On horizon constraints and Hawking radiation

0.12 MB·English

by  David Mattingly

On horizon constraints and Hawking radiation
David Mattingly
Department of Physics, University of California-Davis∗
Questions about black holes in quantum gravity generally presuppose the presence of a horizon.
RecentlyCarliphasshownthatenforcinganinitialdatasurfacetobeahorizonleadstothecorrect
form for the Bekenstein-Hawking entropy of the black hole. Requiring a horizon also constitutes
fixed background geometry, which generically leads to non-conservation of the matter stress tensor
atthehorizon. Inthiswork,IshowthatthegeneratedmatterenergyfluxforaSchwarzschildblack
hole is in agreement with thefirst law of black hole thermodynamics, 8πG∆Q=κ∆A.
INTRODUCTION is a horizon. In a recent work, Carlip has shown that
imposition of “horizon constraints” yields the appropri-
6 ate number of microstates for black hole entropy [5]. In
In the search for a theory of quantum gravity, black
0 this approach H is a spacelike initial data surface con-
holes play an important role. Since black holes behave
0
strained to be a stretched horizon. For a review of both
2 as thermodynamic objects, with an entropy given by
S =A/4~G [1, 2], they presumably are connectedto the these approaches see also [6] and references therein.
n
a underlying microstates of quantum gravity. Indeed, al- The notionofa stretchedhorizonis importantfor this
J most every modern approach to quantum gravity makes work, so I now take a little time to explain what it is

1 attempts, with varying degrees of success, to show that and why it is useful. Naively, one would want H to be a
1 the number of microstates for a black hole matches that true black hole event horizon. This is problematic how-

predictedbyblackholethermodynamics(see[3]forare- ever as the event horizon is a global object and requires
1
view). With a top-down approach this is feasible - one knowledgeabouttheentirespacetimeasopposedtoalo-
v
4 hasthefundamentaltheoryandcandefinetheparticular cal picture for microstates. A local definition of horizon
4 solutions that represent a black hole in the macroscopic which still captures black hole thermodynamics is that
0 limit. These solutions have a statistical mechanics de- of an isolated horizon [7] (see also [8, 9] for overviews
1
termined by the fundamental theory, and the resulting of the isolated horizons program). An isolated horizon
0
macroscopic thermodynamics can be compared with the is essentially just a null hypersurface with vanishing ex-
6
0 laws of black hole thermodynamics. pansion of the null normal.1 This is still not quite good
/ Unfortunately, we don’t have a complete theory of enough, as the approachof Carlip requiresthat H be an
c
q quantum gravity. It is therefore useful to analyze the initial data surface and hence spacelike. If, however,one
- statistical mechanics of black holes from a bottom up takes H to be a spacelike stretched horizon, which is de-
r
g approach as well, in the hopes that information might finedasanearlynullspacelikehypersurfacethattracesa
: be derived about the physics the statistical mechanics true isolated horizon, then one can use the Hamiltonian
v
i of the black hole microstates should follow. In doing so formulation of general relativity.
X
we immediately run into a puzzling problem. Black hole
AssumingthatH isastretchedhorizon,Carliplooksat
r thermodynamics is well defined in the semiclassical ap-
a the algebraofdiffeomorphismsonH in2-ddilatongrav-
proximation: weknowwhatspacetimerepresentsablack
ity. Theusualgeneratorsoftranslationsalongthesurface
hole and can investigate the behavior of quantum fields
andlocalLorentzboostsingeneralrelativitydo notpre-
and small gravitational fluctuations on top of the back-
serve the horizon constraints. If H is to be a horizon,
ground spacetime. In a full theory of quantum gravity
however, then the generators must preserve the horizon
this splitting is unacceptable. There is no background
constraints, which means that they should be modified
spacetime available and metric fluctuations due to the
fromtheirusualform. The necessarymodificationofthe
uncertainty principle will prevent the localization of a
generatorsleadstoageneratoralgebraequivalenttothat
horizon. Now our problem becomes clear: if we cannot
of a conformal field theory with non-zero central charge
even identify the states that represent a black hole, how
andacalculationofthedensityofstatesusingtheCardy
can we possibly analyze their statistical mechanics?
formula yields the standard result for the entropy of a
Two approaches have been developed for dealing with
this issue. The first is the “horizon as boundary” ap-
proach[4]whichtreatstheblackholehorizonasabound-
ary H. Imposing boundary conditions on the path inte- 1 Isolated horizons also have some other properties, one of which
gral for the metric inside and outside the horizon such is that all the field equations are satisfied on the horizon. In
that H is a horizon yields the appropriate number of theapproachpresentedhere,theclassicalmatterfieldequations
are deliberately not satisfied on H. Therefore H is not a true
boundary states. Another, more recent, alternative is to
isolated horizon, however the geometric characteristics are the
impose constraints directly on the theory such that H same.
2
black hole. is no guarantee, of course, that the flux generated from
This result is intriguing as it tells us about the den- requiringH tobeadynamicalhorizonshouldmatchthis
sity of states from the classical symmetries of the ac- law. However, I show that for Schwarzschild geometry
tion, rather than some specific quantum gravity model. thisactuallyisthecaseuptoanundeterminedconstant,
However, it also suffers some limitations. The primary and that the exact first law is recovered if this constant
shortcomingisthattheconstraintapproachintheHamil- ischosentobe unity. Inowturntothederivationofthis
tonian formulation cannot handle dynamics (as the ini- result.
tial data surface is the horizon) and therefore cannot
yieldinformationaboutblackholeevolution. Incontrast,
MODIFIED EINSTEIN EQUATIONS
the first law of black hole thermodynamics is specifically
about dynamics. In this paper, I show how the imposi-
tionofhorizonconstraintsforadynamicalSchwarzschild Consider a local region D of a manifold M in a co-
black hole generates a matter energy-momentum flux at ordinate system xβ = (u,v,y,z). (Eventually spherical
the horizon that agrees with the first law of black hole symmetry will be assumed and y,z will be the angular
thermodynamics. directions.) I can define a hypersurface H by the condi-
tionu=0andavectorfieldhα suchthathα generates
Intuitivelyhowdoesthishappen? Letusstepbackand |H
H. One should think of hα as the (approximate) Killing
consider more generally what happens to the structure
fieldforwhichH istobeaKillinghorizon. Furthermore,
of general relativity when we require H to be a horizon.
defineacovectorfieldi ,proportionalto∂/∂v,suchthat
Relativityisoftenlabelledas“diffeomorphisminvariant” α
on H I have i h = 1. This construction is metric in-
and,ofcourse,itis. However,specialrelativityisalsodif-
· −
dependent since I have not yet imposed any restrictions
feomorphism invariant (as is almost any reasonable the-
ong . Iremindthe readerthathα andi arenotto be
ory of physics that we have) since it produces the same αβ α
considereddynamicalfields,butratherfixedbackground
observableresultsnomatterwhatcoordinatesystemone
geometrical objects.
uses to calculate in. The real way in which general rela-
I now must decide how to impose the constraints that
tivityisuniqueisthattherearenobackgroundgeometri-
make H a horizon. A horizon emitting Hawking radia-
cal objects, unlike special relativity where the spacetime
tion is not quite null, but instead slightly timelike [12].
metric is fixed to be the Minkowski metric. The fact
Therefore the Hamiltonian formulation of Carlip is not
that everyfieldin the actionis dynamicalhas important
directly applicable as H cannot be an initial data sur-
consequences;the most relevant of which for this paper
face. Instead I shall work in the Lagrangianformulation
is that it leads to the conservation of the matter stress
and enforce the constraints via Lagrange multipliers in
tensor via Noether’s theorem.
the action. I take H to be a timelike, instead of space-
The constraint approach violates this tenet of general
like, stretched horizon (a timelike membrane in the lan-
relativity. The requirement that H is a horizon consti-
guageof[8]). Aswell,IwillnotrequiretheexpansionΘ
tutes a fixed backgroundgeometryandhence will gener-
of hα to identically vanish but instead be infinitesimally
ally lead to a conservation anomaly - the matter stress
small,whichwill give an infinitesimal change in the cross
tensor cannot be conserved at the horizon. 2 If the the-
sectional area of the horizon. Hence the constraints are
ory is completely classical then this is a problem as the
matter stress tensoris conservedas a consequence ofthe 1. Φ =hαh +L2 =0
h α
matter equations of motion. The only way out is for
the matter fields to be off shell, i.e. there must be some 2. ΦΘ =Pαβ αhβ Θ=0
∇ −
purelyquantummechanicaleffectinthemattersectorre-
whereL,ΘareinfinitesimalonH andPαβ is the projec-
sponsible for generating matter flux at the horizon. For-
tion tensor Pαβ =gαβ hαhβ/(h h).
tunately, in a black hole spacetime we have exactly such − ·
SofarIhavenotsaidanythingaboutthesurfacegrav-
an effect: Hawking radiation!
ity of the horizon, which also needs to be specified when
Hawking radiationfroma spherically symmetric black
analyzing the dynamics of a black hole. If H was a true
holereducestheareaofthehorizonaccordingtothefirst
nullsurface then L wouldbe zeroandconstantalongH,
lawofblackholethermodynamics,8πG∆Q=κ∆A. ∆Q which implies that the the directional derivative of h2
is the (negative) energy flux of the current T ξβ across
αβ would be of the usual form
the horizon H (where ξβ is the Killing field for which H
is a Killing horizon),κ is the surface gravityofthe black h2 =2κh (1)
α α
∇
hole,andAistheareaofthehorizoncrosssection. There
where κ is the surface gravity. Since hα is not quite null
h2 is not proportional to hα. In spherical symmetry
α
∇
h2 can be written as
α
∇
2 For other examples of how anomalies can be used inblack hole
physicssee[10,11]. αh2 =2κhα+ρiα (2)
∇
3
where ρ is a function that depends on the evolution of the fourvolumeelement√ g does,however,enforcethe
−
the surface. Contracting with iα gives constraintsas√ g is non-vanishingeverywhere(as long
−
as the metric is non-singular).
iα αh2 = 2κ+ρi2. (3) Varying (5) with respect to the Lagrange multipliers
∇ −
enforcesthe constraints. Varying(5)withrespectto gαβ
IfIimposetheconstrainti2 =L2 thenasL 0thei2 and using the identity hα δ(u)=0 (since hα is a gen-
→ α
termdropsoutof(3). So,finally,Ifixthesurfacegravity ∇
erator of H) yields the modified Einstein equation
κ by imposing two additional constraints on H,
(m)
1. Φi =iαiα L2 =0 Gαβ −Cαβ =8πGTαβ . (7)
−
2. Φ =iα h2+2κ=0 Cαβ, the constraint “stress tensor”, is given by
κ α
∇
which gives the right value for κ on the constraint sur- Cαβ =δ(u)[λiiαiβ λhhαhβ +λκi(α β)h2 (8)
− ∇
face. There is still an ambiguity in iα as the above con- λ 1
+ Θ( h h˙ + h h h h˙γ)]
ditions do not uniquely fix iα on D. I use this freedom h2 − (α β) h2 α β γ
to further choose iα such that it satisfies the geodesic 1
equation, iβ βiα = 0. Note that this last choice for iα +hαhβ∇γ(λκδ(u)iγ)+δ(u)2∇γ(λΘhγPαβ)
∇
is not a constraintas it is not a consequence of the pres-
ence of a horizon with surface gravity κ. Therefore it where h˙α = hβ βhα. Note that in (8) I have imposed
∇
will not be implemented with a specific constraint term the constraints and dropped terms that vanish (such as
in the action. Instead, given a solution for g and i those from the variation of √ g) when the constraints
αβ α −
on H I simply choose iα to satisfy the geodesic equation are satisfied. Cαβ can be further simplified by noting
everywhere. that in the limit of L 0, hα is the generator of a null
The constraints are to be imposed only on H. Using hypersurface and so h˙→α = fhα where f is some scalar
the Dirac delta function, defined by function. In this limit the λΘ terms in the secondline of
(8) cancel, leaving the simpler expression
dnxduF(u,x)δ(u)= dnxF(0,x) (4)
Z Z
C =δ(u)[λ i i λ h h +λ i h2 (9)
αβ i α β h α β κ (α β)
the constraints can be enforced only on H by adding − ∇
1
them to the action, + (λ hγP )]+h h (λ δ(u)iγ).
γ Θ αβ α β γ κ
2∇ ∇
S = √ gd4x[R+8πG (m)(Ψ) (5) I now discuss how the presence of Cαβ in the field equa-
Z − L tions affectsthe conservationofthe matter stresstensor.
δ(u)(λ Φ +λ Φ +λ Φ +λ Φ )]
i i h h Θ Θ κ κ
−
where m(Ψ) is the matter Lagrangian(as a function of GENERATED MATTER FLUX
L
matter fields Ψ) and the λ’s are Lagrange multipliers.
The choice of volume element for the constraint terms
Let us consider a simple black hole geometry, that of
requires some explanation. Integrating over u, the extra
a Schwarzschild black hole. Hence from this point on
constraint terms can be rewritten as
spherical symmetry is assumed and all quantities are in-
dependent of y,z. G vanishes and so contracting (7)
αβ
−Z d3x√−g(λiΦi+λhΦh+λΘΦΘ+λκΦκ). (6) with hβ and taking the divergence yields
H
At first glance,this term seems wrong as the integration
α(C hβ)=8πG α(T(m)hβ). (10)
overH naturallyinvolvesthedeterminantoftheinduced −∇ αβ ∇ αβ
metric q onH,ratherthan√ g. However,letuscon-
αβ
− Inowintegrate(10)overaninfinitesimallywideregion
sider more carefully the effect on, for example, the sur-
R that contains a portionof H (Fig. 1). Since the terms
face gravity constraint if the volume element was chosen
in (10) are total divergences Stokes’ theorem gives
to be √ q. On H, the equation of motion derived from
−
varying λ is √ qΦ = 0 which enforces Φ = 0 only if
√−q 6= 0κ. In th−e limκit as L → 0, H becomκes null and −Z∂R√γnαCαβhβ =8πGZ∂R√γnαTα(mβ)hβ (11)
√ q 0 as well. In this limit the constraintΦ is com-
κ
− →
pletely lost as the λ equation of motion is satisfied for where nα is the normal to the boundary ∂R of R and
κ
any value of Φκ. Therefore the natural volume element √γ is determinant of the induced metric γαβ on ∂R. I
onH isnot theappropriatequantitytouseforenforcing choose ∂R such that iα generates ∂R across H. Since
constraints. Integrating the constraints with respect to i is proportional to ∂/∂v and I have the constraints
α
4
ia H is approximately parallel to iα) to H− (where the two
a vectors are approximately anti-parallel). The D term
h γ
in (14) can be integrated over ∂R by parts and becomes
H+
δ(u)λ iγD ((n h)h2). (15)
κ γ
− ·
d R There is no boundary term due to the presence of the
delta function. The entire integral in (11) is then pro-
portional to δ(u). Integrating over the delta function
u v (11) becomes
R
1
8πG∆Q=−Z √γ[λhh2(n·h)− 2λκnα∇αh2(16)
H- H+,H−
λ iγD ((n h)h2)]
κ γ
− ·
wherethe notation signifiesthatthe resultisthe
FIG. 1: Horizon geometry and the surface R. Each point sumoftheintegralsRoHve+r,HH−+andH−. Againreplacingnα
represents a 2-spherein (y,z). by ıα/√i2oneseesthattheλ termvanishesash2 0.
h
± →
ApplyingtheconstraintΦ ,h2 0,and(again)thefact
κ
→
that iα satisfies the geodesic equationthe lasttwo terms
i2,h2 0,i h = 1, iα generates translations in the
≈ · − combine and I have
u direction. I will therefore be integrating over u and
so can easily evaluate the integral over δ(u). The right κ κ
8πG∆Q= √γλ √γλ (17)
hand side of (11) is simply 8πG times the total matter ZH+ κ√i2 −ZH− κ√i2
flux ∆Q emanating from the horizon through the sur-
face ∂R. Without the constraints this would be zero, Now note that in this coordinate system √γ is equal to
corresponding to no anomalous flux for the approximate √i2√γ2,whereγ2 istheinducedmetriconH+,H−. The
Killing field hα. factors of √i2 cancel and I am finally left with
I now evaluate the left hand side of (11). C hβ is
αβ
Cαβhβ =δ(u)[ λhhαh2 λiiα (12) 8πG∆Q=κ(Z √γ2λκ−Z √γ2λκ). (18)
− − H+ H−
1 1
+ λ (i hβ h2 h2) λ P h˙β]
2 κ α ∇β −∇α − 2 Θ αβ Ifλκ variesbetweenH+ andH− thenthereisanon-zero
+h h2 (λ δ(u)iγ) ∆Qevenif the geometrydoesnotchange. However,this
α γ κ
∇ doesnotcorrespondtothecaseofinterest,wheretheflux
whereIhaveappliedthe Liebnizruletoputtheλ term isduetothetimedependenceofthegeometryandnotthe
Θ
intheaboveform. Inthelimitashα isnull,theλ term time dependence of the Lagrange multipliers. Therefore
Θ
vanishes. The presence of the delta functions implies λκ should be chosento be a constant. If I chooseλκ =1
that there is only a contribution to the integral when thenIrecoverthatthegeneratedmatterfluxsatisfiesthe
∂R crosses H (the two dimensional surfaces H−,H+ in first law of black hole thermodynamics,
Figure1). ThereforeIcansafelyusetheidentityn i=0.
·
Contracting with nα and applying n i=0, I have 8πG∆Q=κ( √γ √γ )=κ∆A (19)
· Z 2−Z 2
H+ H−
nαC hβ = δ(u)λ h2(n h) (13)
αβ h
− · which is the promised result. In summary, I have shown
1
δ(u)λ nα h2+(n h)h2 (δ(u)λ iγ). thatrequiringthepresenceofahorizonwithsurfacegrav-
κ α γ κ
−2 ∇ · ∇
ity κ by imposing constraints in the gravitationalaction
I now rewrite the last term in (13) to move the delta also generates a (necessarily quantum mechanical) mat-
function outside the derivative. Using the identity δβ = terfluxwhich,atleastforSchwarzschildgeometry,agrees
α
γβ n nβ the last term can be rewritten as with the first law of black hole thermodynamics.
α− α
(n h)h2 (δ(u)λ iγ) (14)
γ κ
· ∇
=(n h)h2[D (δ(u)λ iγ) δ(u)λ nαnβ i ] DISCUSSION
γ κ κ β α
· − ∇
where Dα = γαβ∇β. The last term in (14) vanishes in Therearesomeopenissueswiththeaboveresult. The
the limit L 0 as nα iα/√i2 and (as chosen previ- most pressing is obviously the value of λ . While the
κ
→ ≈ ±
ously)iα satisfiesthegeodesicequation. Theplus/minus choice λ = 1 is not an exceptional or fine tuned value,
κ
is present as nα changes orientation from H (where nα there is no a priori reason why this choice is the correct
+
5
one. Thedifficultyindeterminingλ stemsfromthefact Acknowledgements
κ
thatIamusingthe variationofthe horizontodetermine
the generated matter flux without reference to any mat-
I heartily thank Steve Carlip and Damien Martin for
ter equations of motion. If the (quantum mechanical)
many helpful discussions and for reading drafts of this
matter response was known then it is possible that the
manuscript. This work was funded under DOE grant
value of λ could be fixed by applying the constraint.
κ
DE-FG02-91ER40674.
However,withoutthematterfieldequationsthismethod
of determining λ is not applicable.
κ
Another open issue is the connection with Carlip’s
method for calculating black hole entropy. One would
hope that a connection can be made since the underly- ∗ Electronic address: [email protected]
ing approach, applying constraints to force a horizon, is [1] J. D.Bekenstein, Phys. Rev.D 7, 2333 (1973).
the same. Ifaconnectioncanbe madeandinthe frame- [2] S. W. Hawking, Nature 248, 30 (1974).
workpresentedheretheentropycanbeshowntobeA/4, [3] R. M. Wald, Living Rev. Rel. 4, 6 (2001)
thenthefirstlawisatruethermodynamicequationsince [arXiv:gr-qc/9912119].
in this derivation it relates the outgoing energy flux to [4] S. Carlip, Nucl. Phys. Proc. Suppl. 57, 8 (1997)
[arXiv:gr-qc/9702017].
the number of microstates of the full quantum theory.
[5] S. Carlip, Class. Quant. Grav. 22, 1303 (2005)
There are a few obstacles that still need to be overcome
[arXiv:hep-th/0408123].
in order to relate the two approaches. Since the hori-
[6] S. Carlip, Horizons, Constraints, and Black Hole En-
zon surface is a timelike hypersurface here (as it must tropy,in preparation.
be to emit Hawking radiation), it is somewhat unclear [7] A. Ashtekar, C. Beetle and S. Fairhurst, Class. Quant.
how to proceed with the Hamiltonian framework. Fur- Grav. 16, L1 (1999) [arXiv:gr-qc/9812065].
thermore, the constraints on the surface gravity do not [8] A. Ashtekar and B. Krishnan, Living Rev. Rel. 7, 10
(2004) [arXiv:gr-qc/0407042].
appear in the formalismof Carlip,so there is no guaran-
[9] I. Booth, arXiv:gr-qc/0508107.
tee that the algebra of constraints is identical. If these
[10] S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95,
technical problems can be overcome,then the constraint
011303 (2005) [arXiv:gr-qc/0502074].
programmight yield not only the number of microstates [11] S. M. Christensen and S. A. Fulling, Phys. Rev. D 15,
of a black hole, but also more directly tie Hawking radi- 2088 (1977).
ation and the laws of black hole thermodynamics to the [12] A. Ashtekar and M. Bojowald, Class. Quant. Grav. 22,
microscopic theory of quantum gravity. 3349 (2005) [arXiv:gr-qc/0504029].