Abstract

In numerous media (nonlinear material, moving dielectrics, superfluids, Bose–Einstein condensates, and others) and different in vacuo states (nontrivial quantum electrodynamics in vacuo) matter or vacuum fluctuations modify light propagation in the same way that an effective gravitational field does. This nonlinear optical behavior affects not only the energy paths but also the form of the energetic invariant. However, such a function plays a key role when we try to develop a phenomenological kinetic theory for participating media. I analyze how modification of light propagation transforms the energetic invariant and modifies its transport inside a participating medium. A semianalytical method is presented to solve the radiative transfer equation for any spherically symmetric problems.

© 2001 Optical Society of America

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References

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  1. L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
    [CrossRef]
  2. U. Leonhardt, P. Piwnicki, “Optics of nonuniformly moving media,” Phys. Rev. A 60, 4301–4312 (1999).
    [CrossRef]
  3. A. Burrows, T. Young, P. A. Pinto, R. Eastman, T. A. Thompson, “Supernova neutrinos and a new algorithm for neutrino transport,” Astrophys. J. 539, 865–932 (2000).
    [CrossRef]
  4. M. Sikora, M. C. Begelman, M. J. Rees, “Comptonization of diffuse ambient radiation by a relativistic jet: the source of gamma rays from blazars?,” Astrophys. J. 421, 153–162 (1994).
    [CrossRef]
  5. M. Novello, V. A. De Lorenci, J. M. Salim, R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001–45011 (2000).
    [CrossRef]
  6. W. Dittrich, H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 25004–25017 (1998).
    [CrossRef]
  7. F. Dalfovo, S. Giorgini, L. P. Pitaevski, S. Stringari, “Theory of trapped Bose-condensed gases,” Rev. Mod. Phys. 71, 463–532 (1999).
    [CrossRef]
  8. W. G. Unruh, “Sonic analog of black holes and the effects of high frequencies on black hole evaporation,” Phys. Rev. D 51, 2827–2838 (1995).
    [CrossRef]
  9. M. Visser, “Acoustic black holes: horizons, ergospheres, and hawking radiation,” Class. Quantum Grav. 15, 1767–1791 (1998).
    [CrossRef]
  10. L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
    [CrossRef] [PubMed]
  11. J. Plebanski, Lectures on Nonlinear Electrodynamics (Nordita, Copenhagen, 1968).
  12. U. Leonhardt, P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
    [CrossRef] [PubMed]
  13. M. Novello, V. A. De Lorenci, E. Elbaz, “Some aspects of geometrical confinement,” Int. J. Mod. Phys. A. 13, 4539–4552 (1998).
    [CrossRef]
  14. M. Novello, J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
    [CrossRef]
  15. M. M. Novak, “The effect of a nonlinear medium on electromagnetic waves,” Fortschr. Phys. 37, 125–159 (1989).
    [CrossRef]
  16. V. A. De Lorenci, M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
    [CrossRef]
  17. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1964).
  18. G. C. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon, London, 1973).
  19. P. Ben-Abdallah, V. Le Dez, “Energetic and optical consequences of light distortion inside an expanding isotropic sphere,” J. Opt. Soc. Am. B (to be published).
  20. P. Ben-Abdallah, M. Sakami, V. Le Dez, J. B. Saulnier, A. Charrette, “Optical remote sensing inside an inhomogeneous axisymmetric medium: the absorption field measurement,” Appl. Opt. 39, 411–417 (2000).
    [CrossRef]

2001 (2)

M. Novello, J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
[CrossRef]

V. A. De Lorenci, M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
[CrossRef]

2000 (5)

P. Ben-Abdallah, M. Sakami, V. Le Dez, J. B. Saulnier, A. Charrette, “Optical remote sensing inside an inhomogeneous axisymmetric medium: the absorption field measurement,” Appl. Opt. 39, 411–417 (2000).
[CrossRef]

L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[CrossRef] [PubMed]

U. Leonhardt, P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

A. Burrows, T. Young, P. A. Pinto, R. Eastman, T. A. Thompson, “Supernova neutrinos and a new algorithm for neutrino transport,” Astrophys. J. 539, 865–932 (2000).
[CrossRef]

M. Novello, V. A. De Lorenci, J. M. Salim, R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001–45011 (2000).
[CrossRef]

1999 (3)

F. Dalfovo, S. Giorgini, L. P. Pitaevski, S. Stringari, “Theory of trapped Bose-condensed gases,” Rev. Mod. Phys. 71, 463–532 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[CrossRef]

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

1998 (3)

W. Dittrich, H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 25004–25017 (1998).
[CrossRef]

M. Novello, V. A. De Lorenci, E. Elbaz, “Some aspects of geometrical confinement,” Int. J. Mod. Phys. A. 13, 4539–4552 (1998).
[CrossRef]

M. Visser, “Acoustic black holes: horizons, ergospheres, and hawking radiation,” Class. Quantum Grav. 15, 1767–1791 (1998).
[CrossRef]

1995 (1)

W. G. Unruh, “Sonic analog of black holes and the effects of high frequencies on black hole evaporation,” Phys. Rev. D 51, 2827–2838 (1995).
[CrossRef]

1994 (1)

M. Sikora, M. C. Begelman, M. J. Rees, “Comptonization of diffuse ambient radiation by a relativistic jet: the source of gamma rays from blazars?,” Astrophys. J. 421, 153–162 (1994).
[CrossRef]

1989 (1)

M. M. Novak, “The effect of a nonlinear medium on electromagnetic waves,” Fortschr. Phys. 37, 125–159 (1989).
[CrossRef]

Anglin, J. R.

L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[CrossRef] [PubMed]

Begelman, M. C.

M. Sikora, M. C. Begelman, M. J. Rees, “Comptonization of diffuse ambient radiation by a relativistic jet: the source of gamma rays from blazars?,” Astrophys. J. 421, 153–162 (1994).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[CrossRef]

Ben-Abdallah, P.

P. Ben-Abdallah, M. Sakami, V. Le Dez, J. B. Saulnier, A. Charrette, “Optical remote sensing inside an inhomogeneous axisymmetric medium: the absorption field measurement,” Appl. Opt. 39, 411–417 (2000).
[CrossRef]

P. Ben-Abdallah, V. Le Dez, “Energetic and optical consequences of light distortion inside an expanding isotropic sphere,” J. Opt. Soc. Am. B (to be published).

Born, M.

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

Burrows, A.

A. Burrows, T. Young, P. A. Pinto, R. Eastman, T. A. Thompson, “Supernova neutrinos and a new algorithm for neutrino transport,” Astrophys. J. 539, 865–932 (2000).
[CrossRef]

Charrette, A.

Cirac, J. I.

L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[CrossRef] [PubMed]

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevski, S. Stringari, “Theory of trapped Bose-condensed gases,” Rev. Mod. Phys. 71, 463–532 (1999).
[CrossRef]

De Lorenci, V. A.

V. A. De Lorenci, M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
[CrossRef]

M. Novello, V. A. De Lorenci, J. M. Salim, R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001–45011 (2000).
[CrossRef]

M. Novello, V. A. De Lorenci, E. Elbaz, “Some aspects of geometrical confinement,” Int. J. Mod. Phys. A. 13, 4539–4552 (1998).
[CrossRef]

Dittrich, W.

W. Dittrich, H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 25004–25017 (1998).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[CrossRef]

Eastman, R.

A. Burrows, T. Young, P. A. Pinto, R. Eastman, T. A. Thompson, “Supernova neutrinos and a new algorithm for neutrino transport,” Astrophys. J. 539, 865–932 (2000).
[CrossRef]

Elbaz, E.

M. Novello, V. A. De Lorenci, E. Elbaz, “Some aspects of geometrical confinement,” Int. J. Mod. Phys. A. 13, 4539–4552 (1998).
[CrossRef]

Garay, L. J.

L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[CrossRef] [PubMed]

Gies, H.

W. Dittrich, H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 25004–25017 (1998).
[CrossRef]

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevski, S. Stringari, “Theory of trapped Bose-condensed gases,” Rev. Mod. Phys. 71, 463–532 (1999).
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[CrossRef]

Klippert, R.

M. Novello, V. A. De Lorenci, J. M. Salim, R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001–45011 (2000).
[CrossRef]

Le Dez, V.

P. Ben-Abdallah, M. Sakami, V. Le Dez, J. B. Saulnier, A. Charrette, “Optical remote sensing inside an inhomogeneous axisymmetric medium: the absorption field measurement,” Appl. Opt. 39, 411–417 (2000).
[CrossRef]

P. Ben-Abdallah, V. Le Dez, “Energetic and optical consequences of light distortion inside an expanding isotropic sphere,” J. Opt. Soc. Am. B (to be published).

Leonhardt, U.

U. Leonhardt, P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

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

Novak, M. M.

M. M. Novak, “The effect of a nonlinear medium on electromagnetic waves,” Fortschr. Phys. 37, 125–159 (1989).
[CrossRef]

Novello, M.

M. Novello, J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
[CrossRef]

M. Novello, V. A. De Lorenci, J. M. Salim, R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001–45011 (2000).
[CrossRef]

M. Novello, V. A. De Lorenci, E. Elbaz, “Some aspects of geometrical confinement,” Int. J. Mod. Phys. A. 13, 4539–4552 (1998).
[CrossRef]

Pinto, P. A.

A. Burrows, T. Young, P. A. Pinto, R. Eastman, T. A. Thompson, “Supernova neutrinos and a new algorithm for neutrino transport,” Astrophys. J. 539, 865–932 (2000).
[CrossRef]

Pitaevski, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevski, S. Stringari, “Theory of trapped Bose-condensed gases,” Rev. Mod. Phys. 71, 463–532 (1999).
[CrossRef]

Piwnicki, P.

U. Leonhardt, P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

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

Plebanski, J.

J. Plebanski, Lectures on Nonlinear Electrodynamics (Nordita, Copenhagen, 1968).

Pomraning, G. C.

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

Rees, M. J.

M. Sikora, M. C. Begelman, M. J. Rees, “Comptonization of diffuse ambient radiation by a relativistic jet: the source of gamma rays from blazars?,” Astrophys. J. 421, 153–162 (1994).
[CrossRef]

Sakami, M.

Salim, J. M.

M. Novello, J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
[CrossRef]

M. Novello, V. A. De Lorenci, J. M. Salim, R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001–45011 (2000).
[CrossRef]

Saulnier, J. B.

Sikora, M.

M. Sikora, M. C. Begelman, M. J. Rees, “Comptonization of diffuse ambient radiation by a relativistic jet: the source of gamma rays from blazars?,” Astrophys. J. 421, 153–162 (1994).
[CrossRef]

Souza, M. A.

V. A. De Lorenci, M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
[CrossRef]

Stringari, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevski, S. Stringari, “Theory of trapped Bose-condensed gases,” Rev. Mod. Phys. 71, 463–532 (1999).
[CrossRef]

Thompson, T. A.

A. Burrows, T. Young, P. A. Pinto, R. Eastman, T. A. Thompson, “Supernova neutrinos and a new algorithm for neutrino transport,” Astrophys. J. 539, 865–932 (2000).
[CrossRef]

Unruh, W. G.

W. G. Unruh, “Sonic analog of black holes and the effects of high frequencies on black hole evaporation,” Phys. Rev. D 51, 2827–2838 (1995).
[CrossRef]

Visser, M.

M. Visser, “Acoustic black holes: horizons, ergospheres, and hawking radiation,” Class. Quantum Grav. 15, 1767–1791 (1998).
[CrossRef]

Wolf, E.

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

Young, T.

A. Burrows, T. Young, P. A. Pinto, R. Eastman, T. A. Thompson, “Supernova neutrinos and a new algorithm for neutrino transport,” Astrophys. J. 539, 865–932 (2000).
[CrossRef]

Zoller, P.

L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[CrossRef] [PubMed]

Appl. Opt. (1)

Astrophys. J. (2)

A. Burrows, T. Young, P. A. Pinto, R. Eastman, T. A. Thompson, “Supernova neutrinos and a new algorithm for neutrino transport,” Astrophys. J. 539, 865–932 (2000).
[CrossRef]

M. Sikora, M. C. Begelman, M. J. Rees, “Comptonization of diffuse ambient radiation by a relativistic jet: the source of gamma rays from blazars?,” Astrophys. J. 421, 153–162 (1994).
[CrossRef]

Class. Quantum Grav. (1)

M. Visser, “Acoustic black holes: horizons, ergospheres, and hawking radiation,” Class. Quantum Grav. 15, 1767–1791 (1998).
[CrossRef]

Fortschr. Phys. (1)

M. M. Novak, “The effect of a nonlinear medium on electromagnetic waves,” Fortschr. Phys. 37, 125–159 (1989).
[CrossRef]

Int. J. Mod. Phys. A. (1)

M. Novello, V. A. De Lorenci, E. Elbaz, “Some aspects of geometrical confinement,” Int. J. Mod. Phys. A. 13, 4539–4552 (1998).
[CrossRef]

Nature (London) (1)

L. V. Hau, S. E. Harris, Z. Dutton, C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature (London) 397, 594–598 (1999).
[CrossRef]

Phys. Lett. B (1)

V. A. De Lorenci, M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Phys. Lett. B 512, 417–422 (2001).
[CrossRef]

Phys. Rev. A (1)

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

Phys. Rev. D (4)

M. Novello, V. A. De Lorenci, J. M. Salim, R. Klippert, “Geometrical aspects of light propagation in nonlinear electrodynamics,” Phys. Rev. D 61, 45001–45011 (2000).
[CrossRef]

W. Dittrich, H. Gies, “Light propagation in nontrivial QED vacua,” Phys. Rev. D 58, 25004–25017 (1998).
[CrossRef]

M. Novello, J. M. Salim, “Effective electromagnetic geometry,” Phys. Rev. D 63, 083511 (2001).
[CrossRef]

W. G. Unruh, “Sonic analog of black holes and the effects of high frequencies on black hole evaporation,” Phys. Rev. D 51, 2827–2838 (1995).
[CrossRef]

Phys. Rev. Lett. (2)

U. Leonhardt, P. Piwnicki, “Relativistic effects of light in moving media with extremely low group velocity,” Phys. Rev. Lett. 84, 822–825 (2000).
[CrossRef] [PubMed]

L. J. Garay, J. R. Anglin, J. I. Cirac, P. Zoller, “Sonic analog of gravitational black holes in Bose–Einstein condensates,” Phys. Rev. Lett. 85, 4643–4647 (2000).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevski, S. Stringari, “Theory of trapped Bose-condensed gases,” Rev. Mod. Phys. 71, 463–532 (1999).
[CrossRef]

Other (4)

J. Plebanski, Lectures on Nonlinear Electrodynamics (Nordita, Copenhagen, 1968).

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

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

P. Ben-Abdallah, V. Le Dez, “Energetic and optical consequences of light distortion inside an expanding isotropic sphere,” J. Opt. Soc. Am. B (to be published).

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Figures (3)

Fig. 1
Fig. 1

Critical velocity u c O(c/ 2) within a relativistic expanding shell. Beyond this velocity many light rays fall into a vortex and are dragged inexorably down to the center. This relativistic effect is similar to a black hole effect.

Fig. 2
Fig. 2

Trapping of light in a quasi-Newtownian potential. The future of a photon that moves toward greater r values from an origin at r 0 is shown. If the difference E 0 2 - V eff(r) between the total energy and the effective potential has two roots that encircle r 0, the photon is confined between the two disks of radius r* and . When this equation has no root on ]r 0;R[, it reaches the boundary.

Fig. 3
Fig. 3

Generalized Bouguer formula [Eq. (16)] relating the local orientation of a ray at the current point to the one at the origin point M 0.

Equations (48)

Equations on this page are rendered with MathJax. Learn more.

Fμν=Fμν exp(iS),
S=- kνdxν,
dxν=cdt, dx,
kν=k0, -k=-νS,
gμνkμkν=0.
δ  g¯¯ dMdλ, dMdλdλ=0,
δ  gttτ˙2+2grtr˙τ˙+grrr˙2-r2θ˙2+sin2 θφ˙2dλ=0,
θ=π/2
r2dφdλ = constant=h0,
gτττ˙+grτr˙=E0,
gttτ˙2+2grtr˙τ˙+grrr˙2-r2φ˙2=0.
r˙2=E02-Vr,
Vr=E021+1det g-h02r2gttdet g
E02=Vr.
dφdr=±1r2h0E02-Vr1/2.
dφdr=±-det g1/2rηr2-gtt1/2.
Ω=drdλ-1drdλ=r˙2+r2φ˙2-1/2r˙er+rφ˙eφ.
Ω·eφ=sin ψ=r˙2+r2φ˙2-1/2rφ˙,
sin ψ=h0r2E02-Vr+h021/2=1det ggtt-r2E02h02+1-1/2.
η=E02h02=1r02gttr0-det gr0tan2ψ0.
η=1r02gttr0-det gr0tan2ψ0=1r2gttr-det grtan2ψr.
δ  1neff2r τ˙2-r˙2-r2θ˙2+sin2 θφ˙2dλ=0.
dφdr=±1rηneff2rr2-11/2,
η-1=h02E02=neff2r2 sin2 ψ.
neff2=1ηr21det ggtt-ηr2+1,
neff2r=1+tan2 ψrgttrtan2 ψr-det gr
Veffr=E021-neff2+h02r2.
dds Gνx, Ω+κνGνx, Ω=Sνx, Ω,
Sνx, Ω=Qνx+0+4π σsx, ν  ν,Ω  ΩGνx, ΩdΩdν
dIνds+κνx-d logneff2dsIν=neff2Sνx, Ω,
k=ωn/c,
keff=ωneffc.
vg=dωdkeff=cneff+ω dneffdω.
vph=keffω=cneff.
tan2 ψ < - 1 + det g1-gtt.
tan2 ψ < 1 + det g1-gtt.
gαβ=ηαβ+1εμ-1uαuβ,
uνγ1,-urc,0,0,
det g=- 1εμ,  gττ=1+χγ2,
ε=χ1+χ2E+χ3E2+,
gμν=εημν-εEEμEν-E2δtμδtν,
EμE0, E1, 0, 0
gμν=ε+εE12E-1L-εE0E1EL00-εE0E1EL-ε+εEL0000-1ε0000-1ε,
L=ε2+εεE+εεE12E+ε2E12+ε2E02E12E2.
det g=- KE2L2ε2,  gττ=ε+εE12E-1L.
ψsr, t=ρr1/2 expiϑr-μt/,
gμν=c2-ν2ν1ν2ν3ν1-100ν20-10ν300-1,
det g=-c2,  gττ=c2-ν2.

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