Abstract

I repeat my (very short and easy) disproof of the recurring main claim of Ardavan et al. [J. Opt. Soc. Am. A 21, 858 (2004) ] that a smooth source of electromagnetic fields moving in a confined region can generate an intensity decaying more slowly than the inverse square of the distance away (“nonspherical” decay). The field is not isotropic, so energy conservation is not enough to dismiss the claim. Instead my disproof follows directly from Maxwell’s equations, supplying an upper bound with inverse square decay on the intensity. It therefore applies under all circumstances, quite irrespective of any fast or slow motion of the source. Despite the falsity of the main claim, the derivation of the uniform approximation to the Green function for superluminal circulation, which was needed for the claim and is based on the previous work of the first author, is valid. Its validity, importantly, extends significantly beyond the regime envisaged by the authors, and it stands as a basic result of superluminal circulation.

© 2006 Optical Society of America

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References

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  1. H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 21, 858-872 (2004).
    [CrossRef]
  2. H. Ardavan, "Generation of focused, nonspherically decaying pulses of electromagnetic radiation," Phys. Rev. E 58, 6659-6684 (1998).
    [CrossRef]
  3. H. Ardavan, "Method of handling the divergences in the radiation theory of sources that move faster than their waves," J. Math. Phys. 40, 4331-4336 (1999).
    [CrossRef]
  4. H. Ardavan, "The mechanism of radiation in pulsars," Mon. Not. R. Astron. Soc. 268, 361-392 (1994).
  5. H. Ardavan, "The near-field singularity predicted by the spiral Green's function in acoustics and electrodynamics," Proc. R. Soc. London, Ser. A 433, 451-459 (1991).
  6. J. H. Hannay, "Bounds on fields from fast rotating sources, and others," Proc. R. Soc. London, Ser. A 452, 2351-2354 (1996). (The factor half in Eqs. 1.1, 1.5, 1.6 should be replaced by two instead, the rest of the equations being unaffected.)
  7. J. H. Hannay, "Comment II on 'Generation of focused, nonspherically decaying pulses of electromagnetic radiation'," Phys. Rev. E 62, 3008-3009 (2000). (The paper immediately after this one is a reply by H. Ardavan.)
    [CrossRef]
  8. J. H. Hannay, "Comment on 'Method of handling the divergences in the radiation theory of sources that move faster than their waves'," J. Math. Phys. 42, 3973-3974 (2001).
    [CrossRef]
  9. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  10. H. Ardavan, "Asymptotic analysis of the radiation by volume sources in supersonic rotor acoustics," J. Fluid Mech. 226, 33-68 (1994).
  11. C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descents," Proc. Cambridge Philos. Soc. 54, 599-611 (1957).
  12. R. Burridge, "Asymptotic evaluation of integrals related to time-dependent fields near caustics," SIAM J. Appl. Math. 55, 390-409 (1995).
    [CrossRef]
  13. M. V. Berry, "Uniform approximations for glory scattering and diffraction peaks," J. Phys. B 2, 381-392 (1969).
    [CrossRef]
  14. J. H. Hannay and M. R. Jeffrey, "The electric field of synchrotron radiation," Proc. R. Soc. London, Ser. A 461, 3599-3610 (2005).
  15. A. Hofmann, The Physics of Synchrotron Radiation (Cambridge U. Press, 2004).
    [CrossRef]

2005 (1)

J. H. Hannay and M. R. Jeffrey, "The electric field of synchrotron radiation," Proc. R. Soc. London, Ser. A 461, 3599-3610 (2005).

2004 (1)

2001 (1)

J. H. Hannay, "Comment on 'Method of handling the divergences in the radiation theory of sources that move faster than their waves'," J. Math. Phys. 42, 3973-3974 (2001).
[CrossRef]

2000 (1)

J. H. Hannay, "Comment II on 'Generation of focused, nonspherically decaying pulses of electromagnetic radiation'," Phys. Rev. E 62, 3008-3009 (2000). (The paper immediately after this one is a reply by H. Ardavan.)
[CrossRef]

1999 (1)

H. Ardavan, "Method of handling the divergences in the radiation theory of sources that move faster than their waves," J. Math. Phys. 40, 4331-4336 (1999).
[CrossRef]

1998 (1)

H. Ardavan, "Generation of focused, nonspherically decaying pulses of electromagnetic radiation," Phys. Rev. E 58, 6659-6684 (1998).
[CrossRef]

1996 (1)

J. H. Hannay, "Bounds on fields from fast rotating sources, and others," Proc. R. Soc. London, Ser. A 452, 2351-2354 (1996). (The factor half in Eqs. 1.1, 1.5, 1.6 should be replaced by two instead, the rest of the equations being unaffected.)

1995 (1)

R. Burridge, "Asymptotic evaluation of integrals related to time-dependent fields near caustics," SIAM J. Appl. Math. 55, 390-409 (1995).
[CrossRef]

1994 (2)

H. Ardavan, "Asymptotic analysis of the radiation by volume sources in supersonic rotor acoustics," J. Fluid Mech. 226, 33-68 (1994).

H. Ardavan, "The mechanism of radiation in pulsars," Mon. Not. R. Astron. Soc. 268, 361-392 (1994).

1991 (1)

H. Ardavan, "The near-field singularity predicted by the spiral Green's function in acoustics and electrodynamics," Proc. R. Soc. London, Ser. A 433, 451-459 (1991).

1969 (1)

M. V. Berry, "Uniform approximations for glory scattering and diffraction peaks," J. Phys. B 2, 381-392 (1969).
[CrossRef]

1957 (1)

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descents," Proc. Cambridge Philos. Soc. 54, 599-611 (1957).

Ardavan, A.

Ardavan, H.

H. Ardavan, A. Ardavan, and J. Singleton, "Spectral and polarization characteristics of the nonspherically decaying radiation generated by polarization currents with superluminally rotating distribution patterns," J. Opt. Soc. Am. A 21, 858-872 (2004).
[CrossRef]

H. Ardavan, "Method of handling the divergences in the radiation theory of sources that move faster than their waves," J. Math. Phys. 40, 4331-4336 (1999).
[CrossRef]

H. Ardavan, "Generation of focused, nonspherically decaying pulses of electromagnetic radiation," Phys. Rev. E 58, 6659-6684 (1998).
[CrossRef]

H. Ardavan, "The mechanism of radiation in pulsars," Mon. Not. R. Astron. Soc. 268, 361-392 (1994).

H. Ardavan, "Asymptotic analysis of the radiation by volume sources in supersonic rotor acoustics," J. Fluid Mech. 226, 33-68 (1994).

H. Ardavan, "The near-field singularity predicted by the spiral Green's function in acoustics and electrodynamics," Proc. R. Soc. London, Ser. A 433, 451-459 (1991).

Berry, M. V.

M. V. Berry, "Uniform approximations for glory scattering and diffraction peaks," J. Phys. B 2, 381-392 (1969).
[CrossRef]

Burridge, R.

R. Burridge, "Asymptotic evaluation of integrals related to time-dependent fields near caustics," SIAM J. Appl. Math. 55, 390-409 (1995).
[CrossRef]

Chester, C.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descents," Proc. Cambridge Philos. Soc. 54, 599-611 (1957).

Friedman, B.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descents," Proc. Cambridge Philos. Soc. 54, 599-611 (1957).

Hannay, J. H.

J. H. Hannay and M. R. Jeffrey, "The electric field of synchrotron radiation," Proc. R. Soc. London, Ser. A 461, 3599-3610 (2005).

J. H. Hannay, "Comment on 'Method of handling the divergences in the radiation theory of sources that move faster than their waves'," J. Math. Phys. 42, 3973-3974 (2001).
[CrossRef]

J. H. Hannay, "Comment II on 'Generation of focused, nonspherically decaying pulses of electromagnetic radiation'," Phys. Rev. E 62, 3008-3009 (2000). (The paper immediately after this one is a reply by H. Ardavan.)
[CrossRef]

J. H. Hannay, "Bounds on fields from fast rotating sources, and others," Proc. R. Soc. London, Ser. A 452, 2351-2354 (1996). (The factor half in Eqs. 1.1, 1.5, 1.6 should be replaced by two instead, the rest of the equations being unaffected.)

Hofmann, A.

A. Hofmann, The Physics of Synchrotron Radiation (Cambridge U. Press, 2004).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Jeffrey, M. R.

J. H. Hannay and M. R. Jeffrey, "The electric field of synchrotron radiation," Proc. R. Soc. London, Ser. A 461, 3599-3610 (2005).

Singleton, J.

Ursell, F.

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descents," Proc. Cambridge Philos. Soc. 54, 599-611 (1957).

J. Fluid Mech. (1)

H. Ardavan, "Asymptotic analysis of the radiation by volume sources in supersonic rotor acoustics," J. Fluid Mech. 226, 33-68 (1994).

J. Math. Phys. (2)

H. Ardavan, "Method of handling the divergences in the radiation theory of sources that move faster than their waves," J. Math. Phys. 40, 4331-4336 (1999).
[CrossRef]

J. H. Hannay, "Comment on 'Method of handling the divergences in the radiation theory of sources that move faster than their waves'," J. Math. Phys. 42, 3973-3974 (2001).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. B (1)

M. V. Berry, "Uniform approximations for glory scattering and diffraction peaks," J. Phys. B 2, 381-392 (1969).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

H. Ardavan, "The mechanism of radiation in pulsars," Mon. Not. R. Astron. Soc. 268, 361-392 (1994).

Phys. Rev. E (2)

H. Ardavan, "Generation of focused, nonspherically decaying pulses of electromagnetic radiation," Phys. Rev. E 58, 6659-6684 (1998).
[CrossRef]

J. H. Hannay, "Comment II on 'Generation of focused, nonspherically decaying pulses of electromagnetic radiation'," Phys. Rev. E 62, 3008-3009 (2000). (The paper immediately after this one is a reply by H. Ardavan.)
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

C. Chester, B. Friedman, and F. Ursell, "An extension of the method of steepest descents," Proc. Cambridge Philos. Soc. 54, 599-611 (1957).

Proc. R. Soc. London, Ser. A (3)

J. H. Hannay and M. R. Jeffrey, "The electric field of synchrotron radiation," Proc. R. Soc. London, Ser. A 461, 3599-3610 (2005).

H. Ardavan, "The near-field singularity predicted by the spiral Green's function in acoustics and electrodynamics," Proc. R. Soc. London, Ser. A 433, 451-459 (1991).

J. H. Hannay, "Bounds on fields from fast rotating sources, and others," Proc. R. Soc. London, Ser. A 452, 2351-2354 (1996). (The factor half in Eqs. 1.1, 1.5, 1.6 should be replaced by two instead, the rest of the equations being unaffected.)

SIAM J. Appl. Math. (1)

R. Burridge, "Asymptotic evaluation of integrals related to time-dependent fields near caustics," SIAM J. Appl. Math. 55, 390-409 (1995).
[CrossRef]

Other (2)

A. Hofmann, The Physics of Synchrotron Radiation (Cambridge U. Press, 2004).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

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Figures (1)

Fig. 1
Fig. 1

A hypothetical superluminal circulating point source leaves in its wake a bent Mach cone or “Schott caustic” surface shown here in the orbit plane cross section. The caustic surface is the envelope of the spheres of constant retardation and rotates rigidly with the source as it circulates. Unlike an ordinary Mach cone, the caustic also has a cusp: a point in the cross section shown, but a curved line in three dimensions. This cusp is the leading feature in many of the papers of H. Ardavan.

Equations (23)

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

2 B c 2 2 B t 2 = μ 0 j ,
B = μ 0 4 π [ j ] R d 3 R ,
B const . R 0 .
M = [ cos ω t sin ω t 0 sin ω t cos ω t 0 0 0 1 ] .
B ( x P , t P ) = μ 0 4 π t P d t d 3 x J ( x , t ) x x P δ ( x x P c ( t P t ) )
= μ 0 4 π t P d t d 3 x 1 x x P δ ( x x P c ( t P t ) ) MJ ( M 1 x , 0 )
= d 3 x ̂ { μ 0 4 π t P d t d 3 x 1 x x P δ ( x x P c ( t P t ) ) δ ( x M x ̂ ) } M J ( x ̂ , 0 ) .
B ( x P , x P + Δ t P ) μ 0 4 π x P t P d t d 3 x δ ( x x P x P c ( Δ t P t ) ) M J ( M 1 x , 0 ) .
R = ( z z P ) 2 + r 2 + r P 2 2 r r P cos ( φ P φ )
B = d 3 x ̂ { μ 0 4 π c d φ 1 R δ ( ω c 1 R φ P + φ ̂ P + φ φ ̂ ) M ( ω 1 φ ω 1 φ ̂ ) } J ( x ̂ , 0 ) .
Δ = ( r 2 ω 2 c 2 1 ) ( r P 2 ω 2 c 2 1 ) ω 2 c 2 ( z z P ) 2 ,
R ± = ( z z P ) 2 + r 2 + r P 2 2 c 2 ω 2 ( 1 Δ ) ,
ϕ ± = 2 π arccos [ ( 1 Δ ) ( r r P ω 2 c 2 ) ] + ω c 1 R ± .
ω c 1 R + φ φ P = 1 3 ν 3 c 1 2 ν + c 2 ,
c 1 = ( 3 4 ) 1 3 ( ϕ + ϕ ) 1 3 , c 2 = ( 1 2 ) ( ϕ + + ϕ ) .
B d 3 x ̂ { μ 0 4 π c d ν d φ d ν 1 R δ ( 1 3 ν 3 c 1 2 ν + c 2 + φ ̂ P φ ̂ ) M ( ω 1 φ ( ν ) ω 1 φ ̂ ) } J ( x ̂ , 0 ) .
p 1 = 1 2 ( 1 R d φ d ν ) a t ν = c 1 + 1 2 ( 1 R d φ d ν ) a t ν = c 1
q 1 = 1 2 c 1 ( 1 R d φ d ν ) a t ν = c 1 1 2 c 1 ( 1 R d φ d ν ) a t ν = c 1
d ν ( p + q ν ) δ ( 1 3 ν 3 c 1 2 ν + c 2 + φ ̂ P φ ̂ ) = 2 c 1 2 ( 1 χ 2 ) 1 2 [ p cos ( 1 3 arcsin χ ) c 1 q sin ( 2 3 arcsin χ ) ]
= c 1 2 ( χ 2 1 ) 1 2 [ p sinh ( 1 3 arccosh χ ) + c 1 q sgn ( χ ) sinh ( 2 3 arccosh χ ) ]
χ = φ ̂ φ ̂ P 1 2 ( ϕ + + ϕ ) 1 2 ( ϕ + ϕ ) = 3 2 φ ̂ φ ̂ P c 2 c 1 3 .
d ν ( p + q ν ) δ ( 1 3 ν 3 + c 1 2 ν + c 2 + φ ̂ P φ ̂ ) = c 1 2 ( χ 2 + 1 ) 1 2 [ p cosh ( 1 3 arcsinh χ ) + c 1 q sgn ( χ ) sinh ( 2 3 arcsinh χ ) ] ,
J ( β 3 ) d β = J ( α ) δ ( α β 3 ) d α d β = J ( α ) 3 α 2 3 d α .

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