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Additional info for Astrophysical and Cosmological Conseq. of Dynamical Localization of Gravity [thesis]
84) where µ is an arbitrary subtraction mass linked with the renormalization energy. 85) r r3 where α = 4 · 32π(a1 + a2 ). 84) for different spins give [47, 22] ai (s = 1) = 4ai (s = 1/2) = 12ai (s = 0) = 1 (−2, 3). 87) we have that the one-loop correction to the Newtonian potential is V (r) = GN M r 1+ 2N 2 GN 3πr 2 . 75), which was found considering a five dimensional classical braneworld scenario. Chapter 4 Stars in the braneworld The first step in considering infinitely large extra dimensions is the discovery that gravity can be localized at low energies .
At large distances, by contrast, the extra-dimensional gravity becomes more and more important. This modifies, at large distances, the effective four-dimensional gravity. The fact that particles can be trapped gravitationally in a submanifold, was argued already by Visser . 1) 18 CHAPTER 3. A DYNAMICAL ALTERNATIVE TO COMPACTIFICATION (ξ is the extra-dimensional coordinate) is exponentially trapped by the scalar field Φ(ξ) > 0. In 1999 Randall and Sundrum  used a similar mechanism to localize gravity.
In this way we don’t have to deal with distributions. 2), we are left with a variational problem for the scalar field σ(y) in the bulk and a variational problem for the metric γµν on the brane. Then we use the constraint γµν = ηµν . 4) where R is the Ricci scalar associated with γµν and K = 21 g µν £n gµν , is the trace of the extrinsic curvature orthogonal to the brane. 5) where the boundary is at y = 0. The last term is called the Gibbons-Hawking term . Since the degrees of freedom of the four-dimensional Ricci scalar and the degrees of freedom of the scalar field σ are now completely separate, we can set R = 0 before the variation.