Gravitational field of static $p$-branes in linearized ghost-free gravity

We study the gravitational field of static $p$-branes in $D$-dimensional Minkowski space in the framework of linearized ghost-free (GF) gravity. The concrete models of GF gravity we consider are parametrized by the nonlocal form factors $\exp (-\square /{\mu }^2)$ and $\exp ({\square }^2/{\mu }^4)$, where ${\mu }^{-1}$ is the scale of nonlocality. We show that the singular behavior of the gravitational field of $p$-branes in general relativity is cured by short-range modifications introduced by the nonlocalities, and we derive exact expressions of the regularized gravitational fields, whose geometry can be written as a warped metric. For large distances compared to the scale of nonlocality, $\mu r\to \infty$, our solutions approach those found in linearized general relativity.

Figure 1

Dimensionless Green functions ${\mu }^{2-m}{D}_m(r)$ for linearized GR, ${\mathrm{GF}}_1$, and ${\mathrm{GF}}_2$ theory in the cases $m=1$, 2, 3, 4. The solid line corresponds to linearized GR, the dashed line corresponds to linearized ${\mathrm{GF}}_1$ theory, and the dotted line represents linearized ${\mathrm{GF}}_2$ theory. All three theories agree for larger values of $\mu r$, that is, when the distance $r$ is larger than the scale of nonlocality ${\mu }^{-1}$. For short distances, $\mu r\ll 1$, there are significant deviations between GR (which is singular at the origin) and ${\mathrm{GF}}_1$ and ${\mathrm{GF}}_2$ (which, in turn, are perfectly well behaved).

Figure 2

Curvature invariants $I_2=-{\mu }^{-2}△D_2(r)$ and $J_2={\mu }^{-4}[{\partial }_i{\partial }_j{D}_2(r){]}^2$ for ${\mathrm{GF}}_1$ theory (dashed lines) and ${\mathrm{GF}}_2$ theory (dotted lines), which are relevant quantities for the cosmic string in $D=4$ as well as the angle deficit solutions in general $D$. The curvature invariants are smooth and well behaved at $r=0$. Note that the invariants of type $I_m$ exhibit fluctuations to negative values at $\mu r\sim 5$, which might correspond to an “anti-screening” effect in GF gravity. For the plots of the other invariants, see Appendix pp2.

Figure 3

Linear curvature invariant $I_m$ evaluated for ${\mathrm{GF}}_1$ and ${\mathrm{GF}}_2$ theory in the cases $m=1$, 2, 3, 4.

Figure 4

Quadratic curvature invariant $J_m$ evaluated for ${\mathrm{GF}}_1$ and ${\mathrm{GF}}_2$ theory in the cases $m=1$, 2, 3, 4.