Induced Matter and Particle Motion in Non-Compact Kaluza-Klein Gravity

Abstract

We examine generalizations of the five-dimensional canonical metric by including a dependence of the extra coordinate in the four-dimensional metric. We discuss a more appropriate way to interpret the four-dimensional energy-momentum tensor induced from the five-dimensional space-time and show it can lead to quite different physical situations depending on the interpretation chosen. Furthermore, we show that the assumption of five-dimensional null trajectories in Kaluza-Klein gravity can correspond to either four-dimensional massive or null trajectories when the path parameterization is chosen properly. Retaining the extra-coordinate dependence in the metric, we show the possibility of a cosmological variation in the rest masses of particles and a consequent departure from four-dimensional geodesic motion by a geometric force. In the examples given, we show that at late times it is possible for particles traveling along 5D null geodesics to be in a frame consistent with the induced matter scenario.Comment: 29 pages, accepted to GR

On the 5D Extra-Force according to Basini–Capozziello–Ponce De Leon Formalism and five important features: Kar–Sinha Gravitational Bending of Light, Chung–Freese Superluminal Behaviour, Maartens–Clarkson Black Strings, experimental measures of Extra Dimensions on board International Space Station (ISS) and the existence of the Particle Z due to a higher dimensional spacetime

“…Also see eqs 2.2a to 2.3 in [15] and we will have a similar result .A constant Warp Field for the ansatz pointed above will have null derivatives of the Warp Field and null derivatives of the spacetime metric components in the 3 + 1 spacetime and also the Ricci Tensors in 5D or 4D will be both equal to each other and both null.Again this reinforces the point of view that in flat spacetime a 5D spacetime reduces to a 4D one and this is again the reason why we cannot see the Extra Dimension.No compactation factors are needed.This will be better explored in the Chung-Freese Superluminal Section.…”

“…In the equations above the term Ω(y) is the so-called Warp Factor and the term Φ(x 3+1 , y) is the Warp Field 4 that is responsible for all the masses generated in the 4D spacetime as we will see later.The term dS 2 is the 5D line-element and the term ds 2 is the 3 + 1 line-element.The term y accounts for the Extra Dimension.Note that the Ponce De Leon formalism is a generic formalism that encompasses particular equations such as dS 2 = y 2 ds 2 − Φ 2 dy 2 (See eq 1 in [9]) where the Warp Factor Ω(y) = y or dS 2 = y 2 L 2 ds 2 − dy 2 (See eq 2.1 to 2.3 in [14] and eq 1.1 to 1.3 in [15] ) where the Warp Factor Ω(y) = y 2 L 2 . According to Ponce De Leon a particle with a 5D mass-component M 5 non-null moving in a 5D spacetime obeys the following relation([2] eq 22):…”

“…∂Φ(x 3+1 , y) ∂x u ( dy ds ) 2 (15) For massless particles in 5D M 5 is null although m 0 may be null or not and this requires a null-like 5D geodesics dS 2 = 0 concerning equation 7. 7 The rest mass in 4D is given by( [2] eq 27 and 28):…”

FLRW Universes From "Wave-Like" Cosmologies in 5d

“…where k 2 (5) is a constant introduced for dimensional considerations, (5) T AB is the energy-momentum tensor in 5D and the non-vanishing components of the Einstein tensor G AB are…”

“…where Λ (5) is the cosmological constant in the bulk. We will see that, in this case the effective energy-momentum tensor in 4D is conserved.…”