Radial and circular motion of photons and test particles in the Schwarzschild black hole with quintessence and string clouds

Abstract The null and timelike geodesic motion in the vicinity of the Schwarzschild black hole in the presence of the string cloud parameter a and the quintessence field parameter q is studied. The ranges for both the parameters a and q are determined, which allow the existence of the black hole. In the radial motion of photon, the coordinate time t first decreases with the increasing values of both the parameters a and q and then in the close proximity of the horizon of the black hole, there is a turning point, after which the effect of the quintessence field is just opposite on the time t . For the massive particles, the proper time $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> decreases with increasing values of the parameter a and increases with increase in the value of the parameter q . In the same case of the massive particles, the coordinate time t decreases with increase in the values of both the parameters a and q . Further, it is found that for test particles, the stable circular orbits exist in this spacetime for small values of both the parameters i.e., for $$0&lt;a\ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>a</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and $$0&lt;q\ll 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>q</mml:mi> <mml:mo>≪</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . It is observed that the radii of the null circular orbits increase as the values of the parameters a and q increase. While in the case of the timelike geodesics, the radii of the circular orbits increase as the value of the parameter a increases, and they decrease as the value of the parameter q increases.

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