Spinoptics in a curved spacetime
Abstract: In this paper we study propagation of the high frequency electromagnetic waves in a curved spacetime. We discuss a so call spinoptics approach which generalizes a well known geometric optics approximation and allows one to take into account spin-dependent corrections. A new element proposed in this work is the use of effective action which allows one to derive spinoptics equations. This method is simpler and more transparent than the earlier proposed methods. It is explicitly covariant and can be applied to an arbitrary spacetime background. We also demonstrate how the initial value problem for the high frequency electromagnetiv wave can be soled in the developed spinoptics approximation.
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- The leading order contribution to the field strength (6\@@italiccorr) is Fμν(0)=−iωa(lμmν−lνmμ)exp(iωS)subscriptsuperscript𝐹0𝜇𝜈𝑖𝜔𝑎subscript𝑙𝜇subscript𝑚𝜈subscript𝑙𝜈subscript𝑚𝜇𝑒𝑥𝑝𝑖𝜔𝑆F^{(0)}_{\mu\nu}=-i\omega a(l_{\mu}m_{\nu}-l_{\nu}m_{\mu})\mathop{exp}% \nolimits(i\omega S)italic_F start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = - italic_i italic_ω italic_a ( italic_l start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT - italic_l start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ) start_BIGOP italic_e italic_x italic_p end_BIGOP ( italic_i italic_ω italic_S ). It is easy to check that Fμν(0)\tmspace∗−.1667em\tmspace−.1667em≡12eμναβF(0)αβ=isFμν(0)superscriptsubscriptsuperscript𝐹0𝜇𝜈superscript\tmspace.1667𝑒𝑚\tmspace.1667𝑒𝑚12subscript𝑒𝜇𝜈𝛼𝛽superscript𝐹0𝛼𝛽𝑖𝑠subscriptsuperscript𝐹0𝜇𝜈{}^{{}^{*}\tmspace-{.1667em}\tmspace-{.1667em}}F^{(0)}_{\mu\nu}\equiv\genfrac{% }{}{}{0}{1}{2}e_{\mu\nu\alpha\beta}F^{(0)\alpha\beta}=isF^{(0)}_{\mu\nu}start_FLOATSUPERSCRIPT start_FLOATSUPERSCRIPT ∗ end_FLOATSUPERSCRIPT - .1667 italic_e italic_m - .1667 italic_e italic_m end_FLOATSUPERSCRIPT italic_F start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ≡ divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_e start_POSTSUBSCRIPT italic_μ italic_ν italic_α italic_β end_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT ( 0 ) italic_α italic_β end_POSTSUPERSCRIPT = italic_i italic_s italic_F start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT. That is in the leading order 𝑭𝑭{\boldsymbol{F}}bold_italic_F is self-dual (for s=1𝑠1s=1italic_s = 1) or anti self-dual for s=−1𝑠1s=-1italic_s = - 1.
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