By C. Germani
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Extra resources for Astrophysical and Cosmological Conseq. of Dynamical Localization of Gravity [thesis]
We suppose that the bulk spacetime is governed by the Einstein equations and then we impose the junction conditions. The normal to the brane is nA . The induced metric on the brane is gAB = g˜AB − nA nB . 16) where 1 KAB = £n g˜AB 2 is the extrinsic curvature of the brane. 18) where we use the notation K A A = K. 19) where we introduced ˜ C DF G nC nF g D g G . 21) 3 I will use the notation with tilde for five-dimensional objects and without for four-dimensional ones. 22 CHAPTER 3. A DYNAMICAL ALTERNATIVE TO COMPACTIFICATION where T˜AB is the five dimensional energy-momentum tensor.
Now Σ is a freely falling surface in both metrics. Therefore the first condition comes from the comparison of the geodesic equations of the two metrics. The geodesic equation for a radial trajectory seen from the exterior is calculated as follows. For a time-like geodesic we have ˙2 ds2 2 ˙2 + R . 77) dτ 2 A Now since we are requiring that the exterior metric is static, it exist a time-like Killing vector ξ α such that £ξ gαβ = ξα;β + ξβ ;α = 0 , ξ α ξα < 0 50 CHAPTER 4. STARS IN THE BRANEWORLD where £ denotes the Lie derivative.
54), give  4 γµν 3 + ∂z2 γµν − ∂z γµν = 0. 66) (1,2) where (because of the traceless conditions) the polarization tensors eµν (k) are trans(1,2) verse and traceless, and where H2 (ikw) are the Hankel functions of first and second kind of order 2. 3. NEWTONIAN LIMIT: LOCALIZATION OF GRAVITY (1) (1) (2) (2) e0i (k)H1 (ikl) + e0i (k)H1 (ikl) = 0 , ki kj κ ˜2 M 1 1 (1) (1) (2) (2) δij − . 67) We are now interested in the solution which converges in the limit z → ∞. This is (2) possible by choosing eµν = 0.
Astrophysical and Cosmological Conseq. of Dynamical Localization of Gravity [thesis] by C. Germani