Nonminimal coupling for the gravitational and electromagnetic fields: Traversable electric wormholes

We discuss new exact solutions of a three-parameter nonminimal Einstein-Maxwell model. The solutions describe static spherically symmetric objects with and without center, supported by an electric field nonminimally coupled to gravity. We focus on a unique one-parameter model, which admits an exact solution for a traversable electrically charged wormhole connecting two universes, one asymptotically flat the other asymptotically de Sitter ones. The relation between the asymptotic mass and charge of the wormhole and its throat radius is analyzed. The wormhole solution found is thus a nonminimal realization of Wheeler’s idea about charge without charge and shows that, if the world is somehow nonminimal in the coupling of gravity to electromagnetism, then wormhole appearance, or perhaps construction, is possible.

Figure 1

Plots of the rescaled electric field $Z(y)=E/E_Q$ as function of $y=Y/r_Q$ for $a<0$. All curves have a vertical asymptote at $y=\sqrt{-3a}$. (a) When $-\infty <a<-\frac{16}{9}$, the curve crosses the asymptote twice. (b) When $a=-\frac{16}{9}$, the curve touches the asymptote at one point. (c) When $-\frac{16}{9}<a<0$, the curve does not cross the asymptote. For all cases the function $Z(y)$ is infinite, the singularity point being situated at $y=0$, when $a=0$ only. When $a\to -0$, the vertical asymptote coincides with the coordinate line $y=0$ and the right branch of the curve (c) converts into the standard Coulombian curve $Z(y)=1/y^2$.

Figure 2

Plots of the rescaled electric field $Z(y)$ displaying the behavior of the Coulombian branch and the two non-Coulombian branches for $a>0$. The interval $0<a<+\infty$ is split into three regions with respect to two specific values of $a$, $a=1$ and $a=\frac{256}{243}$. The curve $Z(y)$ is also drawn for the cases $a=1$ and $a=\frac{256}{243}$. Dashed lines indicate upper ($Z=1/\sqrt{a}$) and lower ($Z=-1/\sqrt{a}$) horizontal asymptotes. (a) When $0<a<1$, $Z_{+}(y,a)>1/\sqrt{a}$. (b) When $a=1$, the second non-Coulombian branch degenerates into a straight line $Z_{+}(y,1)=1$. (c) When $1<a<\frac{256}{243}$, $Z_{+}(y,a)<1/\sqrt{a}$. (d) When $a=\frac{256}{243}$, the second non-Coulombian branch has a common point with the Coulombian one at $y=0$. (e) When $a>\frac{256}{243}$, these branches have a common point at $y>0$.

Figure 3

Plot of the metric function $\sigma (\rho )$ at $a=1$. This function has only one root ${\rho }_s=0.289$.

Figure 4

Plots of the function $N(\rho )$ for $a=1$ for various values of the rescaled asymptotic mass $m=M/r_Q$. We split the interval $0<m<+\infty$ into three regions with respect to two specific values of mass: $m=m^{*}=0.7913$ and $m=m_0=0.7924$. All curves except (b) possess a vertical asymptote at $\rho ={\rho }_s=0.289$. When $m=m_0$ [plot (d)] $N(0)$ is finite, contrary to the other cases.

Figure 5

Plots of the function $g_{00}(\rho )={\sigma }^{2}N$ for $a=1$ for various values of the rescaled asymptotic mass $m=M/r_Q$. We split the interval $0<m<+\infty$ into three regions with respect to two specific values of mass: $m=m^{*}=0.7913$ and $m=m_0=0.7924$. (a) When $m<m^{*}$, the function $g_{00}$ is positive and tends to infinity at origin. (b) When $m=m^{*}$, the curve touches the horizontal axis at $\rho ={\rho }_s=0.289$. (c) When $m^{*}<m<m_0$, the curve intersects the horizontal axis at two different points; the function $g_{00}$ is negative between these points. (d) When $m=m_0$, the curve intersects the horizontal axis at one point; the function $g_{00}$ tends to a finite negative value at origin. (e) When $m>m_0$, the function $g_{00}$ tends to negative infinity at origin.

Figure 6

Plot of the dimensionless radius of the wormhole throat $b$ as a function of the parameter $a$. It starts at $a=\frac{256}{243}$, i.e., if the parameter $a$ is less than this value a throat does not exist.

Figure 7

Plot of the rescaled electric field $Z(\rho )$ for $a=4$. The region $\rho >0$ is the Coulombian branch [$Z(+\infty )=0$], the region of negative values of $\rho$ is the second non-Coulombian one [$Z(-\infty )=\frac{1}{2}$]. The junction point $\rho =0$ is the point of inflection on this curve.

Figure 8

Plot of the metric function $\sigma (\rho )$ for $a=4$. This function is positive and tends to unity at both infinities. The curve is nonsymmetric with respect to the vertical axis.

Figure 9

Plots of $g_{00}$ (upper row) and $N(\rho )$ (lower row) for the value $a=4$, when there is only one horizon. We split the interval $0<m<+\infty$ into two regions with respect to a specific value of the rescaled mass $\tilde{m}=1.0081$. (a) When $m<\tilde{m}$, the curves cross the horizontal axis at negative values of $\rho$, so that the horizon is located behind the throat. (b) When $m=\tilde{m}$, then $g_{00}(0)=0$ and $N(0)=0$, thus, the horizon is located on the throat. (c) When $m>\tilde{m}$, the horizon is located in front of the throat. Plot (a) illustrates the case in which the horizon belongs to an asymptotically de Sitter region of the combined spacetime, it is a standard cosmological apparent horizon. In this sense such a wormhole is traversable.