There is ample evidence that visible matter and detectable radiation make up only a small fraction of the mass of the universe, perhaps only a few percent.
Therefore, an urgent problem is to search for physical systems (models) for which the presence of dark matter is an intrinsic property.
One of such physical systems can be a ``null string'' gas (gas of thin tubes of a massless scalar field).
Earlier it was shown that the gravitational interaction between the elements of such a gas should lead to the appearance of mass in the scalar field (primary particles with nonzero rest mass are formed in the gas). By interacting gravitationally, primary particles can combine into more complex formations -- ``macro'' objects.

In this model, dark matter is formed by extremely numerous and spatially diverse structures with nonzero rest mass, which are formed in a gas of null strings as a result of gravitational interaction. An interesting problem is to study the gravitational field of such structures.

In the proposed work, the asymptotics of the gravitational field of primary particles with nonzero rest mass, which are formed in a gas of null strings as a result of gravitational interaction, is investigated.
It is shown that on time scales much larger than the time of one full cycle of oscillation of null strings forming a particle, or at distances much larger than the dimensions of the region inside which oscillations of interacting null strings occur, the gravitational field of such a particle is described by the Minkowski metric.