A class of explicit solutions disproving the spacetime positive energy conjecture in all dimensions
aa r X i v : . [ m a t h - ph ] F e b A class of explicit solutions disproving the spacetime positiveenergy conjecture in all dimensions
The Cang Nguyen ∗ March 2, 2021
Abstract
In this article, we construct a class of explicit, smooth and spherically symmetricsolutions to the asymptotically flat vacuum constraint equations which have ADM massof arbitrary sign ( −∞ , negative, zero, positive). As a direct consequence, there existasymptotically flat vacuum initial data sets whose metrics are exactly negative massSchwarzschild outside a given ball. We emphasize that our result does not contradictthe spacetime positive energy theorem proven by Eichmair [12], instead it shows that thedecay rate at infinity of the symmetric (0 , k stated in the theorem is sharp. Thekey argument we use in the article is classical, based on the conformal method, in whichthe conformal equations are equivalently transformed into a single nonlinear equation offunctions of one variable. An asymptotically flat (AF) initial data set for the Cauchy problem in general relativity isan AF manifold (
M, g ) of n dimensions, with n ≥
3, coupled with a symmetric (0 , − tensor k such that ( M, g, k ) satisfies the system R g − | k | g + (tr g k ) = µ [Hamiltonian constraint ] (1.1a)div g (cid:0) k − (tr g k ) g (cid:1) = J, [ Momentum constraint ] (1.1b)where R g is the scalar curvature of g and where µ is a non-negative scalar field and J isa 1 − form on M, representing the energy and momentum densities of the matter and non-gravitational fields, respectively. These equations are called the Einstein constraint equations and the study of solutions to (1.1) has been a topical issue for many decades.One of major achievements in the constraint equations is the spacetime positive energy the-orem (PET) proven by Schoen–Yau [20, 21], Witten [22] and later Chrusciel–Maerten [8],Eichmair [12], which roughly states that every AF initial data (
M, g, k ) with decay rates atinfinity | g ij − δ ij | + | x || ∂g ij | + | x || k ij | = O (cid:0) | x | − n − − ǫ (cid:1) (1.2)and satisfying the dominant energy µ ≥ | J | has positive ADM mass, unless ( M, g, k ) is Cauchyinitial data for Minkowski space, i.e. (
M, g ) can be isometrically embedded in Minkowski ∗ Institut Montpelli´erain Alexander Grothendieck, Universit´e de Montpellier. E-mail: [email protected]. k . This theorem is very important in generalrelativity since it practically denies existence of negative mass in reality. However, this featureis also a challenge in physics because the repulsion of negative mass has a key role in theaccelerated expansion of the universe as well as the dark energy, but finding such a modelis clearly difficult due to the theorem. From the modern physical point of view, a naturalquestion to ask is whether there is an AF initial data set that has negative mass withoutviolation of the dominant energy condition. In this article, we will give an answer to thisquestion by considering the simple setting where M ≡ R n and µ ≡ | J | ≡
0, that is thevacuum case on R n . Our first main result is the following theorem that affirms existence ofsmooth solutions with negative mass Schwarzschild metric outside a ball. In what follows, westate only typical results and we refer to Section 4 for our precise statements. Main Theorem 1.
Given a constant m > , let ϕ be a radial and increasing function in R n satisfying ϕ ( x ) = 1 − m | x | n − , for all | x | ≥ m / ( n − . Then there exists a symmetric (0 , − tensor k with | k | = O ( | x | − n ) at infinity such that ( R n , ϕ n − δ Euc , k ) is a solution to the vacuum constraint equations (1.1) . Clearly, the ADM mass of solutions in Main Theorem 1 is − m . In regard to (1.2), weemphasize that this result does not contradict the spacetime PET because the decay rate of k in our result is lightly reduced to be critical, i.e. | k ij | = O ( | x | − n ) at infinity. This reductionmakes k not be square integrable any longer, and so, as mentioned in the letter of Chrusciel[7], that’s why the arguments of Witten or Eichmair cannot work. To see more clearly howthe decay rate of k impacts on the sign of ADM mass, we prove the following result. Main Theorem 2.
Given a constant c > and a decay exponent q ∈ ( n +24 , n ) , let τ be aradial function in R n satisfying | τ | ∼ c | x | − q at infinity. Then there exists an AF vacuuminitial data ( R n , g, k ) such that tr g k = τ and at infinity | k ij | = O ( | x | − q ) , | g ij − δ ij | + | x || ∂ k g ij | = O (cid:0) | x | − q +2 (cid:1) . Moreover,- if q < n , the ADM mass is −∞ ,- if q = n , the ADM mass is negative,- if q > n , the ADM mass is zero, and hence ( R n , g, k ) is an entire spacelike hypersurfaceof mean curvature τ in Minkowski space-time provided that the rigidity part of thespacetime PET holds. It is worth noting that all solutions stated in our results are explicit, spherically symmetricin some sense and quite simple (see. Theorem 4.1). It provides a variety of models ingeneral relativity, and so helps us understand better structure and behavior of the initialdata, especially in numerical general relativity.The outline of this article is as follows. In Section 2, we present the conformal method These are nothing but 2 q − > n − , and hence the mass of g is a geometric invariant (see. Bartnik [2],Chrusciel [6]). In this section, we revisit the conformal method, a traditional way to generate Einsteinsolutions from scratch. For our purpose, we only focus on the Euclidean space ( R n , δ Euc ) with n ≥
3. For a treatment of the general case, the interested reader is referred to [11] and [18].The given data set on ( R n , δ Euc ) consists of a function τ and a TT-tensor σ (i.e. a symmetric,trace-free, divergence-free (0 , − tensor), and one is required to find a positive function ϕ tending to 1 at infinity and a 1 − form W satisfying − n − n − ϕ + n − n τ ϕ N − = | σ + LW | ϕ − N − [ Lichnerowicz equation ] (2.1a)div( LW ) = n − n ϕ N dτ, [ vector equations ] (2.1b)where N = nn − and L is the conformal Killing operator defined by( LW ) ij = ∇ i W j + ∇ j W i − δ ij n (div W ) . (2.2)These equations are called the vacuum Einstein conformal constraint equations , or simply the conformal equations . Once such a solution ( ϕ, W ) exists, it follows that g = ϕ N − δ Euc k = τn ϕ N − δ Euc + ϕ − ( σ + LW ) , (2.3)is a solution to the AF vacuum constraint equations. In this situation, we keep in mind allalong the article that tr g k = τ . We next review some standard facts on elliptic operators used for studying the conformalequations. For the proofs, we refer the reader to [10, 11].Given an integer l ≥
0, a H¨older exponent α ∈ [0 , β >
0, we will usethe weighted H¨older spaces C l,α − β to capture asymptotic of functions and tensors near infinity.For α = 0, we will write C l − β instead of C l, − β . The weighted norm convention we are using isthat the C s,α − β norm is given by k f k l,α, − β := X | s |≤ l sup R n (cid:0) ρ | s | + β | ∂ s f | (cid:1) + X | s | = l sup R n ρ l + β + α sup < | y − x |≤ ρ (cid:16) | ∂ s f ( y ) − ∂ s f ( x ) || y − x | α (cid:17)! ρ is a positive function which equals | x | outside the unit ball and s is amulti-index. It will be clear from the context if the notation refers to a space of functions on R n , or a space of sections of some bundle over R n . Proposition 2.1 (Compact embedding for weighted H¨older spaces) . If l + α > l + α and β > β then the inclusion C l ,α − β ⊂ C l ,α − β is compact. Proposition 2.2 (Weighted elliptic regularity for Laplacian) . Let V ≥ be a function in C l − ,α − − ǫ with l ≥ , α ∈ (0 , and ǫ > .(a) ∆ − V : C l,α − β → C l − ,α − β − is an isomorphism if and only if < β < n − .(b) If u ∈ C β and ∆ u − V u ∈ C l,α − β − , then k u k l,α, − β ≤ c ( k u k C − β + k ∆ u − V u k l − ,α, − β − ) for some constant c > independent of u . Similarly, we also have the following proposition for the operator div L appearing in thevector equations (2.1b), where L is the conformal Killing operator defined in (2.2). Proposition 2.3 (Weighted elliptic regularity for vector Laplacian) . div L : C l,α − β → C l − ,α − β − isan isomorphism if and only if < β < n − . Finally, we give the theorem of existence and uniqueness of solutions to Lichnerowicz’sequation on the Euclidean space, which is one of two main parts of the conformal equations: − n − n − u + n − n τ u N − = w u − N − . (2.4) Theorem 2.4 (Existence and uniqueness of solution to the Lichnerowicz equation) . If τ and w are in C l − ,α − − β/ with l ≥ , α ∈ (0 , and β ∈ (0 , n − , then the Lichnerowicz equation (2.4) admits a unique positive solution u satisfying u − ∈ C l,α − β . We now seek a class of solutions to the vacuum constraint by solving the conformal equationsin a simple setting where σ ≡ τ is a radial function. In this case, the conformalequations (2.1) are rewritten as − n − n − ϕ + n − n τ ϕ N − = | LW | ϕ − N − (3.1a)∆ W i + n − n ∂ i (cid:16) n X j =1 ∂ j W j (cid:17) = n − n ϕ N τ ′ x i r , (3.1b)Here and subsequently, r is the usual Euclidean distance, we denote by f ′ the derivative of f with respect to r and we call a radial function increasing if it is increasing with respectto r . Clearly, these equations consist of ( τ, ϕ, W ) and when we know two of them, we canfind the third. Hence, for simplicity of expression, we may say such as ( ϕ, W ) or ( ϕ, τ ) or ϕ τ is fixed is a solution to (3.1). The idea behind our use of this data is that in view ofSchwarzschild metrics and spherically symmetric solutions obtained by the gluing method [9],we want to look for ϕ in the space of radial functions, which, by the Lichnerowicz equation,will be guaranteed as long as τ and | σ + LW | are radial. Therefore, a simple way to thinkof this is restricting ourselves to the data of null σ and radial τ and expecting that | LW | isradial. The following result fulfills our desire and plays a central role in the article. Theorem 3.1.
Assume that ϕ > and ( ϕ − , τ ) ∈ C ,α − β +2 × C ,α − β with α ∈ (0 , and β > . Assume furthermore that ϕ and τ are radial functions. Then ( ϕ, τ ) is a solution tothe conformal equations (3.1) if and only if ϕ is increasing and | τ ( r ) | = p nN ϕ − N +1 ϕ ′′ if ϕ ′ ( r ) = 0 , (cid:16) r n − (cid:0) ϕ ( N +2) / (cid:1) ′ (cid:17) ′ ( N +2) r n/ ϕ N − √ ϕ ′ ( r n − ϕ ) ′ otherwise. (3.2) Moreover, when ( ϕ, τ ) is the solution, the − form W is computed by W i = − x i r n Z r s n − (cid:16) Z + ∞ s ϕ N ( s ) τ ′ ( s ) ds (cid:17) ds. Remark 3.2.
Since ϕ ′ ≥ , by (3.2) we have τ ( r ) = 0 as long as ϕ ′ ( r ) = 0 and r > .Therefore, when τ does not change sign, ϕ ′ > in R n \ { } . Remark 3.3.
It follows from Theorem 3.1 that for any constant c > , if ( τ ( r ) , ϕ ( r )) is aradial solution to the conformal equations then so is ( cτ ( cr ) , ϕ ( cr )) .Proof of Theorem 3.1. We will divide the proof into three steps.
Step 1.
Solving the vector equations.
Consider the vector equations∆ W i + n − n ∂ i (cid:16) n X j =1 ∂ j W j (cid:17) = n − n ϕ N τ ′ x i r . Letting f ( r ) = − Z + ∞ r ϕ N τ ′ ds, the system becomes ∆ W i + n − n ∂ i (cid:16) n X j =1 ∂ j W j (cid:17) = n − n ∂ i f. (3.3)Differentiating (3.3) with respect to i and summing all equations of the new system, we obtain∆ (cid:16) n X j =1 ∂ j W j (cid:17) = 12 ∆ f. Therefore, by Proposition 2.2(a) we have n X j =1 ∂ j W j = f W i = 12 ∂ i f. Since f is radial, it follows by computations that W i = x i r n Z r s n − f ds. Thus, we have by definition( LW ) ij = − (cid:16) δ ij n − x i x j r (cid:17)(cid:16) f − nr n Z r s n − f ds (cid:17) = − (cid:16) δ ij n − x i x j r (cid:17) Z r s n f ′ ds = − (cid:16) δ ij nr n − x i x j r n +2 (cid:17) Z r s n ϕ N τ ′ ds and so | LW | = 1 r n r n − n (cid:12)(cid:12)(cid:12) Z r s n ϕ N τ ′ ds (cid:12)(cid:12)(cid:12) , (3.4)which is a radial function as we have expected. Step 2.
Solving the Lichnerowicz equation.
It simplifies the argument, and causes no lossof generality, to assume ϕ ′ = 0 almost everywhere. We first take (3.4) into the Lichnerowiczequation, it then follows that ( ϕ, τ ) is a solution of the conformal equations (3.1) if and onlyif they satisfy − nn − (cid:16) ϕ ′′ + ( n − ϕ ′ r (cid:17) + τ ϕ N − = 1 r n (cid:16) Z r s n ϕ N τ ′ ds (cid:17) ϕ − N − . (3.5)Integrating by parts (3.5) we have − nn − (cid:16) ϕ ′′ + ( n − ϕ ′ r (cid:17) + τ ϕ N − = 1 r n (cid:16) r n ϕ N τ − Z r ( s n ϕ N ) ′ τ ds (cid:17) ϕ − N − , equivalently,2 r n ϕ N τ Z r ( s n ϕ N ) ′ τ ds − (cid:16) Z r ( s n ϕ N ) ′ τ ds (cid:17) = 2 N r n ϕ N +1 (cid:16) ϕ ′′ + ( n − ϕ ′ r (cid:17) . (3.6)Next, multiplying (3.6) by ( r n ϕ N ) ′ / ( r n ϕ N ) , we obtain r n ϕ N (cid:16) Z r ( s n ϕ N ) ′ τ ds (cid:17) ! ′ = 2 N ( r n ϕ N ) ′ ϕ N − (cid:16) ϕ ′′ + ( n − ϕ ′ r (cid:17) . (3.7)We observe here that the equations (3.6) and (3.7) are equivalent as long as ϕ ′ ≥
0, whichwill be proven later, so we can continue our process without undue worry about equivalenceamong equations. Now, sincelim r → r n ϕ N (cid:16) Z r ( s n ϕ N ) ′ τ ds (cid:17) ! = 0 , (cid:16) Z r ( s n ϕ N ) ′ τ ds (cid:17) = 2 N (cid:0) r n ϕ N (cid:1)(cid:16) Z r ( s n ϕ N ) ′ ϕ N − (cid:16) ϕ ′′ + ( n − ϕ ′ s (cid:17) ds (cid:17) . (3.8)Therefore, assuming for the moment that Z r ( s n ϕ N ) ′ ϕ N − (cid:16) ϕ ′′ + ( n − ϕ ′ s (cid:17) ds > R n , (3.9)we obtain | τ | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:0) r n ϕ N (cid:1)(cid:16) R r s n ϕ N ) ′ ϕ N − (cid:16) ϕ ′′ + ( n − ϕ ′ s (cid:17) ds ds (cid:17)! ′ ( r n ϕ N ) ′ r N (cid:0) r n ϕ N (cid:1)(cid:16) R r s n ϕ N ) ′ ϕ N − (cid:16) ϕ ′′ + ( n − ϕ ′ s (cid:17) ds (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) N ( rϕ ) (cid:0) r n − ϕ ′ (cid:1) ′ + N R r s n ϕ N ) ′ ϕ N − (cid:16) ϕ ′′ + ( n − ϕ ′ s (cid:17) ds (cid:12)(cid:12)(cid:12)r(cid:0) r n ϕ N (cid:1)(cid:16) N R r s n ϕ N ) ′ ϕ N − (cid:16) ϕ ′′ + ( n − ϕ ′ s (cid:17) ds (cid:17) . To simplify the formula, we calculate Z r ( s n ϕ N ) ′ ϕ N − (cid:16) ϕ ′′ + ( n − ϕ ′ s (cid:17) ds = Z r ( nϕ + N sϕ ′ )( r n − ϕ ′ ) ′ ds, and so by integration by parts Z r ( s n ϕ N ) ′ ϕ N − (cid:16) ϕ ′′ + ( n − ϕ ′ s (cid:17) ds = ( nϕ + N rϕ ′ ) r n − ϕ ′ − N Z r (cid:16) s n ( ϕ ′ ) (cid:17) ′ ds = n r n − ( ϕ ) ′ + N r n ( ϕ ′ ) . (3.10)Taking into account, we have | τ | = (cid:12)(cid:12) N ( rϕ ) (cid:0) r n − ϕ ′ (cid:1) ′ + nN r n − ( ϕ ) ′ + N r n ( ϕ ′ ) (cid:12)(cid:12) q(cid:0) r n ϕ N (cid:1)(cid:0) nN r n − ( ϕ ) ′ + N r n ( ϕ ′ ) (cid:1) = 2 (cid:12)(cid:12)(cid:12)(cid:16) r n − (cid:0) ϕ ( N +2) / (cid:1) ′ (cid:17) ′ (cid:12)(cid:12)(cid:12) ( N + 2) r n/ ϕ N − p ϕ ′ ( r n − ϕ ) ′ . (3.11) Step 3. ϕ is increasing. We remind the reader that ϕ ′ was assumed to be different from 0a.e in R n for simplicity. In view of (3.8)–(3.10), we see that the necessary condition for ( ϕ, τ )to be a solution to (3.1) is nr n − ( ϕ ) ′ + N r n ( ϕ ′ ) = r n − ϕ ′ (2 nϕ + N rϕ ′ ) > R n . (3.12)We will show that this condition is equivalent to the fact that ϕ is increasing. In fact, if ϕ ′ > R n , (3.12) is obvious. Conversely, assume that (3.12) holds. Since ϕ (0) > nϕ + N rϕ ′ > ϕ ′ ≥ ϕ ′ < { r m } such that ϕ ′ ( r m ) < ϕ ′ ( r m ) →
0. This leads to the contradiction that0 ≤ ϕ ′ ( r m ) (cid:0) nϕ ( r m ) + N r m ϕ ′ ( r m ) (cid:1) < . Therefore, we have ϕ ′ > Counterexamples to the spacetime positive energy conjec-ture
In this section, we will apply Theorem 3.1 to smoothing out the singular of the negative massSchwarzschild metric and so disprove the spacetime positive energy conjecture in some sense.We will also explain why the decay rate k = O ( r − n − ǫ ) at infinity is not merely a simpleway to guarantee existence of mass, but it also plays an important role in the positive massinequality. We begin our discussion by recalling the spacetime positive energy conjecture on R n . Let ( R n , g ) be an AF manifold with g − δ Euc ∈ C ,α − n − − ǫ (4.1)for some ǫ >
0. The ADM mass of ( R n , g ) is defined by m ADM ( g ) := 12( n − ω n − lim r → + ∞ Z | x | = r n X i,j =1 ( g ij,i − g ii,j ) x j r d M n − , where M n − is the ( n − − dimensional Euclidean Hausdorff measure and ω n − is the volumeof the standard unit sphere in R n . In particular, when g = ϕ N − δ Euc with ϕ radial, theformula becomes m ADM ( g ) = − n − n −
2) lim r → + ∞ ( r n − ϕ ′ ) . (4.2)Bartnik in [2] showed that under the decay condition (4.1), the mass is a geometric invariant.A long-standing conjecture in general relativity states that the ADM mass of an AF initialdata set satisfying the dominant condition µ ≥ | J | is positive unless it is a Cauchy hypersurfaceof Minkowski space. This conjecture was proven to be true under suitable decay assumptionsof ( g, k ) at infinity. The most recent progress on such results is the spacetime PET proven byEichmair [12], which is expected to be true for all dimension n ≥
3, but so far we have onlyachieved it in dimensions less than eight. The theorem states for a general AF manifold, butfor our purpose, we just recall it on R n . We also note that for n = 3, the proof in [21, 12]requires additionally the assumption tr g k = O ( r − − ǫ ) at infinity, see also the second remarkon page 19 in [13]. However, in the statement below, since R n is spin for all n ≥
3, it isremoved thanks to the spacetime PET version for spin manifolds of dimension 3 shown in [8].
Spacetime Positive Energy Theorem (Chrusciel–Maerten [8] and Eichmair [12]) . Let ≤ n ≤ . Let ( R n , g, k ) be an AF initial data set satisfying the dominant energy condition µ ≥ | J | . Assume that ( g, k ) ∈ C ,α − n − − ǫ × C ,α − n − ǫ and ( µ, J ) ∈ ( C ,α − n − ǫ ) for some ǫ > . Thenthe ADM mass is non-negative. Moreover, if m ADM ( g ) = 0 , then ( R n , g, k ) is Cauchy initialdata for Minkowski space. Nowadays, negative mass is no longer viewed as a bizarre situation due to its central rolein interpretation on the accelerated expansion of the universe. Physicists do not explicitlyexclude the existence of negative mass, but the dominant energy condition and the spacetimePET are clearly difficult to reconcile with a negative ADM mass. From physical point ofview, a natural question to ask is whether there exists an AF initial data set of negativeADM mass without violation of the dominant energy condition. In what follows, we will givea positive answer to the question by considering the vacuum case on R n . In fact, in view ofthe conformal method presented in Subsection 2.1, we first express Theorem 3.1 in terms ofthe constraint equations as follows. 8 heorem 4.1 (Freely specified conformal factor) . Let ϕ > be a radial and increasingfunction in R n . Assume that ϕ − ∈ C ,α − β +2 with α ∈ (0 , and β > . We define τ ∈ C ,α − β by | τ ( r ) | = p nN ϕ − N +1 ϕ ′′ if ϕ ′ ( r ) = 0 (cid:12)(cid:12)(cid:12)(cid:16) r n − (cid:0) ϕ ( N +2) / (cid:1) ′ (cid:17) ′ (cid:12)(cid:12)(cid:12) ( N +2) r n/ ϕ N − √ ϕ ′ ( r n − ϕ ) ′ otherwise. (4.3) Then g ij = ϕ N − δ ij k ij = τn ϕ N − δ ij − ϕ − (cid:16) δ ij nr n − x i x j r n +2 (cid:17) Z r s n ϕ N τ ′ ds (4.4) is a solution to the AF vacuum constraint equations (1.1) . As the reader may have noticed, the surprising point of this result is that the increasingproperty of ϕ makes the ADM mass non-positive. In particular, this property agrees withnegative mass Schwarzschild metrics instead of the positive mass ones, therefore, we canconstruct smooth solutions of negative mass Schwarzschild metric outside a given ball, andso a class of spherically symmetric initial data sets violating the spacetime positive energyconjecture is given. We observe that these smooth solutions are no longer time-symmetriclike Schwarzschild ones. This is certain since otherwise the spacetime PET will lead to thecontradiction that the ADM mass of a negative mass Schwarzschild metrics is non-negative.We will discuss this feature in more detail after stating explicitly what we say above aboutsmoothing out the singular of negative mass Schwarzschild metric. The following result is adirect consequence of Theorem 4.1. Corollary 4.2 (Smoothing out negative mass Schwarzschild metric) . Given a constant m > , let ϕ be a radial and increasing function in C ∞ ( R n ) satisfying ϕ ( r ) = 1 − m r n − , ∀ r ≥ m / ( n − . Then ( g, k ) defined in (4.3) - (4.4) is a smooth solution to the AF vacuum constraint equations (1.1) .Proof. The proof is straightforward by Theorem 4.1.We now explain why Theorem 4.1 and Corollary 4.2 do not contradict the spacetime PET.In fact, for instance when ϕ = 1 − m r n − outside a ball, i.e. g is negative mass Schwarzschildat large distance, we can calculate by (4.3) that | tr g k | = | τ | ∼ cr − n as r → + ∞ for some constant c >
0. Since R g = R Sch = 0 near infinity and since ( g, k ) is an AF vacuumsolution, it follows by the Hamiltonian constraint (1.1a) that | k | g = | tr g k | ∼ cr − n as r → + ∞ . This means the decay rate of k at infinity is exactly r − n , which is critical in the decay as-sumption of symmetric (0 , − tensors in the spacetime PET, and that’s why the theorem failsin this situation.In order to understand better how the decay rate of k at infinity drives the sign of mass,we would like to give the following result. 9 heorem 4.3 (Freely specified mean curvature) . Let τ be an arbitrary radial function in C ,α − β ( R n ) with α ∈ (0 , and β ∈ (cid:0) , n (cid:1) . There exists a solution ( g, k ) to the vacuum con-straint equations (1.1) such that ( g − δ Euc , k ) ∈ C ,α − β +2 × C ,α − β and tr g k = τ . Moreover, givena constant c > and a decay exponent q ∈ ( n +24 , n ) , assume that | τ | ∼ cr − q at infinity. Thenwe have ( g − δ Euc , k ) ∈ C ,α − q +2 × C ,α − q (4.5) and furthermore(i) if q < n , then m ADM ( g ) = −∞ ,(ii) if q = n , then −∞ < m ADM ( g ) < ,(iii) if q > n , then m ADM ( g ) = 0 , and hence ( R n , g, k ) is Cauchy initial data for Minkowskispace provided that ≤ n ≤ . The main tool we use for dealing with the theorem is Leray–Schauder’s fixed point. Forconvenience and ease of presentation, we would like to recall its statement.
Leray–Schauder’s Fixed Point (Gilbarg–Neil [15, Theorem 11.6]) . Let X be a Banachspace and assume that T : [0 , × X → X is a continuous compact operator satisfying T (0 , x ) =0 for all x ∈ X . If the set K := (cid:8) ( t, x ) ∈ [0 , × X (cid:12)(cid:12) T ( t, x ) = x (cid:9) is bounded, then T (1 , . ) has a fixed point.Proof of Theorem 4.3. We first define the operator T : [0 , × L ∞ → L ∞ as follows. For any φ ∈ L ∞ , by Proposition 2.3, there exists a unique W ∈ C ,α − β +1 satisfyingdiv( LW φ ) = n − n | φ | N dτ, (4.6)and hence, thanks to Theorem 2.4, there exists a unique ϕ > ϕ − ∈ C ,α − β +2 and − n − n − ϕ + n − n t N τ ϕ N − = | LW | ϕ − N − . (4.7)We define T ( t, φ ) := tϕ. It is clear that a fixed point of T (1 , . ) is a solution to the conformal equations (3.1). In thespirit of the previous section, we will look for a fixed point of T (1 , . ) in the subspace RL ∞ := { f ∈ L ∞ | f is radial } . The following observations are the key in our arguments:- Let V : L ∞ → C ,α − β +1 and L : [0 , × C ,α − β +1 → C ,α − β +2 be defined by V ( φ ) := W, L ( t, W ) := ϕ − , W and ϕ are determined by (4.6) and (4.7) respectively. Let I : C ,α − β +2 → L ∞ be the compact weighted H¨older embedding map given by Proposition 2.1. It is clearthat T = t (cid:0) I ◦ (1 + L ) ◦ V (cid:1) . (4.8)We have shown in Section 3 that if φ is radial, then so is | L V ( φ ) | . On the other hand,since Laplace’s operator ∆ is invariant under rotations, we deduce from existence anduniqueness of solutions to the Lichnerowicz equation guaranteed by Theorem 2.4 thatif the source ( τ, | LW | ) is radial, then so is L ( t, W ). Therefore, we can conclude by (4.8)that T ( t, . ) maps the subspace RL ∞ into itself.- If T ( t, φ ) = φ with ( t, φ ) ∈ (0 , × RL ∞ , then φ/t is a radial solution to the conformalequations (3.1) associated with the seed data ( δ Euc , t N τ ). Therefore, it follows fromTheorem 3.1 that φ/t must be increasing, and so k φ/t k L ∞ = 1. In particular, the set K = (cid:8) ( t, φ ) ∈ (0 , × RL ∞ (cid:12)(cid:12) T ( t, φ ) = φ (cid:9) is bounded.From these observations, once T is proven to be continuous and compact in [0 , × RL ∞ ,we can ensure by Leray–Schauder’s fixed point that T (1 , . ) has a fixed point in RL ∞ whichis what we have desired. Observing furthermore that in view of (4.8), since V is continuousand since I is continuous compact, the fact that T is continuous and compact will followimmediately once we obtain the continuity of L . Therefore, the task is now to prove that L is a continuous operator. The argument we give here is essentially the same as in Maxwell[17], which is the equivalent result for compact manifolds.In fact, we define F ( t, W, ψ ) : [0 , × C ,α − β +1 × C ,α − β +2 → C ,α − β by F ( t, W, ψ ) := − n − n − ψ + 1) + n − n t N τ ( ψ + 1) N − − | LW | ( ψ + 1) − N − It is clear that F is C map and F ( t, W, L ( t, W )) = 0 for all ( t, W ) ∈ [0 , × C ,α − β +1 . Astandard computation shows that the Fr´echet derivative of F with respect to ψ is given by F ψ ( t, W )( u ) = − n − n − u + ( n − N − n t N τ ( ψ +1) N − u +( N +1) | LW | ( ψ +1) − N − u It follows that F ψ ∈ C (cid:0) [0 , × C ,α − β +1 , L ( C ,α − β +2 , C ,α − β ) (cid:1) , where we denote (cid:0) L ( C ,α − β +2 , C ,α − β ) (cid:1) the Banach space of all linear continuous maps from C ,α − β +2 into C ,α − β . In particular, setting ψ = L ( t, W ) we have F ψ ( t, W )( u ) = − n − n − u + ( n − N − n t N τ ( ψ +1) N − +( N +1) | LW | ( ψ +1) − N − ! u Since ( n − N − n t N τ ( ψ + 1) N − + ( N + 1) | LW | ( ψ + 1) − N − ≥ , it follows by Proposition 2.2(a) that F ψ ( t, W ) : C ,α − β +2 → C ,α − β is an isomorphism. There-fore, the implicit function theorem implies that L is a C -function in a neighborhood of ( t, W ),11hich deduces T is continuous compact in [0 , × L ∞ , and so T (1 , . ) admits a fixed point in RL ∞ .On account of what we have shown above, let ϕ ∈ RL ∞ be a fixed point of T (1 , . ). For theconvenience of the reader, we summarize the properties of ϕ as follows:- As claimed earlier, ϕ > ϕ − ∈ C ,α − β +2 .- Since ( τ, ϕ ) is a radial solution to the conformal equations (3.1), it follows by Theorem3.1 that ϕ is increasing and ( τ, ϕ ) satisfies (4.3).- Let ( g, k ) ∈ C ,α − β +2 × C ,α − β be defined in (4.4). Similarly to Theorem 4.1, we deducefrom the conformal method that ( g, k ) is a solution to the vacuum constraint equations(1.1) with tr g k = τ .Therefore, the rest of the proof is devoted to the decay rate (4.5) and assertions (i–iii). Infact, assume that | τ | ∼ cr − q , then we have by (4.3)lim r → + ∞ (cid:12)(cid:12)(cid:12)(cid:16) r n − (cid:0) ϕ ( N +2) / (cid:1) ′ (cid:17) ′ (cid:12)(cid:12)(cid:12) ( N + 2) r n − q ϕ N − p ϕ ′ ( r n − ϕ ) ′ ! = c. (4.9)On the other hand, by straightforward calculations we have2 (cid:12)(cid:12)(cid:12)(cid:16) r n − (cid:0) ϕ ( N +2) / (cid:1) ′ (cid:17) ′ (cid:12)(cid:12)(cid:12) ( N + 2) r n − q ϕ N − p ϕ ′ ( r n − ϕ ) ′ = (2 n − r − ϕ N/ ϕ ′ + N ϕ ( N − / ( ϕ ′ ) + ϕ N/ ϕ ′′ r − q ϕ N − p ( ϕ ′ ) + ( n − r − ϕϕ ′ . Since ϕ − ∈ C ,α − β +2 , this gives uslim r → + ∞ (cid:12)(cid:12)(cid:12)(cid:16) r n − (cid:0) ϕ ( N +2) / (cid:1) ′ (cid:17) ′ (cid:12)(cid:12)(cid:12) ( N + 2) r n − q ϕ N − p ϕ ′ ( r n − ϕ ) ′ ! = 1 √ n − r → + ∞ (cid:12)(cid:12)(cid:12) ( r n − ϕ ′ ) ′ (cid:12)(cid:12)(cid:12) r n − − q p r n − ϕ ′ = 2( n − q ) √ n − r → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0)p r n − ϕ ′ (cid:1) ′ ( r n − q ) ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.10)Combined with (4.9), we get lim r → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0)p r n − ϕ ′ (cid:1) ′ ( r n − q ) ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = c √ n − n − q )Now, by L’Hˆopital’s rule it then follows thatlim r → + ∞ (cid:16) ϕ ′ r − q +1 (cid:17) = lim r → + ∞ p r n − ϕ ′ r n − q ! = lim r → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0)p r n − ϕ ′ (cid:1) ′ ( r n − q ) ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! = c √ n − n − q ) . (4.11)12herefore, we have ϕ − − Z + ∞ r ϕ ′ ds ∈ C − q +2 and hence thanks to the Lichnerowicz equation and Proposition 2.2(b), we deduce from (4.4)that ( g − δ Euc , k ) ∈ C ,α − q +2 × C ,α − q .Finally, taking (4.11) into the formula (4.2) of the ADM mass yields m ADM ( g ) = − c √ n − n − q ) lim r → + ∞ r n − q , which implies (i–iii) except the statement about entire spacelike hypersurfaces in the assertion(iii). However, since ( g, k ) now satisfies assumptions in the spacetime PET, this follows bythe rigidity part of the theorem. The proof is completed. Remark 4.4.
We observe that our argument for entire spacelike hypersurfaces stated in theassertion (iii) of Theorem 4.3 only uses the rigidity part of the spacetime PET. Therefore,the restriction on n can be omitted once the rigidity holds in arbitrary dimensions. In thiscontext, since R n is spin for all n ≥ , the assumption ≤ n ≤ in this assertion can beremoved as long as q > n − due to the version of the spacetime PET for spin manifoldsproven in [8]. Acknowledgments.
The author would like to thank Bao Tran Nguyen for useful discus-sions. The author is a postdoc fellow of the French National Research Agency (ANR) atInstitut Montpelli´erain Alexander Grothendieck during 2020–2021.
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