A Characterization of GRW Spacetimes

Ibrahim Al-Dayel, Sharief Deshmukh and Mohd. Danish Siddiqi
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Ibrahim Al-Dayel: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, P.O. Box 65892, Riyadh 11566, Saudi Arabia
Sharief Deshmukh: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Mohd. Danish Siddiqi: Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia

Mathematics, 2021, vol. 9, issue 18, 1-9

Abstract: We show presence a special torse-forming vector field (a particular form of torse-forming of a vector field) on generalized Robertson–Walker (GRW) spacetime, which is an eigenvector of the de Rham–Laplace operator. This paves the way to showing that the presence of a time-like special torse-forming vector field ? with potential function ? on a Lorentzian manifold ( M , g ) , dim M > 5 , which is an eigenvector of the de Rham Laplace operator, gives a characterization of a GRW-spacetime. We show that if, in addition, the function ? ( ? ) is nowhere zero, then the fibers of the GRW-spacetime are compact. Finally, we show that on a simply connected Lorentzian manifold ( M , g ) that admits a time-like special torse-forming vector field ? , there is a function f called the associated function of ? . It is shown that if a connected Lorentzian manifold ( M , g ) , dim M > 4 , admits a time-like special torse-forming vector field ? with associated function f nowhere zero and satisfies the Fischer–Marsden equation, then ( M , g ) is a quasi-Einstein manifold.

Keywords: generalized Robertson–Walker spacetime; special torse-forming vector fields; de Rham–Laplace operator; quasi-Einstein manifold (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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