굴절률 물질에서의 특수상대론

김휘 2008.09.22 19:05 조회 수 : 13974 추천:37

올렸던 질문들이 다 지워져 아쉽긴 하지만...

최근 질문중에 굴절률이 1보다 큰 매질에서 빛의 속도는 c/n
인데 이와 같은 환경에서 물체 (또는 정보가) 빛보다 빨리 가는
것이 가능한가?
라는 물음이 있었습니다.

이 물음이 떠오른 것은  
David Miller 가 OpticsExpress에 쓴
active control을 이용한 perfect cloarking이라는 논문을
훑어 보다가 Pendry가 Science에서 제시했던 유며한 cloarking
구조가 특수상대론(causality)에 의해 반드시 단일 frequency
에서만 가능한 구조다 라고 쓴 대목을 읽었기 때문입니다.

Pendry 의 구조에서 spherical shell 껍데기 환의 근방은 refractive index는 1보다 작은 값을 갖는 부분이 있고 또 spherical shell과
내부영역의 경계에서는 정확히 refractive index = 0 입니다.

단일 frequency 일때 phase velocity가 c보다 큰 것은 특수상대론을
위해하지 않는 "그럴 수도 있는" 일입니다.

그런데, pulse가 pendry의 cloarking shell을 지나갈때 shell의 표면을
둘러가는 pulse 와  공간을 지나가는 pulse가 결이 맞기 위해서는
논리적으로 shell을 둘러가는 pulse의 속도가 c보다 빠를 수 밖에 없습니다.  상대론을 고려하지 않은 일반 맥스웰방정식에서야 수학적 해가
나오지만,,, pulse 가 빛보다 빠르는 다는 것은 특수상대론을 위배하는
일입니다.  그래서 David Miller의 논문에서는 active control을 해야하는
이야기를 하고 있습니다.

이 문제와 관련해서,,, refractive material 내부에서 정보가 과연 빛보다
빨리 갈 수 있느냐? 하는 의문을 갖게 되었던 것이죠.

인터넷에서 이런저런 검색을 해보니, 06년에 Physics Forum이라는 사이트에서 누군가 같은 질문을 했었더군요.

아래 답이 있습니다.

Special Relativity in Refractive Media Share It   Thread Tools    
May31-06, 04:00 AM                   #1  
John Bell
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John Bell is
Posts: n/a Special Relativity in Refractive Media

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The following (probably) dumb question has recently occurred to me:
Is the velocity of light in water
a) constant
b) dependent on the velocity of the water relative to the observer
c) dependent on both the velocity of the water and the velocity of the
light source relative to the observer

Options b and c would seem to create a problem because although less
dense, the atmosphere is also a refractive medium. That implies the
Michelson-Morley experiment did not establish experimentally that the
speed of light is constant, as usually claimed. Instead it merely
confirmed the more obvious fact that it is constant for a stationary
source in a stationary medium, relative to the observer.

On the other hand, the more radical option (a) would seem to imply that
for special relativistic transforms under water, c should be replaced
with the speed of light in water. That too would create a problem
because we know that massive particles can travel faster than light in
water (to produce Cherenkov radiation)

I'm probably missing something obvious here, but don't yet see what
it is.

John Bell
http://global.accelerators.co.uk
(Change John to Liberty to respond)


    

John Bell

Jun3-06, 04:00 AM                   #2  
Tom Roberts
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Tom Roberts is
Posts: n/a Re: Special Relativity in Refractive Media

--------------------------------------------------------------------------------

John Bell wrote:
> The following (probably) dumb question has recently occurred to me:
> Is the velocity of light in water
> a) constant

No.

> b) dependent on the velocity of the water relative to the observer

Yes. Specifically, light travels with speed c/n in the (local) rest
frame of the water, and for moving water one uses the Lorentz
composition of velocities to obtain the speed of the light relative to
an (inertial) observer.

> c) dependent on both the velocity of the water and the velocity of the
> light source relative to the observer

No. See above.

> Options b and c would seem to create a problem because although less
> dense, the atmosphere is also a refractive medium. That implies the
> Michelson-Morley experiment did not establish experimentally that the
> speed of light is constant, as usually claimed. Instead it merely
> confirmed the more obvious fact that it is constant for a stationary
> source in a stationary medium, relative to the observer.

Yes. Indeed this is true. When light enters a medium it takes a certain
distance to speed-up/slow-down to c/n in the medium; this distance is
called the "extinction length" (the usual reference is Born and Wolf,
_Principles_of_Optics_). The extinction length depends on the properties
of the medium and the frequency of the light; in air for visible light
the extinction length is a few mm, and in glass it is <1 micron.

Fortunately Joos performed the MMX in vacuum and obtained a result
consistent with the prediction of SR [reference in the FAQ]. And also,
Brillet and Hall performed a much more accurate experiment in which the
rotating wavelength-determining cavity was vacuum [reference in the FAQ].

FAQ: http://math.ucr.edu/home/baez/physic...periments.html

Tom Roberts


    

Tom Roberts

Jun3-06, 04:00 AM                   #3  
sigoldberg1@gmail.com
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sigoldberg1@gmail.com is
Posts: n/a Re: Special Relativity in Refractive Media

--------------------------------------------------------------------------------

See also
http://www.paradox-paradigm.nl/%5CTh...f%20Fizeau.htm
as well as
http://math.ucr.edu/home/baez/physic...%20experiments
subsection 7 "The Fizeau Experiment" for references.


    

sigoldberg1@gmail.com

Jun3-06, 04:00 AM                   #4  
Timo A. Nieminen
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Timo A. Nieminen is
Posts: n/a Re: Special Relativity in Refractive Media

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On Tue, 30 May 2006, John Bell wrote:

> The following (probably) dumb question has recently occurred to me:
> Is the velocity of light in water
> a) constant
> b) dependent on the velocity of the water relative to the observer
> c) dependent on both the velocity of the water and the velocity of the
> light source relative to the observer
>
> Options b and c would seem to create a problem because although less
> dense, the atmosphere is also a refractive medium. That implies the
> Michelson-Morley experiment did not establish experimentally that the
> speed of light is constant, as usually claimed. Instead it merely
> confirmed the more obvious fact that it is constant for a stationary
> source in a stationary medium, relative to the observer.
>
> On the other hand, the more radical option (a) would seem to imply that
> for special relativistic transforms under water, c should be replaced
> with the speed of light in water. That too would create a problem
> because we know that massive particles can travel faster than light in
> water (to produce Cherenkov radiation)
>
> I'm probably missing something obvious here, but don't yet see what
> it is.

(b) is correct. Fresnel-Fizeau drag and all that. Basically, the usual
relativistic transformation for 4-vectors holds in water. In the rest
frame of the water, we know the speed of light in water - transformation
to a moving frame of the velocity of light in some particular direction in
the water gives what would be measured in a relatively moving coordinate
system. This means that the permittivity and permeability of a material
medium are dependent on choice of coordinate system, and in a coordinate
system where the medium is moving, the medium is anisotropic. Van Bladel's
book on relativity (Relativity and Engineering or a similar title) covers
this very well.

--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/...,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html


    

Timo A. Nieminen

Jun3-06, 04:00 AM                   #5  
sigoldberg1@gmail.com
Guest


sigoldberg1@gmail.com is
Posts: n/a Re: Special Relativity in Refractive Media

--------------------------------------------------------------------------------


John Bell wrote:
> The following (probably) dumb question has recently occurred to me:
> Is the velocity of light in water
> a) constant
> b) dependent on the velocity of the water relative to the observer
> c) dependent on both the velocity of the water and the velocity of the
> light source relative to the observer
>

It also occurred to
Fizeau and Foulcault.
Try http://galileo.phys.virginia.edu/cla...ding_vels.html, near
the bottom of the page.
Aside: I recall that this was the experiment of which Einstein was
aware in creating special relativity, not apparently Michaelson-Morely.


    

sigoldberg1@gmail.com

Jun3-06, 04:00 AM                   #6  
Alf P. Steinbach
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Alf P. Steinbach is
Posts: n/a Re: Special Relativity in Refractive Media

--------------------------------------------------------------------------------

* John Bell:
> The following (probably) dumb question has recently occurred to me:
> Is the velocity of light in water
> a) constant
> b) dependent on the velocity of the water relative to the observer
> c) dependent on both the velocity of the water and the velocity of the
> light source relative to the observer
>
> Options b and c would seem to create a problem because although less
> dense, the atmosphere is also a refractive medium. That implies the
> Michelson-Morley experiment did not establish experimentally that the
> speed of light is constant, as usually claimed. Instead it merely
> confirmed the more obvious fact that it is constant for a stationary
> source in a stationary medium, relative to the observer.
>
> On the other hand, the more radical option (a) would seem to imply that
> for special relativistic transforms under water, c should be replaced
> with the speed of light in water. That too would create a problem
> because we know that massive particles can travel faster than light in
> water (to produce Cherenkov radiation)
>
> I'm probably missing something obvious here, but don't yet see what
> it is.

Consider the speed of light constant with respect to the surrounding
water. That speed is still very high. So relativistic effects enter
the picture: you effectively get some "drag" from the water, but not a
simple addition of velocities: it's the relativistic addition.

A.P. French, in "Special Relativity" (M.I.T Introductory phsyics series
of books, I think the edition I have is the 1986 edition) explains on
pages 131...132 that the light velocity V a stationary observer finds
for light moving through water that has velocity v wrt. the observer,
where the speed of light wrt. the water is c/n (n the refractive index
of water), is the relativistic addition of c/n and v, yielding

V = (c/n + v)/(1 + v/nc)

which when expanded in powers of v/c yields the approximation

V ~= c/n + (1 - 1/n^2)v

where (1 - 1/n^2) is what before special relativity was known as the
Fresnel "drag coefficient", postulated to explain the behavior.

I gather that this is the commonly accepted view.

It means that the commonly accepted view is your option (b), but that
the speed of light through air wrt. a stationary observer depends very
little on the motion of the air, because the refractive index of air is
is 1.008, very close to 1 (the refractive index of water is 1.330).

Perhaps someone else can answer whether that means that within the
experimental limits M&M could have observed the drag effect in air if
they'd invited the Big Bad Wolf to blow on the apparatus?

--
A: Because it messes up the order in which people normally read text.
Q: Why is it such a bad thing?
A: Top-posting.
Q: What is the most annoying thing on usenet and in e-mail?


    

Alf P. Steinbach

Jun6-06, 04:00 AM                   #7  
Timo A. Nieminen
Guest


Timo A. Nieminen is
Posts: n/a Re: Special Relativity in Refractive Media

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On Fri, 2 Jun 2006, Tom Roberts wrote:

> When light enters a medium it takes a certain
> distance to speed-up/slow-down to c/n in the medium; this distance is
> called the "extinction length" (the usual reference is Born and Wolf,
> _Principles_of_Optics_). The extinction length depends on the properties
> of the medium and the frequency of the light; in air for visible light
> the extinction length is a few mm, and in glass it is <1 micron.

While this is the usual story used to reconcile "emission theories" of
light, where the speed of light is dependent on the speed of the source,
with most observations (some problems with Fresnel-Fizeau drag remain),
is this at all correct in classical electrodynamics? All that is needed is
a sufficient thickness to have enough atoms/molecules to take a
macroscopic average over. It isn't as if incoming photons are billiard
ball-like things passing between atoms until they hit one; in a plane wave
mode they interact with them all the way across the width of the wave.

--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/...,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html


    

Timo A. Nieminen

Jun7-06, 04:01 AM                   #8  
Chalky
Guest


Chalky is
Posts: n/a Re: Special Relativity in Refractive Media

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Timo A. Nieminen wrote:
> On Fri, 2 Jun 2006, Tom Roberts wrote:
>
> > When light enters a medium it takes a certain
> > distance to speed-up/slow-down to c/n in the medium; this distance is
> > called the "extinction length" (the usual reference is Born and Wolf,
> > _Principles_of_Optics_). The extinction length depends on the properties
> > of the medium and the frequency of the light; in air for visible light
> > the extinction length is a few mm, and in glass it is <1 micron.
>
> While this is the usual story used to reconcile "emission theories" of
> light, where the speed of light is dependent on the speed of the source,
> with most observations (some problems with Fresnel-Fizeau drag remain),
> is this at all correct in classical electrodynamics? All that is needed is
> a sufficient thickness to have enough atoms/molecules to take a
> macroscopic average over. It isn't as if incoming photons are billiard
> ball-like things passing between atoms until they hit one; in a plane wave
> mode they interact with them all the way across the width of the wave.
>
I certainly found Tom Roberts' comment helpful in relation to trying to
understand the statements by Cl. Massé and Igor Khavkine under my
discussion title Light and Gravity.
I took it to imply this is an indication of the mean distance travelled
by a photon (at speed c) before a scattering (Igor Khavkine) or an
absorption/re-emission event (Cl. Massé). The photon/next photon then
travels this same mean distance again (again at speed c) before the
next interaction event.

Have I got this right?

C


    

Chalky

Jun19-06, 04:00 AM                   #9  
John (Liberty) Bell
Guest


John (Liberty) Bell is
Posts: n/a Re: Special Relativity in Refractive Media

--------------------------------------------------------------------------------


Tom Roberts wrote:
> John Bell wrote:
> > The following (probably) dumb question has recently occurred to me:
> > Is the velocity of light in water
> > a) constant
>
> No.
>
agreed

> > b) dependent on the velocity of the water relative to the observer
>
> Yes.

>agreed

> Specifically, light travels with speed c/n in the (local) rest
> frame of the water, and for moving water one uses the Lorentz
> composition of velocities to obtain the speed of the light relative to
> an (inertial) observer.
>
> > c) dependent on both the velocity of the water and the velocity of the
> > light source relative to the observer
>
> No. See above.

How do you know?

Fizeau's experiment confirming (b) used a light source in vacuum (or
air {which amounts to the same thing to the experimentally verified
accuracy of the experiment [10%]}). Obviously, if such a source was
moving, this would make no difference. However, that does not answer
the question of what happens when the source is moving in the
refractive medium. Perhaps the easiest way to test this would be with a
LED on a rotating arm, fed with electricity via an appropriately
designed commutator, because:

a) the LED is solid state
b) it is encased in glass
c) this prevents wires getting tangled, and ensures the lit LED is
always moving in the same direction.

Has such an experiment ever been performed?

If not,
1) why not?
2) how do you know the answer to the question?

John (Liberty) Bell
http://global.accelerators.co.uk
(Change John to Liberty to respond by email)

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