Abstract

The propagation of light inside a moving inhomogeneous medium is modified by two distortion phenomena. The phase function of light changes as a result of both the spatial variation of the refractive index and the motion of matter. We deal with the expanding or collapsing motions of a dielectric and study propagation of light in such a medium within the geometrical optics approximation. A correction of the famous Clausius principle and Bouguer’s law is given to take into account the distortion of light and to correct the classic form of the radiative-transfer equation. We prove that the expression of a new energetic invariant is a direct consequence of local dilatations of space–time that are due to both the Doppler shift that arises from the problem of unsteady optics and the phase distortion phenomena assigned to the velocity field.

© 2001 Optical Society of America

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References

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  1. W. Gordon, “Zur Lichtfortpflanzung nach der relativitätstheorie,” Ann. Phys. (Leipzig) 72, 421–456 (1923).
    [CrossRef]
  2. U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
    [CrossRef]
  3. M. Novello, V. A. De Lorenci, and E. Elbaz, “Some aspects of geometrical confinement,” Notas de Física CBPF-NF-033/98 (Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, Brazil, 1998).
  4. M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001 (2000).
    [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).
  6. G. C. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon, New York, 1973).
  7. L. Landau and E. Lifchitz, Electrodynamique des Milieux Continus (Mir, Moscow, 1990).
  8. J. Hadamard, Leçons sur la Propagation des Ondes (Collège de France, Hermann, 1903).
  9. Y. Choquet-Bruhat, C. De Witt-Morette, and M. Dillard-Bleick, in Analysis, Manifolds and Physics (North-Holland, New York, 1977).
  10. Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
    [CrossRef]
  11. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, New York, 1973).
  12. P. H. Hauschildt and R. Wehrse, “Solution of the special relativistic equation of radiative transfer in rapidly expanding spherical shells,” J. Quant. Spectrosc. Radiat. Transfer. 46, 81–98 (1991).
    [CrossRef]
  13. A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
    [CrossRef]
  14. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  15. C. Marle, “Sur l’établissement des équations de l’hydrodynamique des fluides relativistes dissipatifs: l’équation de Boltzmann relativiste,” Ann. Inst. Henri Poincaré A 1, 67–126 (1969).
  16. D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics (Oxford U. Press, New York, 1984), pp. 300–316.

2000 (1)

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001 (2000).
[CrossRef]

1999 (1)

U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

1991 (1)

P. H. Hauschildt and R. Wehrse, “Solution of the special relativistic equation of radiative transfer in rapidly expanding spherical shells,” J. Quant. Spectrosc. Radiat. Transfer. 46, 81–98 (1991).
[CrossRef]

1969 (1)

C. Marle, “Sur l’établissement des équations de l’hydrodynamique des fluides relativistes dissipatifs: l’équation de Boltzmann relativiste,” Ann. Inst. Henri Poincaré A 1, 67–126 (1969).

1959 (1)

Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

1923 (1)

W. Gordon, “Zur Lichtfortpflanzung nach der relativitätstheorie,” Ann. Phys. (Leipzig) 72, 421–456 (1923).
[CrossRef]

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Aharonov, Y.

Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Bohm, D.

Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

De Lorenci, V. A.

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001 (2000).
[CrossRef]

Gordon, W.

W. Gordon, “Zur Lichtfortpflanzung nach der relativitätstheorie,” Ann. Phys. (Leipzig) 72, 421–456 (1923).
[CrossRef]

Hauschildt, P. H.

P. H. Hauschildt and R. Wehrse, “Solution of the special relativistic equation of radiative transfer in rapidly expanding spherical shells,” J. Quant. Spectrosc. Radiat. Transfer. 46, 81–98 (1991).
[CrossRef]

Klippert, R.

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001 (2000).
[CrossRef]

Leonhardt, U.

U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

Marle, C.

C. Marle, “Sur l’établissement des équations de l’hydrodynamique des fluides relativistes dissipatifs: l’équation de Boltzmann relativiste,” Ann. Inst. Henri Poincaré A 1, 67–126 (1969).

Novello, M.

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001 (2000).
[CrossRef]

Piwnicki, P.

U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

Salim, J. M.

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001 (2000).
[CrossRef]

Schuster, A.

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

Wehrse, R.

P. H. Hauschildt and R. Wehrse, “Solution of the special relativistic equation of radiative transfer in rapidly expanding spherical shells,” J. Quant. Spectrosc. Radiat. Transfer. 46, 81–98 (1991).
[CrossRef]

Ann. Inst. Henri Poincaré A (1)

C. Marle, “Sur l’établissement des équations de l’hydrodynamique des fluides relativistes dissipatifs: l’équation de Boltzmann relativiste,” Ann. Inst. Henri Poincaré A 1, 67–126 (1969).

Ann. Phys. (Leipzig) (1)

W. Gordon, “Zur Lichtfortpflanzung nach der relativitätstheorie,” Ann. Phys. (Leipzig) 72, 421–456 (1923).
[CrossRef]

Astrophys. J. (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1–22 (1905).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer. (1)

P. H. Hauschildt and R. Wehrse, “Solution of the special relativistic equation of radiative transfer in rapidly expanding spherical shells,” J. Quant. Spectrosc. Radiat. Transfer. 46, 81–98 (1991).
[CrossRef]

Phys. Rev. (1)

Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Phys. Rev. A (1)

U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
[CrossRef]

Phys. Rev. D (1)

M. Novello, V. A. De Lorenci, J. M. Salim, and R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001 (2000).
[CrossRef]

Other (9)

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1964).

G. C. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon, New York, 1973).

L. Landau and E. Lifchitz, Electrodynamique des Milieux Continus (Mir, Moscow, 1990).

J. Hadamard, Leçons sur la Propagation des Ondes (Collège de France, Hermann, 1903).

Y. Choquet-Bruhat, C. De Witt-Morette, and M. Dillard-Bleick, in Analysis, Manifolds and Physics (North-Holland, New York, 1977).

M. Novello, V. A. De Lorenci, and E. Elbaz, “Some aspects of geometrical confinement,” Notas de Física CBPF-NF-033/98 (Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, Brazil, 1998).

C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, New York, 1973).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

D. Mihalas and B. W. Mihalas, Foundations of Radiation Hydrodynamics (Oxford U. Press, New York, 1984), pp. 300–316.

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Figures (8)

Fig. 1
Fig. 1

Geometrical configuration and coordinates: A photon that is propagating inside an expanding sphere is framed on a canonical basis by polar angle φ and azimuthal angle θ. The orientation of unit tangent vector Ω is given in a local spherical framework (er, eθ, eφ) by ζ and γ.

Fig. 2
Fig. 2

Photon trajectories within an expanding refractive sphere are plane. In particular, they obey Kepler’s second law (equal areas are swept out in equal times). Here we have made the assumption of purely specular boundary conditions (i.e., that reflecting angles γ+ and γ- are equal).

Fig. 3
Fig. 3

Within a static homogeneous sphere the potential is monotonic and the paths are straight lines. The sphere behaves as a barrier for the photon, and consequently the turning point of the orbit occurs where a horizontal line of height E02 crosses V.

Fig. 4
Fig. 4

Within a static inhomogeneous sphere the potential generally exhibits many extrema. Either of two configurations might occur. Case I corresponds to the emerging trajectories. The photon reaches unique extremum along a curved trajectory before touching the boundary. Moreover, the presence of a local extremum in the potential curve is directly responsible for the existence of confined trajectories. In such a configuration (Case II) the light path propagates between two spheres defined by the lowest root r2 of V-E02 in the interval ]r0; R] and its highest root r1 in ]0; r0]. These types of trajectory never reach the boundary of the sphere, so the corresponding flux line is isolated from the external medium.

Fig. 5
Fig. 5

Critical speeds for expansion of the homogeneous spheres. There are two distinct regimes. The first, called the low-velocity regime, has an optical behavior similar to that encountered inside inhomogeneous static spheres. The second regime is analogous to relativistic gravitational behavior.

Fig. 6
Fig. 6

Within an expanding homogeneous sphere for which the condition γ>g is fulfilled for a given refractive index n0, there are many light rays that fall into a vortex and are dragged inexorably to the center. This procedure is similar to a black-hole effect.

Fig. 7
Fig. 7

Illustration of the local dilatation of an elementary surface in presence of distortion induced by a refractive index. There is a simple temporal dilatation.

Fig. 8
Fig. 8

Geometric illustration of dilatation inside an expanding sphere. There is simultaneously a dilatation of time as inside the refractive medium and a spatial dilatation that is intrinsic to motion.

Equations (66)

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Fμν=Eμvν-Eνvμ+ημνρσvρHσ,
Pμν=Dμvν-Dνvμ+ημνρσvρBσ,
vFμν*=0,vPμν=0,
Fμν*=½ημντσFτσ.
[λEμν]Σ=kλeμν,
[λDμν]Σ=kλdμν,
[λHμν]Σ=kλhμν,
[λBμν]Σ=kλbμν,
Dα=εEα,Bα=Hα/μ
kμkν[ημν+(εμ-1)vμvν]=0,
gμν[ημν+(εμ-1)νμνν],
uνγ1, u(r)c, 0, 0,
uνγ1,-u(r)c,0, 0,
γ[1-(u/c)2]-1/2.
gμν=Cημν+Duμuν
δμν=gμβ gβν,
gμν=1+χγ2-γ2βχ00-γ2βχχγ2β2-10000-10000-1,
ds2=gμνdxμdxν=[1+χγ2](cdt)2-2γ2βχdrdt+[χγ2β2-1]dr2-r2dθ2-r2 sin2 θdφ2.
δ (gτττ˙2+2grtr˙τ˙+grrr˙2-r2θ˙2-r2 sin2 θφ˙2)dλ,
r2 sin2 θ dφdλ=Cte
r2 dθdλ=Cte=h0.
gτττ˙+grτr˙=E0,
dsdλ2=gτττ˙2+2grτr˙τ˙+grrr˙2-r2θ˙2=0.
r˙2=E02grτ2-grrgττ+h02r2 gττgrrgττ-grτ2.
r˙2=E02-V(r),
V(r)=E02(1-n2)+h02r2[(1-n2)γ2+n2].
dθdr=±1r2 h0[E02-V(r)]1/2.
θ-θ0=±h0 r0r 1s2 ds[E02-V(r)]1/2
E02=V(r).
Ω=drdλ-1 drdλ=(r˙2+r2θ˙2)-1/2(r˙er+rθ˙eθ).
Ωeθ=sin φ=(r˙2+r2θ˙2)-1/2rθ˙,
sin φ=h0{r2[E02-V(r)]+h02}1/2=1E02h02n2r2+(1-n2)(1-γ2)1/2.
E02h02=1n2(r0)r02 1sin φ02+[1-n2(r0)][γ2(r0)-1].
dθdr=±1r2 h0[E02-V(r)]1/2=±h0r 1(E02n2(r)r2-h02{[1-n2(r)]γ2+n2(r)})1/2,
dθdr=±h0r 1(E02n02r2-h02)1/2.
θ-θ0=±r0r h0s ds(E02n02s2-h02)1/2=1|h0| arccosh0E0n0r-arccosh0E0n0r0,
V(r)=E02[1-n2(r)]+h02r2,
dVdr=0n n dndrr3=-h0E02.
θ-θ0=±r0r h0s ds[E02n2(s)s2-h02]1/2.
E02n2(r*)r*2=h02.
V(r)=E02(1-n02)+h02r2[(1-n02)γ2(r)+n02].
Vs(r)=ε-2ε GMr+L2r2-2 GML2r3,
η10000-10000-10000-1.
Iν(x, Ω)=hνN(x, Ω, ν)dSdtdνdΩ,
gμν=1/n20000-10000-10000-1,
dSdt=n2dSdt
Iν(x, Ω)ν3=hνN(x, Ω, ν)ν3n2dSdtdνdΩ=Cte.
Iν(x, Ω)ν3=Iν(x, Ω)ν3n2=Cte
λ0,1=½{χγ2(1+β2)±[χ2γ4(1+β2)2+4(1+χ)]1/2},λ2,3=-1,
e0=(K+L)(2βγ2χ)-1100,
e1=(K-L)(2βγ2χ)-1100,e2e2,e3e3,
K=-(2+χ),L=[4(1+χ)+γ4χ2(1+β2)2]1/2.
A=dx1e1=a0e0+a1e1=dx1[(e1, e0)e0+(e1, e1)e1],
B=dte0=b0e0+b1e1=dt[(e0, e0)e0+(e0, e1)e1],
dΣ=dx2AB=dx1dx2dt.
A=a0λ0e0+a1λ1e1,B=b0λ0e0+b1λ1e1
A=[a0λ0(e0, e0)+a1λ1(e1, e0)]e0+[a0λ0(e0, e1)+a1λ1(e1, e1)]e1
=dx1{[λ0(e0, e1)(e0, e0)+λ1(e1, e1)(e1, e0)]e0+[λ0(e0, e1)2+λ1(e1, e1)2]e1}=A0e0+A1e1,
B=[b0λ0(e0, e0)+b1λ1(e1, e0)]e0+[b0λ0(e0, e1)+b1λ1(e1, e1)]e1
=dt{[λ0(e0, e0)2+λ1(e1, e0)2]e0+[λ0(e0, e0) (e0, e1)+λ1(e1, e0)×(e1, e1)]e1}=B0e0+B1e1.
dΣ=dx2AB=dx1dx2dt|A0B1-A1B0|=J-1dΣ,
J|λ0-1λ1-1|[(e0, e1)(e1, e0)-(e0, e0)(e1, e1)]-2.
Gν(x, Ω)Iν(x, Ω)ν3J(x)=Cte.
ddsGν(x, Ω)+κνGν(x, Ω)=Sν(x, Ω),
Sν(x, Ω)=Qν(x)+1J(x) 0+ 1ν3 4π σs(x, νν, ΩΩ)Iν(x, Ω)dΩdν
dIνds+κν(x)-d log(ν3J)dsIν=ν3J(x)Sν(x, Ω),

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