# Boundedness of general relativity Hamiltonian

When one consider a lagrangian and construct hamiltonian, we expect to be bounded below. While looking to the Hamiltonian formulation of general relativity, I have difficulties to see how it can be bounded.

1. How this can be shown?

2. Is it related to the positive energy theorem?

## migrated from math.stackexchange.comJul 20 '18 at 11:05

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• What Lagranian and Hamiltonian are you considering? – md2perpe Jul 19 '18 at 10:19
• The Lagrangian of general relativity $L=\sqrt{-g}R$ and the ADM hamiltonian – anubis Jul 19 '18 at 11:14

Let $(\Sigma, h_{ab}, K_{ab})$ be an initial data set that is geodesically complete and asymptotically flat. Assume that the energy-momentum tensor satisfies the dominant energy condition. Then $E_{ADM} \geq \sqrt{P_i P_i}$, with equality if and only if the initial data set arises from a surface in Minkowski spacetime.