Classical defocussing of world lines in higher dimensions

Abstract

A five-dimensional gravity theory, motivated by the brane-world picture, with Kaluza scalar in the 5 -dimensional metric as g 55 (r); r = √ x 2 + y 2 + z 2 , is considered near the possible singularity (small distance scales where gravity is strong) and is shown to give rise to a positive contribution to the Raychaudhuri equation. This inhibits the focusing of world lines and contributes to non -focusing of the worldlines in the 5-dimensional space. It is also shown that the results extend to time dependent cases such as those relevant for black hole interiors and cosmology.

Classical defocussing of world lines—Cosmological Implications

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We have extended our result on defocussing of world lines [1] by modifying gravity in the early epoch by a 5-d theory with scalar ψ(r). The acceleration term in the Raychaudhuri equation has been shown to be positive for flat FRW metric.

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“…In our earlier communication [1], we have shown classical defocussing of world lines in 5-dimensional Kaluza theory by modifying gravity in the early universe epoch by 5-d gravity with Kaluza scalar. The result obtained was general in the sense that no specific metric, other than spherical symmetry or explicit form for the Kaluza scalar ψ(r) were used.…”

“…From these investigations the consensus is that certain modifications of Einstein theory are expected during the early universe epoch where gravity is strong. We have considered in [1] one such modification, namely, during the early universe the spacetime is 5-dimensional and the modified gravity is taken to be 5-dimensional Kaluza gravity with the metric scalar ψ(r). We point out here that we do not introduce a potential for the scalars and they are massless.…”

“…In Section.2, we briefly review our earlier [1] results. In Section.3, we give the results using Friedmann -Walker -Robertson flat metric and in Section.4, we present the differential equations for the Hubble parameter H(t) and the FRW scale factor a(t).…”

Raychaudhuri Equation, Geometrical Flows and Geometrical Entropy

“…For a static spherical symmetric metric, − a 2 2 g µρ D µ (∂ ρ 1 g 55 ) is always positive [19]. This indicates that a scalar field will always defocus world lines.…”