Stable Lorentzian wormholes in dilatonic Einstein-Gauss-Bonnet theory

We discuss the properties of Lorentzian wormholes in dilatonic Einstein-Gauss-Bonnet theory in four spacetime dimensions. These wormholes do not need any form of exotic matter for their existence. A subset of these wormholes is shown to be linearly stable with respect to radial perturbations. We perform a comprehensive study of their domain of existence, and derive a generalized Smarr relation for these wormholes. We also investigate their geodesics, determining all possible particle trajectories, and perform a study of the acceleration and tidal forces that a traveler crossing the wormhole would feel.

Figure 1

(a) The metric function $f(l)$, (b) the metric function $\nu (l)$, (c) the dilaton function $\varphi (l)$, and (d) the scaled GB term $\alpha R_{\mathrm{GB}}^2$ versus $l$ for wormhole solutions with the parameter values $f_0=1.1,1.5,2.0$ and $\alpha /r_0^2=0.02,0.05$.

Figure 2

(a) Isometric embedding of the wormhole solution for $\alpha /r_0^2=0.02$ and $f_0=1.1$ (taken from 37), (b) $z$ versus $\eta$ for $\alpha /r_0^2=0.02$ and $f_0=1.1,2.0,20.0$.

Figure 3

(a) Scaled area $A/16\pi M^2$ of the throat versus the scaled dilaton charge $D/M$ for several values of $\alpha /r_0^2$. The boundaries represent EGBd black holes (asterisks), limiting solutions with $f_0\to \infty$ (crosses), and solutions with curvature singularities (dots). (b) Metric function $f_0$ at the throat (determining the curvature radius $R_0=r_0{f}_0$) versus the scaled dilaton charge $D/M$; (c) dilaton function $\varphi (0)$ at the throat versus the scaled dilaton charge $D/M$; (d) redshift function $\nu (0)$ at the throat versus $1/f_0$.

Figure 4

(a) Profile function $b/r_0$ and (b) dilaton function $\varphi$ versus $r/r_0$ for $f_0=1.1,1.01,1.001,1.0001$ and $\alpha /r_0^2=0.02$ together with the limiting black hole functions.

Figure 5

(a) Metric function $f(l)$ and (b) scaled GB term $\alpha R_{\mathrm{GB}}^2$ versus $l$ for $f_0=10.0,15.0,20.0$ and $\alpha /r_0^2=0.02,0.1$.

Figure 6

(a) Scaled GB term $\alpha R_{\mathrm{GB}}^2$ versus $l$, (b) metric function $f(l)$, (c) derivative $f^{\prime }(l)$, and (d) derivative of the dilaton function ${\varphi }^{\prime }(l)$ for several values of $f_0$ and $\alpha /r_0^2=0.1$ demonstrating the emergence of a curvature singularity.

Figure 7

Demonstration of the violation of the null energy condition for $\alpha /r_0^2=0.02$ and several values of $f_0$.

Figure 8

Check of the Smarr relation for several sets of wormhole solutions: (a) mass $M$ versus $f_0$ for $\alpha /r_0^2=0.1$, $\gamma =1$, (b) mass $M$ versus $\gamma$ for $\alpha /r_0^2=0.02$, $f_0=1.4$.

Figure 9

(a) Scaled entropy-analogue throat quantity $S_{\mathrm{th}}/4\pi M^2$ versus the scaled dilaton charge $D/M$ for several values of $\alpha /r_0^2$. The boundaries represent EGBd black holes (asterisks), limiting solutions with $f_0\to \infty$ (crosses), and solutions with curvature singularities (dots). The shaded areas indicate linear stability (lilac or lower), instability (red or upper), undecided yet (white) with respect to radial perturbations (Sec. 4). (b) Scaled surface gravity $\kappa r_0$ at the throat versus the scaled dilaton charge $D/M$; (c) scaled first term of the mass formula; (d) scaled third term of the mass formula.

Figure 10

(a) Eigenvalue ${\sigma }^2$ versus $f_0$ for $\alpha /r_0^2=0.02$. (b) Domain of existence: scaled area $A/16\pi M^2$ of the throat versus the scaled dilaton charge $D/M$ for several values of $\alpha /r_0^2$. The shaded areas indicate linear stability (lilac or lower), instability (red or upper), undecided yet (white) with respect to radial perturbations (taken from 37).

Figure 11

Nonsymmetric and symmetric extension of (a) the metric function $f$ and (b) the dilaton function $\varphi$ for $\alpha /r_0^2=0.02$ and $f_0=1.1$ versus $l$.

Figure 12

(a) Energy density $\rho$, (b) pressure $p$, (c) dilaton charge density ${\rho }_{\mathrm{dil}}$ and (d) interaction potential versus ${\varphi }_0$.

Figure 13

(a) Energy density $\rho$ and (b) $\lambda ={\lambda }_0={\lambda }_1$ versus ${\varphi }_0$ for dust.

Figure 14

Effective potential $V_{\mathrm{eff}}$ versus ${log}_{10}(r/r_0)$ for several values of the angular momentum $L$ for the wormhole solution with $\alpha /r_0^2=0.05$ and $f_0=1.1$.

Figure 15

Minimal energy $E_0=\sqrt{V_{\mathrm{eff}}(r_0,0)}$ of test particles versus the scaled dilaton charge $D/M$ for several values of $\alpha /r_0^2$. The black line on the left denoted by ${\sigma }^2=0$ indicates the stability change of the wormhole solutions.

Figure 16

Bound orbits of massive test particles with angular momentum $L=2$ and energy (a) $E=0.9$ and (b) $E=0.98$ in the wormhole spacetime with $\alpha /r_0^2=0.05$ and $f_0=1.1$. The colors red (solid) and blue (dotted) indicate motion on the first and the second asymptotically flat part of the manifold, respectively. The throat of the wormhole is shown by the black circle. The right panel of (a) shows the orbit on the isometric embedding of the wormhole.

Figure 17

(a) Same as Fig. 16 for unbound orbits of massive test particles and energies $E=1.02$, 1.05. (b) Normalized effective potential $v_{\mathrm{eff}}$ for massless test particles versus ${log}_{10}(r/r_0)$ for the wormhole solution with $\alpha /r_0^2=0.05$ and $f_0=1.1$.

Figure 18

The dimensionless acceleration $\widehat{a}=|a|/(c^2/r_0)$ that a traveler would feel traversing the wormhole as a function of $D/M$ for a variety of values of $\alpha /r_0^2$.