Radiation transport equations in non-Riemannian space-times

The transport equations for polarized radiation transfer in non-Riemannian, Weyl-Cartan type space-times are derived, with the effects of both torsion and nonmetricity included. To obtain the basic propagation equations we use the tangent bundle approach. The equations describing the time evolution of the Stokes parameters, of the photon distribution function and of the total polarization degree can be formulated as a system of coupled first-order partial differential equations. As an application of our results we consider the propagation of the cosmological gamma-ray bursts in spatially homogeneous and isotropic spaces with torsion and nonmetricity. For this case the exact general solution of the equation for the polarization degree is obtained, with the effects of the torsion and nonmetricity included. The presence of a non-Riemannian geometrical background in which the electromagnetic fields couple to torsion and/or nonmetricity affect the polarization of photon beams. Consequently, we suggest that the observed polarization of prompt cosmological gamma-ray bursts and of their optical afterglows may have a propagation effect component, due to a torsion/nonmetricity induced birefringence of the vacuum. A cosmological redshift and frequency dependence of the polarization degree of gamma-ray bursts also follows from the model, thus providing a clear observational signature of the torsional/nonmetric effects. On the other hand, observations of the polarization of the gamma-ray bursts can impose strong constraints on the torsion and nonmetricity and discriminate between different theoretical models.

Figure 1

Variation, as a function of the redshift $z$, of the parameter $\delta$ (in a logarithmic scale) for gamma-ray bursts propagating in a Weyl-Cartan geometry with constant contorsion tensor $K_A$, for different values of $K_A$: $K_A={10}^{-14}{\mathrm{s}}^{-1}$ (solid curve), $K_A={10}^{-15}{\mathrm{s}}^{-1}$ (dotted curve), $K_A={10}^{-16}{\mathrm{s}}^{-1}$ (dashed curve) and $K_A={10}^{-17}{\mathrm{s}}^{-1}$ (long dashed curve). For the cosmological parameters we have adopted the values $a_{\mathrm{rec}}=1$, ${\Omega }_{m,0}=0.3$ and ${\Omega }_{\Lambda }=0.7$. The frequency at the detector of the gamma-ray bursts is assumed to be ${\nu }_{\mathrm{rec}}=3\times {10}^{14}{\mathrm{s}}^{-1}$, corresponding to the optical afterglow emission.

Figure 2

The parameter $\delta$ as a function of the redshift $z$ for gamma-ray bursts propagating in a Weyssenhoff spinning fluid filled universe, with the torsion tensor proportional to the spin density, for different values of the cosmological parameters: ${\Omega }_{m,0}=0.25$, ${\Omega }_{\Lambda }=0.7$ and ${\Omega }_{s,0}=0.05$ (solid curve), ${\Omega }_{m,0}=0.20$, ${\Omega }_{\Lambda }=0.7$ and ${\Omega }_{s,0}=0.1$ (dotted curve), ${\Omega }_{m,0}=0.15$, ${\Omega }_{\Lambda }=0.7$ and ${\Omega }_{s,0}=0.15$ (short dashed curve) and ${\Omega }_{m,0}=0.20$, ${\Omega }_{\Lambda }=0.6$ and ${\Omega }_{s,0}=0.20$ (long dashed curve). The frequency at the detector of the gamma-ray bursts is assumed to be ${\nu }_{\mathrm{rec}}=3\times {10}^{14}{\mathrm{s}}^{-1}$, corresponding to the optical afterglow emission. For the constant $K_B$ we have adopted the value $K_B=2.5\times {10}^{-16}{\mathrm{s}}^{-1}$

Figure 3

The parameter $\delta$ as a function of the redshift $z$ for gamma-ray bursts propagating in a Riemann-Cartan space-time, with vanishing spin density and the modified double duality ansatz satisfied, for different values of the constant $K_C$: $K_C=2\times {10}^{-15}{\mathrm{s}}^{-1}$ (solid curve), $K_C={10}^{-15}{\mathrm{s}}^{-1}$ (dotted curve), $K_C=8\times {10}^{-16}{\mathrm{s}}^{-1}$ (dashed curve) and $K_C=5\times {10}^{-16}{\mathrm{s}}^{-1}$ (long dashed curve). For the cosmological parameters we have adopted the values $a_{\mathrm{rec}}=1$, ${\Omega }_{m,0}=0.3$ and ${\Omega }_{\Lambda }=0.7$. The frequency at the detector of the gamma-ray bursts is assumed to be ${\nu }_{\mathrm{rec}}=3\times {10}^{14}{\mathrm{s}}^{-1}$, corresponding to the optical afterglow emission.