Positivity of Quasilocal Mass

Positivity of Quasilocal Mass

Chiu-Chu Melissa Liu* and Shing-Tung Yau^†

Harvard University, Department of Mathematics, One Oxford Street, Cambridge, Massachusetts 02138, USA (Received 2 March 2003; published 10 June 2003)

Motivated by the important work of Brown and York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with a positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface.

DOI: 10.1103/PhysRevLett.90.231102 PACS numbers: 04.20.Cv

Introduction.—There have been many attempts to de- fine quasilocal energy in general relativity (see [1] for a historical survey and a sample of relevant literature). In [1,2], Brown and York obtained definitions of surface stress-energy-momentum density, quasilocal energy, and conserved charges from a Hamiltonian-Jacobi analysis of the gravitational action. Hawking and Horowitz gave a similar derivation in [3].

The Brown-York quasilocal energy and angular mo- mentum are defined for a spacelike 2-surface which bounds a compact spacelike hypersurface in a time ori- entable spacetime. They have desirable properties such as specializing to Arnowitt-Deser-Misner (ADM) energy momentum and Bondi-Sachs energy momentum in suit- able limits [1,4]. Brown and York also justified their definitions with thermodynamics of black-hole physics.

Motivated by the Brown-York work [1,2], we propose definitions of quasilocal energy and surface momentum density for spacelike 2-surfaces with positive intrinsic curvature. Our definition of quasilocal mass also arises naturally from calculations in the second author’s recent work [5] on black holes as a candidate for positivity, which is an essential property for any definition of mass. (Our definition of quasilocal mass was found ini- tially by reasons motivated by understanding the second author’s calculations in [5]. However, the works of Brown and York certainly make it well motivated from the point of view of physics.)

Quasilocal mass is important as it has several potential applications. It can be used to define binding energy of stars. It can also be useful for numerical relativity as it tells us how to cut off the data on a noncompact region to a compact region where the quasilocal mass of the com- pact region approximates the ADM mass of the noncom- pact region. When we evolve the data according to Einstein equations, we need local control of energy to see how the space changes. The positivity of quasilocal mass is essential for such an investigation. This could be regarded as a generalization of the energy method in the theory of a nonlinear hyperbolic system.

Brown and York’s definitions.—Let be a compact spacelike hypersurface in a time orientable spacetime

M. We do not assume the boundary @ is connected.

Let g_ij denote the positive definite metric on, and let K_ijdenote the extrinsic curvature ofinM. Letbe a connected component of@, and letkbe the trace of the extrinsic curvature k_ab of in with respect to the outward unit normalvⁱ. This definition ofkis the nega- tive of the definition in [2]. Define

₁ 1

8Gk; jⁱ₁ 1

8Gv_jKg^ijK^ij; whereGis Newton’s constant, andKg^ijK_ijis the trace of the extrinsic curvature of inM. The three vector field jⁱ₁ along can be decomposed into components tangential and normal to . The tangential component can be viewed as a vector fieldj^a₁ on, and the normal component is given by p=8Gvⁱ, where p KK^ijv_iv_j.

Suppose thathas positive intrinsic curvature so that is topologically a 2-sphere. By Weyl’s embedding theo- rem,can be isometrically embedded into the Euclidean three spaceR³ such that the extrinsic curvature k_0ab is positive definite. Let be the region inR³enclosed by. The embedding R³R^3;1 gives ₀ k₀=8G and jⁱ₀ 0, whereR^3;1 is the Minkowski spacetime. The isometric embeddingR³is unique up to isometry of R³, sok₀is determined by the metric of.

The energy surface density and the momentum sur- face densityj^a are defined by

₁₀ 1

8Gkk₀; j^aj^a₁j^a₀ 1

8G^a_iv_jKg^ijK^ij; where^a_i is the projection to the tangent space of. The quasilocal energy is defined by

EZ

1 8G

Z

k₀k:

The angular momentum with respect to a Killing vector field^aonis defined by

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JZ

j^a_a 1 8G

Z

_a^a_iv_jK^ijKg^ij: The above quasilocal energyEand angular momentum Jdepend on the spacelike hypersurface. Let us denote them byE;andJ;; ^a.

Definition of quasilocal energy.—Let u denote the future timelike unit normal of in the time orientable spacetimeM, and viewvⁱas a four vector fieldvdefined along. Then kupv is a four vector field defined along and normal to . The vector kupv can be expressed in terms of null normals. We use the notation in [6]. Letl,n be outward and inward null normals in the sense thatlv>0 andnv<0. Let 2and 2 denote the traces of the extrinsic curvature of with respect to l and n, respectively. The product de- pends only onln, which we fix to be1. Then

kupv2nl: (1) We assumek >jpjso thatkupv is future timelike.

We have8k²p²>0. It is clear from (1) that, if ~

is another spacelike hypersurface such that is a connected component of@~ and the corresponding four vector field kk~~uupp~~vv is future timelike, then kk~~uu

~ p

p~vvkupv.

The embeddingR³ R^3;1gives similarly defined quantities₀and₀. We have8₀₀k²₀ >0. We define the quasilocal energy ofto be

E 1

8 p G

Z

p₀₀

p

(2)

1 8G

Z

k₀ p8

: (3) Note thatE; E.

The quantity p8

k²p²

p is the boost invariant mass defined by Lau in [7]. The quantityalso appears in the definition of Hawking mass [8]. Our definition of quasilocal energy should be compared with the invariant quasilocal energy (IQE) proposed by Epp in [9], where we chose the reference energy differently. (We found Epp’s paper after we completed the first version of this Letter.) Definitions of momentum surface density and angular momentum.—Letj^abe the momentum surface density of defined by Brown and York. There exist functionsfand gon, unique up to addition of a constant, such that

j^aabf_bg^a:

In the above decomposition,g^adepends on the spacelike hypersurface, butjj^aabf_b does not. We definejj^ato be the momentum surface density of. The field strength defined by Lau in [7] is given by

F_{ab a}j_{b b}j_{a a}jj_{b b}jj_a f_ab; which is boost invariant. We haveF f.

EmbedinR³ as before. Let ⁱ be a Killing vector field on R³ which generates a rotation. We define the angular momentum with respect toⁱto be

J; ⁱ Z

jj_a0^a_iⁱ;

where₀^a_i is the projection ofR³to the tangent space of R³.

Positivity of quasilocal energy.—Let be a compact spacelike hypersurface in a time orientable spacetimeM.

Let g_ij denote the metric of , and let K_ij denote the extrinsic curvature of inM, as before. The local mass densityand the local current densityJⁱonare related tog_ij andK_ijby the constraint equations

¹₂RK^ijK_ijK²; JⁱD_jK^ijKg^ij; where Kg^ijK_ij is the trace of the extrinsic curvature, andR is the scalar curvature of the metricg_ij.

Theorem 1.—Let,,Jbe as above. We assume that andJⁱsatisfy the local energy condition

JⁱJ_i q

;

and the boundary @ has finitely many connected com- ponents ₁;. . .;_l, each of which has positive intrinsic curvature. Then the quasilocal energy

E@ X^l

#1

E_#:

of @ is strictly positive unless M is a flat spacetime along . In this case, @ is connected and will be embedded intoR³ R^3;1 by Weyl’s embedding theorem.

When the extrinsic curvatureK_ijof inMvanishes, the local energy condition reduces toR0, and we have E_#; E_#. In this case, Shi and Tam proved in [10] thatE_# 0for each#, andE_# 0for some

#if and only if@is connected andis a domain inR³. We now briefly describe Shi and Tam’s proof. Each_# can be isometrically embedded toR³as a strictly convex hypersurface diffeomorphic to a 2-sphere. Let N be the complete three-dimensional manifold obtained by gluing the regionE_#exterior to _# inR³ to along_#. The region E_# is foliated by dilations_#;r of_#, whereris the distance from _#. Shi and Tam showed that the Euclidean metric on E_# can be deformed radially to a metric with zero scalar curvature such that its restriction to_#is unchanged and the mean curvature of_#inE_# coincides with that in. This gives an asymptotically flat metric on N which is smooth away from @ and Lefschitz near@. Shi and Tam proved that the positive mass theorem holds for such a metric. The quasilocal energy of _#;r is nonincreasing in r and tends to the ADM mass of the end E_# as r! 1, from which one derives the result.

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We shall reduce Theorem 1 to Shi and Tam’s result by a procedure used by Schoen and the second author in [11]

and also by the second author in [5].

As in [11], we consider the following equation pro- posed by Jang [12]:

g^ij fⁱf^j 1 jrfj²

f_ij 1 jrfj² p K_ij

0: (4) We first assume that there is no apparent horizon in. Here an apparent horizon is a smoothly embedded 2- sphere S in satisfying k_sp_s0 or k_sp_s0, where k_s is the extrinsic curvature ofS with respect to the inward unit normalvⁱ_s, andp_sKK_ijvⁱ_sv^js. Under this assumption, there exists a solutionfto (4) onsuch thatfj_@ 0. The induced metric of the graph_f off inR; g_ijdxⁱdx^jdt² is gg_ij g_ijf_if_j. Let R

Rbe the scalar curvature of the metricgg_ij, and leth_ij be the extrinsic curvature of_f inR. Then

R RX

h_ijK_ij²2X

i

h_i4K_i4² 2X

i

D

D_ih_i4K_i4; (5) where the index 4 corresponds to the downward unit normal to_f, DD_iis the covariant derivative of gg_ij. The above inequality implies that there is a unique solution to

u¹₈RRu 0 (6) on such that uj_@1. The solution is everywhere positive, sogg^_ij u⁴gg_ij is a metric with zero scalar cur- vature and coincides withgg_ij on@. Letkk andkk^denote the traces of the extrinsic curvatures kk_ab andkk^_ab of@ with respect to gg_ij and gg^_ij, respectively. Then kk^kk 4u_ivvⁱ, where vvⁱ is the outward unit normal of @ in

;gg_ij. Applying Stokes’ theorem and using (6), (5), we have

Z

@

k^ kZ

@

k k4Z

u_iuⁱRR 8u²

Z

@

kk h_i4K_i4vvⁱ;

where the equality holds if and only ifRR 0,gg^_ij gg_ij, andh_ij K_ij.

Letuu be the downward unit normal of_f inR. Letwⁱand vvⁱbe the unit outward normals of@ in₀ (the graph of the zero function) and_f, respectively. We viewwⁱandvvⁱas four vector fieldswandvvalong@.

It was computed in [5], Section 5, that k

k h_i4K_i4vvⁱ uuw v

vwp 1 v

vwk: (7) Using uuw² vvw²ww 1, one can check that the right-hand side of (7) is greater or equal to

k²p²

p . Hence,

Z

@

kk h_i4K_i4vvⁱ Z

@

k²p² q

Z

@

8 p :

By Shi and Tam’s result, Z

@

k^ k Z

@

k₀;

where the equality holds if and only if@ is connected and;gg^_ijis a domain inR³. We conclude that

E@ 1

8G Z

@

k₀ p8

0;

and the equality holds if and only ifis diffeomorphic to a domain₀ R³and can be isometrically embedded in R^3;1 as a graphfx; fx jx2₀g R^3;1 with extrinsic curvatureK_ij, wherefis a smooth function on₀which vanishes on@₀. This completes the proof of Theorem 1 if there is no apparent horizon in.

In general, solutions to (4) which vanish on @ are defined only on the exterior of a finite family of apparent horizons, but the above proof can be modified as in [11] to give Theorem 1.

Quasilocal mass of the black-hole.—When we are on the apparent horizon, the second term in (3) vanishes;

only the reference extrinsic curvaturek₀ comes in. In this case, a well-known Minkowski inequality says that

Z

S

k₀

16A p

;

for a convex surfaceSin the Euclidean spaceR³, wherek₀ is the trace of the extrinsic curvature ofSinR³, andAis the area ofS. Hence, our quasilocal mass must be greater

than or equal to

A=4G² p .

Based on the second author’s work in [5], we conjecture that, when the quasilocal mass of a surface with positive curvature is greater than a universal constant times the square root of its area, a black hole must form in its vicinity. The proof of such a statement will then give qualitatively a necessary and sufficient condition for a black hole to form.

Remark.—Booth and Mann showed in [13] that, with- out the assumption that the timelike boundary is orthog- onal to the foliation of the spacetime, the Brown-York derivation yields boost invariant quantities.

We wish to thank R. Bousso, L. Motl, S. F. Ross, and A. Strominger for helpful conversations. We also wish to thank I. S. Booth and R. B. Mann. The second author is supported in part by the National Science Foundation under Grant No. DMS-9803347.

*Electronic address: ccliu@math.harvard.edu

†Electronic address: yau@math.harvard.edu

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