Wormhole geometries in $f(R)$ modified theories of gravity

In this work, we construct traversable wormhole geometries in the context of $f(R)$ modified theories of gravity. We impose that the matter threading the wormhole satisfies the energy conditions, so that it is the effective stress-energy tensor containing higher order curvature derivatives that is responsible for the null energy condition violation. Thus, the higher order curvature terms, interpreted as a gravitational fluid, sustain these nonstandard wormhole geometries, fundamentally different from their counterparts in general relativity. In particular, by considering specific shape functions and several equations of state, exact solutions for $f(R)$ are found.

Figure 1

The stress-energy tensor satisfying the WEC, for the specific case of the traceless stress-energy tensor equation of state, and for the values $C_1=0$ and $C_2=-1$. We have considered the dimensionless quantities $\mathrm{WEC}1=r_0^2\rho$, $\mathrm{WEC}2=r_0^2(\rho +p_r)$ and $x=r/r_0$.

Figure 2

The specific form of $f(R)$, for the specific case of the traceless stress-energy tensor equation of state, by imposing the values $C_1=0$ and $C_2=-1$. The range is given by $0\le R/R_0\le 1$.

Figure 3

The stress-energy tensor satisfies the WEC, for the specific case of the equation of state $p_t=\alpha \rho$ and considering the form function $b(r)=r_0^2/r$. We have imposed the values $C_1=-1$ and $\alpha =-1$, and considered the dimensionless quantities $\mathrm{WEC}1=r_0^2\rho$, $\mathrm{WEC}2=r_0^2(\rho +p_r)$ and $x=r/r_0$.

Figure 4

The profile of $f(R)$ is depicted for the specific case of the equation of state $p_t=\alpha \rho$ and considering the form function $b(r)=r_0^2/r$. The values $C_1=-1$ and $\alpha =-1$ have been imposed, with the range given by $0\le R/R_0\le 1$.