Scalar wormholes with nonminimal derivative coupling

We consider static spherically symmetric wormhole configurations in a gravitational theory of a scalar field with a potential $V(ϕ)$ and nonminimal derivative coupling to the curvature describing by the term $(εg_{μν} + κG_{μν}) ϕ^{,μ}ϕ^{,ν}$ in the action. We show that the flare-out conditions providing the geometry of a wormhole throat could fulfilled both if $ε=-1$ (phantom scalar) and $ε=+1$ (ordinary scalar). Supposing additionally a traversability, we construct numerical solutions describing traversable wormholes in the model with arbitrary $κ$, $ε=-1$ and $V(ϕ)=0$ (no potential). The traversability assumes that the wormhole possesses two asymptotically flat regions with corresponding Schwarzschild masses. We find that asymptotical masses of a wormhole with nonminimal derivative coupling could be positive and/or negative depending on $κ$. In particular, both masses are positive only provided $κ<κ_1\le0$, otherwise one or both wormhole masses are negative. In conclusion, we give qualitative arguments that a wormhole configuration with positive masses could be stable.

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