We extend Brill’s positive mass theorem to a large class of asymptotically flat, maximal, U(1)2-invariant initial data sets on simply connected four dimensional manifolds Σ. Moreover, we extend the local mass angular momenta inequality result [A. Alaee and H. K. Kunduri, Classical Quantum Gravity 32(16), 165020 (2015)] for U(1)2 invariant black holes to the case with nonzero stress energy tensor with positive matter density and energy-momentum current invariant under the above symmetries.
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The replaced preprint version contains an improved discussion of the geometry of Σ.
This condition is asymptotic flatness15 for s ≥ 2 and when we write f = os(rl) it means ∂β1⋯∂βpf = o(rl−p) for 0 ≤ p ≤ s.
It may be possible to prove this assumption is unnecessary (see Ref. 5 for the three-dimensional case).
We will refer to this as the “mass” hereafter.
There is a sign mistake in Ref. 10 because of the orientation. The sign of summation term over rods should be positive.
