Energy, entropy and the Ricci flow
aa r X i v : . [ g r- q c ] D ec Energy, Entropy and the Ricci Flow
Joseph Samuel and Sutirtha Roy Chowdhury
Raman Research Institute, Bangalore-560 080 (Dated: October 25, 2018)
Abstract
The Ricci flow is a heat equation for metrics, which has recently been used to study the topologyof closed three manifolds. In this paper we apply Ricci flow techniques to general relativity. Weview a three dimensional asymptotically flat Riemannian metric as a time symmetric initial dataset for Einstein’s equations. We study the evolution of the area A and Hawking mass M H of atwo dimensional closed surface under the Ricci flow. The physical relevance of our study derivesfrom the fact that, in general relativity the area of apparent horizons is related to black holeentropy and the Hawking mass of an asymptotic round 2-sphere is the ADM energy. We beginby considering the special case of spherical symmetry to develop a physical feel for the geometricquantities involved. We then consider a general asymptotically flat Riemannian metric and derivean inequality ddτ A / ≤ − π / M H which relates the evolution of the area of a closed surface S to its Hawking mass. We suggest that there may be a maximum principle which governs the longterm existence of the asymptotically flat Ricci flow. PACS numbers: 02.40.-k,04.70.Dy . INTRODUCTION The Ricci flow [1, 2, 3, 4] has been used by mathematicians to understand the topologyof three manifolds. It appears likely that these mathematical developments will also beuseful in physics in the study of geometric theories like general relativity. Several papers[5, 6, 7, 8, 9, 10, 11] have appeared dealing with physical applications of the Ricci flow. TheRicci flow (RF) is a (degenerate) parabolic differential equation, and is very similar to theheat equation. In a previous paper [12] we explored an analogy between the Ricci flow andthermodynamics. The analogy is based on the observation that the Ricci flow (like the heatequation) loses memory of initial conditions, just as a physical system loses memory of itsinitial state as it approaches thermal equilibrium. As was noticed there, a slight modificationof the Ricci flow yields Schwarzschild space as a fixed point. In this paper we look at theunmodified Ricci flow to see how some physically interesting quantities evolve with the flow.Energy and entropy are quantities of physical interest from the thermodynamic point ofview. In general relativity these quantities take on a purely differential geometric meaning:the entropy is related to the area of black hole horizons and the energy to the ADM mass atinfinity. We investigate the the evolution of these quantities under the Ricci flow and derivesome inequalities relating them.The subject of entropy bounds [13, 14] has been of wide and sometimes controversial[15] interest to physicists. It is believed that such bounds contain clues towards quantumgravity. Entropy bounds motivate ideas like “the holographic hypothesis”. There is ageometric entropy bound proposed by Penrose, which connects two geometrical quantities:the area of the outermost horizon and the ADM mass. In the seventies, Roger Penrose[16],in an attempt to test the idea of cosmic censorship (which is part of the “establishmentview” of gravitational collapse), used physical arguments to deduce from cosmic censorshipan inequality relating the ADM mass of an initial data set for GR and the area of itsoutermost apparent horizon: M ADM ≥ √ π A . This inequality, which is saturated by theSchwarzschild space, has a clear thermodynamic interpretation. It states that Schwarzschildspacetime maximises geometric entropy for a given energy. Maximising entropy for fixedenergy is a property which characterises a thermal state in statistical mechanics. The thermalcharacter of Hawking radiation from a Schwarzschild black hole is entirely consistent withthis interpretation. Penrose’s inequality appears to capture something deep about general2elativity, with a thermodynamic statement expressed in geometric terms. In this respect, itis similar to the the area theorem[17], which represents the second law of thermodynamicsin geometric form.A counterexample to Penrose’s inequality would imply a flaw in the establishment view.No counterexample has so far been found. Special cases of Penrose’s inequality have beenproved[18, 19, 20] using geometric flow techniques. Jang and Wald [21] showed that if oneassumes the existence of the inverse mean curvature (IMC) flow, (a particular diffeomor-phism of a spatial slice), one could address Penrose’s inequality in the special case of timesymmetric initial data. Their work was based on the monotonicity of the Hawking massunder the inverse mean curvature flow, which follows from earlier work by Geroch[22]. Thissuggests a proof of the Penrose inequality [21]. The only gap in this proof is the existenceof the IMC flow. This gap was filled by Huisken and Ilmanen [20] who showed that theHawking mass remained monotonic under a discontinuous version of the IMC flow.The motivation behind the present work is to explore whether the Ricci flow could lead toa new proof of the time symmetric Penrose inequality. The general Penrose inequality is stillan open conjecture and new lines of attack (even in the time symmetric case) are certainlyof interest. Is it possible that a smooth combination of the Ricci flow plus diffeomorphismsexists so that the evolution of Hawking mass is monotonic? The hope underlying the useof the RF is that the discontinuities (like those of the IMC flow) will be smoothed out bythe RF term. There are known examples in fluid mechanics where discontinuities of inviscidflow are smoothed out by viscous effects. With this motivation of developing a Ricci flowapproach to the Penrose inequality, we have made a beginning in this paper by studying theevolution of area and Hawking mass under the Ricci flow.In section II we give a brief review of the Ricci flow. For a more complete and rigorousaccount the reader is referred to the mathematical literature[23, 24]. Section III lists thegeometrical quantities of physical interest in this paper. Section IV paves the way for thegeneral treament by treating the special case of spherical symmetry. This special case isuseful since one can explicitly work out geometrical quantities of interest and develop aphysical feel for them. Spherical symmetry is a good source of physical examples and coun-terexamples which guide the general study. In Section V we give up spherical symmetry andtreat the evolution of area of a closed surface under the Ricci flow and derive an inequalityrelating the rate of change of area and the Hawking mass. Section VI treats the evolution of3awking mass under Ricci flow and puts forward a conjectured maximum principle whichgoverns the long time behaviour of the flow. Section VII is a concluding discussion. Ourmetric conventions are from Ref.[25]. II. THE RICCI FLOW
Let (Σ , h ab ) be an asymptotically flat, three dimensional Riemannian manifold. ( a, b runover 1,2,3. We restrict our discussion to three dimensional manifolds.) Our interest is inasymptotically flat spaces since we are interested in the energy and entropy of black holes.The definitions of energy, entropy and black holes all need an asymptotic region. The totalenergy or ADM mass of an initial data set is only well defined if an asymptotic structure(either flat or AdS) is fixed. Black holes are defined as regions of spacetime from whichescape to infinity is impossible and thus refer to an asymptotic structure. We require thatthe metric tend to a fixed flat metric δ ab at infinity h ab → δ ab + O (1 /r ). Given an initialmetric h ab , the Ricci flow evolves the metric according to its Ricci tensor. The evolutionparameter is τ and the family of metrics on (Σ, h ab ( τ )) satisfies the Ricci flow equation ∂h ab ∂τ = − R ab . (1)In the neighborhood of a point p ∈ Σ, we can introduce a Riemann normal co-ordinate systemand then the form of (1) becomes parabolic ( ∇ is the Laplacian in local co-ordinates) ∂h ab ∂τ = ∇ h ab (2)and looks like a heat equation for the metric coefficients. However, in a general co-ordinatesystem, the PDE (1) is a degenerate parabolic equation, because of its diffeomorphisminvariance.More generally, we will be interested in the Ricci flow modified by a diffeomorphism ∂h ab ∂τ = − R ab + D a ξ b + D b ξ a , (3)where ξ a is any vector field on Σ which vanishes at infinity. Calculationally, it is convenientto consider the two terms separately, defining a pure Ricci flow (1) and a pure diffeo ∂h ab ∂τ = D a ξ b + D b ξ a . (4)4n the mathematical literature on Ricci flows (which deals with compact spaces) the term λh ab is sometimes added to the RHS of (1) to define a “normalised Ricci flow”. Such a termis inadmissible in the present physical context as it would rescale the metric at infinity andviolate our asymptotic requirement that the metric tends to a fixed flat metric at infinity.In the standard initial value formulation [25] of general relativity, the basic variables arethe induced metric h ab on a spatial slice Σ in the space-time manifold M , Σ ∈ M and theextrinsic curvature k ab of Σ. We also use the notation k = h ab k ab . These variables aresubject to constraints: D b ( k ab − h ab k ) = 8 πj b (5)and R + k − k ab k ab = 16 πρ, (6)where j b is the matter current and ρ is the matter density. The matter is required to satisfy“energy conditions”, the dominant, weak or strong energy condition.A three dimensional manifold (Σ , h ab ) can be viewed as a time symmetric intitial dataset for Einstein’s equations. By “time symmetric” we mean that the extrinsic curvature k ab of Σ has been set to zero, so that (Σ , h ab ) is totally geodesic. (This is similar to choosinginitial data in classical mechanics so that all the momenta vanish. Dropping N particles fromrest is an example.) With k ab (and the matter current j a ) set to zero, the diffeomorphismconstraint (5) is automatically satisfied and the Hamiltonian constraint reduces to R = 16 πρ .A physically important constraint on initial data for general relativity is that the data aresubject to an energy condition. The dominant, weak and strong energy conditions all implythe local energy condition, which states that the local energy density is non-negative. Thistranslates into the geometrical statement that the scalar curvature R of h ab be non-negative.The Ricci flow has the appealing property that it preserves the non-negativity of scalarcurvature[24]. This follows from the “maximum principle” for the scalar curvature. UnderRicci flow(3), the scalar curvature satisfies the non linear heat type equation ∂R∂τ = ∇ R + 2 R ab R ab + L ξ R, (7)which implies that ∂R∂τ is positive at a minimum of R . Eq.(3) provides us with a flow onthe space of initial data to Einstein’s equations which remains within physically allowed(non-negative scalar curvature) data. 5 II. GEOMETRIC QUANTITIES OF INTEREST
Let S be a closed surface in Σ, γ ij , ( i, j = 1 ,
2) the pull back or induced metric on S , R the scalar curvature of ( S, γ ij ) and K the trace of its extrinsic curvature. We will beinterested in the evolution of some geometric properties of S under the Ricci flow. Theseare the area of S , A ( S ) = Z S dA = Z S d x √ γ (8)and the Hawking mass of S M H ( S ) = p A ( S )64 π / Z S dA (2 R − K ) . (9)Our interest in these quantities stems from their physical significance. The area of apparenthorizons is related to the entropy of Black Holes and the Hawking Mass is related to theEnergy. The Hawking Mass of a surface S can be physically interpreted as the mass containedwithin the surface S . While there are some problems with this interpretation (positivity isnot always assured), the Hawking mass is an useful notion[21, 22] of quasilocal mass. Itvanishes in the limit that S shrinks to a point and becomes the ADM energy for a roundsphere at infinity. In fact, the supremum of M ( S ) over S is the ADM mass as one can seefrom [21, 22]. Unlike the ADM energy, which is only well-defined for asymptotic spheres,the Hawking mass is defined for any closed surface S . Under the RF, a surface S , which isinitially asymptotic may shrink into the interior of Σ. For this reason Hawking mass is amore convenient object to study than the ADM mass.Rather than the Hawking mass, it is more convenient to deal with the related dimension-less quantity the “compactness” of S C ( S ) = Z S dA (2 R − K ) , (10)which is a combination of the Hawking mass and the area[26]. The quantity C ( S ) has beenused to good effect by Geroch, Jang and Wald [21, 22] in their approach to positive masstheorem and the Penrose inequality. In fact their work forms the base for recent progress[20] on the Riemannian Penrose inequality. We will see that C ( S ) tends to zero as S tendsto a round sphere of infinitesimal radius and also as S tends to an asymptotic round sphere.6 V. SPHERICAL SYMMETRY
The Ricci flow is a tensor evolution equation and therefore commutes with diffeomor-phisms. It follows that isometry groups are preserved under the Ricci flow. One way toapproach the Ricci flow is to start with symmetric situations so that the complexity of theflow is reduced. If we impose so much symmetry that the spaces of interest are homoge-neous, the RF becomes an ODE rather than a PDE. Such situations have been studied[24].However, this assumption is too restrictive from our present physical motivation. We wouldlike to deal with asymptotically flat (or AdS ) spaces since energy is defined with respectto an asymptotic structure. Homogeneous spaces which are asymptotically flat would beeverywhere flat and not very interesting. Spherical symmetry has the advantage that there are non trivial asymptotically flat spaces. The Ricci flow reduces to a PDE with just twoindependent variables, the τ and r co-ordinates. As we will see, spherically symmetric spaceswill provide us with an analytical as well as numerical testing ground and pave the way forthe general treatment. This special case is useful since one can develop a physical feel for thegeometrical quantities of interest and easily produce physical examples and counterexamplesas a guide to intuition.While setting up the spherically symmetric initial data set, we will work with two formsof the metric. Each of these has its use and its limitation. We will refer to them as “a-form”and “b-form”. They correspond to different choices of co-ordinate gauge. • The a-form:
In this form of the initial data set, the 3-metric is taken as ds = a ( r ) dr + r ( dθ + sin θdφ ) . (11)With this form of the metric we calculate the Ricci tensor and the scalar curvature:The nonzero components of the Ricci tensors are (a prime means differentiation withrespect to r ) R rr = a ′ ra ,R θθ = a ′ r a + 1 − a ,R φφ = sin θ ( R θθ ) . The scalar curvature is R = 2 r + 2 a ′ ra − ar . M H . However,this form of the metric is not useful if there is an apparent horizon because in thatcase a ( r ) blows up at the apparent horizon. To study the evolution of the area of theapparent horizon under RF we use the “b-form” of the metric discussed next. • The b-form:
In this form of the initial data set, the 3-metric is taken as ds = dr + b ( r )( dθ + sin θdφ ) . (12)With this metric we again calculate the Ricci tensor and the scalar curvature: Thenonzero components of the Ricci tensor are R rr = b ′ − bb ′′ b ,R θθ = 1 − b ′′ ,R φφ = sin θ ( R θθ ) . The scalar curvature is R = b ′ − b ( b ′′ − b . (13) a-form and Hawking Mass: Let us start with the a-form (11) of the metric and evaluate theHawking mass functional (9) for S chosen to be a sphere r = constant . For this sphericaltopology, the first term in (10) gives 16 π by the Gauss-Bonnet theorem and M H ( S ) := √A π (cid:18) π − Z S K dA (cid:19) . (14)Let ˆ n a be a unit normal to the surface S . The normalization h ab ˆ n a ˆ n b = a − ( r )ˆ n r ˆ n r = 1 (15)fixes ˆ n r = √ a (16)and so ˆ n a = ( 1 √ a , , . (17)The trace of the extrinsic curvature is K = D a ˆ n a = 2 r √ a , (18)8o we have Z K dA = 16 πa ( r ) , (19)and M H ( S ) = √A π / (cid:20) − a ( r ) (cid:21) . (20)For flat space a ( r ) = 1 and we have M H ( S ) = 0 for round spheres in flat space as expected.If we take the example of the exterior Schwarzschild space a ( r ) = (cid:18) − Mr (cid:19) − (21)and then for any r > M , M H ( S ) = √ πr √ π (cid:20) − (cid:18) − Mr (cid:19)(cid:21) = M (22)For a general a-form metric we can write a ( r ) as (cid:18) − M ( r ) r (cid:19) − . We find that[25] M H ( S ) = M ( r ) (23)and the “compactness” works out to C ( r ) = 32 π M ( r ) r . (24)In order to get a better physical feel for what these geometrical quantities mean, letus consider some typical distributions of matter. Let us choose the matter density ρ ( r ) =(1 / π ) R ( r ) positive and plot the functions ρ ( r ), C ( r ) and M H ( r ). Figure(1) displays theforms of these functions (in arbitrary units) for a spherical shell of matter. C ( r ) increasesto a maximum value and then decreases to 0 at infinity. For a matter distribution of twomomentarily static shells of matter, the slightly more complex behaviour of C ( r ) is shownin Fig. (2). For a star C ( r ) attains its maximum near the surface of the star. From (24),in the Newtonian limit C ( r ) is a constant times the dimensionless Newtonian potential, orthe mass to radius ratio. Hence the name “compactness” is justified.Note that M H ( r ) monotonically increases with r to attain its asymptotic value. Thisis due to the local energy condition, which implies positive scalar curvature R ≥
0. Thiscondition is conveniently stated in terms of the function M ( r ). The scalar curvature of(11) is given by R = 4 M ′ ( r ) /r and so the constraint of positivity of scalar curvaturesimply states that M ( r ) is a non-decreasing function of r . Assuming that the form (11)9olds all the way to the origin and that the scalar curvature R is finite, we have M (0) =0 , M ′ (0) = 0 , M ′′ = 0 , M ′′′ (0) ≥ M ( r ) increases from zero and tends to an asymptoticvalue M ADM = lim r − > ∞ M ( r ) which is the ADM mass of the metric. In a sense[25], M H ( r )measures the total mass contained within a sphere of areal radius r . M H ( r ) is non-negativefor all r and and from (20), we conclude that a ( r ) ≥ r .Note that from (19) it follows that if the space contains an apparent horizon ( K = 0), a ( r ) must diverge. This is why the a-form is unsuitable for treating apparent horizons. Theb-form does not suffer from this problem. Geometric quantities in the b-form:
We now consider a round sphere S ⊂ Σ given by r = constant in the spherically symmetric “b-form” (12) of the metric. The unit normalsatisfies h ab ˆ n a ˆ n b = ˆ n r ˆ n r = 1 (25)and ˆ n r = (1 , , A of S is given as A ( r ) = Z S √ γdθdφ = 4 πb ( r ) , (26)where γ = b sin θ is the determinant of the induced metric γ ij on S . The trace of theextrinsic curvature is given by K := D a ˆ n a = b ′ b , (27)(where a prime indicates differentiation with respect to r ). The general formula for thecompactness reduces in spherical symmetry to, C ( r ) = 16 π − Z S √ γdθdφK = 16 π − πb ′ b (28)and the Hawking mass is M H ( r ) = p A ( r )64 π / C ( r ) = √ b − b ′ b ] . (29)Note that M H depends on b ( r ) and its derivative b ′ ( r ), in constrast to the simple algebraicrelation (20) we had in the a-form of the metric. Area under Ricci flow:
We consider a metric initially in the b-form, evolving under a pureRicci flow (without a diffeomorphism term). Under this evolution, the b-form may not bepreserved. We view S as a fixed surface of Σ and so the co-ordinate location of the surface S does not change and hence drdτ = 0. From the Ricci flow we have ∂h θθ ∂τ = − R θθ d A dτ = 4 π ∂b∂τ = − π (2 − b ′′ )for the instantaneous rate of change of area of S .Using the scalar curvature R for the “b-form” of the metric (13) we see that, in sphericalsymmetry, d A dτ = − Z S √ γdθdφR − C (30)and so we arrive at the inequality (since R ≥ d A dτ ≤ − C. (31)In the case of the Schwarzschild space, R = 0 and so the first integral in (30) vanishes andwe have d A dτ = − C ( S ) . (32)Thus Schwarzschild space saturates our inequality (31), just as it saturates the Penroseinequality. Area of apparent horizons under Ricci Flow:
Let S be a minimal surface (or apparenthorizon, they coincide in the case of time symmetric data) in Σ i.e S is a closed two manifoldembedded in Σ with the property that the trace of the extrinsic curvature vanishes. We wantto see how the area of S varies under the RF. We start with the spherically symmetric “b-form” of the metric (12). Let the location of the apparent horizon be at r = r . From (27),we have that K = b ′ /b | r = 0 The condition that the surface r = r be an apparent horizonis b ′ | r = r = 0 . (33)Can a minimal surface spontaneously appear if none was present initially? The answeris no, as the following argument shows by contradiction. Let us consider the b form of themetric and evolve it by the Ricci flow supplemented by a suitable radial diffeomorphism ξ a chosen to maintain the b -form ∂h rr ∂τ = − R rr + 2( Dξ ) rr = 0 , (34) ∂h θθ ∂τ = ∂b∂τ = − R θθ + 2( Dξ ) θθ . (35)11e may write ξ r = ∂ r f and using (34) and choose f so that f ′′ = b ′ − bb ′′ b . (This leaves some freedom in f , but this does not affect the following.) (35) then gives usthe evolution equation for b ( r, τ ) in the b form of the metric ∂b∂τ = b ′′ − b ′ f ′ . Differentiating this equation we arrive at an evolution equation for b ′ ∂b ′ ∂τ = b ′′′ + b ′′ f ′ + b ′ f ′′ , (36)which gives us a maximum principle for b ′ . Suppose that b ′ > τ < τ and that for thefirst time τ = τ , a minimal surface appears ( b ′ ( r ) = 0) at r = r . We have ∂b ′ ∂τ ( r ) <
0, since b ′ decreased to zero. On the other hand, since r is a minimum of b ′ , we have b ′′ ( r ) = 0 and b ′′′ ( r ) ≥
0. A glance at (36) shows that ∂b ′ ∂τ ≥
0, which is a contradiction. Thus a minimalsurface cannot spontaneously appear under RF if it was not initially present.Regions where b ′ < b ′ = 0). Fromthe last paragraph it is clear that trapped regions cannot spontaneously appear under RFsince they are accompanied by minimal surfaces. However, trapped regions can continuouslyshrink to zero and disappear. When this happens, the two minimal surfaces that form theboundary of the trapped region merge and disappear. More descriptively, a minimal surface( b ′ = 0 , b ′′ >
0) merges with a maximal surface ( b ′ = 0 , b ′′ <
0) and disappears.Setting aside such mergers, let us study how the area of minimal surfaces evolves underthe pure RF (1). During the RF the metric changes and the location of the horizon maychange and so r = r ( τ ) where τ is the parameter of the RF. Also the geometry of S willchange. In principle both these effects could lead to change of area. The area is given by(26) A ( r ) = 4 πb ( r ). The total rate of change of area, therefore, is4 π dbdτ = 4 π (cid:20) ∂b∂r (cid:12)(cid:12)(cid:12)(cid:12) r = r dr dτ + ∂b∂τ (cid:12)(cid:12)(cid:12)(cid:12) r = r (cid:21) . (37)The first term vanishes because of the apparent horizon condition (33). The second term isevaluated by specialising eq.(31) to an apparent horizon. So the area A = 4 πb satisfies ∂ A ∂τ ≤ − π. τ . Since thearea was finite to begin with, we find that (if the horizon persists) b evaluated at the horizongoes to zero in a finite τ .Next we see that as b → R is finite and will arrive at a contradiction. As b → R = 2(1 − b ′′ ) b is finite only if b ′′ = 1. We Taylor expand in powers of r − r about r , the location of theapparent horizon b ( r ) = b ( r ) + b ′ ( r )( r − r ) + 12 b ′′ ( r )( r − r ) + .... As b ( r ) → b ′′ ( r ) → b ′ ( r ) = 0 due to the apparent horizon condition (33), we have b ( r ) = 12 ( r − r ) , so the metric is ds = dr + 12 ( r − r ) ( dθ + sin θdφ ) . Shifting the r co-ordinate r → ( r − r ) gives the form ds = dr + 12 r ( dθ + sin θdφ ) . So the volume of a ball of radius r is Z dθdφdr √ r sin θ (cid:18) πr (cid:19) . We then find from the expression for the volume of a ball of radius r (as r →
0) centered atpoint p that volume B ( p, r ) = (cid:18) π (cid:19)(cid:18) r − R ( p ) r (cid:19) and so the scalar curvature blows up, R ( p ) ∼ r − , which is a contradiction to the assumptionof finite R that we started with. So an apparent horizon which persists under RF results ina singularity in a finite amount of τ parameter τ ≤ τ , where τ = A (0) / (4 π ). Compactness under RF:
We have previously derived the relation (24) for the compactnessof a sphere r = const from which follows C ( r ) = 16 π [1 − /a ] . (38)13 calculation reveals that under RF dCdτ = − M ′ ( r ) r + 2 M ′′ r . (39)While the first term here is of definite sign, the second term is not. As a result, the rateof change of C with τ is not monotonic. Note however that for Schwarzschild M ( r ) is aconstant M and so dC/dτ = 0.The Hawking mass is given by M H ( r ) = rC ( r ) / (32 π ) and to work out its rate of change d M H dτ under the RF, we need to find dr/dτ . This is easily read off from (30) and we find drdτ = − M ′ r − C πr . (40)Putting these equations together we can work out d M H dτ and it turns out to be a linearcombination of M ′ and M ′′ . As a result, the Hawking mass is not monotonic along the pureRicci flow in the a-form of the metric. It may be that adding a suitable diffeomorphism willresult in monotonic behaviour for M H . Maximum principle for compactness:
However, we can make a diffeo invariant statementabout the behaviour of compactness. We have already seen that if a minimal surface isinitially present this may result in a finite τ singularity. Let us suppose that no minimalsurface is initially present. None develops under RF, as we saw earlier. We can thereforeuse the a-form of the metric. We have seen that C ( r ) starts from 0 at r = 0, reaches amaximum (perhaps several local maxima) and then decays to zero as r → ∞ . Let us focuson C max the value of C at its absolute maximum. Just as for apparent horizons, the valueof C max is not affected by a diffeomorphism: moving the surface does not affect the valueof C max since we are at a maximum. We choose a diffeomorphism to preserve the a-form ofthe metric and then the evolution equation for a ( r ) reads ∂a ( r ) ∂τ = a ′′ ( r ) a ( r ) − a ′ ( r )) a ( r ) − a ( r ) −
1) + ra ′ ( r )(1 − a ( r )) /a ( r ) r , (41)where a prime denotes differentiation with respect to r . We have studied the evolution of(41) using numerical techniques. Some of the results presented here were initially suggestedby the numerical evidence.Let us focus on the maximum value of a ( r ). Recall that a ( r ) ≥
1. At the maximum valueof a ( r ), we have a ′ ( r ) | max = 0 and a ′′ ( r ) | max ≤
0. So from equation (41) we see that the14aximum value of a ( r ) is monotone non-increasing as the flow parameter τ increases i.e., ∂a ( r ) max ∂τ = a ′′ ( r ) a ( r ) − a ( r ) max − r ≤ . (42)A maximum of C corresponds via (38) to a maximum of a ( r ). Clearly, a maximum principlefor a ( r ) implies a maximum principle for compactness C ( r ) and we have the inequality dC ( r ) max dτ ≤ . (43)We can use this maximum principle to comment on the long time existence of the sphericallysymmetric asymptotically flat Ricci flow. Let us suppose that no minimal surface is initiallypresent. From (42), we see that for the LHS of (42) to vanish we must have a ( r ) max = 1,which implies a ( r ) = 1 identically. This describes flat space. If the initial metric is notflat space, its a ( r ) max decreases with the flow and finally attains the flat space fixed point a ( r ) = 1. Our argument shows clearly that in spherical symmetry, the only asymptoticallyflat fixed point of the flow is flat space. This recovers results obtained by other methods[7, 27].To summarise, the maximum principle for compactness leads to a criterion for the exis-tence of the Ricci flow in the asymptotically flat case. If there are no apparent horizons, theflow exists for all τ and converges to flat space. If there are apparent horizons, the Ricciflow either terminates in finite time singularity or removes the horizons by mergers. V. AREA UNDER RICCI FLOW
We now drop the assumption of spherical symmetry and deal with a general asymptot-ically flat manifold Σ, with one end at infinity and a fixed closed orientable surface S ofarbitrary topology embedded in Σ. The induced metric of S is written γ ij and extrinsiccurvature tensor of S is written K ij , where i, j are two dimensional indices in the spacetangential to S . We will sometimes use projected a, b indices for these. The trace of theextrinsic curvature is written K = γ ij K ij . As the metric h ab evolves according to the Ricciflow (1), how does the area change with τ ? Since the metric of Σ is changing, we have toremember that the unit normal to S is also changing with the metric.Let us define the surface S as the level set of a function η on Σ which is strictly increasingoutward from S . Quite independent of any metric, the normal η a = D a η is a well defined15o-vector. η a is non-zero (since we assumed η is not locally constant). (Choosing a differentfunction η will result in multiplication of η a by a positive function on S .) The unit normalˆ n a = η a ( η.η ) / (44)depends on the metric. As the metric changes the unit normal can only change by a multipleof itself d ˆ n a /dt = α ˆ n a . By differentiating ˆ n. ˆ n = 1, we arrive at α = 12 dh ab dτ ˆ n a ˆ n b , (45)(where in our notation, dh ab dτ is defined as dh ab dτ with its indices raised using the metric tensor h ab ).Starting from the definition (8) we compute d A dτ d A dτ = Z S d x d √ γdτ . (46)We easily see that d √ γdτ = 12 √ γγ ab dh ab dτ (47)and we arrive at d A dτ = 12 Z S √ γd x ( h ab − ˆ n a ˆ n b ) dh ab dτ . (48) Area under ricci flow:
Using the form (1) of the Ricci flow, we find d A dτ = Z S √ γd x [ˆ n a ˆ n b R ab − R ] . (49)From the Gauss-Codazzi equation[25], we haveˆ n a ˆ n b R ab − R = − / R + R + ( K ij K ij − K )] , (50)which can be rearranged to give d A dτ = − / Z S d x √ γ (cid:20) R + ( K ij − Kγ ij )( K ij − Kγ ij ) (cid:21) − Z S d x √ γ (2 R − K ) . (51)The second integral in (51) is identified as −C ( S ) /
4, minus one fourth the compactnessintegral of S and the first integral, which is of definite sign can be dropped to arrive at theinequality d A dτ ≤ − C ( S )4 . (52)16his inequality is one of the main results of this paper. This result can be reexpressed interms of the Hawking Mass: d A / dτ ≤ − π / M H ( S ) . (53)Thus the rate of decrease of area of a closed 2-surface under Ricci flow is bounded by theHawking mass. As we mentioned earlier, the inequality (52) is saturated in the case ofthe spheres of Schwarzschild space (which is given by (11) with a ( r ) = (1 − M/r ) − ). Inthis case R = 0 and the spheres are shear free ( K ij = Kγ ij ), so the first integral in (51)vanishes. We conjecture that round spheres in Schwarzschild are the only surfaces whichsaturate this bound.As a simple application of this inequality, let us consider flat space. Since the Riccitensor vanishes we have that d A /dτ = 0 and so the LHS of (53) vanishes. We arrive atthe conclusion that for all surfaces in flat space, the Hawking mass is non positive! Thisfact has also been noticed in [28], where a direct proof is given. In fact, the converse of thisstatement is also true: Given positive scalar curvature, flat space is the only one for whichthe Hawking mass is non-positive. To see this, note that the supremum over S ∈ Σ of theHawking mass is the ADM mass and if this supremum vanishes, it follows from the positivemass theorem that the space must be flat.
Area under diffeos:
Under a diffeo, the metric changes as in (4) and so we have from (48) d A dτ = 12 Z √ γd x ( h ab − ˆ n a ˆ n b )2 D a ξ b = Z √ γd x [ D a ξ a − n a n b D a ξ b ] (54)If we suppose that ξ a is tangent to S , then with ˜ D a denoting covariant derivative intrinsicto ( γ ab , S ), ˜ D a ξ b = γ a ′ a γ b ′ b D a ′ ξ b ′ (55)gives us ˜ D a ξ a = γ ab ˜ D a ξ b = ( h ab − ˆ n a ˆ n b ) D a ξ b (56)So, as one would expect, d A dτ = Z √ γd x [ ˜ D a ξ a ] = 0 , (57)since this is a divergence over a closed surface S . It is therefore enough to consider thecomponent of ξ normal to S . Let ξ a = u ˆ n a . (58)17hen d A dτ = Z √ γd x ( h ab − ˆ n a ˆ n b ) D a ( u ˆ n b ) , (59)which works out to the standard answer d A dτ = Z √ γd x uγ ab D a n b = Z √ γd x uK. (60) Area of horizons under ricci flow: If S is a minimal surface, we find from (10) that since K = 0, the area changes according to d A ( S ) dτ ≤ C ( S ) = − πχ ( S )4 , (61)where χ ( S ) is the Euler characteristic of S . For a minimal surface of spherical topology, wehave d A dτ ≤ − π, (62)which we had seen earlier for the special case of spherical symmetry. This result is unaffectedby adding a diffeo to the RF because of (60) and the minimal surface condition K = 0.An interesting special case is one for which S is the outermost horizon with respect toasymptotic infinity. S is defined as the boundary of the region having trapped surfaces. Sis a minimal surface and is known to have spherical topology. According to (62), the areaof S will shrink. Under the Ricci flow, the trapped region cannot disappear suddenly, butevolves continuously, because trapped surfaces remain trapped under small perturbations.Near S there will be a new minimal surface with the same area. If the initial area is A , itdissappears within a time A / π , either by merger or by shrinking to zero. VI. HAWKING MASS UNDER RICCI FLOW
The Hawking mass of a closed surface S given by (9) is a combination of the “com-pactness” of S and its area. To study the evolution of the Hawking mass, it is enough tounderstand the evolution of the compactness, the evolution of area being already treated inthe last section. From the formula (10) for compactness, we see that the first term R d x √ γ R drops out on differentiation since it is a topological invariant by the Gauss-Bonnet theorem.The second term gives dC ( S ) dτ = − ddτ Z K √ γd x = − Z K √ γ dKdτ d x − Z K d √ γdτ d x. (63)18sing (47) for the second term in (63) and the formula dKdτ = ddτ ( D a ˆ n a ) = d Γ aam dτ ˆ n m + D a d ˆ n a dτ , (64)we find after a straightforward calculation that dCdτ = − Z dAK (cid:20) h ab ˆ n c D c dh ab dτ − D a ( dh ab dτ ˆ n b ) + ˆ n a D a ( dh cd dτ ˆ n c ˆ n d ) (cid:21) (65) − Z dA K (cid:20) dh cd dτ ˆ n c ˆ n d + h ab dh ab dτ (cid:21) . (66)Equations (66) give the general evolution of the compactness under for any one parameterfamily h ab ( τ ) of metrics. We will specialise these equations to the two cases of interest: thepure Ricci flow and a pure diffeo. Compactness under ricci flow:
Substituting dh ab dτ = − R ab into (66) we find after simpli-fication using the contracted Bianchi identity that dCdτ = Z S dA { K ( R + ˆ n a ˆ n b R ab ) − K [2 R ab D a ˆ n b − ˆ n a D a ( R cd ˆ n c ˆ n d )] } . (67) Compactness under diffeos:
How does the compactness change under a pure diffeo gen-erated by ξ a ? As we saw for the area, the tangential component of ξ a does not cause anychange in the integral. The diffeo vector field can be characterised by its normal component ξ a = u ˆ n a . Substituting (4) into (66) and simplifying and using the Gauss-Codazzi equation[25] − n a n b R ab + R = R + ( k ij k ij − k ) (68)gives us the formula [22] dCdτ = Z S [2 K ˜ D a ˜ D a u + uKσ ij σ ij + ukR − uK (2 R − K )] √ γd x (69)where ˜ D a denotes the intrinsic covariant derivative operator within the surface and σ ij = K ij − / γ ij K . If there exists a diffeo such that uK = 1 (this is the inverse mean curvatureflow (IMC)) one can conclude [22] that under this diffeo dCdτ ≥ − C/ M H ( S ) = p A ( S )64 π / C ( S ) . (71)19e then have ddτ M H = π / ! √A d A dτ C + √A dCdτ ! (72)Knowing d A dτ and dCdτ for a flow, can calculate ddτ M H . (60)and Geroch’s inequality (70) thenimply that the Hawking mass is monotonic under the IMC flow. A maximum principle for compactness?
The compactness C ( S ) of a closed surface S tends to zero for S tending to a small round sphere, and also tends to zero for a roundspheres at asymptotic infinity. We would expect based on our experience with sphericalsymmetry, that somewhere in between, there is a surface S for which C ( S ) attains its globalmaximum. If this maximum is equal to 16 π , then S is a minimal surface and conversely. Ifthe maximal compactness is less than 16 π , we may expect from the spherically symmetriccase that this value C ( S ) | max will monotonically decrease to zero. dC max dτ ≤ , (73)with the equality holding only for flat space. A zero value for C ( S ) | max would also imply anon-positive value for M H ( S ). Supremising this over S tells us that the ADM mass of thespace vanishes, which implies (by the positive energy theorem[29]) that space is flat. Weconjecture that (73) is true generally, but have not been able to establish the truth of thisconjecture. If this conjecture is true, it would imply that in the long time, an initial metricwithout minimal surfaces would approach flat space. VII. CONCLUSION
We have described some applications of Ricci flow techniques to asymptotically flat spacesin general relativity. Our main result is that under the Ricci flow, the rate of change of areaof a closed surface is bounded by its Hawking mass. This inequality is saturated by theSchwarzshild space. Since the Schwarzschild space saturates the Penrose inequality as wellas Geroch’s (70), our inequality may be related to these. We have also studied the behaviourof compactness under the Ricci flow. Our work in spherical symmetry suggests that theremay be a maximum principle for the compactness. The compactness is a functional onclosed two dimensional surfaces. The conjecture is that for positive scalar curvature, themaximum value of this functional decreases under the Ricci flow plus diffeomorphisms. Thisproposal is a functional maximum principle unlike the more usual ones which are which are20ormulated on functions. If such a principle does exist, it would be of great interest as adiagnostic for the long term existence of the asymptotically flat Ricci flow. Most of themathematical work in the area of Ricci flows is concerned with closed three manifolds. Oneof our main points here is that there may be interesting physical applications of these ideasto asymptotically flat spaces. For instance, [5] poses the question of stability of Euclideanflat space IR under RF. If a maximum principle for compactness exists, it would imply thatflat space is indeed stable under RF.In general a minimal surface can have any topology. Special interest attaches to thecase where S is an outermost horizon. In this case one can physically identify S as theboundary of a black hole region. Outermost horizons have the property that the surface hasminimum area (and not just stationary area). This immediately implies [30] (when curvatureis positive) that the outermost horizon has spherical topology. Our general result showsthat outermost horizons always shrink under the Ricci flow. This result is diffeomorphisminvariant. If one identifies the area of an outermost horizon as black hole entropy, wearrive at the conclusion that entropy is monotonically decreasing along the RF. A similarconclusion has been reached by [11] in a slightly different context: entanglement entropy ofa two dimensional black hole is monotonically decreasing along the RF.If one uses the Ricci flow to model the approach of a system to thermal equilibrium,one would expect that the entropy increases along the flow. However, we have seen thatoutermost horizons have spherical topology and their area decreases under the flow. In fact,it has been observed [12] that while the Ricci flow does have something in common (memoryloss) with approach to equilibrium, the analogy is not perfect. A slight modification [12] ofthe Ricci flow is necessary in order for the entropy to be identified with black hole entropy. Itmay be that such a modification of the RF is necessary for application to black hole physics.This paper was concerned exclusively with the case of asymptotically flat spaces. Itwould be interesting to generalise the treatment to allow for asymptotically AdS spaces.This appears to be a straightforward generalisation and some of the necessary changes arementioned in [12, 30]. 21
20 40 60 80
Radial coordinate (r) M , C a nd ρ MC ρ FIG. 1: Note that M H ( r ) increases with r to attain its asymptotic ADM value. But C ( r ) increasesto a maximum value and then decreases to 0 at infinity. Radial coordinate (r) M , C a nd ρ MC ρ FIG. 2: Two shells of matter. cknowledgements: We thank Javed Ahmad for his collaboration and Harish-Chandra Research Institute fortheir hospitality in the early stages of this work. One of us (JS) thanks Harish Seshadri fordiscussions on Ricci flows.
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