Abstract
We study Lorentzian wormholes in the ghost-free bigravity theory described by two metrics, g and f. Wormholes can exist if only the null energy condition is violated, which happens naturally in the bigravity theory since the graviton energy-momentum tensors do not apriori fulfill any energy conditions. As a result, the field equations admit solutions describing wormholes whose throat size is typically of the order of the inverse graviton mass. Hence, they are as large as the universe, so that in principle we might all live in a giant wormhole. The wormholes can be of two different types that we call W1 and W2. The W1 wormholes interpolate between the AdS spaces and have Killing horizons shielding the throat. The Fierz-Pauli graviton mass for these solutions becomes imaginary in the AdS zone, hence the gravitons behave as tachyons, but since the Breitenlohner-Freedman bound is fulfilled, there should be no tachyon instability. For the W2 wormholes the g-geometry is globally regular and in the far field zone it becomes the AdS up to subleading terms, its throat can be traversed by timelike geodesics, while the f-geometry has a completely different structure and is not geodesically complete. There is no evidence of tachyons for these solutions, although a detailed stability analysis remains an open issue. It is possible that the solutions may admit a holographic interpretation.
Highlights
It follows that for the wormhole to be a solution of the Einstein equations, the matter should violate the null energy condition (Tμνvμvν ≥ 0 for any null vμ)
This corresponds to a Killing horizon of the f-geometry and, as we shall see below, the curvature diverges at the horizon
A knowledge of such solutions would allow one to construct a maximal extension of the spacetime geometry in order to find out if the wormhole can be traversed by geodesics or not
Summary
The theory is defined on a four-dimensional spacetime manifold endowed with two Lorentzian metrics gμν and fμν with the signature (−, +, +, +). The metrics and all coordinates are assumed to be dimensionless, with the length scale being the inverse graviton mass 1/m. The parameters bk can apriori be arbitrary, but if one requires the flat space to be a solution of the theory and m to be the Fierz-Pauli mass of the gravitons in flat space, the five bk are expressed in terms of two arbitrary parameters, sometimes called c3 and c4, as b0 = 4c3 + c4 − 6, b1 = 3 − 3c3 − c4, b2 = 2c3 + c4 − 1, b3 = −(c3 + c4), b4 = c4. This is important in what follows: test g-particles will follow geodesics of the g-metric and will not directly feel the f-metric
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