| Summary: | We consider the generalized Tolman solution of general relativity, describing the evolution of a spherical dust cloud in the presence of an external electric or magnetic field. The solution contains three arbitrary functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>τ</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>R</i> is a radial coordinate in the comoving reference frame. The solution splits into three branches corresponding to hyperbolic (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>), parabolic (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>) and elliptic (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo><</mo><mn>0</mn></mrow></semantics></math></inline-formula>) types of motion. In such models, we study the possible existence of wormhole throats defined as spheres of minimum radius at a fixed time instant, and prove the existence of throats in the elliptic branch under certain conditions imposed on the arbitrary functions. It is further shown that the normal to a throat is a timelike vector (except for the instant of maximum expansion, when this vector is null), hence a throat is in general located in a T-region of space-time. Thus, if such a dust cloud is placed between two empty (Reissner–Nordström or Schwarzschild) space-time regions, the whole configuration is a black hole rather than a wormhole. However, dust clouds with throats can be inscribed into closed isotropic cosmological models filled with dust to form wormholes which exist for a finite period of time and experience expansion and contraction together with the corresponding cosmology. Explicit examples and numerical estimates are presented. The possible traversability of wormhole-like evolving dust layers is established by a numerical study of radial null geodesics.
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