The mass of light-cones

0.17 MB·

by  Piotr T. Chruściel

The mass of light-cones
∗
Piotr T. Chru´sciel† and Tim-Torben Paetz‡
Gravitational Physics, University of Vienna
Boltzmanngasse 5, 1090 Vienna, Austria
4 June 12, 2014
1
0
2
n Abstract
u
Wegiveanelementaryproofofpositivityoftotalgravitationalenergy
J
in space-times containing complete smooth light-cones.
1
1 PACS: 04.20.Cv, 04.20.Ex,04.20.Ha

]
c Oneofthe deepestquestionsarisingingeneralrelativityisthatofpositivity
q of total energy. Years of attempts by many authors have led to an affirmative
- answer in the milestone papers of Schoen and Yau [13, 14] and Witten [17]; see
r
g also [6, 10, 15]. These proofs use sophisticated PDE techniques, with positivity
[ resultingfromtheanalysisofsolutionsofseeminglyunrelatedpartialdifferential

2 equations. The aim of this letter is to show that an elementary direct proof of
v positivity can be givenfor a largeclassof space-times,namely those containing
9 globally smooth light-cones. As a bonus, our proof gives an explicit positive-
8
definite expression for the mass, equation (31), in terms of physically relevant
7
fields, such as the shear of the light-cone.
3
. Thus, consider a globally smooth, null-geodesically complete light-cone in
1
an asymptotically Minkowskian space-time. The formula (47) below for the
0
mass in this context has been derived by Bondi et al. [2, 12], compare [16].
4
1 We show how to rewrite this formula in terms of geometric data on the light-
: cone,Equation(37)below. TheconstraintequationsinducedbyEinstein’sfield
v
i equationsonthelight-conearethenusedtoobtainourmanifestlypositivemass
X
formula (31) by elementary manipulations.
r The initial data on the light-cone comprise a pair (N ,gˇ), where N =R3
a 0 andgˇisasmoothfieldofsymmetrictwo-covarianttensorsonN ofsignatur\e
{(0,}+,+)suchthatgˇ(∂ , )=0.1 ThevertexO ofthelight-coneC :=N O
r O
islocatedattheoriginof·R3,andthehalf-raysissuedfromtheorigincorres∪p{ond}
to the generators of C . For simplicity we assume throughout that the initial
O
data leadto a smoothspace-timemetric, cf.[5]. The requirementsofregularity
∗Preprint UWThPh-2014-2. Supported in part by the Austrian Science Fund (FWF): P
24170-N16. PTCisgratefultotheStanfordMathematicsResearchCenterforhospitalityand
supportduringworkonthispaper.
†URLhomepage.univie.ac.at/piotr.chrusciel,[email protected]
‡[email protected]
1To avoid an ambiguity in notation we write gˇfor what was denoted by g˜ in [4], as g˜ is
usuallyusedfortheconformallyrescaledmetricwhendiscussingI+.
1
at the origin, asymptotic flatness, and global smoothness, lead to the following
restrictions on gˇ:
Letting (r,xA), A 2,3 , denote spherical coordinates on R3, and writing
∈{ }
s dxAdxB fortheunitroundmetriconS2,regularityconditionsatthevertex
AB
imply that the coordinate r can be chosen so that for small r we have
gˇ g dxAdxB =r2(s +h )dxAdxB , (1)
≡ AB AB AB
h =O(r2), ∂ h =O(r2), ∂ h =O(r), (2)
AB C AB r AB
see[4,Section4.5]foradetaileddiscussion,includingpropertiesofhigherderiva-
tives; the latter are assumedimplicitly whenever needed below. Here, and else-
where, an overline denotes a space-time object restricted to the light-cone.
Existence of a Penrose-type conformal completion implies that the coordi-
nate r can be chosen so that for large r we have
gAB =r2(gAB)−2+r(gAB)−1+ψAB , (3)
ψ =O(1), ∂ ψ =O(1), ∂ ψ =O(r−2), (4)
AB C AB r AB
∂2ψ =O(r−3), ∂ ∂ ψ =O(1), (5)
r AB C D AB
∂ ∂ ψ =O(r−2), ∂ ∂ ∂ ψ =O(r−2), (6)
r C AB r C D AB
∂2∂ ψ =O(r−3), ∂2∂ ∂ ψ =O(r−3), (7)
r C AB r C D AB
for some smooth tensors (gAB)−i =(gAB)−i(xC), i=1,2, on S2.
Equations(3)-(7) arenecessaryfor existence of a smoothI+, but certainly
not sufficient: our conditions admit initial data sets which might lead to a
polyhomogeneous but not smooth I+; see [1, 7, 8, 11].
Wedenotebyτ thedivergence,sometimescalledexpansion,ofthelight-cone:
1
τ :=χ A , where χ B := gBC∂ g . (8)
A A 2 r AC
Conditions (1)-(7) imply
τ = 2r−1+τ r−2+O(r−3) for large r , (9)
2
τ = 2r−1+O(r) for small r . (10)
In particular τ is positive in both regions. Now, standard arguments show
that if τ becomes negative somewhere, then the light-cone will either fail to be
globallysmooth,orthespace-timewillnotbenull-geodesicallycomplete. Soour
requirement of global smoothness of the light-cone together with completeness
of generators imposes the condition2
τ >0. (11)
We further require
det(gAB)−2 >0, (12)
which excludes conjugate points at the intersection of the light-cone with I+.
Both conditions will be assumed to hold from now on.
2The Raychaudhuri equation (13) below with κ = 0 implies that τ is monotonous non-
increasing, which yields (11) directly in any case after noting that the sign of τ is invariant
underorientation-preservingchanges ofparametrisationofthegenerators.
2
Theinequality(11)impliesthattheconnectioncoefficientκ,definedthrough
the equation
∂ =κ∂ ,
∇∂r r r
and measuring thus how the parameter r differs from an affine-one, can be
algebraically calculated from the Raychaudhuriequation:
τ2
∂rτ κτ + + σ 2+8πTrr N =0
− 2 | | |
1 τ2
κ= ∂rτ + + σ 2+8πTrr N . (13)
⇐⇒ τ 2 | | |
(cid:0) (cid:1)
Here σ is the shear of the light-cone:
1
σ B =χ B τδ B , (14)
A A A
− 2
which satisfies
σ B =O(r) for small r , σ B =O(r−2) for large r. (15)
A A
Assuming that
Trr N =O(r−4) for large r, (16)
|
we find from (13) and our previous hypotheses
κ=O(r) for small r, κ=O(r−3) for large r. (17)
Fortheproofofpositivityofthe Trautman-Bondimassitwillbeconvenient
to changer toa new coordinatesothat(gAB)−2 in(3) isthe unit roundsphere
metric and that the resulting κ vanishes (i.e. the new coordinate r will be an
affine parameter along the generators of the light-cone). Denoting momentar-
ily the new coordinate by r , and using the fact that every metric on S2 is
as
conformal to the unit round metric s , the result is achieved by setting
AB
r
r (r,xA) = Θ(xA) eH(rˆ,xA)drˆ, (18)
as
Z
0
∞
H(r,xA) = κ(r˜,xA)dr˜, (19)
−Z
r
Θ(xA) = det(gAB)−2 1/4 . (20)
dets
(cid:16) AB (cid:17)
The functions r r (r,xA) are strictly increasing with r (0,xA) = 0. Equa-
as as
7→
tion (17) shows that there exists a constant C such that for all r we have
e−C eH eC , (21)
≤ ≤
which implies that limr→∞ras(r,xA)=+ . We conclude that for eachxA the
function r r (r,xA) defines a smooth∞bijection from R+ to itself. Conse-
as
7→
quently, smooth inverse functions r r(r ,xA) exist.
as as
7→
We have normalised the affine parameter r so that
as
ras(r,xA) = Θ(xA)r+(ras)∞(xA)+O(r−1), (22)
3
for large r, which implies
r (r,xA) = (r ) (xA)r+O(r3) for small r , (23)
as as 0
where
∞
(ras)∞(xA) = Θ(xA) eH(r,xA) 1 dr , (24)
Z −
0 (cid:0) (cid:1)
(ras)0(xA) = Θ(xA)e−R0∞κ(r˜,xA)dr˜. (25)
After some obvious redefinitions, for r large (3) becomes
as
gAB =ra2ssAB+ras(gAB)−1+ψAB . (26)
The boundary conditions (4)-(7) remain unchanged when r is replaced by r
as
there. On the other hand, (1)-(2) will not be true anymore. However, we note
for further use that the coordinate transformation (18) preserves the behavior
of τ and σ near the vertex. Indeed, inserting r = r(r ) into (3) and using the
as
definitions (8) and(14)one finds that(9)-(10) and(15)continue to hold with r
replaced by r .
as
A key role in what follows will be played by the equation [4, Equations
(10.33) and (10.36)],3
1
(∂ +τ +κ)ζ+Rˇ ξ 2+gAB˜ ξ =S , (27)
r A B
− 2| | ∇
where ξ 2 := gABξ ξ . In coordinates adapted to the light-cone as in [4] the
A B
| |
space-time formula for the auxiliary function ζ is
ζ = 2gABΓr +τgrr , (28)
AB N
(cid:0) (cid:1)(cid:12)
andζ isinfactthedivergenceofthefamilyofsuita(cid:12)blynormalizednullgenerators
normal to the spheres of constant r and transverse to N . Here ˜ denotes the
∇
Levi-Civita connection of gˇ viewed as a metric on S2 (more precisely, an r-
dependent family of metrics). The symbol Rˇ denotes the curvature scalar of
gˇ. The connection coefficients ξA ≡ −2ΓrrA|N are determined by [4, Equation
(9.2)]:
1 1
2(∂r+τ)ξA−∇˜BσAB + 2∂Aτ +∂Aκ=−8πTrA|N . (29)
Finally,
S := 8π(gABTAB gµνTµν)N
− |
= 8π grrTrr+2grATrA+2gurTur N , (30)
− |
(cid:0) (cid:1)
with T =T(∂ ,∂ ), where ∂ is transverse to N . The first equality makes it
ur u r u
clear that S does not depend upon the choice of coordinates away from N . In
a coordinate system where grr = grA = 0 we have S = 16πgurTur N which,
− |
with our signature ( ,+,+,+), is non-negative for matter fields satisfying the
−
dominant energy condition when both ∂ and ∂ are causal future pointing, as
r u
will be assumed from now on.
3On the right-hand-side of the second equality in [4, Equation (10.36)] a term τg11/2 is
missing.
4
Letting dµ = detg dx2dx3, we derive below the following surprising
gˇ AB
formula for the Traputman-Bondi [2, 12, 16] mass mTB of complete light-cones:
∞
mTB = 1 1 ξ 2+S+(σ 2+8πTrr N)eRr∞ r˜τ2−r˜2dr˜ dµgˇdr. (31)
16π Z0 ZS2(cid:18)2| | | | | (cid:19)
The coordinate r here is an affine parameter along the generators normalised
so that r = 0 at the vertex, with (26) holding for large r. Positivity of m
TB
obviously follows in vacuum. For matter fields satisfying the dominant energy
condition we have S 0, T 0 and positivity again follows.
rr
Notethatsincem≥ decrea≥seswhensectionsofI+ aremovedtothefuture,
TB
(31)providesanaprioriboundontheintegralsappearingtherebothonN and
for all later light-cones, which is likely to be useful when analysing the global
behaviour of solutions of the Einstein equations.
Toprove(31),weassume(16). Wechangecoordinatesvia(18),andusenow
the symbol r for the coordinate ras. Thus κ = 0, we have (3) with (gAB)−2 =
s , and further (4)-(7), (9)-(10) and (15) hold. For r large one immediately
AB
obtains
detg = r2 dets 1 τ r−1+O(r−2) . (32)
AB AB − 2
p p (cid:0) (cid:1)
Let us further assume that, again for large r,
TrA N =O(r−3), S =O(r−4). (33)
|
It then follows from (29) and our remaining hypotheses that ξ satisfies
A
ξ =(ξ ) r−1+o(r−1) and ∂ ξ =O(r−1), (34)
A A 1 B A
for some smooth covector field (ξ ) on S2. An analysis of (27) gives
A 1
ζ(r,xA) = 2r−1+ζ (xA)r−2+o(r−2), (35)
2
−
with a smooth function ζ . Regularity at the vertex requires that for small r
2
ξ =O(r2), ζ =O(r−1), (36)
A
whereristheoriginalcoordinatewhichmakestheinitialdatamanifestlyregular
atthevertex. Usingthe transformationformulaeforconnectioncoefficientsone
checks that this behavior is preserved under (23).
We will show shortly that if the light-cone data arise from a space-time
with a smooth conformal completion at null infinity I+, and if the light-cone
intersects I+ in a smooth cross-section S, then the Trautman-Bondi mass of
S equals
1
m = (ζ +τ )dµ , (37)
TB 2 2 s
16π ZS2
where dµ = √dets dx2dx3. Note that this justifies the use of (37) as the
s AB
definition ofmassofaninitialdatasetonalight-conewithcompletegenerators,
regardless of any space-time assumptions.
It follows from (9), (32) and (35) that for large r we have
ζdµ = ( 2r+ζ +o(1))(1 τ r−1+O(r−2))dµ
gˇ 2 2 s
ZS2 ZS2 − −
= 8πr+ (ζ +2τ )dµ +o(1). (38)
2 2 s
− ZS2
5
This allows us to rewrite (37) as
16πm = lim ζdµ +8πr τ dµ . (39)
TB r→∞(cid:0)ZS2 gˇ (cid:1)−ZS2 2 s
To establish (31), first note that from (27) with κ = 0 and the Gauss-Bonnet
theorem we have, using ∂ detg =τ detg ,
r AB AB
p p
1
∂ ζdµ = 8π+ ξ 2+S dµ . (40)
r gˇ gˇ
ZS2 − ZS2(cid:0)2| | (cid:1)
Integrating in r and using (33)-(36) one obtains
∞
1
lim ζdµ +8πr = ξ 2+S)dµ dr .
r→∞(cid:0)ZS2 gˇ (cid:1) Z0 ZS2(cid:0)2| | gˇ
Next,let τ :=2/r, δτ :=τ τ . Itfollowsfromthe Raychaudhuriequation
1 1
−
with κ=0 that δτ satisfies the equation
dδτ τ +τ
+ 1δτ = σ 2 8πTrr N .
dr 2 −| | − |
Letting
r r˜τ 2
Ψ=exp − dr˜ , (41)
Z 2r˜
(cid:0) 0 (cid:1)
and using (9)-(10) and (15)-(16) one finds
r
δτ(r) = r−2Ψ−1 (σ 2+8πTrr N)Ψr2dr
− Z | | |
0
= τ r−2+o(r−2), (42)
2
where
r
τ2 =−rl→im∞Ψ−1Z (|σ|2+8πTrr|N)Ψr2dr ≤0. (43)
0
Inserting this into (39) gives (31) after noting that
dµgˇ =e−Rr∞ r˜τr˜−2dr˜r2dµs . (44)
To continue, suppose that m vanishes. It then follows from (43) that
TB
Trr N =0=σ. In vacuum this implies [3] that the metric is flat to the future
|
of the light-cone. In fact, for many matter models the vanishing of T on the
rr
light-cone implies the vanishing of T to the future of the light-cone [3], and
µν
the same conclusion can then be obtained.
It remains to establish (37). We decorate with a symbol “Bo” all fields
arisinginBondicoordinates. ConsidercharacteristicdatainBondicoordinates,
possibly defined only for large values of r . The space-time metric on N =
Bo
uBo =0 can be written as
{ }
g =gBodu2 +2νBodu dr +2νBodu dxA +gˇBo . (45)
00 Bo 0 Bo Bo A Bo Bo
UndertheusualasymptoticconditionsonνBo andνBo onehas(see,e.g.,[7,12])
A 0
2M(xA )
gBo = 1+ Bo +O(r−2), (46)
00 − r Bo
Bo
6
and the Bondi mass is then defined as
1
m = Mdµ . (47)
TB s
4π ZS2
In Bondi coordinates (28) becomes
ˇAνBo (gBo)rr
ζBo =2 ∇ A , (48)
(cid:16) ν0Bo − rBo (cid:17)
which allows us to express (gBo)rr in terms of ζBo, leading eventually to
gBo = (gBo)ABνBoνBo (νBo)2(gBo)rr
00 A B − 0
= 1+ ζ2Bo−2∇˚A(νABo)0 +O(r−2), (49)
− 2r Bo
Bo
where ˚ is the Levi-Civita connection of the metric s dxAdxB, and (νBo)
∇ AB A 0
is the r-independent coefficient in an asymptotic expansion of νBo. Comparing
A
with (46), we conclude that
1
m = ζBodµ . (50)
TB 16π ZS2 2 s
To finish the calculation we need to relate ζBo to the characteristic data. In
2
Bondi coordinates we have
2
τ = ,
Bo
r
Bo
and the Raychaudhuri equation implies that
σBo 2+8πT
κBo =r | | rr .
Bo
2
The equation κ Γr = 0 together with the usual transformation law for
≡ rr
connection coefficients gives the equation
∂r ∂r
∂ Bo +κBo Bo =0
rBo ∂r ∂r
(cid:16) (cid:17)
= ∂r =e−Rr∞BoκBo =1+O(r−2),
⇒ ∂r Bo
Bo
where we have used the asymptotic condition limr→∞ ∂r∂Bro =1. Hence
r = rBoe−Rr∞BoκBo
Z
0
∞
= rBo+Z e−Rr∞BoκBo −1 +O(rB−o1).
0 (cid:0) (cid:1)
=:−(rBo)0
| {z }
To continue, we note that xA =xA and
Bo
gBo (r ,xA)=g r(r ,xA),xA ,
AB Bo AB Bo
(cid:0) (cid:1)
7
which implies
2 2 τ
= τBo =τ∂ r = + 2 +O(r−3) 1+O(r−2)
r rBo r r2
Bo (cid:0) (cid:1)(cid:0) (cid:1)
2 τ
= + 2 +O(r−3),
r r2
equivalently
τ τ
r =r 2 +O(r−1) = (r ) = 2 . (51)
Bo Bo 0
− 2 ⇒ − 2
We are ready now to transform ζ as given by (28) to the new coordinate
system:
ζBo = 2(gBo)AB(ΓBo)rBo +τBo(gBo)rBorBo
AB
∂r ∂xi ∂xj ∂r ∂2r
= 2(gBo)AB Bo Γk + Bo
(cid:16) ∂xk ∂xABo∂xBBo ij ∂r ∂xABo∂xBBo(cid:17)
∂r ∂r ∂r
+τ Bo Bogij
∂r ∂xi ∂xj
Bo
∂r ∂r
= Boζ+2 Bo∆ r+O(r−3),
∂r ∂r gˇ Bo
where ∆ is the Laplace operator of the two-dimensional metric gˇ dxAdxB.
gˇ AB
From this one easily obtains
ζBo =ζ +τ +∆ τ . (52)
2 2 2 s 2
Inserting (52) into (50) one obtains (37), which completes the proof.
In fact, the calculations just made show that the integral (37) is invariant
under changes of the coordinate r of the form r r+r (xA)+O(r−1).
0
LetS be a sectionofI+ arisingfromasmoo7→thlight-cone asabove,andlet
S′ be any section of I+ contained entirely in the past of S. The Trautman-
′
Bondi mass loss formula shows that m (S ) will be larger than or equal to
TB
m (S) [9, Section 8.1]. So, our formula (31) establishes positivity of m for
TB TB
′
all such sections S .
Inmanycasesthepastlimitofm istheADMmass,andoneexpectsthisto
TB
be true quite generally for asymptotically Minkowskianspace-times. Whenever
thisand(31)hold,weobtainanelementaryproofofnon-negativityoftheADM
mass.
It is tempting to use a density argument to remove the hypothesis of non-
existence of conjugate point precisely at I+ along some generators; such gen-
erators will be referred to as asymptotically singular. For this one could first
rewrite (31) as
∞
1 1
m = ξ 2+S
TB
16π Z0 ZS2(cid:18)2| |
+(σ 2+8πTrr N)eRr∞ r˜τ2−r˜2dr˜ r2e−Rr∞ r˜τr˜−2dr˜dµsdr, (53)
| | | (cid:19)
where we have used (44). One might then consider an increasing sequence of
tensors σ which converge to σ as i so that σ 2 converges to σ 2 from
i i
→ ∞ | | | |
below, and such that for each i the associated solution τ of the Raychaudhuri
i
8
equation leads to a metric satisfying (12). It is easy to see that τ is then
i
decrasing to zero along the asymptotically singular generators as i tends to
infinity, monotonically in i, which leads to an infinite integral ∞ r˜τ−2dr˜on
− r r˜
those generators. It is however far from clear if and when this diRvergence leads
to a finite volume integral after integrating over the generators, and we have
not been able to conclude along those lines.
Suppose, finally, that instead of a complete light-cone we have a smooth
characteristic hypersurface N with an interior boundary S diffeomorphic to
0
S2, and intersecting I+ transversally in a smooth cross-section S as before.
Let r be an affine parameter on the generators chosen so that S = r = r
0 0
{ }
for some r > 0 and such that (3)-(7) hold. The calculations above give the
0
following formula for m :
TB
mTB =r.h.s. of (53)+ 1 8πr0+ ζ+(2r−1 τ)eRr∞0 rτ2−r2dr dµgˇ , (54)
16π(cid:16) Zr=r0(cid:0) − (cid:1) (cid:17)
with the range r [0, ) replaced by r [r , ) in the first integral symbol
0
∈ ∞ ∈ ∞
appearing in (53). (For outgoing null hypersurfaces issued from any sphere of
symmetry in the domain of outer-communications in Schwarzschild, the term
multiplyingtheexponentialin(54)andtheright-handsideof(53)vanish,while
the remaining terms add to the usual mass parameter m.)
Equation (54) leads to the following interesting inequality for space-times
containing white hole regions. We will say that a surface S is smoothly visi-
ble from I+ if the null-hypersurface generated by the family of outgoing null
geodesics is smooth in the conformally rescaledspace-time. Assume, then, that
S is smoothly visible and weakly past outer trapped:
ζ 0. (55)
S
| ≥
Smooth visibility implies that τ > 0 everywhere, in particular on S. By a
translation of the affine parameter r of the generators of N we can achieve
2
r = . (56)
0
sup τ
S
All terms in (54) are non-negative now, leading to the interesting inequality
1
m . (57)
TB
≥ sup τ
S
Further, equality implies that S and T vanish along N , and that we have
rr
2 1
ξ =σ =0 along N , ζ =0, τ = . (58)
S S
| | ≡ r m
0 TB
Integrating the Raychaudhuri equation gives
2
τ = , r r . (59)
0
r ≥
The equations τg =2χ =∂ g together with the asymptotic behaviour
AB AB r AB
of the metric imply
g =r2s , r r . (60)
AB AB 0
≥
9
ItfollowsthattheoutgoingnullhypersurfaceissuedfromS canbeisometrically
embeddedintheSchwarzschildspace-timewithm=m asanullhypersurface
TB
emanatingfromasphericallysymmetriccross-sectionofthe pasteventhorizon.
Time-reversal and our result provide, of course, an inequality between the
mass of the past directed null hypersurface issuing from a weakly future outer
trapped surface S and sup ζ in black hole space-times.
S
We finish this paper by noting that formulae such as (54), together with
monotonicity of mass, might be useful in a stability analysis of black hole solu-
tions, by chosing the boundary to lie on the initial data surface, for then one
obtainsanaprioriL2-weightedboundon σ 2 and ξ 2 onallcorrespondingnull
| | | |
hypersurfaces.
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