Summary
The generally covariant «Dirac-spinor form» of Maxwell's equations is considered in the presence of external gravitational fields. In the presence of the weak gravitational fields, arising from neutral matter currents, we exhibit solutions which contain a gravitational effect which rotates the plane of polarization and changes the phase of plane and circularly polarized light beams, respectively. This new, calculational approach to the gravitational «Faraday-rotation effect» offers new insight into the physical processes involved in the interaction of light beams with gravitational fields.
Riassunto
Si considera la «forma spinoriale di Dirac» generalmente covariante delle equazioni di Maxwell in presenza di campi gravitazionali esterni. In presenza di campi gravitazionali deboli, che nascono da correnti neutre di materia, si mostrano soluzioni che contengono un effetto gravitazionale che ruota il piano di polarizzazione e cambia la fase dei raggi di luce a polarizzazione piana e circolare, rispettivamente. Questo nuovo approccio di calcolo all'«effetto di rotazione di Faraday» gravitazionale fornisce nuove intuizioni riguardo ai processi fisici coinvolti nell'interazione dei raggi di luce con i campi gravitazionali.
Резюме
Рассматривается ковариантная «Дираковская спинорная форма» уравнений Максвелла во внешних гравитационных полях. При наличии слабых гравитационных полей, связанных с потоками нейтрального вещества, мы получаем решения, в которых содержится гравитационный эффект, приводящий к вращению плоскости поляризации и изменению фазы плоско и циркулярно поляризованных пучков света. Новый подход к гравитационному «эффекту фарадея» дает представление о физических процессах, имеющих место при взаимодействии световых пучков с гравитационными полями.
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References
H. Dehnen:Intern. Journ. Theor. Phys.,7, 467 (1973).
D. Leiter:Nuovo Cimento,68, 53 (1970). In that paper we exhibit a nonminimal coupling of the Dirac-spinor form of Maxwell's equation. To specialize it to minimal coupling, we introduce the spinor vector potentiala defined asΦ = -iγ μ ϖ μ a (this implies that\(\bar a = (a^0 ,0, - a^3 , - (a^1 - ia^2 ))\) and also∂ μ a μ = 0). Then the action which yieldsiγ μ ∂ μ Φ =J and
, upon variation of the\(\delta \bar a\) and the\(\delta \bar \Phi \) independently, is
. The term\((\bar Ja + \bar aJ)/2\) is simply the Lorentz-force invariantJ
μ
a
μ, and yields the usual Lorentz force, when the associated mechanical trajectory variables (inherent in the definition of the current spinor) are varied in the usual manner to get the equation of motion of the associated charges.For an elementary development seeD. Leiter andT. C. Chapman:Amer. Journ. Phys.,44, 858 (1976).
A clear and detailed discussion of the relativistic-covariance properties of the Dirac-spinor equations, in general, is given inS. Schweber:Introduction to Relativistic Quantum Field Theory (New York, N. Y., 1962).
Ibid. A clear and detailed discussion of the relativistic-covariance properties of the Dirac-spinor equations, in general, is given inS. Schweber:Introduction to Relativistic Quantum Field Theory (New York, N. Y., 1962).
This is because a correspondence between the minimal-coupling invariants and solutions of the tensor form of Maxwell's equations and the Dirac-spinor form of Maxwell's equations can be formed. Hence, in the context of minimal coupling, the predictions of the two classical electromagnetic formalisms are the same, only the calculational techniques are different. For example, operating on (5) with
and equating components yields∂
μ
J
μ = 0.This is because (13) implies (8) with ϱ=0 and\( \bar J = 0 \) . Hence we are led to the usual form of the wave zone radiation field solutions forE andB.
That is we will show that gravitational effects generate an effect which can be associated with a matrix operator like this, and this identification of it in flat space will help us to pick out the effect in curved space later.
J. K. Lawrence, D. Leiter andG. Szamosi:Nuovo Cimento,17 B, 113 (1973).
Ibid..
Note that the term involvingS integrates to zero since it is equal to\(I_3 \equiv \int\limits_{--\infty }^\infty {dx^\mu } S_{,\mu } S^{--1} = \int\limits_{--\infty }^\infty {d(ln} S) = ln S(\infty )--ln S(--\infty ) = ln I--ln I = 0.\)
Again the term involvingS does not contribute since, for a closed spatial pathC x begining and ending atx,\(\int\limits_{c_x } {dx^\mu } S_{,\mu } S^{ - 1} = \int\limits_{c_x } {d(ln S)} = ln S(x, T)---ln S(x,0) = 0\), in quasistatic metrics. Moreover, in the weak-field approximation, we can setS −1≅I in (41) before performing the integration. (Note that we use the fact thatSS −1=I implies that dS S −1=S −1dS.)
SeeN. L. Balazs andB. Bertotti:Nuovo Cimento,27, 1087 (1963).
B. Mashoon:Nature,250, 316 (1974).
, upon variation of the
. The term
and equating components yields∂
μ
J
μ = 0.Author information
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Leiter, D. A new approach to the physical optics of polarized light beams in gravitational fields. Nuov Cim B 44, 275–288 (1978). https://doi.org/10.1007/BF02726793
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DOI: https://doi.org/10.1007/BF02726793