Strings and Quantum Gravity

THEORETICAL HIGH ENERGY PHYSICS
Strings and Quantum Gravity

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RESEARCH > A brane in infinite-volume > 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

$\begin{displaymath} r_c = \left(\frac{m}{\mu^4 \bar M^2} \right)^{1/5}\end{displaymath}$ (9)

where $\mu$ is the mass of the graviton and $\bar M$ is the Planck mass. A similar solution was shown to exist in the DGP model [63] (cosmological solution). Dr. Siopsis and C. Middleton tackled the problem of a point souce in the DGP model (Schwarzschild solution) which seems to be considerably more complicated than a cosmological solution. By adding an off-diagonal term to the five-dimensional metric, they showed [65] that two different expansions of the solutions to the non-linear Einstein equations can be obtained, similar to the massive gravity case [62]. The two expansions were valid in two different regions separated by the scale parameter (cf. (9))

$\begin{displaymath} r_c = \left(\frac{m}{\mu^2\bar M^2} \right)^{1/3}\end{displaymath}$ (10)

where $\mu = M^3/\bar M^2$ 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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