The positive energy theorem for asymptotically anti-de Sitter spacetimes

We establish the inequality for Henneaux-Teitelboim's total energy-momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary \(t\)-slice in anti-de Sitter spacetime. In particular, when \(t=0\), it generalizes Chru\'{s}ciel-Maerten-Tod's inequality in the center of AdS mass coordinates. We also show that the determinant of energy-momentum endomorphism \({\bf Q}\) is the geometric invariant of asymptotically anti-de Sitter spacetimes.