Upper Bound for Diameter of Cosmological Black Holes and Nonexistence of Black Strings

The diameter of the apparent horizon, defined by the distance between furthest points on the horizon, in spacetimes with a positive cosmological constant $\Lambda$ has been investigated. It is established that the diameter of the apparent horizon on the totally umbilic partial Cauchy surface cannot exceed $2\pi \mathrm{/}\sqrt{3\Lambda }$. Then, it is argued that arbitrarily long black strings cannot be formed in our universe.