Gravitational Lensing by Lemaître–Tolman–Bondi Wormholes in a Friedmann Universe

The Lemaître–Tolman–Bondi (LTB) solution to the Einstein equations describes the dynamics of a self-gravitating spherically symmetric dust cloud with an arbitrary density profile and any distribution of initial velocities, encoded in three arbitrary functions f(R), F(R), and τ0(R), where R is a radial coordinate in the comoving reference frame. A particular choice of these functions corresponds to a wormhole geometry with a throat defined as a sphere of minimum radius at a fixed time instant. In this paper we explore LTB wormholes and discuss their possible observable appearance, studying in detail the effects of gravitational lensing by such objects. For this aim, we study photon motion in wormhole space-time inscribed in a closed Friedmann dust-filled universe and find the wormhole shadow as it could be seen by a distant observer. Because the LTB wormhole is a dynamic object, we analyze the dependence of its shadow size on the observation time and on the initial size of the wormhole region. We reveal that the angular size of the shadow exhibits a non-monotonic dependence on the observation time. At early times, the shadow size decreases as photons with smaller angular momentum gradually reach the observer. At later times, the expansion of the Friedmann universe becomes a dominant factor that leads to an increase in the shadow size.