By C. Germani
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Additional info for Astrophysical and Cosmological Conseq. of Dynamical Localization of Gravity [thesis]
Thanks to the four-dimensional Bianchi identities we get differential equations for the projection of the Weyl tensor, Eµν . These effective gravitational equations do not in general close the system. Indeed the full determination of the projected Weyl tensor requires in general the study of the five-dimensional gravitational problem. As we will see, we can sometimes get around that by introducing particular symmetries for the metric. Then this formalism becomes a very powerful tool. 24 CHAPTER 3.
The main idea of this thesis is to study the classical astrophysics and cosmology of braneworlds. Therefore these sections aim to give the reader only a flavour of the quantum counterpart of the correspondence. 3. 79) where the superscript c means “classical” and the superscript q means “quantum”. The quantum spin-two tensor hqµν must vanish with the vanishing of the classical perturbations. We have then two other ingredients, the graviton propagator and the self-energy of the gravitons. These two objects must be represented by tensors of rank 4.
Eµν gives a non-local contribution in the sense that it codifies all the bulk geometrical back-reaction on the brane. The decomposition of the matter-correction is Sµν = 1 ρ [ρuµ uν + (ρ + 2p) hµν ] . 38) where we introduced an effective “dark” radiative energy-momentum on the brane, with energy density U, pressure U/3, momentum density Qµ and anisotropic stress Πµν . The brane-world corrections can conveniently be consolidated into an effective total energy density, pressure, momentum density and anisotropic stress : ρeff = ρ 1 + 5 2 κ = 8πGN .
Astrophysical and Cosmological Conseq. of Dynamical Localization of Gravity [thesis] by C. Germani