Area bound for surfaces in generic gravitational field
Keisuke Izumi, Yoshimune Tomikawa, Tetsuya Shiromizu, Hirotaka Yoshino
PProg. Theor. Exp. Phys. , 00000 (24 pages)DOI: 10.1093 / ptep/0000000000 Area bound for surfaces in generic gravitationalfield
Keisuke Izumi , , Yoshimune Tomikawa , Tetsuya Shiromizu , , and HirotakaYoshino Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan Department of Mathematics, Nagoya University, Nagoya 464-8602, Japan Faculty of Economics, Matsuyama University, Matsuyama 790-8578, Japan Advanced Mathematical Institute, Osaka City University, Osaka 558-8585, Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
We define an attractive gravity probe surface (AGPS) as a compact 2-surface S α withpositive mean curvature k satisfying r a D a k/k ≥ α (for a constant α > − /
2) in thelocal inverse mean curvature flow, where r a D a k is the derivative of k in the outwardunit normal direction. For asymptotically flat spaces, any AGPS is proved to satisfy theareal inequality A α ≤ π [(3 + 4 α ) / (1 + 2 α )] ( Gm ) , where A α is the area of S α and m isArnowitt-Deser-Misner (ADM) mass. Equality is realized when the space is isometric tothe t =constant hypersurface of the Schwarzschild spacetime and S α is an r = constantsurface with r a D a k/k = α . We adapt the two methods, the inverse mean curvatureflow and the conformal flow. Therefore, our result is applicable to the case where S α hasmultiple components. For anti-de Sitter (AdS) spaces, the similar inequality is derived,but the proof is performed only by using the inverse mean curvature flow. We alsodiscuss the cases with asymptotically locally AdS spaces.
1. Introduction
Strong gravitational field can make any objects trapped in some region. Even photons cannotescape from such a region, which leads to the existence of a black hole. The strong gravita-tional field would be created by compactly concentrated gravitational sources, and thus thetrapped region must be small. The Penrose inequality [1] is an argument about the area ofits boundary; the area of an apparent horizon A AH must satisfy A AH ≤ π (2 Gm ) , where m is the Arnowitt-Deser-Misner (ADM) mass and G is Newton’s gravitational constant (weuse the unit c = 1). The proof for the Penrose inequality has been done for time-symmetricinitial data [2–4]. To be more precise, their results are applicable for a minimal surface (MS)in an asymptotically flat space with a Ricci scalar that is nonnegative everywhere. Thisversion of the inequality is called the Riemannian Penrose inequality.The Riemannian Penrose inequality may seem to provide a way to test the theory of generalrelativity. However, if the cosmic censorship holds, an apparent horizon A AH is hidden by anevent horizon, inside of which we cannot observe by the definition of a black hole. It is desiredto derive an inequality that is applicable to a surface outside the event horizon for the purposeof testing general relativity. The authors in this paper have succeeded in deriving such an © The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. a r X i v : . [ g r- q c ] J a n nequality [5]; We introduced the loosely trapped surface (LTS) as a surface with positivemean curvature k > k in the outwardunit normal direction r a D a k ≥
0, and proved that the LTS satisfies an areal inequality, A LTS ≤ π (3 Gm ) , where A LTS is the area of the LTS. The upper bound is realized if andonly if the space is isometric to the time-constant slice of the Schwarzschild spacetime andthe surface is the photon sphere. The photon sphere exists outside the horizon, and thus anLTS is observable for a distant observer. The proof for the areal inequality for the LTS isdone with the inverse mean curvature flow in Ref. [5].In this paper, we further generalize the Riemannian Penrose inequality so that it is appli-cable to a surface in a region where the gravity is moderately strong or even weak. Thesurface that we consider in this paper is a compact 2-surface S α with positive mean cur-vature k that satisfies r a D a k/k ≥ α (for a constant α > − /
2) in the local inverse meancurvature flow, which we name the attractive gravity probe surface (AGPS) . We show thatan AGPS with α satisfies A α ≤ π [(3 + 4 α ) / (1 + 2 α )] ( Gm ) , where A α is the area of theAGPS. For α → ∞ , the AGPS becomes the MS and the Riemannian Penrose inequality isobtained. Moreover, for α = 0, we have the theorem for the LTS in Ref. [5]. Therefore, theinequality for the AGPS is the generalization of the Riemannian Penrose inequality and theareal inequality for the LTS. We derive the inequality in two different ways; one is by usingthe inverse mean curvature flow [2, 3, 6] and the other is by using Bray’s conformal flow[4]. Both methods were applied for proving the Riemannian Penrose inequality. Since Bray’sproof is applicable to a surface with multiple components, our inequality can be applied toan AGPS with multiple components as well.We also show areal inequalities for AGPSs in asymptotically (locally) AdS spaces. SinceBray’s proof based on the conformal flow is not applicable to asymptotically (locally) AdSspaces, our inequality is only derived by using the inverse mean curvature flow for thesesetups.The rest of this paper is organized as follows. In Sect. 2, we show the proof by usage ofthe inverse mean curvature flow in the case for asymptotically flat and AdS spaces. We alsodiscuss the case for asymptotically locally AdS spaces. In Sect. 3, we show another proof ofthe areal inequality for asymptotically flat spaces, by applying Bray’s method [4]. We willgive a summary and discussions in Sect. 4. An example of the matching function used inSect. 3 is presented in Sect. A.
2. Inverse mean curvature flow
One of the ways to prove the Riemannian Penrose inequality is applying the monotonicityof the Geroch energy in the inverse mean curvature flow. The idea of the monotonicity wasintroduced in order to prove the positivity of the ADM mass by Geroch [6]. By applyingGeroch’s monotonicity, the Riemannian Penrose inequality was proved by Wald and Jang [2]under the assumption of the existence of a global inverse mean curvature flow. This methodwas extended by Huisken and Ilmanen [3] to resolve the possible singularity formation in theflow. Hence, the monotonicity exists even if a global inverse mean curvature flow cannot be For a Schwarzschild spacetime, r a D a k + k / (3) R ab r a r b = 2 m/r > aken. Geroch’s monotonicity was generalized to spaces with negative cosmological constantsby Boucher, Horowitz and Gibbons [7, 8].We first review the monotonicity with and without the negative cosmological constant,based on Ref. [8] in Sect. 2.1. Then, we show the generalized areal inequality in Sect. 2.2. Itis applicable also for asymptotically AdS spaces. The areal inequality is further generalizedto asymptotically locally AdS spaces, which is shown in Sect. 2.3. We consider a three-dimensional space Σ where the scalar curvature (3) R satisfies (3) R − ≥ ≤ . (1)Let us introduce an inverse mean curvature flow. We first consider the foliation of { S y } y ∈ R .The lapse function is denoted by ϕ . Geroch’s quasilocal energy of S y is defined as E ( y ) := A / ( y )64 π / G (cid:90) S y (cid:18) (2) R − k − (cid:19) dA, (2)where A ( y ) is the area, dA is the areal element, and (2) R is the scalar curvature of S y . Underthe condition of the inverse mean curvature flow kϕ = 1, the first derivative of E ( y ) is shownto satisfy the nonnegativity dE ( y ) dy = A / ( y )64 π / (cid:90) S y (cid:104) ϕ − ( D ϕ ) + ˜ k ab ˜ k ab + (3) R − (cid:105) dA ≥ , (3)where ˜ k ab is the traceless part of the extrinsic curvature k ab , that is,˜ k ab := k ab − kg ab , (4)with g ab being the metric of S y . Here, we used the assumption that the topology of S y does notchange under the flow, and thus, (cid:82) (2) RdA is a constant for all S y because it is the topologicalinvariant. Therefore, E ( y ) is an increasing function, which leads to E ( ∞ ) ≥ E ( y ). SinceGeroch’s energy at infinity coincides with the ADM mass in an asymptotically flat spacetimeand with the Abbott-Deser mass [9] or the Ashtekar-Magnon mass [10] in asymptoticallyAdS spacetimes, the inequality implies m ≥ E ( y ). If E ( y ) is expressed with the area of S y ,we have an areal inequality as shown below. In an asymptotically flat space without the cosmological constant, E ( y ) for a minimal surface(MS) and for a loosely trapped surface (LTS) [5] is evaluated as E ( y ) = 12 G (cid:114) A ( y )4 π and E ( y ) ≥ G (cid:114) A ( y )4 π , (5)respectively. Together with the inequality m ≥ E ( y ), they give the inequalities A ( y ) ≤ π (2 Gm ) for an MS and A ( y ) ≤ π (3 Gm ) for an LTS. Let us show similar areal inequalitiesby considering a broader class of surfaces, which includes MSs and LTSs. The extension including the singularity resolution in the inverse mean curvature flow was done byHuisken and Ilmanen [3] under the assumption (3) R ≥
0. It may be true in the case with the negativecosmological constant as pointed out in Ref. [8].3/24 e consider a surface S α satisfying k > r a D a kk ≥ α, (6)where α is a constant satisfying α > − , (7) r a is the outward unit normal vector to S α , namely r a = ϕD a y , and the derivative of k istaken for the foliation of the inverse mean curvature flow. We call a surface satisfying theabove conditions an attractive gravity probe surface (AGPS). The limit α → ∞ gives thecondition for an MS, i.e. k = 0, while the case with α = 0 is corresponding to an LTS [5].Thus, this surface is a generalization of MSs and LTSs. Note that in the limit α → − / S α coincides with the S surface at spacelike infinity. In this sense, the concept of the AGPScan cover from the surfaces in extremely strong gravity regions to those in very weak gravityregions by choosing the value of α appropriately. We show that its area A α satisfies aninequality similar to the Penrose inequality.On S α , we have a geometrical identity r a D a k = − ϕ − D ϕ − (3) R + 12 (2) R −
12 ( k + k ab k ab ) . (8)Integrating this relation over S α , we have12 (cid:90) S α (2) RdA = (cid:90) S α (cid:20) r a D a k + ϕ − ( D ϕ ) + 12 (3) R + 12 ˜ k ab ˜ k ab + 34 k (cid:21) dA ≥ (cid:90) S α (cid:20)(cid:18)
34 + α (cid:19) k + Λ (cid:21) dA, (9)where we used the inequalities of Eqs. (1) and (6). Note that in the case without a cosmo-logical constant, Λ = 0, this inequality leads to the positivity of its left-hand side. Thus, as aconsequence of the Gauss-Bonnet theorem, the topology of S α is S . Moreover, for a genericnegative cosmological constant, Λ ≤
0, the Gauss-Bonnet theorem gives us (cid:90) S α k dA ≤ (cid:18)
34 + α (cid:19) − [2 πχ − Λ A α ] , (10)where A α is the area of S α and χ is the Euler characteristic of S α . The Geroch energy E ( y )for the surface S α is estimated as E ( y ) | S α = A / α π / G (cid:90) S α (cid:18) (2) R − k − (cid:19) dA ≥ α α χ G (cid:18) A α π (cid:19) / − α α ) Λ G (cid:18) A α π (cid:19) / . (11)Under the assumption of the global existence of the inverse mean curvature flow, the topologyof S α should be S even in the case with negative cosmological constant, because the flowcannot change the topology of all leaves of the foliation. Then, the inequality m ≥ E ( y ) givesus Gm ≥ α α (cid:18) A α π (cid:19) / − α α ) Λ (cid:18) A α π (cid:19) / . (12)Equality in the inequality of Eq. (12) occurs if and only if equalities hold in the inequalitiesin Eqs. (3) and (9), that is, all D a ϕ , ˜ k ab and (3) R −
2Λ vanish on all of S y and r a D a k/k = α olds on S α . Hence, the metric of Σ is dl = (cid:18) − Gmr − Λ3 r (cid:19) − dr + r d Ω , (13)which corresponds to the metric of the maximal slice of an AdS-Schwarzschild spacetime(or, for Λ = 0, a Schwarzschild spacetime), and S α is located at r = constant on which r a D a k/k = α is satisfied.Since Λ is nonpositive, the right-hand side of the inequality of Eq. (12) is an increasingfunction with respect to A α , and the maximum occurs when equality holds. Hence, A α isbounded by the area of the r = constant surface satisfying r a D a k/k = α on the maximalslice of the AdS spacetime for Λ <
0, and the Schwarzschild spacetime for Λ = 0.In asymptotically flat spacetimes without the cosmological constant ( i.e.
Λ = 0), theinequality of Eq. (12) can be rewritten as A α ≤ π (cid:18) α α Gm (cid:19) . (14)For α → ∞ (the MS) and for α = 0 (the LTS), the quantity in the bracket of the aboveinequality becomes 3 + 4 α α Gm = (cid:40) Gm ( α → ∞ )3 Gm ( α = 0) . (15)Hence, the inequality of Eq. (14) includes the known results for the MS and for the LTS. One often considers asymptotically locally AdS spacetimes. Simple examples are given bythe metric ds = − f ( r ) dt + f − ( r ) dr + r dω κ , (16)with f ( r ) = κ − Gmr − Λ3 r , (17)where κ = ± dω κ denotes the metric of a two-dimensional maximally symmetricsurface with the Gauss curvature κ : dω κ = dθ + sin θdφ ( κ = +1) dθ + dφ ( κ = 0) dθ + sinh θdφ ( κ = − . (18)Note that two-dimensional section of this space can be compactified into a torus in thecase of κ = 0, and into a multi-torus in the case of κ = −
1, and such black holes are calledtopological black holes. See Ref. [11] for the procedure for making topological black holes.Based on these vacuum solutions, it is natural to consider asymptotically locally AdSspaces whose metrics asymptote to dl = f − ( r ) dr + r dω κ (19)at infinity. Here, the two-dimensional sections with the metric r dω κ are compactified so thatit has genus g = 0 for κ = +1, g = 1 for κ = 0 and g = 2 , , . . . for κ = −
1. Supposing the xistence of a global inverse mean curvature flow, the topology of S α satisfying the inequalityof Eq. (6) is the same as that of the compactified two-dimensional sections. We modify thedefinition of the Geroch energy as E ( y ) := A / ( y )(4 ω κ ) / G (cid:90) S y (cid:18) (2) R − k − (cid:19) dA, (20)where ω κ is the area of the compact two-dimensional surface with the metric dω κ . Then, E ( y ) converges to m in the limit y → ∞ , and we have the inequality Gm ≥ α α κ (cid:18) A α ω κ (cid:19) / − α α ) Λ (cid:18) A α ω κ (cid:19) / , (21)where we used κω κ = 2 πχ . Here, the area of the unit compact two-dimensional surface ω κ is given by ω κ = π ( κ = +1)arbitrary ( κ = 0)4 π ( g −
1) ( κ = − . (22)
3. Conformal Flow
There is another approach for proving the Riemannian Penrose inequality based on theconformal flow (Bray’s theorem [4]). It works for the case of a minimal surface with multiplecomponents, and thus it is more generic than the approach based on the inverse meancurvature flow. In this section, with the aid of Bray’s theorem, we prove our inequality ofEq. (14) to make it applicable to an AGPS with multiple components. Since Bray’s approachcan only be applied to asymptotically flat spaces (not asymptotically AdS spaces), our arealinequalities for AGPSs are only applicable to asymptotically flat spaces as well.In Sect. 3.1, we present our theorem explicitly. Remaining subsections are devoted to theproof. In Sect. 3.2, we show the sketch of the proof. The main idea is as follows. Our initialdata Σ with boundaries S α (each of which is supposed to be one component of an AGPS)is glued at S α to certain manifolds ¯Σ with inner boundaries that are minimal surfaces S .¯Σ is constructed so that Bray’s method works for Σ ∪ ¯Σ. There, we find that the proof iscompleted if the following two statements hold: that the minimal surface S is the outermostsurface and that the extension of the manifold is smooth. The former is proved under certainconditions in Sect. 3.3. With respect to the latter, the extended manifold is genericallynonsmooth ( C -class) on the gluing surface S α . However, we can show the existence ofa sequence of smooth manifolds, which uniformly converges to the nonsmooth extendedmanifold. Therefore, we will be able to apply Bray’s theorem by taking the limit. This issuewill be discussed in Sect. 3.4. The areal inequality that we will prove is for attractive gravity probe surfaces (AGPSs),which we have introduced in Sect. 2.2. To make our theorem clear, we explicitly give thedefinition of the AGPS here.
Definition 1.
Suppose Σ to be a smooth three-dimensional manifold with a positive definitemetric g . A smooth surface S α in Σ is an attractive gravity probe surface (AGPS) with aparameter α ( α > − / ) if the following conditions are satisfied everywhere on S α : i) The mean curvature k is positive.(ii) With the local mean curvature flow in the neighborhood of S α , r a D a k ≥ αk (23) is satisfied, where r a is the outward unit normal vector to S α and D a is the covariantderivative of the metric g . Note that the mean curvature flow always can be taken at least locally around a smoothsurface [3]. Now, we present our main theorem:
Theorem 1.
Let Σ be an asymptotically flat three-dimensional smooth manifold with non-negative Ricci scalar. The boundaries of Σ are composed of asymptotic infinity and an AGPS S α , which can have multiple components, with a parameter α . Suppose that no minimalsurface satisfying one of the following conditions exists:(i) It encloses (at least) one component of S α .(ii) It has a boundary on S α and its area is less than π (2 Gm ) .Then, the area of S α has an upper bound; A α ≤ π (cid:18) α α Gm (cid:19) , (24) where m is the ADM mass of the manifold, G is Newton’s gravitational constant. Equalityholds if and only if Σ is a time-symmetric hypersurface of a Schwarzschild spacetime and S α is a spherically symmetric surface with r a D a k = αk . The proof of the theorem is given in the following subsections.
The local inverse mean curvature flow gives a metric ds = ϕ dr + g ab dx a dx b , (25)where each r -constant surface is a leaf of the foliation in the local inverse mean curvatureflow with ϕk = 1. Here, we take r such that r = 0 on S α and r increases to the outwarddirection. Since ϕ is supposed to have the dimension of the length, the coordinate r is anondimensional quantity.On (each component of) S α , we attach a manifold ¯Σ with a metric in the range r ≤ d ¯ s = 1 − (cid:0) − r (cid:1) − (cid:0) − r + r (cid:1) exp ( r ) ϕ ( x a ) dr + exp ( r ) g ,ab ( x a ) dx a dx b , (26)where g ,ab is g ab | r =0 in the metric of Eq. (25) and ϕ = ϕ | r =0 . The r -constant surface isumbilical, that is, the extrinsic curvature is¯ k ab = 12 ¯ k ¯ g ab . (27) ö Îö Î S ë S ë S S Fig. 1
Extension of the manifold: Σ is an asymptotically flat space with a boundary S α , which is an AGPS with the parameter α (with multiple components, in general). Theshaded region is the neighborhood of S α in Σ, where the local inverse mean curvature flowof Eq. (25) is taken. On S α , the manifold ¯Σ with the metric of Eq. (26) is glued to Σ. In ¯Σ,a minimal surface S exists at r = − r + 2 log 2 ( <
0) in our current setup.Hereafter, quantities with bar indicate that they are associated with the metric of Eq. (26).The mean curvature and its derivative are¯ k = ϕ − (cid:115) − (cid:0) − r + r (cid:1) − (cid:0) − r (cid:1) exp ( − r ) , (28)¯ r a ¯ D a ¯ k = −
12 ¯ k − (cid:0) − r + r (cid:1) − (cid:0) − r + r (cid:1) . (29)Note that the lapse function in the metric of Eq. (26) has been tuned so that ¯ k | S α = k | S α issatisfied. Meanwhile, r is tuned so that¯ r a ¯ D a ¯ k | S α = α ¯ k | S α (30)holds, which is equivalent to exp (cid:16) r (cid:17) = 3 + 4 α α . (31)The manifold ¯Σ is continuously glued to Σ at r = 0 because the induced metrics on the gluingsurface S α ( r = 0) are the same. However, the metric is generally C -class there because S α may not be umbilical, i.e. the extrinsic curvature k ab may have nonvanishing traceless part.The three-dimensional Ricci scalar of ¯Σ is expressed with the two-dimensional quantitiesas (3) ¯ R = (2) ¯ R − ϕ − ¯ D ¯ ϕ − r a ¯ D a ¯ k −
32 ¯ k , (32)where ¯ ϕ is the lapse function in the metric of Eq. (26), that is,¯ ϕ = ϕ ( x a ) (cid:115) − (cid:0) − r (cid:1) − (cid:0) − r + r (cid:1) exp ( r ) . (33) ach term in the right-hand side of Eq. (32) can be explicitly calculated as (2) ¯ R = exp( − r ) (2) R , ¯ ϕ − ¯ D ¯ ϕ = exp( − r ) ϕ − D ϕ , r a ¯ D a ¯ k + 32 ¯ k = exp( − r ) (cid:18) α + 32 (cid:19) ϕ − . (34)Therefore, we have (3) ¯ R = exp( − r ) (cid:26) (2) R − ϕ − D ϕ − (cid:18) α + 32 (cid:19) ϕ − (cid:27) = exp( − r ) (3) ¯ R , (35)where (3) ¯ R is defined by (3) ¯ R := lim r (cid:37) ¯ R. (36)Since ϕ − = k | r =0 holds in the inverse mean curvature flow and r a D a k | r =0 ≥ αk | r =0 holdsfrom the definition of the AGPS, (3) ¯ R turns out to be nonnegative, (3) ¯ R ≥ (2) R − ϕ − D ϕ − r a D a k | r =0 − k (cid:12)(cid:12) r =0 − k ab (cid:12)(cid:12) r =0 = (3) R ≥ , (37)where (3) R is the Ricci scalar of Σ on S α ( r = 0), that is, (3) R := R | r =0 . Thus, the three-dimensional Ricci scalar of the glued manifold Σ ∪ ¯Σ is nonnegative everywhere. Hence, itis expected that Brays’ proof to the Riemannian Penrose inequality [4] by the conformalflow would be applicable for Σ ∪ ¯Σ. However, it requires the smoothness of the manifold asan assumption, while our manifold Σ ∪ ¯Σ is generally C -class, not C ∞ -class at the gluingsurface. This problem will be fixed later (see Sect. 3.4), and in the rest of the presentsubsection, we assume that Bray’s theorem is applicable to the manifold Σ ∪ ¯Σ.The manifold ¯Σ has a minimal surface at r = − r + 2 log 2 (the value of − r + 2 log 2 isalways negative because of Eq. (31) and α > − / S .From the metric of Eq. (26), we find the direct relation between the area of S and that of S α as A = exp( − r + 2 log 2) A α = (cid:20) α )3 + 4 α (cid:21) A α , (38)where A is the area of S . Supposing S is the outermost minimal surface, the requiredconditions for which will be discussed in Sect.3.3, the Riemannian Penrose inequality [4]gives A ≤ π (2 Gm ) . (39)Combining this with Eq. (38), we have A α ≤ π (cid:18) α α Gm (cid:19) . (40)When equality in the inequality of Eq. (40) holds, that for the Riemannian Penrose inequal-ity of Eq. (39) holds as well. Then, the manifold has the metric of the time-symmetric slice of Schwarzschild spacetime. Hence, each of the manifolds Σ and ¯Σ is isometric to a part of it,and in particular, the surface r = − r + 2 log 2 is a spherically symmetric minimal surface.Hence, because of the construction of the manifold ¯Σ whose metric is given in Eq. (26), S α should also be spherically symmetric and must satisfy r a D a k = αk . As a result, equalityin the inequality of Eq. (40) holds if and only if Σ is the time-symmetric hypersurface of aSchwarzschild spacetime and S α is the spherically symmetric surface with r a D a k = αk .As stated before, the remaining task is to show that the manifold Σ ∪ ¯Σ can be madesmooth at the gluing surface S α and S is the outermost minimal surface. We will discussthe latter in Sect. 3.3, and then, prove the former in Sect. 3.4. In the previous subsection, we used the assumption that the minimal surface S at r = − r + 2 log 2 in the extended manifold ¯Σ is outermost. Our Theorem 1 is carefully stated toguarantee this assumption, as we explain below.The minimal surface at r = − r + 2 log 2 is not outermost if an outermost minimal surfaceexists in r >
0, in r <
0, or across r = 0. We examine these three possible cases, one by one.The first possibility is prohibited by the assumption of the theorem that any part of S α isnot covered by a minimal surface.The second case is impossible, that is, we cannot take minimal surfaces in the extendedmanifold ¯Σ not touching with the boundary of ¯Σ except the minimal surface r = − r +2 log 2. To show this, let us assume that the second case is possible. Such a surface hasmaximum r at some points. However, we can show that the mean curvature k there becomespositive (that is, non-zero), and thus, it is not a minimal surface.Let us begin with a generic discussion of geometry. Suppose we have a codimension-onesurface σ in a manifold and we take the Gaussian normal coordinates in the neighborhoodof σ , dl = dy + h ij dx i dx j , (41)where the surface σ is supposed to exist at y = 0 and its unit normal vector is n µ = ∂ µ y. (42)We take another surface ˜ σ characterized with ˜ y = 0 and˜ y = y − δy ( x i ) . (43)Suppose that this surface has maximum y at x i = 0 and its value is y = 0. Hence, ˜ σ is tangentto σ at x i = 0 and Hessian matrix of δy ( x i ) is negative-semidefinite there, i.e. we have ∂ µ δy | x i =0 = 0 , h ij ∂ i ∂ j δy (cid:12)(cid:12) x i =0 ≤ . (44)The unit normal vector of ˜ σ directing to the same side as n µ is˜ n µ = α∂ µ ˜ y. (45)with α = (cid:0) h ij ( ∂ i δy )( ∂ j δy ) (cid:1) − . (46) ë S ë S ë S S S Î Îö Îö Îö Fig. 2
Examples of the third case where S is not the outermost minimal surface: Thethick solid curves show the surface S α . The outside of S α is Σ and the shaded region is ¯Σ. Theother boundary of ¯Σ inside of S α is the minimal surface S . Minimal surfaces in Σ touchingwith S α , which possibly exist, are shown by solid curves. They might be further extendedinto ¯Σ as described with dashed curves. If there exists such a smooth closed minimal surfacewith a vanishing mean curvature, S is not the outermost minimal surface.The induced metric of ˜ σ is ˜ h µν = g µν − ˜ n µ ˜ n ν , (47)and (˜ h µν , ˜ n µ ) coincides with ( h µν , n µ ) at x i = 0. The mean curvature of ˜ σ , ˜ k , is calculatedas ˜ k := g µν ∇ µ ˜ n ν = g µν ( ∂ µ α ) ( n ν − ∂ ν δy ) + αg µν (cid:0) ∇ µ n ν − ∂ µ ∂ ν δy − Γ αµν ∂ α δy (cid:1) . (48)Since we have α | x i =0 = 1 , g µν ∂ µ ∂ ν δy | x i =0 = h ij ∂ i ∂ j δy,∂ µ α | x i =0 = − α (cid:2) h ij ( ∂ i δy )( ∂ µ ∂ j δy ) + ( ∂ µ h ij )( ∂ i δy )( ∂ j δy ) (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) x i =0 = 0 , (49)the mean curvature ˜ k is related to k as˜ k | x i =0 = ( k − h ij ∂ i ∂ j δy ) | x i =0 ≥ k | x i =0 , (50)where we used Eq. (44).We apply the fact expressed in Eq. (50) to surfaces on ¯Σ; σ , ˜ σ and the point x i = 0correspond to r -constant surface, a minimal surface σ m other than r = − r + 2 log 2 whichis assumed to exist, and the point p M where the maximum of r (= r M ) occurs on the minimalsurface, respectively. Hence, the r -constant surface which we take here is indicated by r = r M .Then, from Eq. (50), at p M the mean curvature k m of σ m satisfies k m ≥ k M , (51)where k M is the mean curvature of the r = r M surface. Since r M is larger than − r + 2 log 2,the mean curvature k M is positive and then k m should be positive at p M . This is inconsistentwith the assumption that the surface σ m is minimal. Therefore, there exists no minimalsurface in ¯Σ not touching with the boundary of ¯Σ other than r = − r + 2 log 2.The last possibility may occur in the following situation. In general, there exist minimalsurfaces touching with the boundary of ¯Σ. If one of them can be extended to a compact inimal surface in Σ, we could have a minimal surface enclosing the minimal surface at r = − r + 2 log 2 (see Fig.2). If this occurs, we cannot apply Bray’s theorem to the minimalsurface at r = − r + 2 log 2.In our theorem, however, we assume that no minimal surface exists in Σ whose area isless than 4 π (2 Gm ) and which has a boundary on S α . It prevents the situation in Fig. 2 forthe following reason. If we have a minimal surface crossing S α like Fig. 2, its area has to besmaller than or equal to 4 π (2 Gm ) by Bray’s theorem. However, by assumption, the areaof its part existing in the domain Σ is equal to or larger than 4 π (2 Gm ) . Hence, the areaof the connected minimal surface becomes larger than 4 π (2 Gm ) , which contradicts Bray’stheorem. This means that such a surface cannot exist.Summarizing all above discussions together, the minimal surface at r = − r + 2 log 2 isguaranteed to be the outermost one under the assumptions of Theorem 1. To apply Bray’s proof for the Riemannian Penrose inequality, the manifold should besmooth [4]. However, the constructed manifold Σ ∪ ¯Σ in Sect. 3.2 is C -class at the glu-ing surface, generally. Here, we show that there is a sequence of smooth manifolds with anonnegative Ricci scalar (3) R which uniformly converges to Σ ∪ ¯Σ. On each smooth mani-fold, we can apply the Riemannian Penrose inequality, and taking the limit, we achieve theinequality of Eq. (39) on Σ ∪ ¯Σ.First, without details, we give a brief sketch on the smooth deformation and extensionof the manifold (see Fig. 3). We take the foliation of the local inverse mean curvature flowwith the metric of Eq. (25) in the neighborhood of S α ( δ > r > ϕ to ˆ ϕ in δ > (cid:15) > r > ϕ, g ). The second procedure is to consider the C -extension of the manifold inthe similar way to that in Sect. 3.2, but the position at which the two manifolds are gluedis slightly shifted: We choose the gluing position to be r = ¯ δ ( (cid:15) > ¯ δ > r = 0. Theextended manifold is denoted by ˚Σ with the metric (˚ ϕ, ˚ g ), and it uniformly converges to ¯Σ.As seen in Sect. 3.2, ˚Σ has the nonnegative Ricci scalar everywhere. The third procedure isto introduce a manifold ˇΣ in the domain ¯ δ > r > ¯ δ − ¯ (cid:15) >
0, where the metric ( ˇ ϕ, ˇ g ) is givenby the mixture of ( ˆ ϕ, g ) of ˆΣ and (˚ ϕ, ˚ g ) of ˚Σ so that ˇΣ is smoothly connected to ˆΣ and ˚Σ at r = ¯ δ and r = ¯ δ − ¯ (cid:15) , respectively. Then, we can show that ˇΣ with the metric ( ˇ ϕ, ˇ g ) has thepositive Ricci scalar everywhere. Therefore, the obtained manifold, which is the combinationof the original manifold in r ≥ (cid:15) , ˆΣ with the metric ( ˆ ϕ, g ) in (cid:15) > r ≥ ¯ δ , ˇΣ with the metric( ˇ ϕ, ˇ g ) in ¯ δ > r > ¯ δ − ¯ (cid:15) , and ˚Σ with the metric (˚ ϕ, ˚ g ) in ¯ δ − ¯ (cid:15) ≥ r , is smooth, has the positiveRicci scalar everywhere, and uniformly converges to Σ ∪ ¯Σ in the limit (cid:15) →
0. Below, weprovide the detailed description.Since the original manifold Σ is smooth, we can take a smooth local inverse mean curvatureflow [3], where k is also a smooth function in the neighborhood of S α . Because of the positivityof k on S α , k is positive in a sufficiently small neighborhood, 0 ≤ r < δ . Similarly, since r a D a k and k ab are smooth as well, there exists a positive constant β satisfying0 < r a D a k + k + k ab < βk (52) S ë ( '; g ) î Î S ë ( ' ê ; g ) ï Î S ë Î S ( '; g ) Î S ï ö î Î ( '; g ) Î Fig. 3
Smooth deformation and extension of the manifold. There are three procedures.The upper-left is the original manifold Σ. The shaded region is the neighborhood of S α ( δ > r >
0) where the local inverse mean curvature flow can be taken. In the first procedure,the lapse function is smoothly deformed in δ > (cid:15) > r >
0, which is described in the upper-right, to make the Ricci scalar (3) R strictly positive. The obtained manifold is describedby ˆΣ with the metric ( ˆ ϕ, g ). In the second procedure, a manifold ˚Σ with a metric (˚ ϕ, ˚ g ) isattached at r = ¯ δ ( (cid:15) > ¯ δ >
0) to obtain the lower-left manifold. Since the method of gluingthe two manifolds is the same as that in Sect. 3.2 except that the gluing position is slightlyshifted, the glued manifold is C -class at r = ¯ δ . In the third procedure, we smooth the gluedmanifold by introducing a manifold ˇΣ with the metric ( ˇ ϕ, ˇ g ) in the domain ¯ δ > r > ¯ δ − ¯ (cid:15) > ≤ r < δ , where we used our assumption, r a D a k ≥ αk > − (1 / k on S α .On S α , the three-dimensional Ricci scalar (3) R is nonnegative by assumption. The firstprocedure is to deform the metric so that (3) R becomes strictly positive on and near S α .If it is already satisfied, this first procedure is unnecessary and we can skip to the secondprocedure. Otherwise, we introduce another metric in the region 0 ≤ r < (cid:15) < δ as d ˆ s = u ( r ) ϕ dr + g ab dx a dx b =: ˆ ϕ dr + g ab dx a dx b , (53) u ( r ) := 1 − exp (cid:18) − Cδ(cid:15) − r (cid:19) , (54)where C is a constant. At r = (cid:15) , this metric is smoothly connected with the original metric ofEq. (25). The connected smooth manifold is denoted by ˆΣ. This metric uniformly converges o the metric of Eq. (25) in the limit (cid:15) →
0. The geometrical quantities with the metric ofEq. (53) on a r -constant surface are written as (2) ˆ R = (2) R, ˆ ϕ − ˆ D ˆ ϕ = ϕ − D ϕ, ˆ k ab = u − k ab , ˆ k = u − k, ˆ r a ˆ D a ˆ k = − u − k ( ∂ r log u ) + u − r a D a k, (55)where we used ϕk = 1 in the last equation. Here, the variables with and without hat, such as ˆ k and k , indicate those of the metrics of Eqs. (53) and (25), respectively. The three-dimensionalRicci scalar of the metric of Eq. (53) can be expressed as (3) ˆ R = (2) ˆ R − ϕ − ˆ D ˆ ϕ − r a ˆ D a ˆ k − ˆ k − ˆ k ab = (3) R + (cid:0) − u − (cid:1) (cid:0) r a D a k + k + k ab (cid:1) + 2 u − k ∂ r log u. (56)The functions involving u in the above equation are estimated as0 > − u − = u − (cid:20) − (cid:18) − Cδ(cid:15) − r (cid:19) + exp (cid:18) − Cδ(cid:15) − r (cid:19)(cid:21) > − u − exp (cid:18) − Cδ(cid:15) − r (cid:19) , (57) ∂ r log u = u − Cδ ( (cid:15) − r ) exp (cid:18) − Cδ(cid:15) − r (cid:19) > Cδ exp (cid:18) − Cδ(cid:15) − r (cid:19) . (58)These relations, together with Eq. (52), imply (3) ˆ R > (3) R + 2 k u (cid:18) Cδ − β (cid:19) exp (cid:18) − Cδ(cid:15) − r (cid:19) . (59)Therefore, if we take C > βδ > (3) ˆ R is strictly positive in the region 0 ≤ r < (cid:15) . On S α ,(ˆ r a ˆ D a ˆ k ) / ˆ k is estimated asˆ r a ˆ D a ˆ k ˆ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S α = ( r a D a k ) k (cid:12)(cid:12)(cid:12)(cid:12) S α − ∂ r log u | S α = ( r a D a k ) k (cid:12)(cid:12)(cid:12)(cid:12) S α − Cδ(cid:15) exp (cid:18) − Cδ(cid:15) (cid:19) − exp (cid:18) − Cδ(cid:15) (cid:19) ≥ α + C (cid:15) =: ¯ α, (60)where C is a (negative) constant.We now describe the second procedure. We introduce a manifold ˚Σ with the followingmetric: d ˚ s = 1 − (cid:0) − r ¯ δ (cid:1) − (cid:16) − r − ¯ δ + r ¯ δ (cid:17) exp (cid:0) r − ¯ δ (cid:1) ˆ ϕ δ ( x a ) dr + exp (cid:0) r − ¯ δ (cid:1) g ¯ δ,ab ( x a ) dx a dx b =: ˚ ϕ dr + ˚ g ab dx a dx b , (61)where ˆ ϕ ¯ δ = ˆ ϕ | r =¯ δ and g ¯ δ,ab ( x a ) = g ab | r =¯ δ . Here, this metric has one parameter ¯ δ , which isrequired to satisfy 0 < ¯ δ < (cid:15) , and r ¯ δ is determined later. This manifold ˚Σ uniformly converges o ¯Σ in the limit (cid:15) → δ → r = ¯ δ denoted by S ¯ δ . Since ˆ r a ˆ D a ˆ k and ˆ k are smooth regular functions in 0 < r ≤ ¯ δ ,the relation between the values of ˆ r a ˆ D a ˆ k/ ˆ k on S ¯ δ and on S α is estimated asˆ r a ˆ D a ˆ k ˆ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S ¯ δ ≥ ˆ r a ˆ D a ˆ k ˆ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S α + ¯ C (cid:15) ≥ ¯ α + ¯ C (cid:15) = α + C (cid:15) =: ˚ α, (62)where ¯ C and C (= C + ¯ C ) are constants, and ¯ α is determined in Eq. (60). Note that ˚ α > − / (cid:15) , because C is a constant independentof (cid:15) . Similarly to the case of the metric of Eq. (26) in Sect. 3.2, ˚ ϕ and r ¯ δ are taken so that˚ k = ˆ k and (˚ r a ˚ D a ˚ k ) = ˚ α ˚ k are satisfied on S ¯ δ . Here, the geometric quantities with circles(such as ˚ k ) are with respect to the metric of Eq. (61). At this point, the obtained manifold,˚Σ glued to Σ on S ¯ δ , is still C -class, generally.The third procedure is as follows. Let us deform the glued manifold in the domain ¯ δ − ¯ (cid:15) 4. Summary In this paper, we have introduced the new concept, the attractive gravity probe surface(AGPS), to characterize the strength of the gravitational field. An AGPS is a two-dimensional closed surface that satisfies k > r a D a k/k ≥ α for the local inverse meancurvature flow, where α is a constant satisfying α > − / 2. This concept can be applied tosurfaces in regions with various strength of the gravitational field, including weak-gravityregions: The parameter α controls the strength of the gravitational field. For instance, for r -constant surfaces in a t -constant slice of a Schwarzschild spacetime, r a D a k/k is a mono-tonically decreasing continuous function of r . It converges to ∞ in the limit where the surfaceapproaches the horizon, while it converges to − / α is interpretedas an indicator for the minimum strength of the gravitational field on the surface, becauseit gives the lower bound of r a D a k/k on that surface.Then, we have generalized the Riemannian Penrose inequality to the areal inequality forAGPSs; for asymptotically flat or AdS spaces, the area of an AGPS surface has an upperbound, A α ≤ π [(3 + 4 α ) / (1 + 2 α )] ( Gm ) (the exact statement is presented in Sect. 2.2and as Theorem 1 in Sect. 3.1). In the limit α → ∞ and in the case α = 0, our inequality isreduced to the Riemannian Penrose inequality and its analog for the LTS, respectively. Theproof for asymptotically flat spaces is done by applying Bray’s theorem derived using theconformal flow. This means that our inequality is applicable to the case of an AGPS withmultiple components. For asymptotically AdS spaces, our argument relied on the Gerochmonotonicity in the inverse mean curvature flow. The proof for the areal inequality forAGPSs based on the conformal flow, which requires the generalization of Bray’s method toAdS spaces, will be an important remaining problem.It would be interesting to extend our inequality to that in Einstein-Maxwell systems, as inthe case of the Riemannian Penrose inequality [12–15] and its analog for the LTS [16].Moreover, the extension to the higher-dimensional cases is expected to be possible forasymptotically flat spaces. We leave these topics for future works.It is also important to consider the relation to behavior of photons and the possible connec-tion to observations. For LTSs, the relations to dynamically transversely trapping surfaces Fig. A1 The behavior of the functions f , ( rf ) (cid:48) , and ( r f ) (cid:48)(cid:48) for the matching functiongiven by Eq. (A1).(DTTSs), which is a generalization of the concept of the photon surface for generic space-times based on the photon orbits, have been discussed [16–19]. It would be interesting toexplore the relation between the AGPSs and the photon behavior. Acknowledgement K. I. and T. S. are supported by JSPS Grants-in-Aid for Scientific Research(A)(No. 17H01091). K. I. is also supported by JSPS Grants-in-Aid for Scientific Research(B) (20H01902). H. Y. is supported by the Grant-in-Aid for Scientific Research (C) (No.JP18K03654) from Japan Society for the Promotion of Science (JSPS). The work of H.Y. ispartly supported by Osaka City University Advanced Mathematical Institute (MEXT JointUsage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). A. An example of the matching function f An example of the matching function is f ( r ) = (1 + exp x ( r )) − , x ( r ) := ¯ (cid:15)r + ¯ (cid:15)r + ¯ (cid:15) + 1 , (A1)defined on − ¯ (cid:15) < r < 0. The behavior of f , ( rf ) (cid:48) , [( rf ) ] (cid:48)(cid:48) is shown in Fig. A1, where theprime means the derivative with respect to r (not to x ), that is, f (cid:48) = ddr f (cid:0) x ( r ) (cid:1) . From thisfigure, one can expect that the requirements of Eq. (66) would be satisfied. Here, we providea solid proof for this fact. A.1. Evaluation of the first three equations of conditions of Eq. (66) It is trivial that the first two equations of Eq. (66) are satisfied because oflim r (cid:37) e x = 0 , lim r (cid:38)− ¯ (cid:15) e x = ∞ . (A2)Since the function f converges to 1 and 0 very rapidly in the limit of r (cid:37) r (cid:38) − ¯ (cid:15) ,respectively, the third equation of Eq. (66) is expected to be satisfied. Let us show it exactly. he n th-order derivatives of f ( x ( r )) with respect to r are written by the finite sum ofterms, each of which has the form c l d m fdx m m (cid:89) k =1 d a k xdr a k , (A3)where m is a positive integer, c l is a constant and a k ’s are positive integers satisfying n = m (cid:88) k =1 a k . (A4)Different a k ’s may take the same integer. The third equation of Eq. (66) holds if each termof Eq. (A3) converges to zero in the limits r (cid:37) r (cid:38) − ¯ (cid:15) .The a k th-order derivatives of x ( r ) become d a k xdr a k = ( − a k a k ! (cid:18) r a k +1 + 1( r + ¯ (cid:15) ) a k +1 (cid:19) ¯ (cid:15). (A5)Around r = 0, the absolute value of it is bounded from above as (cid:12)(cid:12)(cid:12)(cid:12) d a k xdr a k (cid:12)(cid:12)(cid:12)(cid:12) < c a k (cid:12)(cid:12)(cid:12)(cid:12) r a k +1 (cid:12)(cid:12)(cid:12)(cid:12) , (A6)where c a k is a positive constant. Thus, the absolute value of the product of the derivativesof x in Eq. (A3) is bounded from above as m (cid:89) k =1 (cid:12)(cid:12)(cid:12)(cid:12) d a k xdr a k (cid:12)(cid:12)(cid:12)(cid:12) < c a (cid:12)(cid:12)(cid:12)(cid:12) r a + m (cid:12)(cid:12)(cid:12)(cid:12) , where c a := m (cid:89) k =1 c k , a := m (cid:88) k =1 a k . (A7)We now estimate the m th-order derivative of f ( x ( r )) with respect to x . The first-orderderivative is calculated as dfdx = f ( f − . (A8)Defining a functional space F as F := { f ( f − p ( f ) | p ( f ) is a polynomial of f } , (A9)we show d m fdx m ∈ F (A10)for m ≥ m = 1, it is true because of Eq. (A8). SupposeEq. (A10) is satisfied for m = q . The assumption indicates that d q f /dx q is written by f ( f − p q ( f ) with a polynomial p q ( f ), and its derivative becomes d q +1 fdx q +1 = f ( f − (cid:20) (2 f − p q ( f ) + dp q ( f ) df (cid:21) ∈ F . (A11)Therefore, Eq. (A10) is proved for arbitrary m ≥ m th-order derivatives is written as d m fdx m = f ( f − p m ( f ) , (A12)with a polynomial p m ( f ). The function f converges to unity in the limit r (cid:37) 0. Therefore, | f p m ( f ) | is bounded from above by a positive constant ˜ c m around r = 0. On the other hand, f − | is estimated around r = 0 as | f − | = exp x x < ¯ c m exp x = ¯ c m exp (cid:18) ¯ (cid:15)r + ¯ (cid:15)r + ¯ (cid:15) + 1 (cid:19) < ¯ c m exp (cid:16) ¯ (cid:15)r (cid:17) , (A13)with a positive constant ¯ c m . Therefore, we have (cid:12)(cid:12)(cid:12)(cid:12) d m fdx m (cid:12)(cid:12)(cid:12)(cid:12) < ˜ c m ¯ c m exp (cid:16) ¯ (cid:15)r (cid:17) =: c m exp (cid:16) ¯ (cid:15)r (cid:17) , (A14)around r = 0. As a result, Eq. (A3) is estimated around r = 0 as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c l d m fdx m l (cid:89) k =1 d a k xdr a k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < | c l | c a c m (cid:12)(cid:12)(cid:12)(cid:12) r a + l (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:16) ¯ (cid:15)r (cid:17) −−−→ r (cid:37) . (A15)This leads to lim r (cid:37) f ( n ) ( r ) = 0 . (A16)In a similar way, we can show the vanishing of f ( n ) ( r ) for the limit r (cid:38) − ¯ (cid:15) . A.2. Estimate of ( rf ) (cid:48) We now show that the condition | ( rf ) (cid:48) | < rf ) (cid:48) as follows:( rf ) (cid:48) = 11 + e x (cid:20) r ¯ (cid:15)e x e x (cid:18) r + 1( r + ¯ (cid:15) ) (cid:19)(cid:21) < 11 + e x < . (A17)Note that f ( r ) is defined on − ¯ (cid:15) < r < i.e. , r is negative.Next, we estimate the lower bound of ( rf ) (cid:48) in the cases of − ¯ (cid:15)/ ≤ r < − ¯ (cid:15) < r ≤− ¯ (cid:15)/ 2, separately. It is useful to express ( rf ) (cid:48) as follows:( rf ) (cid:48) = 11 + e x (cid:40) e x e x (cid:34) x − (cid:18) ¯ (cid:15) ¯ (cid:15) + r (cid:19) − (cid:35)(cid:41) (A18)= 11 + e x (cid:26) e x e x (cid:20) x − (cid:16) x − ¯ (cid:15)r − (cid:17) − (cid:21)(cid:27) . (A19) A.2.1. The case − ¯ (cid:15)/ ≤ r < . In this case, we have the inequalities x ≤ , e x ≤ e, < ¯ (cid:15) ¯ (cid:15) + r ≤ . (A20)From Eq. (A18), ( rf ) (cid:48) is bounded as( rf ) (cid:48) > 11 + e x (cid:20) e x e x ( x − (cid:21) . (A21)The derivative of the right-hand side of the above is ddx (cid:18) 11 + e x (cid:20) e x e x ( x − (cid:21)(cid:19) = e x (1 − e x )(1 + e x ) ( x − . (A22)This is negative for x < < x ≤ 1. Thus, the minimum of the right-handside of Eq. (A21) occurs at x = 0, and hence, we have( rf ) (cid:48) ≥ 11 + 1 (cid:20) − (cid:21) = − > − . (A23) .2.2. The case − ¯ (cid:15) < r ≤ − ¯ (cid:15)/ . In this case, we have x ≥ , e x ≥ e, < − ¯ (cid:15)r ≤ . (A24)From Eq. (A19), the lower bound is evaluated as( rf ) (cid:48) ≥ 11 + e x (cid:26) e x e x (cid:104) x − ( x + 1) − (cid:105)(cid:27) = 1(1 + e x ) (cid:2) − e x (cid:0) x + x + 1 (cid:1)(cid:3) > − e − x ) e − x (cid:0) x + x + 1 (cid:1) . (A25)Since ddx (cid:2) e − x (cid:0) x + x + 1 (cid:1)(cid:3) = − e x x ( x − < e − x (cid:0) x + x + 1 (cid:1) is a decreasing function for x > 1. Meanwhile, 1 / (1 + e − x ) is anincreasing function. Then, ( rf ) (cid:48) is found to be bounded from below as( rf ) (cid:48) > − (cid:20) e − x ) (cid:21) (cid:12)(cid:12)(cid:12) x =2 (cid:2) e − x (cid:0) x + x + 1 (cid:1)(cid:3) | x =1 > − < x < 2, and( rf ) (cid:48) > − (cid:20) e − x ) (cid:21) (cid:12)(cid:12)(cid:12) x →∞ (cid:2) e − x (cid:0) x + x + 1 (cid:1)(cid:3) | x =2 > − < x . A.3. Estimate of (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) The second-order derivative of ( rf ) can be written as (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) = 2(1 + e x ) (cid:40) (1 + e x ) + 2 e x (1 + e x ) (cid:34) x − − (cid:18) ¯ (cid:15) ¯ (cid:15) + r (cid:19) (cid:35) + e x (2 e x − (cid:34) x − − (cid:18) ¯ (cid:15) ¯ (cid:15) + r (cid:19) (cid:35) . (A29)We will show that (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) < e x ≤ / i.e. x ≤ − log 2),1 / < e x ≤ e ( i.e. − / < − log 2 < x ≤ e < e x ( i.e. < x ), separately. A.3.1. The case e x ≤ / (i.e. x ≤ − log 2 ). Since x and (2 e x − 1) are negative, Eq. (A29)gives (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) ≤ e x ) (1 + e x ) < . (A30) .3.2. The case / < e x ≤ e (i.e. − / < − log 2 < x ≤ ). We can express ¯ (cid:15)/r interms of ¯ (cid:15)/ ( r + ¯ (cid:15) ) as ¯ (cid:15)r = ¯ (cid:15)r + ¯ (cid:15) (cid:18) − ¯ (cid:15)r + ¯ (cid:15) (cid:19) − , (A31)and then, we can see a relation between ¯ (cid:15)/ ( r + ¯ (cid:15) ) and x , (cid:18) ¯ (cid:15)r + ¯ (cid:15) (cid:19) = ( x + 1) ¯ (cid:15)r + ¯ (cid:15) − ( x − . (A32)The terms including ¯ (cid:15)/ ( r + ¯ (cid:15) ) in the right-hand side of Eq. (A29) are rewritten as − (cid:18) ¯ (cid:15)r + ¯ (cid:15) (cid:19) = − (cid:0) x + x + 2 (cid:1) ¯ (cid:15)r + ¯ (cid:15) + x − , (A33) (cid:34)(cid:18) ¯ (cid:15)r + ¯ (cid:15) (cid:19) − ( x − (cid:35) = (cid:0) x − x + 3 x + 5 (cid:1) ¯ (cid:15)r + ¯ (cid:15) − x + 3 x − x + 5 . (A34)Solving Eq. (A32) for ¯ (cid:15)/ ( r + ¯ (cid:15) ), it is estimated as75 < ¯ (cid:15)r + ¯ (cid:15) = ( x + 1) + (cid:112) ( x − + 42 ≤ , (A35)where we used 1 < ¯ (cid:15)/ ( r + ¯ (cid:15) ) to choose the sign in the front of the square root and − / 1. As a consequence, we find (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) = 2(1 + e x ) (cid:26) (cid:0) x − x + 9 x − (cid:1) e x + (cid:0) − x + 8 x − x + 7 (cid:1) e x + (cid:2) − (cid:0) x + x + 5 x + 9 (cid:1) + (cid:0) x − x + 4 x + 6 (cid:1) e x (cid:3) e x ¯ (cid:15)r + ¯ (cid:15) (cid:27) = 2(1 + e x ) (cid:26) (cid:0) x − x + 9 x − (cid:1) e x + (cid:0) − x + 8 x − x + 7 (cid:1) e x + (cid:2) − (cid:0) x + 3 x + 6 (cid:1) + (cid:0) x − x + 2 x + 3 (cid:1) (2 e x − (cid:3) e x ¯ (cid:15)r + ¯ (cid:15) (cid:27) < e x ) (cid:26) (cid:0) x − x + 9 x − (cid:1) e x + (cid:0) − x + 8 x − x + 7 (cid:1) e x + (cid:20) − (cid:0) x + 3 x + 6 (cid:1) + 2 (cid:0) x − x + 2 x + 3 (cid:1) (2 e x − (cid:21) e x (cid:27) = 2(1 + e x ) (cid:20) (cid:18) − x − x + 45 x − (cid:19) e x + (cid:0) x − x + 19 (cid:1) e x (cid:21) . (A36)For − / < x ≤ 0, we have e x ≤ (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) is bounded from above as (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) ≤ e x ) (cid:20) (cid:18) − x − x + 45 x − (cid:19) e x + (cid:0) x − x + 19 (cid:1) e x (cid:21) = 2(1 + e x ) (cid:20) (cid:18) x − x − x − (cid:19) e x (cid:21) < e x ) < . (A37) or 0 < x ≤ 1, we have e x ≤ e < 3, and it gives (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) < e x ) (cid:20) (cid:18) − x − x + 45 x − (cid:19) e x + (cid:0) x − x + 19 (cid:1) e x (cid:21) = 2(1 + e x ) + 2 e x e x ) (cid:2) x − x − x + 178 (cid:3) . (A38)The part in the square bracket of the second term in the second line is positive in theinterval 0 < x ≤ x = (6 + 2 √ / 75 ( < 1) anda maximal value at x = (6 − √ / 75 ( < < x ≤ e x e x ) < , (A39)where we used the fact that e x / (1 + e x ) monotonically decreases for x > (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) is evaluated as (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) < e x ) + 178120 < 18 + 178120 < A.3.3. The case e < e x (i.e. < x ). We estimate the first and second lines of Eq. (A36).The terms in the square bracket in the second line is shown to be positive as − (cid:0) x + x + 5 x + 9 (cid:1) + (cid:0) x − x + 4 x + 6 (cid:1) e x = − (cid:0) x + x + 5 x + 9 (cid:1) + 2 (cid:2) x ( x − + x + 3 (cid:3) e x > − (cid:0) x + x + 5 x + 9 (cid:1) + 5 (cid:2) x ( x − + x + 3 (cid:3) = 4 x − x + 5 x + 6 > , (A41)where we used 5 / < e < e x and x > x > x > (cid:15)/ ( r + ¯ (cid:15) ) < x + 1 [see Eq. (A35)], the value of (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) is bounded as (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) < e x ) (cid:8) (cid:0) x − x + 9 x − (cid:1) e x + (cid:0) − x + 8 x − x + 7 (cid:1) e x + (cid:2) − (cid:0) x + x + 5 x + 9 (cid:1) + (cid:0) x − x + 4 x + 6 (cid:1) e x (cid:3) e x (1 + x ) (cid:9) = 2(1 + e x ) (cid:8) − (cid:0) x + x + 7 x + 5 x + 16 (cid:1) e x + (cid:0) x − x + 8 x − x + 13 (cid:1) e x (cid:9) < e x ) + 2 e x (1 + e x ) (cid:0) x − x + 8 x − x + 13 (cid:1) . (A42) ere, the first term has the maximum value 2(1 + e ) − < / 50 at x = 1. Derivative of thesecond term is shown to be negative as ddx (cid:20) e x (1 + e x ) (cid:0) x − x + 8 x − x + 13 (cid:1)(cid:21) = 4 e x (1 + e x ) (cid:2)(cid:0) x + 2 x + 6 x + 12 (cid:1) − e x (cid:0) x − x + 14 x − x + 14 (cid:1)(cid:3) < e x (1 + e x ) (cid:20)(cid:0) x + 2 x + 6 x + 12 (cid:1) − (cid:18) x + 12 x (cid:19) (cid:0) x − x + 14 x − x + 14 (cid:1)(cid:21) = − e x (1 + e x ) (cid:104)(cid:0) x − x − x − (cid:1) + x ( x − x + 4) + 3 x + 1 (cid:105) < , (A43)where we used2 x − x + 14 x − x + 14 = 2( x − x + 5( x − + x + 9 > e x > x + x / 2. Thus, the second term of the right-hand side of Eq. (A42) is adecreasing function. The maximum of the right-hand side of the inequality of Eq. (A42)occurs at x = 1 and its value satisfies 34 e (1 + e ) < . (A45)Therefore, the value of (cid:2) ( rf ) (cid:3) (cid:48)(cid:48) is less than 2. References [1] R. Penrose, Annals N. Y. Acad. 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