Quasi-local mass at axially symmetric null infinity

Total mass and limits of quasi-local mass at future null infinity

“…In [43], Wang-Yau proposed the following definition of quasilocal mass. To evaluate the quasilocal mass of a 2-surface Σ with the physical data (σ, H), one first solves the optimal isometric embedding equation, see (16) below, which gives an embedding of Σ into the Minkowski spacetime with the image surface Σ 0 that has the same induced metric as Σ, i.e. σ.…”

“…In [6], we evaluate the large sphere limit of quasilocal mass at I + which recovers the Bondi-Sachs energy momentum. At a retarded time u = u 0 , we consider the family of large spheres Σ r parametrized by the Bondi-Sachs coordinate r. The positivity of the Bondi mass guarantees the unique solvability of the optimal isometric embedding system (16) with a solution (X r , T r ). Suppose X r and T r admit expansions:…”

“…Both the positivity of Bondi mass (3) and the mass loss formula (5) are global statements about I + that require the information in all direction of (θ, φ) on S 2 ∞ . In [10,13,14,15,16], we study the limit of quasilocal mass along a single direction (θ, φ) and obtain quasilocal quantities at I + . Consider a null geodesic γ with affine parameter s that approaches I + .…”

“…In order to understand such a quasilocal mass at the purely gravitational level, we compute the case for the null infinity of a general spacetime in a Bondi-Sachs coordinate system [15,16]. The 1 s 2 term of the quasilocal mass of a unit sphere approaching null infinity is (up to a constant factor) (18) in which h (−1) and k (−1) are part of the physical data and h (−1) 0 and τ (−1) depend on the solution of the optimal isometric embedding system.…”

“…However, some rather amazing cancellations show that the answer depends only on the shear tensor: Theorem 7. [15,16] The limit of the quasilocal mass of unit-size sphere at I + is a positive quasilocal quantity that depends only on the shear tensor.…”