Violation of the cosmic no hair conjecture in Einstein-Maxwell-dilaton system

The cosmic no hair conjecture is tested in the spherically symmetric Einstein-Maxwell-dilaton~(EMD) system with a positive cosmological constant $\Lambda$. Firstly, we analytically show that once gravitational collapse occurs in the massless dilaton case, the system of field equations breaks down inevitably in outer communicating regions or at the boundary provided that a future null infinity ${\cal I}^+$ exists. Next we find numerically the static black hole solutions in the massive dilaton case and investigate their properties for comparison with the massless case. It is shown that their Abbott-Deser~(AD) mass are infinite, which implies that a spacetime with finite AD mass does not approach a black hole solution after the gravitational collapse. These results suggest that ${\cal I}^+$ cannot appear in the EMD system once gravitational collapse occurs and hence the cosmic no hair conjecture is violated in both the massless and the massive cases, in contrast to general relativity.