C l c 95 second there are reparametrizations of

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Unformatted text preview: he brane coordinate functions X A ðxÞ are essentially Goldstone bosons since they shift under the bulk gauge symmetry, X A ðxÞ ! X A ðxÞ À ÄA ðX ðxÞÞ. We can thus reach a sort of unitary gauge where the X A are fixed to some specified values. We set values so that the brane is the surface X 5 ¼ 0, and the brane coordinates x coincide with the coordinates X  ; thus we set B A B þ @ XA @ X B Ä GAB ðXðxÞÞ; then in transforming GAB , remember that both the function and the argument are changing, Putting all this together, we find Ä GAB ¼ 0. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 ¼ ÀÄ ðXðxÞÞ þ  @ X  ðxÞ ! X ðxÞ¼x À Ä ðX ðxÞÞ þ  ðxÞ ) Ä ðX ðxÞÞ ¼  ðxÞ: (9.12) The residual gauge transformations are bulk gauge transformations that do not move points onto or off of the brane, but only move brane points to other brane points. Furthermore, the brane diffeomorphism invariance is no longer an independent invariance but is fixed to be the diffeomorphisms induced from the bulk. We now fix this gauge in the action (9.10), which is permissible since no equations of motion are lost. This means that the induced metric is now g ðxÞ ¼ G ðx; X 5 ¼ 0Þ: (9.13) We split the action into two regions, region L to the left of the brane, and region R to the right of the brane, with outward pointing normals, as in Fig. 3. We call the fifth coordinate X 5  y. The brane is at y ¼ 0: 3 Z Z pffiffiffiffiffiffiffiffiffi M5 Z d4 xdy ÀGRðGÞ þ d4 xL4 ; þ S¼ 2 L R (9.14) (9.7) Ä GAB ðXðxÞÞ ¼ LÄ GAB ðX ðxÞÞ À ÄC @C GAB : Ä X  ðxÞ þ  X  ðxÞ (9.8) pffiffiffiffiffiffiffiffi 2 where L4  ðM4 =2Þ ÀgRðgÞ þ LM ðg; c Þ is the 4d part of the Lagrangian. To have a well-defined variational principle, we must have Gibbons-Hawking terms on both sides (Dyer and Hinterbichler, 2009), corresponding to the outward pointing normals. Adding these, the resulting action is Kurt Hinterbichler: Theoretical aspects of massive gravity 703 symmetric solution with a flat 5d bulk, which contains a 3 2 de Sitter brane with a 4d Hubble scale H $ M5 =M4 . This is called the self-accelerating branch and has caused much interest because the solution exists even though the brane and bulk cosmological constants vanish. B. Linear expansion To see the particle content of DGP, we expand the action (9.20) to linear order around the flat space solution and then integrate out the bulk to obtain an effective 4d action. We start by expanding the 5d graviton about flat space FIG. 3. GAB ¼ AB þ HAB : Splitting the DGP action. M3 S¼ 5 2 Z þ2 þ Z þ Z L I L We use the lapse, shift, and 4d metric variables, with their expansions around flat space, pffiffiffiffiffiffiffiffiffi d4 xdy ÀGRðGÞ R I pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi d4 x ÀgKL þ 2 d4 x ÀgKR g ¼  þ h ;  N ¼ 1 þ n: (9.22) d4 xL4 ; (9.15) We have the relations, to linear order in h , n , and...
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