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We investigate the redshift and blueshift of light emitted by timelikegeodesic particles in orbits around a Kerr-Newman-(anti) de Sitter (KN(a)dS)black hole. Specifically we compute the redshift and blueshift of photons thatare emitted by geodesic massive particles and travel along null geodesicstowards a distant observer-located at a finite distance from the KN(a)dS blackhole. For this purpose we use the Killing-vector formalism and the associatedfirst integrals-constants of motion. We consider in detail stable timelikeequatorial circular orbits of stars and express their correspondingredshift/blueshift in terms of the metric physical black hole parameters(angular momentum per unit mass, mass, electric charge and the cosmologicalconstant) and the orbital radii of both the emitter star and the distantobserver. These radii are linked through the constants of motion along the nullgeodesics followed by the photons since their emission until their detectionand as a result we get closed form analytic expressions for the orbital radiusof the observer in terms of the emitter radius, and the black hole parameters.In addition, we compute exact analytic expressions for the frame dragging oftimelike spherical orbits in the KN(a)dS spacetime in terms of multivariablegeneralised hypergeometric functions of Lauricella and Appell. We solve theconditions for timelike spherical orbits in KN(a)dS and KN spacetimes. Wepresent new, elegant compact forms for the parameters of these orbits. Last butnot least we derive a very elegant and novel exact formula for the periapsisadvance for a test particle in a non-spherical polar orbit in KNdS black holespacetime in terms of Jacobi's elliptic function sn and Lauricella'shypergeometric function $F_D$.

european physical journal. c, particles and fields

Gravitational redshift/blueshift of light emitted by geodesic test  particles, frame-dragging and pericentre-shift effects, in the Kerr-Newman-de  Sitter and Kerr-Newman black hole geometries