Hadrian was not designated successor to Trajan
It is commonly believed that the reign of Nerva, Trajan, Hadrian, Antoninus Pius and Marcus Aurelius, was a time of stabilization and peaceful transfer of power in the Roman Empire, through adoption and the choice of a sane successor. Hadrian, who took over power after the great Trajan, was not particularly favoured by him, and the adoption document could have been forged.