The Sagnac Effect In Curved Space-times

The effect of rotation on space-time have always been sources of stimulating andfascinating physical issues for the last centuries. The story of the interferometricaldetection of the effects of rotation dates back to the end of theⅩⅨcentury. In 1913Sagnac verified his early predications, using a rapidly rotating lightoptical interferom-eter. In fact, on the ground of classical physics, he predicted the following fringe shift,for monochromatic light waves in vacuum, counter-propagating along a closed pathin a rotating interferometer:ΔZ=4Ω·s/λc. The time difference associated to the fringeturns out to be:Δt=λ/cΔz=4Ω·s/c~2. After him, the effect of rotation on spacetimewas named "Sagnac effect". In 1960, the field of light-optical Sagnac interometryhad a revived interest after the development of laser. As a consequence, there wasan increasing precision in measurements and a growth of technological applications,suah as inertial navigation, where the "fibeer-optical gyro" are used. The research onthe Sagnac effect is of great importance and significance, therefore, attaches a lotof interest recently. In this paper, we concentrate on the Sagnac effect in curvedspacetimes, in order to obtain the general relativity corrections.The derivation of Sakurai is based on a formal analogy between the Aharonov-Bohm effect and Sagnac effect, which is the first order approximation. The Catteno'ssplitting is adopted: it will enable us to describe the geometrodynamics of the rotat-ing frame in a very transparent and powerful way. In particular, Catteneo's splittingallows to generalized the Newtonian elements. We study the effect in Schwarzchild,Kerr and Kerr-Newman-kasuya fields. We can distinguish two contributions:thefirst one is proportional to the angular velocity of the observer, and the other one de-pends on the gravitational mass, angular momentum per unit mass, the electric andmagnetic charges of the gravitation. The gravitational mass is to produce the Sagnaceffect and then to reduce the effect made by electric and magnetic charges. We seethat even whenΩ=0 a time difference appears: this is due to the rotation of thesource of the gravitational field. More recently proposals that the Sagnac effect may be made the basis of grav-itational wave detection have been of considerable recent interest. The experimentaltechnique envision a LISA which round the Sun in the Earth's revolution. The threedetectors float in space forming a triangle. Clockwise and counterclockwise beamson a path C which bounds a triangular∑.It is proposed that the Sagnac time dif-ferenceΔt=4/c~2∫_∑Ω·d∑in the limitλ>>L can be employed to probe the localrotational velocityΩproduced by the gravitational wave. However,if the gravita-tional wavelengthλobeysλ~2>>∑and to lowest order in the gravitational wavecurvature,the Sagnac effect is thereby null.