B. The wave equation In the DGP model, a scalar field that propagates in the bulk is taken to satisfy the wave equation

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B. The wave equation In the DGP model, a scalar field that propagates in the bulk is taken to satisfy the wave equation I a2 a2 a2 a2 a2 Әt2 Ә2 ду? дz2 Ә2 a2 a2 22 +($(w) 22x2 ay2 az2 Ә2 дх2 ду 02 o=s(r,w,). (44) Equation (44) has the form of the wave equation one would expect from the spacetime interval in Eq. (43) only if we set l = 0. The term in Eq. (44) that is proportional to l is respon- sible for the remarkable behavior of the DGP model described above. At the end of this section, we give a physi- cal motivation for the form of this terın by an analogy to the electrostatics of a dielectric sheet. The analogy shows that much as a dielectric sheet will screen a point charge, so also in the DGP model gravity is screened by the brane at short distances. The screening leads to behavior at short distances appropriate to three spatial dimensions, but at long distances gravity is unscreened, yielding behavior appropriate to four spatial dimensions.

- D. Static point source solution We now consider the field produced by a static point source localized on the brane. Thus, the source is $ 18(r)S(w). The resulting field will also be static and we are primarily interested in the field on the brane, (r, w0). We obtain the exact expression for the point source profile from straightforward Fourier analysis of Eq. (44); see Problem 4 of Appendix C in the supplementary material. 17 The result is 2 roo sin(kr) °(r,0) = dk 47²r Jo (48) (1 + (k Ck At short distances (r< 0) 1 21 °(r, 0) 4alr whereas at long distances (r > 1) (49) or,0 2 1 412 72 (50) r a 335 Am. J. Phys., Vol. 86, No. 5, May 2018

Problem 4. DGP Analysis. Here we fill in the steps that lead from Eq. (41) to Eqs. (48), (49) and (50). (a) Let g() be the Fourier transform of g(w). Show that the Fourier transform of g(w)(w) is a constant given by Lehet op dp g(p). 27 (C2)

Thus it follows that the Fourier transform of -S(w)V_$(x, y, z, w) is kaſ(k) where f(x) = L OCK, P) (C3) = and V2 = 22/ar? + a2 ſay? + 02 /az. (b) Use the result of part (a) to rewrite Eq. (44) in Fourier space. You should obtain 12 (k,p) + (C4) = k? + p2 F(k) k2 + p2