Electromagnetic field in causal and acausal Gödel-type space-times
The paper discusses a significant subject in general relativity: the rotating universe. The Gödel metric of Bianchi III type which describes spatially homogeneous universes with expansion and rotation, but zero shear is given by \[ ds^2 = (dt + RH(x)dy)^2 - R^2 (dx^2 + D^2 (x) dy^2 + dz^2), \] where \(R\), \(H = 2\Omega e^{\lambda x}/\lambda\) and \(D = e^{\lambda}\) are functions of the cosmological time \(t\). \(\lambda\) is a parameter which labels the set of isometrically independent metrics such that if \(\lambda \geq 2 \Omega\), the metric describes a causal universe, and if \(\lambda \leq 2 \Omega\), it describes an acausal one. If the vorticity vanishes \((\Omega\rightarrow 0)\), this universe transforms to a flat expanding one. The author solves the Maxwell equations by performing local splitting of 4-space time into time and 3-space orthogonal to each other, in order to take advantage of the fact that all physical variables defined in this selected 3-space are observed by physical devices. An important remark is made at this point: when the polarized electromagnetic radiation propagates in the gravitational field in the Gödel type universe, the right-circular polarized radiation (RCP) decouples from the left circular polarized radiation (LCP) and the photon helicity is conserved. Consequently, the image of a distant object viewed in RCP radiation will be displaced from that observed in LCP radiation. Finally, the exact solution of Maxwell's equations is obtained in terms of Gamma and confluent hypergeometric functions. For the causal case three subcases are possible: A. For low energies the energy spectrum of the photon is discrete and the photons with positive energy can move only to the left along the \(y\)-axis. B. At intermediate energies, the continuous photon spectrum arises from the nondiagonal components of \(g_{ik}\), which accounts for the rotation of the universe. C. At large energies the spectrum is continuous as in case B. In the acausal case there is a large energy region with discrete spectrum and an occurrence of a bifurcation in the frequency (case A) and two regions (B and C) with a continuous spectrum of frequencies. At the end of the paper, the author draws a series of conclusions, one of them being that the frequency spectra of the electromagnetic field in causal and noncausal nonstationary Gödel type metrics are essentially different.
scientific article; zbMATH DE number 798691