The small sphere limit of quasilocal energy in higher dimensions along lightcone cuts

Abstract

The problem of quasilocal energy has been extensively studied mainly in four dimensions. Here we report results regarding the quasilocal energy in spacetime dimension n ≥ 4. After generalising three distinct quasilocal energy definitions to higher dimensions under appropriate assumptions, we evaluate their small sphere limits along lightcone cuts shrinking towards the lightcone vertex. The results in vacuum are conveniently represented in terms of the electromagnetic decompositions of the Weyl tensor. We find that the limits at presence of matter yield the stress tensor as expected, but the vacuum limits are in general not proportional to the Bel-Robinson superenergy Q in dimensions n > 4. The result defies the role of the Bel-Robinson superenergy as characterising the gravitational energy in higher dimensions, albeit the fact that it uniquely generalises. Surprisingly, the Hawking energy and the Brown-York energy exactly agree upon the small sphere limits across all dimensions. The "new" vacuum limit Q, however, cannot be interpreted as a gravitational energy because of its non-positivity. Furthermore, we also give the small sphere limits of the Kijowski-Epp-Liu-Yau type energy in higher dimensions, and again we see Q in place of Q. Our work extends earlier investigations of the small sphere limits [1,2,3,4], and also complements [5].

Outer entropy = Bartnik-Bray inner mass, and the gravitational ant conjecture

“…) which is exactly the result we expect from volume Ω n−2 n−1 times the matter energy density T (e 0 , e 0 )| p . It matches with the non-vacuum small sphere limits of other quasilocal masses in four dimensions [18] and higher dimensions [97]. Note that this result holds for both the AF case (4.6) and the AH case (4.7), because the second term with higher power in (4.7) is irrelevant at the leading order.…”

Outer entropy equals Bartnik-Bray inner mass and the gravitational ant conjecture

“…which is exactly the result we expect from volume Ω n−2 n−1 times the matter energy density Tðe 0 ; e 0 Þj p . It matches with the nonvacuum small sphere limits of other quasilocal masses in four dimensions [18] and higher dimensions [96]. Note that this result holds for both the AF case (14) and the AH case (15), because the second term with higher power in (15) is irrelevant at the leading order.…”

Second-order mass estimates for static vacuum metrics with small Bartnik data

“…Indeed, as reviewed in a moment, the Bartnik mass does not fail to meet this natural expectation, and the present article is motivated by a desire to identify the next most significant contributions. See for example [15,19,20,27,31,52,53,57] for estimates of other quasilocal masses of small spheres in both Riemannian and spacetime settings.…”