Einstein Gravity on a Brane in 5 D Non - compact Flat Spacetime – DGP model revisited –

We revisit the 5D gravity model by Dvali, Gabadadze, and Porrati (DGP). In the model it was shown that even in 5D non-compact Minkowski space, the Newtonian gravity can emerge on a brane at short distance by introducing a brane-localized 4D Ricci scalar δ(z)M2 4 R̄4 in the action. Based on this idea, we construct simple setups in which graviton standing waves can arise, and we introduce a z derivative term δ(z)∂2 z R̄4, which also respects 4D general covariance on the brane. We show that the gravity potential of brane matter becomes − r at long distance, because δ(z)∂2 z R̄4 allows only a smooth graviton wave function near the brane. Since the bulk gravity coupling may be arbitrarily small, strong interacting modes from the 5D graviton do not appear. We note that the brane metric utilized to construct δ(z)M2 4 R̄4 can be relatively different from the bulk metric by a conformal factor, and show that the graviton tensor structure that the 4D Einstein gravity predicts are reproduced in the DGP model. [email protected] Since Kaluza and Klein proposed the five dimensional (5D) theories, it had been believed for a long time that an extra space, if it exists, should be compactified on an extremely small manifold. The Newtonian gravity theory, which explains well the observed gravity interactions, seemingly ensures that our space should be effectively three dimensional. As noted in Refs. [1, 2], however, the size of the extra dimension(s) could be as large as (TeV) scale, and even infinite provided the graviton is effectively localized on a four dimensional (4D) sub-space (brane) embedded in a 5D AdS spacetime. Especially in Ref. [3], Dvali, Gabadadze, and Porrati (DGP) argued that the Newtonian gravity can be compatible even with 5D non-compact flat spacetime, only if the relevant matter fields are localized on a 4D brane, and a 4D Einstein-Hilbert termM 4 R̄4 is additionally introduced on the brane apart from the bulk gravity kinetic term M 5R5. In Ref. [3], it was claimed that the ordinary Newtonian potential arises at short distance, whereas at long distance the potential becomes that of a 5D theory. Thus, M5 should be supposed to be extremely small (<< TeV) so that the 4D gravity potential is modified at longer distances than the Hubble length scale. This setup was employed in the self-tuning model of the cosmological constant [4]. As shown in Ref. [3], however, the graviton tensor structure in the 5D (minimal) DGP model is given by that in tensor-scalar gravity theory rather than that in the Einstein theory. Thus, an extra scalar polarization degree is also involved in 4D gravity interaction. This gives rise to unacceptable deviation from the observation results on light bending around the sun as in the massive gravity case [6]. Moreover, in Ref. [9] the authors criticized the DGP model pointing that extremely small M5 possibly induces strong interactions by hμ5 and h55 (whose kinetic terms were supposed In Ref. [5], it is demonstrated that in D ≥ 6 the brane-localized gravity kinetic term exactly gives the result of the Einstein gravity on the 4D brane. To compensate the additional attractive force by the extra scalar mode, the authors in Ref. [3] suggested to introduce a vector field, which is universally coupled to all matter fields with an U(1) charge. In Ref. [7], it was argued that the resummation of nonlinear effects in massive gravity recovers the result of the Einstein gravity near the sun. This issue in DGP setup is handled in Refs. [8].