How Extra Symmetries Affect Solutions in General Relativity

Abstract

To get exact solutions to Einstein’s field equations in general relativity, one has to impose some symmetry requirements. Otherwise, the equations are too difficult to solve. However, sometimes, the imposition of too much extra symmetry can cause the problem to become somewhat trivial. As a typical example to illustrate this, the effects of conharmonic flatness are studied and applied to Friedmann–Lemaitre–Robertson–Walker spacetime. Hence, we need to impose some symmetry to make the problem tractable, but not too much so as to make it too simple.

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

“…Even in modified gravity theories the situation does not improve much, it ends up resulting into vanishing effective pressure and energy density. We further can add to the findings of the article [22] that instead of conharmonic flatness if one imposes divergence free conharmonic curvature into the geometry, then the Ricci scalar R of the spacetime is constant and the Ricci curvature tensor is of Codazzi type. Therefore, same conclusions as in theorem 4.1 and theorem 4.2 follow from similar arguments.…”

“…Remark 4.1. In [22], the authors rightly showed that conharmonic flatness in a spacetime constrains the standard theory of gravity to a trivial vacuum case with = R 0 ij and R = 0 case. Even in modified gravity theories the situation does not improve much, it ends up resulting into vanishing effective pressure and energy density.…”

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

“…We further can add to the findings of the article [22] that instead of conharmonic flatness if one imposes divergence free conharmonic curvature into the geometry, then the Ricci scalar R of the spacetime is constant and the Ricci curvature tensor is of Codazzi type. Therefore, same conclusions as in Theorem 4.1 and Theorem 4.2 follow from similar arguments.…”

“…Remark 4.1. In [22], the authors rightly showed that conharmonic flatness in a spacetime constrains the standard theory of gravity to a trivial vacuum case with R ij = 0 and R = 0 case. Even in modified gravity theories the situation does not improve much, it ends up resulting into vanishing effective pressure and energy density.…”

Editorial to the Special Issue “Selected Papers from the 17th Russian Gravitational Conference—International Conference on Gravitation, Cosmology and Astrophysics (RUSGRAV-17)”

“…Several articles of this Special Issue are devoted to different problems of classical gravity. An impact of extra symmetries on the exact solutions of general relativity is discussed in [1]. Article [2] considers the problem of universal constants and natural systems of units in the spacetime with an arbitrary number of dimensions.…”