The normalized mode propagator

For the Randall-Sundrum brane world where a positive tension Minkowski brane is embedded in AdS 5 , two candidate propagators have been suggested in the literature, one being based on the normalized mode solutions to the source-free volcano potential fluctuation equation, and the other being the Giddings, Katz and Randall outgoing Hankel function based one. We show that while both of these two propagators have the same pole plus cut singularity structure in the complex energy plane, they behave differently on their respective circles at infinity, as a consequence of which only the Hankel function based propagator proves to be causal, with the normalized mode based one being found to take support outside the AdS 5 lightcone. In addition we show that unlike the Hankel function based propagator, the normalized mode based propagator does not correctly implement the junction conditions which hold in the presence of a perturbative source on the brane. 1 The normalized mode propagator The Randall-Sundrum brane world associated with the embedding of a positive tension M 4 Minkowski brane in an AdS 5 space with constant negative curvature −b 2 can be characterized by a background metric given as [1] ds 2 = dw 2 + e 2A(|w|) (dx 2 + dy 2 + dz 2 − dt 2) (1) where A = −b|w|. (Here w denotes the fifth coordinate which is to accompany the familiar x, y, z, t, with the brane being at w = 0.) In the brane world one is interested in gravitational fluctuations around this background, with the axial gauge, transverse-traceless (T T) ones being found (see e.g. [2] where full bibliographical citations as well as details of the present work are given) to obey the wave equation