Exact two-soliton solutions to the Einstein gravitational field equations

Abstract

Following our previous work on the existence of 1-soliton solution to the Einstein gravitational field equations in the presence of a spherically-symmetric static background field, we have found six sets of analytical 2-soliton solutions to the Einstein field equations under a certain ansatz in the absence of the stated background field. Numerical analysis shows that if the two solitons of the transverse nature are injected at space variable z→±∞, the longitudinal field componentg ₃₃ will acquire non-zero values for a bounded spatial region at later time. The nature of the solitons becomes rather complex when they interact. The amplitudeg _μν of each soliton may change its magnitude resulting from the interaction. We have found that we might interpret the evolution of one field component as the gravitational instanton in our solutions. We must remark also that the total energy of the interacting solitons remains constant, as expected, at all time. These solutions correspond to the situation where the Riemann tensor is in general non-zero and are truly non-trivial solutions.