Ghost-free infinite-derivative dilaton gravity in two dimensions

We present the ghost-free infinite-derivative extensions of the spherically reduced gravity (SRG) and Callan-Giddings-Harvey-Strominger (CGHS) theories in two space-time dimensions. For the case of SRG, we specify the Schwarzschild-type gauge and diagonalize the quadratic action for field perturbations after taking the background fields to be those of the flat-space solution with a linear dilaton. Using the obtained diagonalization, we construct ghost-free infinite-derivative modifications of the SRG theory. In the context of this modified SRG theory we derive a nonlocal modification of the linearized spherically reduced Schwarzschild solution. For the case of CGHS gravity, we work in the conformal gauge and diagonalize the quadratic action associated with this theory for a general background solution. Using these results, we construct the ghost-free infinite-derivative modifications of the CGHS theory and examine nonlocal modifications to the linearized CGHS black-hole solution.

Figure 1

Perturbed metric $\delta f$ radial dependence for the SRG theory as given by Eqs. (3.20) and (3.33) for the local and nonlocal cases respectively. The solid blue curve shows the local case, i.e., $\ell =0$. The remaining three curves correspond to nonlocal solutions with the dashed green, dotted orange, and dash-dotted red curves corresponding to $a\ell =0.05$, $a\ell =0.1$, and $a\ell =0.2$ respectively. In producing these plots, we have set $a=M=1$.

Figure 2

Radial dependence of the Ricci scalar $R/a^2$ for the SRG theory when calculated to first order in $\delta f$ using $R=-\mathrm{\Delta }\delta f+\mathcal{O}(\delta f^2)$. The solid blue curve shows the local case, i.e., $\ell =0$, which is singular at the origin. The remaining three curves correspond to nonlocal solutions with the dashed green, dotted orange, and dash-dotted red curves corresponding to $a\ell =0.05$, $a\ell =0.1$, and $a\ell =0.2$ respectively. The vertical line through the origin is included to show the distributional character of the Ricci scalar for the local case as well as how it is regularized in the nonlocal cases. In producing these plots, we have set $a=M=1$.

Figure 3

Radial dependence of the perturbed dilaton field $\delta \mathrm{\Phi }$ for the local and nonlocal CGHS cases. The solid blue curve corresponds to the local case ($\ell =0$), the dashed green curve corresponds to $\lambda \ell =0.25$, the dotted orange curve corresponds to $\lambda \ell =0.4$, and the red dash-dotted curve corresponds to $\lambda \ell =0.6$. For these plots, we have set $M=10$ and $\lambda =35$.

Figure 4

Radial dependence of the perturbed conformal scalar $\delta w$ for both the local and nonlocal CGHS cases. The solid blue curve corresponds to the local case ($\ell =0$), whereas the dashed green, dotted orange, and red dash-dotted curves correspond to $\lambda \ell =0.25$, 0.4 and 0.6 respectively. For the local case, the vertical line through the origin is included to illustrate the distributional character of $\delta w$ as well as how it is regularized in the nonlocal cases. For these plots, we have set $M=10$ and $\lambda =35$.

Figure 5

Radial dependence of the Ricci scalar $R$ for both the local and nonlocal CGHS cases when computed up to first order, i.e., $R=-2\mathrm{\Delta }\delta w+\mathcal{O}(\delta w^2)$. The solid blue curve corresponds to the local case ($\ell =0$), whereas the dashed green, dotted orange, and red dash-dotted curves correspond to $\lambda \ell =0.25$, 0.4, and 0.6 respectively. For the local case, the vertical line through the origin is included to illustrate the distributional character of the Ricci scalar as well as how it is regularized in the nonlocal cases. For these plots, we have set $M=10$ and $\lambda =35$.