Yamabe and Ricci solitons on Vaidya spacetime

The focus of this paper is to investigate the existence of Yamabe solitons and gradient Yamabe solitons on Vaidya spacetime. Our investigation is motivated by the observation in Khatri et al. [Existence of Ricci soliton vector fields on Vaidya spacetime, Phys. Scripta 98 (2023) 115244] that Vaidya spacetime reduces to the Schwarzschild spacetime when it admits a Ricci soliton and the mass is a nonzero constant. However, in Vaidya spacetime, mass is a function of retarded null coordinate [Formula: see text]. So, the focus is to find a soliton on Vaidya spacetime such that mass is variable and a function of [Formula: see text] instead of being constant. Moreover, in the way to find such a soliton, we see some modification is required in the results of the work by Khatri et al. [Existence of Ricci soliton vector fields on Vaidya spacetime, Phys. Scripta 98 (2023) 115244] and Fasihi-Ramandi [Erratum to existence of Ricci soliton vector fields on Vaidya spacetime (2024), doi:10.13140/RG.2.2.15457.52322]. To this end, we prove a rigidity theorem in a sense that Vaidya spacetime reduces to either the Schwarzschild or Minkowski spacetime, depending on whether the mass is a nonzero constant or zero, respectively, when it admits a Ricci soliton. Furthermore, we discuss the existence of gradient Ricci solitons in each case, addressing overlooked conditions in Khatri et al. [Existence of Ricci soliton vector fields on Vaidya spacetime, Phys. Scripta 98 (2023) 115244] and Fasihi-Ramandi [Erratum to existence of Ricci soliton vector fields on Vaidya spacetime (2024), doi:10.13140/RG.2.2.15457.52322]. Finally, we remark on the existence of conformal Ricci solitons on this spacetime of interest in the light of recent work by Chhakchhuak and Singh [Conformal Ricci solitons on Vaidya spacetime, Gen. Relativ. Gravit. 56(1) (2024) 11] and Fasihi-Ramandi [Erratum to conformal Ricci solitons on Vaidya spacetime (2024), doi:10.13140/RG.2.2.28879.29605]. While Chhakchhuak and Singh had initially proven that such solitons exist, we argue that the study of conformal Ricci solitons is not possible on Vaidya spacetime.