By integrating out the extra dimensions we can write

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Unformatted text preview: ble than those of the Ã5 theory. 2 1=3 Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 DGP is the model of a ð3 þ 1Þ-dimensional brane (the 3brane) floating in a ð4 þ 1Þ-dimensional bulk spacetime. Gravity is dynamical in the bulk and the brane position is dynamical as well, and the action contains both 4d and 5d parts, Kurt Hinterbichler: Theoretical aspects of massive gravity 702 S¼ 3 M5 Z 5 pffiffiffiffiffiffiffiffiffi M2 Z pffiffiffiffiffiffiffi ffi d X ÀGRðGÞ þ 4 d4 x ÀgRðgÞ þ SM : 2 2 (9.1) Here X A , A, B; . . . ¼ 0, 1, 2, 3, and 5 are the 5d bulk coordinates, GAB ðX Þ is the 5d metric, and M5 is the 5d Planck mass. x , , ; . . . ¼ 0, 1, 2, and 3 are the 4d brane coordinates, g ðxÞ is the 4d metric which is given by inducing the 5d metric GAB onto the brane, and M4 is the 4d Planck mass. SM is the matter action, which we imagine to be localized to the brane, Z SM ¼ d4 xLM ðg; c Þ; (9.2) where c ðxÞ are the 4d matter fields. Because of the presence of a brane Einstein-Hilbert term, this scenario is also called brane induced gravity (Gabadadze, 2007) [see Kiritsis, Tetradis, and Tomaras (2001) and Antoniadis, Minasian, and Vanhove (2003) for attempts at string theory realizations]. The dynamical variables are the 5d metric depending on the 5d coordinates, the embedding XA ðxÞ of the brane depending on the 4d coordinates, and the 4d matter fields depending on the 4d coordinates GAB ðX Þ; X ðxÞ; A c ð xÞ : g ðxÞ ¼ @ X A @ X B GAB ðXðxÞÞ: (9.4) Note that the dependence of the action on the X A enters only through the induced metric g . The action (9.1) has a lot of gauge symmetry. First, there are the reparametrizations of the brane given by infinitesimal vector fields  ðxÞ, under which the X A transform as scalars and the matter fields transform as tensors (i.e., with a Lie derivative),  c ¼ L c : (9.5) Second, there are reparametrizations of the bulk given by infinitesimal vector fields ÄA ðX Þ, under which GAB transforms as a tensor and the XA shift, Ä GAB ¼ rA ÄB þ rB ÄA ;  Ä X A ¼ ÀÄA ðX Þ: (9.6) The induced metric g transforms as a tensor under  , and is invariant under Ä 15. 15 To see invariance under Ä , transform Ä GAB ¼ Ä ð@ X A @ X B GAB ðX ðxÞÞÞ ¼ À@ Ä @ X GAB ðXðxÞÞ À @ X @ Ä GAB ðXðxÞÞ A X  ð xÞ ¼ x  ;  ¼ 0; 1; 2; 3; (9.9) X 5 ð xÞ ¼ 0 : (9.10) There are still residual gauge symmetries which leave this gauge choice invariant. Acting with the two gauge transformations  and Ä on the gauge conditions and demanding that the change be zero, we find Ä X 5 ðxÞ þ  X 5 ðxÞ ¼ ÀÄ5 ðX ðxÞÞ þ  @ X 5 ! X5 ðxÞ¼0 À Ä5 ðX ðxÞÞ ) Ä 5 ð X ð xÞ Þ ¼ 0 : (9.11) (9.3) The 4d metric is not independent, but is fixed to be the pullback of the 5d metric,  X A ¼  @ X A ; We first proceed to fix some of this gauge symmetry. In particular, we freeze the position of the brane. Note that t...
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