Ricci-Bourguignon Solitons With Certain Applications to Relativity

This article concerns with the investigation of Ricci-Bourguignon solitons and gradient Ricci-Bourguignon solitons in perfect fluid space-times and generalised Robertson–Walker space-times. First, we deduce the criterion for which the Ricci-Bourguignon soliton in a perfect fluid space-time is steady, expanding or shrinking. Then, we establish that if a perfect fluid space-time of dimension four admits a gradient Ricci-Bourguignon soliton with killing velocity vector, then either the space-time represents phantom era or the gradient Ricci-Bourguignon soliton is expanding or shrinking under some condition. Moreover, we illustrate that a generalised Robertson–Walker space-time represents a perfect fluid space-time if it admits the differential equations of Ricci-Bourguignon solitons. Then, we establish that if a generalised Robertson–Walker space-time allows a Ricci-Bourguignon soliton of gradient type with constant scalar curvature, then it also represents a perfect fluid space-time. Also, several interesting results are obtained as a corollary.