Attractive gravity probe surface, positivity of quasi-local mass and Arnowitt-Deser-Misner mass expression

Under certain conditions, it is shown that the positivity of the Geroch/Hawking quasi-local mass holds for the attractive gravity probe surfaces in any higher dimensions than three. We also comment on the Arnowitt-Deser-Misner mass.


Introduction
Recently the attractive gravity probe surface (AGPS)/loosely trapped surface have been proposed as indicators for the existence of the attractive (strong) gravity [1][2][3] and the areal inequalities have been proven under some assumptions as the Penrose inequality [4].For the Schwarzschild spacetime, it is easy to see that the presence of the AGPS requires the positivity of the mass.In this paper, we will show that, under certain conditions, the Geroch/Hawking mass quasi-locally defined [5,6] is positive on the AGPS for general cases.We will also point out that the existence of the sequence of the AGPSs directly show us the positivity of the Arnowitt-Deser-Misner (ADM) mass in asymptotically flat spacetimes.While one may not be able to show the positivity of the ADM mass using the Geroch mass for higher dimensions than four because the Gauss-Bonnet theorem is crucial [5], our argument for the positivity of the Geroch/Hawking mass works regardless of spacetime dimensions.
The rest of this paper is organized as follows.In Sect.2, we review certain expression for the ADM mass and a natural generalization of the Geroch mass in higher dimensions.In Sect.3, we show the positivity of the Geroch mass on the AGPS.In Sect.4, we shorty comment on the positivity of the Hawking mass.In Sect.5, inspired by these arguments for the AGPS, we present a new expression for the ADM mass and, for the comparison, new expressions of the Geroch/Hawking mass in terms of the Weyl tensor.Finally, we give a summary in Sect.6.In Appendix A we confirm that a certain expression of the ADM mass is equivalent to that of the Geroch mass.In Appendix B, we show that one can immediately see the positivity of the Geroch/Hawking mass for the refined version of the AGPS introduced in Ref. [3].We also discuss the lower bound and positivity of the Bartonik mass through the discussion in Ref. [3].

ADM mass and Geroch mass in higher dimensions
As Ref. [7], it is known that the ADM mass in an asymptotically flat spacetime has the following form1 where S ∞ is the (n − 2)-sphere at spatial infinity, (n−1) R ab is the Ricci tensor of the (n − 1)dimensional asymptotically flat spacelike hypersurface Σ, r a is the outward unit normal vector to S ∞ in Σ and r is the radial coordinate near the spatial infinity.Using the double traced Gauss equation, the ADM mass is also expressed by2 where, in the second equality, we used the fact that the (n − 1)-dimensional Ricci scalar (n−1) R, the traceless part of the extrinsic curvature of S ∞ , kab , rapidly decays near the spatial infinity.Here, (n−2) R is the Ricci scalar and k is the trace of the extrinsic curvature of S ∞ in Σ.
In n-dimensions, one may define the Geroch mass on an (n − 2)-dimensional compact surface S in Σ as3 [5,10] This may be attained through the replacement of S ∞ and r by S and (A n−2 /Ω n−2 ) 1 n−2 in Eq. ( 2) respectively, where A n−2 is the area of S and Ω n−2 is the volume element of the (n − 2)-dimensional unit round sphere, that is, 2 ).

Positivity of Geroch mass for AGPS
Now we note that the following identity holds where D a and D a are the covariant derivatives of Σ and S, respectively.Here, k ab is the extrinsic curvature of S in Σ and ϕ is the lapse function for the radial direction.Using this, we can rewrite the Geroch mass as Then we consider an attractive gravity probe surface (AGPS) defined as an (n − 2)dimensional compact surface satisfying two inequalities at each point on the surface k > 0, (7) where α is a constant satisfying α > −1/(n − 2).If (n−1) R ≥ 0 holds on the AGPS, one can show that the Geroch mass is bounded from below as Therefore, the conditions ( 6) and (7) in the definition of the AGPS with the nonnegative scalar curvature (n−1) R ≥ 0 of Σ guarantee the positivity of the Geroch mass.Note from the Hamiltonian constraint that (n−1) R ≥ 0 holds on the maximal spacelike hypersurface under the assumption that the spacetime geometry satisfies the Einstein equations with the dominant energy condition.
It would be interesting to consider the limit case with α = −1/(n − 2), although this case is out of the definition of AGPS.Suppose that the scalar curvature is nonnegative (n−1) R ≥ 0. When m G vanishes, all inequalities of the above discussion should be equalities, that is, r a D a k + k 2 /(n − 2) = 0, (n−1) R = 0, kab = 0 and D a ϕ = 0 hold on the surface.

Positivity of Hawking mass for AGPS
One may be also interested in the Hawking mass defined by where κ = h ab K ab , h ab is the induced metric of S and K ab is the extrinsic curvature of spacelike hypersurface Σ.Since the definitions of the Geroch/Hawking mass manifestly result in m H ≥ m G , we see that m H of an AGPS is positive on Σ with (n−1) R ≥ 0. Furthermore, under the assumption that the spacetime geometry satisfies the Einstein equations with the dominant energy condition, we have the sharper lower bound for m H .The Hamiltonian constraint gives where ρ is the energy density and κab is the traceless part of κ ab .Using Eqs. ( 4) and ( 10), we have For the AGPS on the maximal slice (K = K (r) + κ = 0), we can prove the positivity of m H as As discussed for the Geroch mass, we consider the surface with α = −1/(n − 2).When m H vanishes, r a D a k + k 2 /(n − 2) = 0, ρ = 0, K ab = 0, kab = 0 and D a ϕ = 0 hold on the surface.

Asymptotic behavior and Weyl tensor expression
In this section, inspired by the arguments in the previous two sections, we present a new expression for the ADM mass.In addition, since the ADM mass is written in terms of the Weyl tensor [7], we will also rewrite the Geroch/Hawking mass in terms of the Weyl tensor.
Since m G coincides with the ADM mass at spatial infinity limit, we have6 where, in the second equality, we used the asymptotic behavior of geometry, . This expression implies that the existence of a sequence of AGPSs near the spatial infinity leads to the positivity of the ADM mass.Conversely, the positivity mass theorem [11,12] tells us the non-negativity of the ADM mass and it guarantees the existence of the sequences of surface S near the spatial infinity satisfying Note that the existence of surface S is seen in an indirect way without using the asymptotic behavior of the metric 7 .
Next, we explicitly write down a new expression of the Geroch/Hawking mass in terms of the Weyl tensor.From the definition of the Weyl tensor, we have where E ab = C acbd n c n d and C abcd is the n-dimensional Weyl tensor.The Gauss equation gives us Since holds, equations ( 16) and ( 17) imply us Using this, Eqs. ( 5) and ( 12) become and respectively.At the spatial infinity limit, one can confirm the ADM mass expression in terms of the Weyl tensor shown in Ref. [7] m ADM = − 1 8π(n − 3) S∞ rE ab r a r b dA.
In the quasi-local mass expressions of Eqs. ( 20) and ( 22), however, there are additional terms which vanish for vacuum, static and spherically symmetric region.Note that the left-hand side of Eq. ( 19) appeared in the integrands of Eqs. ( 5) and ( 12).The condition (6) imposed in the definition of the AGPS controls the signature of this combination, that is, not only E ab r a r b , but also non-trivial terms composed of the n-dimensional Ricci tensor etc through Eq. (19).

Summary
In a summary, we could see that the issue of the existence of the AGPS is strongly related to the positivity of the Geroch/Hawking quasi-local mass.Motivated by the AGPS condition, a new expression for the ADM mass was also presented.As the ADM mass formula with the Weyl tensor, we had a similar formula for the Geroch/Hawking mass.5/7