Exploring black hole phenomena in Kaluza-Klein gravity with massive vector fields discussing the particle dynamics and gravitational deflection of light

An Solution of exact Black Hole (BH) is developed within the framework of Kaluza-Klein (KK) gravity, featuring the existence of massive vector fields in its surroundings. KK gravity in 4D (four Dimension) is significantly noteworthy because it indicates the spectrum of states of particle comprising not only the massless spin-2 gravitons found in context of General Relativity (GR) but also some additional spin-0 and spin-1 graviton modes. This article develop a solution which is encompassed by massive spin-1 gravitons in context of KK gravity. This solution is influenced by spin-1 graviton mass with an extra parameter that represents refinement to Newton’s constant, along with interaction between mass of black hole and massive vector field. In this article, We explained the motion of massive and massless particles by investigating their effective potential V e f f , event horizon r h , photon orbit r p h , and inner circular orbit r I S C O . It has been shown that Black Hole parameters λ and γ produces a prominent impact. We also explore the phenomenon of weak gravitational lensing, employing a detailed approach to observe the deflection angle.The aim behind this work highlights promising directions for further exploration into the fundamental principles underlying gravitational theories.