ =,,, ( ω,c k,c k,c k ) x y z =,,, sinh ρ z sinh ρ z 1 1 sinh ρ z k( ) After the 4-vector transformation, =ω is transverse Doppler shifted to, while ckz= becomes ckz' = - sinh ρ z. (The x-component is unchanged: ckx' = - = ckx and so is y-component: cky' = - = cky.) σ sinh ρ z e ρ ω of k( ) of k( ) x-axis = z-axis sinh ρ z = secσ = sinσ σ sinh ρ z = tanσ secσ tanσ cosσ σ 14

15 Per-spacetime 4-vector (ω,ωx,ωy,ωz) =(ω,ckx,cky,ckz) transformation (a) Laser frame k( ) k( ) k( ) ω c k x c k y c k z = (Lighthouse frame) k( ) CW Laser-pair wavevectors sinh ρ z 1 1 sinh ρ z (b) z-( Moving) ship Suppose starlight in lighthouse frame is straight down x-axis : ω,ck x,ck y,ck z ω k( ) k( ) +ρ z -rapidity ship frame sees starlight Lorentz transformed to : ck x ck y ck z e ρ k( ) = ω coshρ =,,, ( ω,c k,c k,c k ) x y z =,,, sinh ρ z sinh ρ z 1 1 sinh ρ z k( ) e ρ of k( ) of k( ) After the 4-vector transformation, =ω is transverse Doppler shifted to, while ckz= becomes ckz' = - sinh ρ z. (The x-component is unchanged: ckx' = - = ckx and so is y-component: cky' = - = cky.) = sinh ρ z Recall hyperbolic invariant to Lorentz transform: ω 2 -c 2 k 2 =ω 2 -c 2 k 2 (= for 1-CW light) The 4-vector form of this is: ω 2 -c 2 k k=ω 2 -c 2 k k (= ) σ sinh ρ z ω x-axis z-axis = secσ = sinσ σ sinh ρ z = tanσ secσ tanσ cosσ σ 15

16 Fig. 5.1 CW cosmic speedometer. Geometry of Lorentz boost of counter-propagating waves. 16

17 Fig. 5.1 CW cosmic speedometer. Geometry of Lorentz boost of counter-propagating waves. If starlight is horizontal right-moving k wave then ship going u along z-axis sees : ω sinh ρ z ω c k x cosh ρ z sinh ρ z ω e ρ z 1 = = ω c k y 1 = c k z sinh ρ z + sinh ρ z + e ρ z If starlight is horizontal left-moving k wave then ship going u along z-axis sees : ω + sinh ρ z e +ρ z c k x c k y c k z = sinh ρ z 1 1 sinh ρ z The usual longitudinal Doppler blue shifts e +ρ z or Doppler red shifts e ρ z appear on both k-vector and frequency ω. = ω sinh ρ z = e +ρ z 17

18 Epstein s space-proper-time (x,cτ) plots ( c-tau plots) Time contraction-dilation revisited This is also proportional to Lagrangian L= -Mc 2 (1-u 2 /c 2 ) cτ (Age) Proper time cτ= (ct) 2 -x 2 ο Coordinate x=(u/c) ct P σ Coordinate time ct= (cτ) 2 +x 2 Light (never ages) x=ct Particle going u in (x,ct) is going speed c in (x, cτ) γ x (Distance) Fig. 5.8 Space-proper-time plot makes all objects move at speed c along their cosmic speedometer. 18

19 Epstein s space-proper-time (x,cτ) plots ( c-tau plots) Length contraction-dilation revisited A cute Epstein feature is that Lorentz-Fitzgerald contraction of a proper length L to L=L 1-u 2 /c 2 is simply rotational projection onto the x-axis of a length L rotated by σ. cτ Proper time cτ= (ct) 2 -x 2 ο Coordinate x=(u/c) ct P σ σ ct Proper length L Contracted length L=L 1-u 2 /c 2 Particle P going u in (x,ct) is going speed c in (x, cτ) L Comoving particle P ct P x 19

20 Epstein s space-proper-time (x,cτ) plots ( c-tau plots) Twin-paradox revisited τ B way older than A τ B stationary A inbound B older than A A outbound B stationary A outbound x B x B 2

21 Epstein s space-proper-time (x,cτ) plots ( c-tau plots) Twin-paradox revisited τ B way older than A τ τ B way older than A B stationary A inbound B older than A B younger than A A outbound B stationary A outbound x B x B x A 21

22 Epstein s space-proper-time (x,cτ) plots ( c-tau plots) Velocity addition revisited cτ-axis Geometry of relativistic velocity addition tanh(ρ u +ρ v )= tanhρ u +tanhρ v where:tanhρ=sinσ 1+tanhρ u tanhρ v σ u σ v u/c=1/2=tanhρ u =sinσ u u/c=1/2=tanhρ u =sinσ u v/c=2/3=tanhρ v =sinσ v unit circle x-axis 22

23 Epstein s space-proper-time (x,cτ) plots ( c-tau plots) Velocity addition revisited cτ-axis ship1 Geometry of relativistic velocity addition tanh(ρ u +ρ v )= tanhρ u +tanhρ v where:tanhρ=sinσ 1+tanhρ u tanhρ v ship2 ship1 2/ 5=sinhρ v =tanσ v 3/ 5=coshρ v =secσ v 1/ 3=sinhρ u =tanσ u σ u 2/ 3=coshρ u =secσ u σ v ship2 1 u/c=1/2=tanhρ u =sinσ u 3 2 u/c=1/2=tanhρ u =sinσ u v/c=2/3=tanhρ v =sinσ v 3/2=sechρ u =1/coshρ u unit circle In ships frame: Bullet going u= c/2 from ship1 passes ship2 at ship2-time τ=1 and at bullet-time τ= 3/2=sechρ u 1 x-axis In frame where ships go v=2c/3: Bullet going Vu(+)v=?? from ship1 passes ship2 at ship2-time τ=1 and at bullet-time τ= 3/2=sechρ u 23

24 Epstein s space-proper-time (x,cτ) plots ( c-tau plots) Velocity addition revisited cτ-axis Geometry of relativistic velocity addition tanh(ρ u +ρ v )= tanhρ u +tanhρ v where:tanhρ=sinσ 1+tanhρ u tanhρ v 2/ 5=sinhρ v =tanσ v 3/ 5=coshρ v =secσ v 1/ 3=sinhρ u =tanσ u σ u σ v σ uv 2/ 3=coshρ u =secσ u tanhρ u sinhρ u =sinσ u tanσ u tanhρ u =sinσ u 1 u/c=1/2=tanhρ u =sinσ u tanhρ u coshρ u =sinσ u secσ u 3 2 u/c=1/2=tanhρ u =sinσ u sechρ u =1/coshρ u v/c=2/3=tanhρ v =sinσ v unit circle V/c=7/8=tanh(ρ uv )=sinσ uv =tanh(ρ u +ρ v ) In ships frame: Bullet going u= c/2 from ship1 passes ship2 at ship2-time τ=1 and at bullet-time τ= 3/2=sechρ u 1 x-axis In frame where ships go v=2c/3: Bullet going Vu(+)v=7c/8 from ship1 passes ship2 at ship2-time τ=1 and at bullet-time τ= 3/2=sechρ u 24

25 sinh(ρ u +ρ v ) Velocity addition revisited cosh(ρ u +ρ v ) cτ-axis Geometry of relativistic velocity addition tanh(ρ u +ρ v )= tanhρ u +tanhρ v where:tanhρ=sinσ 1+tanhρ u tanhρ v 2/ 5=sinhρ v =tanσ v 1/ 3=sinhρ u =tanφ u 3/ 5=coshρ v =secσ v tanhρ u sinhρ u =sinσ u tanσ u σ u σ v σ uv 2/ 3=coshρ u =secφ u tanhρ u =sinσ u 1 u/c=1/2=tanhρ u =sinσ u tanhρ u coshρ u =sinσ u secσ u 3 2 u/c=1/2=tanhρ u =sinσ u sechρ u =1/coshρ u v/c=2/3=tanhρ v =sinσ v unit circle V/c=7/8=tanh(ρ uv )=sinσ uv =tanh(ρ u +ρ v ) 1/2+2/3 1+(1/2)(2/3) =7/8 In ships frame: Bullet going u= c/2 from ship1 passes ship2 at ship2-time τ=1 and at bullet-time τ= 3/2=sechρ u coshρ u sinhρ v +sinhρ v coshρ v =sinh(ρ u +ρ v )=7/ 15 1 x-axis (coshρ v +tanhρ u sinhρ v ) (coshρ v +tanhρ u sinhρ v )coshρ u =coshρ u coshρ v +sinhρ v sinhρ v =cosh(ρ u +ρ v )=8/ 15 In frame where ships go v=2c/3: Bullet going Vu(+)v=7c/8 from ship1 passes ship2 at ship2-time τ=1 and at bullet-time τ= 3/2=sechρ u 25

26 tanhρ u =u/c=1/2=sinφ u tanhρ v =v/c=2/3=sinφ u tanhρ uv =u/c=7/8=sinφ uv φ u =sin -1 (tanhρ u )=.5236=3 φ v =sin -1 (tanhρ)=.7297=41.8 φ uv =sin -1 (tanhρ uv )=1.6543=61.4 sinh(ρ u +ρ v ) cosh(ρ u +ρ v ) ρ u =tanh -1 (1/2)=.5493 ρ v =tanh(2/3)=.8472 ρ uv =tanh(7/8)= =ρ u +ρ v cτ-axis 2/ 5=sinhρ v =tanφ v 3/ 5=coshρ v =secφ v 1/ 3=sinhρ u =tanφ u φ u φ v φ uv 2/ 3=coshρ u =secφ u tanhρ u sinhρ u =sinφ u tanφ u tanhρ u =sinφ u 1 u/c=1/2=tanhρ u =sinφ u tanhρ u coshρ u =sinφ u secφ u 3 2 u/c=1/2=tanhρ u =sinφ u 5 3 v/c=2/3=tanhρ v =sinφ v unit circle 3/2=sechρ u =1/coshρ u 4 V/c=7/8=tanh(ρ uv )=sinφ uv 5 =tanh(ρ u +ρ v ) In ships frame: Bullet going u= c/2 from ship1 passes ship2 at ship2-time τ=1 and at bullet-time τ= 3/2=sechρ u coshρ u sinhρ v +sinhρ v coshρ v =sinh(ρ u +ρ v )=7/ 15 1 x-axis (coshρ v +tanhρ u sinhρ v ) (coshρ v +tanhρ u sinhρ v )coshρ u =coshρ u coshρ v +sinhρ v sinhρ v =cosh(ρ u +ρ v )=8/ 15 In frame where ships go v=2c/3: Bullet going Vu(+)v=7c/8 from ship1 passes ship2 at ship2-time τ=1 and at bullet-time τ= 3/2=sechρ u 26

27 Lorentz symmetry effects How it makes momentum and energy be conserved 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 T(δ,τ ) in the vacuum). T ψ k,ω = Ae i(kx ω t) e Operator has plane wave eigenfunctions with roots-of-unity eigenvalues. ψ k,ω T i(k δ ω τ ) = ψ k,ω e (5.18a) T ψ k,ω = e i(k δ ω τ ) ψ k,ω i(k δ ω τ ) (5.18b) This also applies to 2-part or 2-particle product states Ψ K,Ω = ψ k1,ω 1 ψ k2,ω 2 where exponents add (k,ω)-values of each constituent to K=k1+k2 and Ω=ω1+ω2, and -eigenvalues also have that form. Ψ K, T(δ,τ ) e i(k δ Ω τ ) Matrix Ω U Ψ K,Ω of T-symmetric evolution U is zero unless K = k 1 + k 2 = K and Ω = ω 1 + ω 2 = Ω. Ψ K, Ω U Ψ K,Ω = Ψ K, Ω T (δ,τ )UT(δ,τ ) Ψ K,Ω (if UT = TU for all δ and τ ) = e i( K δ Ω τ ) e i(k δ Ω τ ) Ψ K, Ω U Ψ K,Ω = unless: K = K and: Ω = Ω That s momentum (P=hK) and energy (E=hW) conservation! 27