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The Schwarzschild solution in the DGP model
The five-dimensional (D=5) case is special and does not share all of the
features of higher-dimensional cases [56]. In particular, the five-dimensional
model appears to be plagued by the van Dam-Veltman-Zakharov (vDVZ) discontinuity of massive gravity [61]. This issue has attracted some attention
recently. In massive gravity the problem was solved by Vainshtein [62] who observed that one may obtain
a Schwarzschild solution to the non-linear Einstein equations (with a point source of mass
m) as two different expansions.
One expansion is valid in the large-r
regime and is obtained by linearizing
the field equations; the other expansion (leading to the Schwarzschild solution)
is valid in the small-r regime.
The two regions are separated by the length scale
where and M is the five-dimensional mass scale. This result is in agreement with ref. [64] where it was argued that the discontinuity can be avoided by bending the brane through a coordinate transformation. This bending can be large even if the matter sources are weak, leading to a breakdown of the standard linearization of the field equations at distances r << r_{c} with r_{c} given by (10). It would be interesting to extend these results to a Randall-Sundrum-type scenario [55] which is also known to be plagued by a vDVZ discontinuity [66]. |
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