How a projectively flat geometry regulates $F(R)$-gravity theory?

TL;DR

This paper investigates how projectively flat and harmonic spacetime geometries influence $F(R)$-gravity, revealing conditions for matter fields and spacetime structures, including generalised Robertson-Walker models, with implications for specific $F(R)$ models.

Contribution

It introduces the effects of projective flatness and harmonic conditions on $F(R)$-gravity solutions, connecting geometric properties to matter content and spacetime evolution.

Findings

Projective flatness leads to constant energy density and isotropic pressure.

Weakening the condition yields a generalised Robertson-Walker spacetime with specific expansion.

Analysis of $F(R)$-models illustrates the geometric effects on gravity solutions.

Abstract

In the present paper we examine a projectively flat spacetime solution of $F(R)$-gravity theory. It is seen that once we deploy projective flatness in the geometry of the spacetime, the matter field has constant energy density and isotropic pressure. We then make the condition weaker and discuss the effects of projectively harmonic spacetime geometry in $F(R)$-gravity theory and show that the spacetime in this case reduces to a generalised Robertson-Walker spacetime with a shear, vorticity, acceleration free perfect fluid with a specific form of expansion scalar presented in terms of the scale factor. Role of conharmonic curvature tensor in the spacetime geometry is also briefly discussed. Some analysis of the obtained results are conducted in terms of couple of $F(R)$-gravity models.