Consequences of the absence of Birkhoff’s theorem in modified-gravity theories: The Dvali-Gabadaze-Porrati model

We consider the consequences of the absence of Birkhoff’s theorem in theories of modified gravity. As an example, we calculate the gravitational force on a test particle due to a spherical mass shell in the Dvali-Gabadadze-Porrati model. We show that unlike in general relativity, the force depends on the mass distribution. In particular, the gravitational force within a spherical mass shell depends on the geometric structure of the bulk, and is likely nonzero.

Figure 1

Mock model of a galactic neighborhood as concentric spherical shells of mass.

Figure 2

Gravitational acceleration as a function of radius. $g$ (solid line) includes only the galaxy as a source, $g+c$ (dashed line) adds the cluster, and $g+c+sc$ (dotted line) adds the supercluster. The Newtonian acceleration is $g_n$ (dot-dashed line). The crossover scale is $r_0={10}^5{r}_g={10}^{10}\mathrm{pc}\approx 1/H_0$.

Figure 3

As in Fig. 2, but with $r_0=r_g={10}^5\mathrm{pc}$.