Weyl Tensor Decomposition to the Formation of Black Hole (original) (raw)

2024, Turkish journal of physics

This work investigates the role of the Weyl tensor in the formation of a black hole. We discuss the development of the Weyl tensor and prove its existence in spacetime during the gravitational collapse of cosmic objects, utilizing the Riemannian curvature tensor, Ricci tensor, Kulkarni-Nomizu product, and Schouten tensor. By decomposing the Weyl tensor, we use theorems and proofs that satisfy the exact solutions of the Einstein field equations. We observe that the Riemann curvature tensor and Weyl tensor share the same symmetric identities, as trW (δ, .)σ = 0 such that W δσγτ = 0 when Riemannian curvature tensor, R δσγτ = 0. Additionally, the Riemann curvature and Weyl scalar tensor invariants are conformally related to each other, as R δσγτ R δσγτ = W δσγτ W δσγτ = 48(GM) 2 r 6 in the Schwarzschild metric. From the Einstein field equations, the Ricci tensor is R στ = 0 ; consequently, the stress-energy tensor, T στ = 0 , indicating that the Einstein field equation is empty space. However, in the Schwarzschild black hole solution, the Ricci tensor vanishes, but the Weyl tensor does not. Additionally, it seems that divergence occurs around the event horizon in a stagnant and uncharged Schwarzschild black hole with proper acceleration. Furthermore, the investigation into the existence of the Weyl tensor in the Schwarzschild black hole reveals its presence. We also explore the Reissner-Nordström, Kerr, and Kerr-Newman black holes by examining the coupling between the Einstein-Maxwell field equations and the Weyl tensor, utilizing small Weyl corrections. We obtain the metric that reduces to the Kerr-Newman black hole solution in Boyer-Lindquist coordinates when α = 0. The same metric equation obtained reduces to Kerr black hole solutions when the electric charge q = 0 and the coupling parameter α = 0. Furthermore, when the parameter of the charged rotating black hole a vanishes, we obtain solutions for the static and spherically symmetric black hole with Weyl corrections. When the terms a = q = 0 , the obtained metric reduces to the Schwarzschild black hole solution.