arXiv:hep-th/0401242v1 30 Jan 2004 Non-linear Brane Dynamics in Six Dimensions B. Cuadros-Melgar a∗ a Instituto de F´ ısica, Universidade de S˜ ao Paulo C.P.66.318, CEP 05315-970, S˜ ao Paulo, Brazil We consider a dynamical brane world in a six dimensional spacetime containing a singularity. Using the Israel conditions we study the motion of a 4-brane embedded in this setup. We analize the brane behavior when its position is perturbed about a fixed point and solve the full non-linear dynamics in the several possible scenarios. We also investigate the possible gravitational shortcuts and calculate the delay between graviton and photon signals and the ratio of the corresponding subtended horizons. 1. The Brane Cosmological Model We consider a six-dimensional model described by the following metric ds 2 = −n 2 (t, y, z )dt 2 + a 2 (t, y, z )dΣ 2 k + +b 2 (t, y, z )dy 2 + c 2 (t, y, z )dz 2 , (1) where dΣ 2 k represents the metric of the three di- mensional spatial sections with k = −1, 0, 1 cor- responding to a hyperbolic, a flat and an elliptic space, respectively. The matter content on the brane is directly re- lated to the jump of the extrinsic curvature tensor across the brane [1,2]. This relation has been de- rived in the case of a static brane in a previous work [3]. Here we generalize our result for the Israel conditions to include the case of a brane moving with respect to the coordinate system, which position at any bulk time t is denoted by z = R(t). The extrinsic curvature tensor on the brane is given by K MN = η L M ▽ L ˜ n N , (2) where ˜ n A is a unit vector field normal to the brane worldsheet ˜ n A =    c ˙ R n 2  1 − c 2 n 2 ˙ R 2 , 0, 0, 0, 0, 1 c  1 − c 2 n 2 ˙ R 2    , (3) * I would like to thank Elcio Abdalla for useful discussions and for reading the manuscript. This work has been sup- ported by Funda¸ c˜ ao de Amparo ` a Pesquisa do Estado de S˜ ao Paulo (FAPESP), Brazil. and η MN = g MN − ˜ n M ˜ n N (4) is the induced metric on the brane, from which we can obtain a relation between dt (the bulk time) and dτ (the brane time), dτ = n(t, R(t))  1 − c 2 (t, R(t)) n 2 (t, R(t)) ˙ R 2 dt ≡ nγ −1 dt , (5) where a dot means derivative with respect to the bulk time t. 1.1. The Israel Conditions The energy-momentum tensor on the brane lo- cated at z 0 can be written as T (b) MN = δ(z − z 0 ) c {(ρ + p)u M u N + pη MN } . (6) We also define a tensor ˆ T AB as ˆ T AB ≡ T AB − 1 4 Tη AB . (7) The Israel junction conditions [4] are given by [K µν ]= −κ 2 (6) ˆ T µν , (8) where the brackets stand for the jump across the brane and K µν = e A µ e B ν K AB , where e A µ form a basis of the vector space tangent to the brane worldvolume. The left-hand side of (8) can be calculated taking into account the mirror sym- metry across the brane. 1