8/18/2019 1505.04753v3

1/8

Entanglement equilibrium and the Einstein equation

Ted Jacobson∗

Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 

 Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742

We establish a link between the semiclassical Einstein equation and a maximal vacuum entanglement hypoth-

esis. The hypothesis asserts that entanglement entropy in small geodesic balls is maximized at fixed volume in

a locally maximally symmetric vacuum state of geometry and quantum fields. For conformal quantum fields,

the hypothesis implies the Einstein equation for first order variations of the local vacuum state, and the Einsteinequation implies that the vacuum entanglement is stationary, in agreement with the hypothesis. For nonconfor-

mal fields, the same conclusions follow modulo a conjecture about the variation of entanglement entropy.

I. INTRODUCTION

When restricted to one side of a spatial partition, the vac-

uum state of a quantum field has entropy because the two sides

are entangled. The entanglement entropy of the restricted state

is dominated by the ultraviolet (UV) field degrees of freedom

near the interface, and hence scales with the area. This is sim-

ilar to the Bekenstein-Hawking black hole entropy,  A/4L2 p,

where   A   is the horizon area and  L p   = ( G/c3

)1/2

is thePlanck length [1–3]. The similarity of these two “area laws” is

striking, and has led to the idea that black hole entropy is just a

special case of vacuum entanglement entropy [4–9]. To match

the Bekenstein-Hawking entropy, vacuum entanglement en-

tropy should be cut off at the Planck scale. Considering the

gravitational back-reaction of vacuum fluctuations, such a cut-

off appears natural [7,  10], but it lies deep in the regime of 

poorly understood quantum gravity effects.

Bekenstein defined the generalized entropy S gen as the sumof the horizon entropy and the ordinary entropy in the exte-

rior. If the horizon entropy is indeed entanglement entropy,

then the (fine-grained) generalized entropy is nothing but the

total von Neumann entropy of the quantum state outside the

horizon [9, 11, 12]. Bekenstein proposed the generalized sec-

ond law (GSL) stating that S gen never decreases [2]. The GSLhas been shown to hold in various regimes [13], the proofs

having been recently strengthened to apply to rapid changes

and arbitrary horizon slices [14, 15]. The validity of the law

depends on the Einstein equation, which relates the curvature

— and therefore the focussing of light rays that determines

the change of horizon area — to the local energy-momentum

density of matter.

The remarkable dovetailing of the Einstein equation with

the GSL calls for a deeper explanation, and suggests that the

Einstein equation might follow directly from the properties of 

entanglement. In fact, we already have evidence that this is the

case. The Einstein equation can be understood as an equationof state of the thermal vacuum [16], and recent work invokes

Anti-de Sitter/conformal field theory (AdS/CFT) duality and

properties of CFT entanglement to derive the linearized Ein-

stein equation for perturbations of AdS spacetime [17–19].

Here I will show that the  nonlinear , semiclassical Einstein

equation is closely related to an equilibrium condition: that

∗  jacobson@umd.edu

vacuum entanglement entropy in a small, maximally sym-

metric causal diamond is   maximal   compared to deformed

configurations with the same spatial volume. The analy-

sis combines the local spacetime setting of the equation of 

state approach, with the CFT entanglement technology of the

AdS/CFT method. It proceeds directly in spacetime, without

the use of holography. In deriving the Einstein equation, three

assumptions are made as input:

(i) A UV cutoff renders the area density of vacuum entan-

glement entropy finite and universal.

(ii) The variation of entanglement entropy of nonconformal

matter fields in a small, maximally symmetric diamond 

is like that for a CFT, but with the energy-momentum

tensor supplemented by an unspecified trace part.

(iii) When the geometry and quantum fields are simultane-

ously varied from maximal symmetry, the entanglement 

entropy in a small diamond is stationary at fixed vol-

ume.

Assumption (i) is supported by the evidence that horizon en-

tropy can indeed be identified with entanglement entropy (see,

e.g. [9, 20, 21], and references therein). However, it involves

UV aspects of quantum gravity that are not currently under-

stood. Assumption (ii) concerns only a property of quantum

fields in conformally flat spacetime. Assumption (iii) is based

on the notion of vacuum as a maximal entropy equilibrium

state, and is ultimately a statement about quantum gravity. If 

instead the Einstein equation is assumed, together with (i) and

(ii), then assumption (iii) is a consequence.

The basic idea is that these assumptions relate the UV part

of the entanglement entropy of a small casual diamond, en-

coded in the area, to the IR part of the entanglement entropy,

encoded in the energy-momentum tensor, thus linking the cur-

vature to the energy-momentum tensor. The diamond must be

taken small enough for the curvature and energy-momentum

tensors to be treated as constant, but larger than the quantum

gravity scale at which the entanglement entropy of the quan-

tum fields is cut off. In the following, we first establish the

relevant geometrical facts, and then proceed to show how the

above assumptions lead to Einstein’s equation.

 a r X i v : 1 5 0 5 . 0

 4 7 5 3 v 3

 [ g r - q c ] 2 1

 D e c 2 0 1 5

mailto:jacobson@umd.edumailto:jacobson@umd.edu

8/18/2019 1505.04753v3

2/8

2

II. AREA DEFICIT AND GENERAL RELATIVITY

Einstein’s field equation,

Gab = 8πG T ab,   (1)

relates the Einstein curvature tensor   Gab   to the energy-momentum tensor of matter,  T ab. Central to our story is the

equivalence of (1) to the statement that the surface area deficitof any small, spacelike geodesic ball of fixed volume is pro-

portional to the energy density in the ball (see Appendix A for

a related statement by Feynman). We begin by demonstrating

this lovely relation.

At any point  o  in a spacetime of dimension  d, choose anarbitrary timelike unit vector   ua, and generate a   (d −  1)-dimensional spacelike ball   Σ   by sending out geodesics of length   from o   in all directions orthogonal to ua. The pointo is the center of the ball, and the boundary  ∂ Σ is the surface(see the grey region of Fig. 1). Choose a Riemann normal co-

ordinate (RNC) system based at o, launched from an orthonor-mal basis formed by ua and d − 1  spacelike vectors tangentto Σ. Let the timelike coordinate be x0, and let the spacelikeones be {xi}. The signature of the spacetime metric is takenhere to be (−+++), and units are chosen with  c = 1.

We will assume the radius of the ball is much smaller than

the local curvature length,

Lcurvature   (2)and work to lowest nontrivial order in their ratio. The volume

variation at fixed radius, relative to flat space, is then given by

δV | = −   Ωd−2d+1

6(d − 1)(d + 1)R,   (3)

whereR

= Rik

ik is the spatial Ricci scalar at o [see AppendixA for details] and the area variation of ∂ Σ is given by dδV/d,i.e.

δA| = −  Ωd−2d

6(d − 1)R.   (4)

We will also be interested in the area variation at fixed vol-

ume, rather than at fixed geodesic radius. When the radius

of the ball varies, the volume and area variations have the

additional contributions   δ rV    =   d−2  

 δr dΩ   and   δ rA   =(d − 2)d−3   δr dΩ. Choosing   δ r dΩ so that the total vol-ume variation vanishes, we obtain the area variation at fixed

volume,

δA|V   = δA −  d − 2

  δV   = −   Ωd−2d

2(d2 − 1)R.   (5)

This is smaller by the factor  3/(d + 1)   than the variation atfixed radius (4).

To connect now with spacetime and the Einstein equation,

note that the spatial Ricci scalar at o is equal to twice the RNC00-component of the spacetime Einstein tensor:

R = Rikik = R − 2R00 = 2(R00 −   12Rg00) = 2G00.   (6)

FIG. 1. Causal diamond, in a maximally symmetric spacetime, for

a geodesic ball  Σ of radius    with center  o  and boundary  ∂ Σ. The

dashed curves are flow lines of  ζ , the conformal Killing vector field,

whose flow preserves the diamond and which vanishes at the top and

bottom vertices and on  ∂ Σ. The vectors show ζ  at four points of Σ.

The area deficit (5) can thus also be expressed as

δA|V   = −Ωd−2d

d2 − 1 G00.   (7)

Then, using the Einstein equation (1), we see that the area

deficit is proportional to the energy density,

δA|V   = −8πGΩd−2d

d2 − 1   T 00.   (8)

Conversely, this simple geometrical relation contains the full

content of Einstein’s equation, if it holds at all spacetime

points and for all timelike unit vectors.

The evidence that the Einstein equation implies maximalvacuum entanglement can now be stated in a qualitative, in-

tuitive fashion. Suppose the ball has a Bekenstein-Hawking

entropy A/4 G, arising from vacuum entanglement, and wetry to increase the entropy by placing an entangled qbit in the

ball. To localize the qbit within a region of size   we mustgive it an energy of at least   /  which, according to (8), willcontribute an area deficit of order   G, hence a surface entropydecrease of order unity, offsetting the added qbit. It would

not help to use a “highly entropic object” with many internal

states, because the existence of such objects makes its mark in

the vacuum as well, diluting the entropic effect of adding the

object to the ball. Indeed, in the context of the Rindler wedge,

it was argued that δS 

 ≤δE/T , where T  is the Unruh temper-

ature  /2π and δE  is the change of boost Killing energy, sincea thermal state maximizes entropy at fixed energy [22, 23]. We

now proceed to make this link between the Einstein equation

and maximal vacuum entanglement more precise.

III. CAUSAL DIAMOND AND CONFORMAL ISOMETRY

The spacetime region causally determined by a spacelike

geodesic ball  Σ   is a called the   causal diamond  D(Σ). I n a

8/18/2019 1505.04753v3

3/8

3

maximally symmetric spacetime, D(Σ)  is the intersection of the future of a past vertex and the past of a future vertex, and

has a conformal isometry and rotational symmetry in the rest

frame defined by these vertices (see Fig. 1).

The Minkowski line element  ds2 = −dt2 + dr2 + r2dΩ2takes the form ds2 = −dudv + r2dΩ2 with null coordinatesu =  t−r and v =  t +r. The Minkowski diamond centered onthe origin consists of the intersection of the regions  u > −and v < . The unique conformal isometry that preserves thediamond, and is spherically symmetric, is generated by the

conformal Killing vector

ζ  =  1

2

(2 − u2)∂ u + (2 − v2)∂ v

  (9)

(for a derivation see Appendix B). Expressed in t  and r  coor-dinates, ζ  is given by

ζ  =  1

2

(2 − r2 − t2)∂ t − 2rt ∂ r

.   (10)

The Lie derivative of the Minkowski metric along  ζ   is

Lζ ηab =

−(2t/) ηab.   (11)

The vector ζ   is tangent to the null generators on the past andfuture null boundaries of the diamond, so those boundaries

are conformal Killing horizons [25]. They meet at the ball

boundary ∂ Σ, where  ζ   vanishes, so  ∂ Σ   is a bifurcation sur-face. The surface gravity κ  of a conformal Killing horizon iswell-defined by the equation ∇aζ 2 = −2κζ a   [26], and withthe normalization of  ζ  in (9) it is equal to unity.

IV. ENTANGLEMENT ENTROPY OF A DIAMOND

Under a simultaneous variation of the geometry and the

state of the quantum fields,  (δgab, δ |ψ), the diamond entan-glement entropy variation will consist of two contributions,a state-independent UV part  δS UV  from the area change in-duced by δgab, and a state-dependent IR part  δS IR  from δ |ψ.

As mentioned above, we are assuming that, as a result of 

the UV physics, the entanglement entropy in a spatial region

is finite in any state, with a leading term  ηA. Here A  is thearea of the boundary of the region and  η   is a universal con-stant with dimensions [length]2−d. The scaling with area is

natural in any theory with a large density of states at short

distances. The assumption that η  is universal is motivated bythe idea that the UV structure of the vacuum is common to all

states in the class being considered.1 Under this assumption,

when the geometry is varied, the contribution to the entangle-

ment entropy in Σ  from the UV degrees of freedom near theboundary ∂ Σ changes by an amount

δS UV =  η δA.   (12)

1 This involves an implicit choice of “conformal frame”  [27] for the metric,

namely, the one for which   η  is constant in spacetime. This metric turns

out to satisfy the Einstein equation, so this frame is the so-called “Einstein

frame”.

The total entropy variation will thus be given by

δS tot  =  η δA + δS IR   (13)

If  η   = 1/4 G,   (12) coincides with the variation of Beken-stein’s generalized entropy, here interpreted as simply the

total entropy in the diamond. The Einstein equation with

G = 1/4 η will follow from the vacuum equilibrium assump-tion, that the total entropy variation (13) vanishes when com-

paring to a maximally symmetric spacetime with the volume

of  Σ held fixed.To motivate this equilibrium condition, we first recall

that, for ordinary thermodynamic systems in equilibrium, the

Helmholtz free energy  F   =   E  − T S  is minimized at fixedvolume. If   E   is fixed, the entropy increases with volume,hence the necessity of fixing the volume. That the free en-

ergy of a diamond has no energy term can be motivated by

comparison with de Sitter spacetime (dS). A small diamond

is conformally isometric to a static patch of dS. The quantum

statistical mechanics of dS was considered long ago by Gib-

bons and Hawking [28]. They examined the thermal partition

function and argued that, since Euclidean dS is closed, the

mass, angular momentum, and electric charge vanish, so thatthe only term in the free energy for pure dS space is the en-

tropy term, F dS = −T S . The minimization of dS free energyis thus equivalent to the maximization of entropy.

Another way to understand this result is to consider the

“first law of event horizons” [29]. For a Schwarzschild black 

hole, this law would read  δM   = (κ/8πG)δA + δE K , whereM   is the mass,  κ  and A   are the surface gravity and horizonarea of the black hole, and  E K  is the Killing energy of a mat-ter fluid surrounding the black hole [30]. For dS, instead the

first law reads

(κ/8πG)δA + δE K  = 0.   (14)

There is no geometric total energy term since there is noasymptotic region. The vanishing of the total entanglement

entropy variation (13) is directly analogous to the first law of 

horizon mechanics in de Sitter space (14). In fact, one can de-

rive a classical first law of flat causal diamonds, which has the

form (14) with δA  replaced by δA|V   (5). The δV   term arisesfrom the fact that the diamond has a conformal Killing vector

rather than a true Killing vector (see Appendix  C).

Our next step is to evaluate δS IR. The vacuum state of anyQFT, restricted to the diamond, can be expressed (formally)

as a thermal density matrix,

ρ =  Z −1 exp(−K/T ), T   =   /2π,   (15)where   K   is the “modular Hamiltonian”. The temperatureT   =    /2π  is factored out here so that  K  will be the gener-ator of Lorentz boosts, i.e. hyperbolic angle shifts, at the edge

of the diamond. For an infinite diamond that coincides with

the Rindler wedge in Minkowski space, T  is the Unruh tem-perature [31, 32].

Since ρ in (15) has the form of a thermal state, it minimizesthe modular free energy,

F K  = K  − T S,   (16)

8/18/2019 1505.04753v3

4/8

8/18/2019 1505.04753v3

5/8

5

VI. DISCUSSION

We have shown, given our assumptions, that the semiclas-

sical Einstein equation holds, for first order variations of the

vacuum, if and only if the entropy in small causal diamonds is

stationary at constant volume, when varied from a maximally

symmetric vacuum state of geometry and quantum fields. We

assumed the diamond size     is much smaller than the local

curvature length, the wavelength of any excitations of the vac-uum, and the scales in the matter field theory, but much larger

than the UV scale at which quantum gravity effects become

strong. Our entanglement variation assumption for noncon-

formal matter (22) (with constant T ab) concerns only stan-dard QFT, and is either true or false.

Strictly speaking the “first order variation” refers to the

derivative with respect to a parameter labeling the state, eval-

uated at the vacuum. To be physically applicable, however,

the result should apply to finite but small variations. The ex-

ample of a coherent state reveals a challenge in this regard

[37]: such a state can have nonzero energy density while leav-

ing entanglement entropy unchanged [38,  39]. That is, not

all energy registers as a change of entanglement. This is con-sistent with the hypothesis of maximal vacuum entanglement,

although the Einstein equation implies that the entropy has de-

creased — relative to vacuum — by more than it needs to in

order to satisfy the hypothesis. Unless a further consequence

of that hypothesis is found, or the hypothesis is refined and

strengthened in some way, the Einstein equation does not ap-

pear to follow from it in all generality.

We close with some questions and remarks concerning the

derivation and its implications.

•  Do graviton fluctuations contribute to the entanglemententropy? The UV part of the entanglement entropy

S  =  ηA  is inscrutable at this level, and the IR part does

not include gravitons. Since the diamond is taken muchsmaller than the wavelength of any ambient gravitons,

they have no gauge-invariant meaning in the diamond.

In the RNC gauge they are absent at first derivative or-

der. Moreover, the full, nonlinear Einstein tensor al-

ready appears on the geometric side of the equation, so

it would be double-counting to include any graviton en-

ergy.

•  Can a gravitational field equation with higher curvaturecorrections be derived along these lines? Maybe. We

neglected terms of order /Lcurv  in the geometry cal-culations, whereas a next higher curvature correction to

the field equation might be of order (1/Lcurv)2, where21   is the relative coefficient of the curvature squaredterm in the action. To capture this within our approxi-

mation would require /1  < 1/Lcurv. The right handside is presumably less than unity, in order for higher

curvature terms not to dominate, so the diamond would

have to be taken smaller than  1. If, say,  1  were thestring length, would classical geometry and quantum

field theory apply at that scale? Probably not. On the

other hand, perhaps with improved accuracy of the ge-

ometric analysis, and the inclusion of sub-leading UV

terms in the entanglement entropy, one could consis-

tently capture higher curvature corrections using a dia-

mond larger than 1.

•  A derivation of Einstein’s equation invoking a quan-tum limit to measurements of the spacetime geometry

of small causal diamonds was given in Ref. [40]. How

are the assumptions used there related to those made

here?

•  According to our derivation the Einstein equation is aproperty of vacuum equilibrium. Does this suggest how

to include non-equilibrium effects?

ACKNOWLEDGMENTS

I am very grateful to H. Casini, W. Donnelly, C.T. Eling,

S. Hollands, J. Maldacena, D. Marolf, M. van Raamsdonk,

V. Rosenhaus, C. Rovelli, A. Satz, A. Speranza, J. Suh, M.Varadarajan, and A. Wall for helpful discussions, comments,

and suggestions. This research was supported in part by the

National Science Foundation under grants No. PHY-1407744,

and PHY11-25915.

Appendix A: Area deficit calculation

For the Riemann normal coordinate (RNC) system based

at a point  o  we let  x0 denote the timelike coordinate and let{xi =  rni} denote the spacelike coordinates, where  r  is thegeodesic distance and  ni is a unit vector at  o,  δ ijninj = 1.

The components of the metric at  o   in this coordinate systemare g00  = −1, g0i = 0, and gij  =  δ ij. The spacelike geodesicball of radius   is denoted Σ. By definition, Σ  lies within thex0 = 0 surface in this coordinate system. Using the standardresult for the RNC metric components, the spatial metric  hijon Σ takes the form

hij  = δ ij −   13r2Rikjl nknl + O(r3)   (A1)

where Rikjl  are the spatial components of the spacetime Rie-mann tensor evaluated at  o. The extrinsic curvature of  Σ  van-ishes at  o, since  Σ   is generated by geodesics from  o. Thecomponents Rikjl  are therefore also equal to the componentsof the spatial Riemann tensor.

To lowest nontrivial order in the ratio  /Lcurvature the vol-ume element of  Σ is

dV   =√ 

h dd−1x = (1 −   16

r2Rikil n

knl)rd−2dr dΩ,   (A2)

where dΩ is the area element on the unit  (d − 2)-sphere. Theintegral over dΩ yields

   dΩ nknl =

 Ωd−2d − 1 δ 

kl,   (A3)

8/18/2019 1505.04753v3

6/8

6

where Ωd−2 is the area of the unit  (d− 2)-sphere. For spheri-cally symmetric integrands, the volume element is therefore

dV   = Ωd−2

1 −   r

2R6(d − 1)

rd−2dr,   (A4)

where R  =  Rikik is the spatial Ricci scalar at o. Integrating(A4) over  r   from 0 to    yields the volume variation at fixed

radius written in the main text,

δV | = −   Ωd−2d+1

6(d − 1)(d + 1)R,   (A5)

where R = Rikik is the spatial Ricci scalar at o.In the main text, the relation (A5) is used to find the area

variation at fixed radius and at fixed volume, and the link to

the Einstein equation is made using the relation R   = 2G00,where Gab  is the Einstein tensor. The viewpoint here is notnew. One can find the area and volume deficits at fixed radius,

and the relation to the Einstein tensor, in Section 17 (“Rie-

mannian coordinates and their applications”) of Pauli’s 1921

review of relativity [41]. Pauli cites Riemann, as well as con-

temporary researchers, for the relevant geometrical relations.

He does not remark that the Einstein equation can be char-

acterized purely in this manner. Feynman (in Section 11.2 of 

[42] and Section 42-3 of [43]) expressed the Einstein equation

for d  = 4 as the statement that, for all timelike directions, theradius excess (equivalently the radius variation at fixed area)

of a small sphere is given by − 

A/4π =  δ|A =  GM/3c2,where M  = T 00V  is the “mass” contained in the sphere. Thiscan be derived from the relations above using [Eq. (4), main

text] together with δA|/A = −(d − 2)δ|A/.

Appendix B: Conformal isometry of a flat causal diamond

The Minkowski line element  ds2 = −dt2 + dr2 + r2dΩ2takes the form ds2 = −dudv + r2dΩ2 with null coordinatesu  =  t − r and  v  = t  + r. The diamond consists of the inter-section of the regions u > −  and  v < . To determine theconformal isometry that preserves the diamond, and is spher-

ically symmetric, note that any vector field of the form

ξ  =  A(u)∂ u + B(v)∂ v   (B1)

is a conformal isometry of the  dudv   factor of the metric,Lξdudv   = [A(u) +  B (v)]dudv  (here Lξ   is the Lie deriva-tive along   ξ ). It will be a conformal isometry of the full

Minkowski metric provided Lξr2

= [A

(u) +  B

(v)]r2

. Us-ing r  = (v − u)/2 we find that in fact Lξr2 = (B − A)r, soξ  is a conformal Killing field if 

[A(u) + B(v)](v − u)/2 =  B(v) − A(u).   (B2)

At  u   =   v   this implies  B(v) =   A(v), hence at  v   = 0   thiscondition becomes [A(u) + A(0)]u/2 =  A(u)− A(0). Thegeneral solution is

A(u) =  B(u) =  a + bu + cu2.   (B3)

The group generated by these vector fields is  S L(2, R). Tomap the diamond onto itself, the flow of ξ  must leave invariantthe boundaries u   = −  and v   =   . This implies A(±) =0, hence  A(u) =   a(1 − u2/2). Fixing the constant  a  byrequiring that  ξ  have unit surface gravity, we thus obtain theconformal Killing vector field

ζ  =  1

2(2 −

u2)∂ u + (2

−v2)∂ v   (B4)

=  1

2

(2 − r2 − t2)∂ t − 2rt∂ r

.   (B5)

Appendix C: First law of causal diamond mechanics

Consider a conformally spherical causal diamond in a con-

formally flat solution to a diffeomorphism invariant gravity

theory. There is a conformal Killing vector ζ   that preservesthe diamond, and for which the edge of the diamond is the

bifurcation surface. We may invoke the variational identity

δH ζ  = δ  ∂ Σ dQζ ,   (C1)

where H ζ   is the Hamiltonian generating evolution along theflow of   ζ   (constructed from the symplectic form evaluatedon the Lie derivative of the fields along  ζ ),  Qζ   is the asso-ciated Noether charge  (d − 2)-form, and the variation is toany neighboring solution [44]. The surface integral yields

−( κ/2π)δS , where  S   is the Wald entropy associated withthe edge. (I include the factor   in the prefactor and the im-

plicit  1/  in the definition of the entropy, but these cancel inthis strictly classical relation.) If  ζ  were a Killing vector, δH ζ would vanish for all fields invariant under the Killing flow. For

matter treated as a fluid, it would not vanish, because the fluid

potentials do not share the symmetry of the metric [45, 46].Thus, for example, if the diamond were the static patch of de

Sitter space in general relativity, we would recover in this way

the first law of de Sitter horizons,  (κ/8πG)δA  +  δE ζ   = 0,where δE ζ  is the matter Killing energy variation.

If  ζ  is only a conformal Killing vector,  δH ζ  does not van-ish. In general it receives contributions both from matter

and from the gravitational field. If the theory is general rel-

ativity, and the background solution is Minkowski spacetime,

then the gravitational field contribution to δH ζ  turns out to be−(d − 2)κδV/8πG  (see below). When combined with thearea variation term −κδA/8πG, this yields −κ/8πG   timesthe area variation δA|V  at fixed volume, [Eq. (5), main text].Moreover, the conformal boost energy of the matter integrated

over Σ  is given by the right hand side of [Eq. (8), main text],when T 00   is constant. In this way we see that (C1) recoversthe statement of [Eq. (8), main text], that addition of matter

energy to the diamond decreases the area of the boundary at

fixed volume.

To verify the result for δH ζ  quoted above, we may use theexpression for the symplectic form in [47],

δH ζ  =  1

16πG

 Σ

wa(g; γ 1, γ 2)dΣa,   (C2)

8/18/2019 1505.04753v3

7/8

7

with

wa = P abcdef (γ 2 bc∇dγ 1 ef  − γ 1 bc∇dγ 2 ef ),   (C3)

P abcdef  = gaegfbgcd −   12

gadgbegfc −   12

gabgcdgef 

−   12

gbcgaegfd +   12

gbcgadgef .   (C4)

In the case of interest the background metric   gab   is theMinkowski metric ηab,  γ 2   is the tangent vector Lζ ηab   to thephase space flow generated by H ζ , and γ 1 is an arbitrary phasespace tangent vector, i.e. a linearized solution about the back-

ground. Now the conformal Killing vector in Minkowski co-

ordinates is given by (B5), so we have

γ 2 ef  = Lζ ηef  = −2t

 ηef .   (C5)

Since this vanishes at  t   = 0, only the ∇dγ 2 ef   = − 2 δ tdηef term of  (C3) contributes if the slice Σ corresponds to the t = 0surface. Then (C3) yields

wt = −d − 2

  hbcγ 1 bc  = −2 d − 2

δ √ 

h√ h

,   (C6)

where hbc

= ηbc

− ub

uc

is the metric on the spatial subspaceorthogonal to ∂ t. Inserting this in (C2) then yields

δH ζ  = −   18πG

d − 2

  δV.   (C7)

The surface gravity is unity for the chosen conformal Killing

field, so this yields the result claimed above.

[1] J. D. Bekenstein, “Black holes and the second law.” Nuovo Ci-

mento Lettere 4, 737–740 (1972).

[2] J. D. Bekenstein, “Black Holes and Entropy,”  Phys. Rev. D  7,

2333–2346 (1973).

[3] S.W. Hawking, “Particle Creation by Black Holes,”   Com-

mun.Math.Phys. 43, 199–220 (1975).

[4] Rafael D. Sorkin, “1983 paper on entanglement entropy: ‘On

the Entropy of the Vacuum outside a Horizon’,” (2014),

arXiv:1402.3589 [gr-qc].

[5] Gerard ’t Hooft, “On the Quantum Structure of a Black Hole,”

Nucl.Phys. B256, 727 (1985).

[6] Luca Bombelli, Rabinder K. Koul, Joohan Lee, and Rafael D.

Sorkin, “A Quantum Source of Entropy for Black Holes,”

Phys.Rev. D34, 373–383 (1986).

[7] Valeri P. Frolov and Igor Novikov, “Dynamical origin of the

entropy of a black hole,”   Phys.Rev.   D48, 4545–4551 (1993),

arXiv:gr-qc/9309001 [gr-qc].[8] Mark Srednicki, “Entropy and area,”  Phys.Rev.Lett.   71, 666–

669 (1993), arXiv:hep-th/9303048 [hep-th].

[9] Sergey N. Solodukhin, “Entanglement entropy of black holes,”

Living Rev.Rel. 14, 8 (2011), arXiv:1104.3712 [hep-th].

[10] Ted Jacobson, “Gravitation and vacuum entanglement entropy,”

Int.J.Mod.Phys.   D21, 1242006 (2012),   arXiv:1204.6349 [gr-

qc].

[11] Rafael D. Sorkin, “Toward a proof of entropy increase in the

presence of quantum black holes,”  Phys.Rev.Lett.   56, 1885–

1888 (1986).

[12] Rafael D. Sorkin, “The statistical mechanics of black hole ther-

modynamics,” (1997), arXiv:gr-qc/9705006 [gr-qc].

[13] Aron C. Wall, “Ten Proofs of the Generalized Second Law,”

JHEP 0906, 021 (2009), arXiv:0901.3865 [gr-qc].

[14] Aron C. Wall, “A Proof of the generalized second law forrapidly-evolving Rindler horizons,”   Phys.Rev.   D82, 124019

(2010), arXiv:1007.1493 [gr-qc].

[15] Aron C. Wall, “A proof of the generalized second law for

rapidly changing fields and arbitrary horizon slices,” Phys.Rev.

D85, 104049 (2012), arXiv:1105.3445 [gr-qc].

[16] Ted Jacobson, “Thermodynamics of space-time: The Ein-

stein equation of state,” Phys.Rev.Lett. 75, 1260–1263 (1995),

arXiv:gr-qc/9504004 [gr-qc].

[17] Nima Lashkari, Michael B. McDermott, and Mark Van Raams-

donk, “Gravitational dynamics from entanglement ’thermody-

namics’,”  JHEP 1404, 195 (2014), arXiv:1308.3716 [hep-th].

[18] Thomas Faulkner, Monica Guica, Thomas Hartman, Robert C.

Myers, and Mark Van Raamsdonk, “Gravitation from En-

tanglement in Holographic CFTs,”   JHEP   1403, 051 (2014),

arXiv:1312.7856 [hep-th].

[19] Brian Swingle and Mark Van Raamsdonk, “Universality of 

Gravity from Entanglement,” (2014),   arXiv:1405.2933 [hep-

th].

[20] Ted Jacobson and Alejandro Satz, “Black hole entangle-

ment entropy and the renormalization group,”  Phys.Rev.  D87,

084047 (2013), arXiv:1212.6824.

[21] Joshua H. Cooperman and Markus A. Luty, “Renormalization

of Entanglement Entropy and the Gravitational Effective Ac-

tion,” JHEP 1412, 045 (2014), arXiv:1302.1878 [hep-th].

[22] Donald Marolf, Djordje Minic, and Simon F. Ross, “Notes

on space-time thermodynamics and the observer dependence

of entropy,”   Phys. Rev.   D69, 064006 (2004),   arXiv:hep-th/0310022 [hep-th].

[23] Donald Marolf, “A Few words on entropy, thermodynamics,

and horizons,” in  General relativity and gravitation. Proceed-

ings, 17th International Conference, GR17, Dublin, Ireland,

 July 18-23, 2004   (2004) pp. 83–103,   arXiv:hep-th/0410168

[hep-th].

[24] See Supplemental Material at [URL will be inserted by pub-

lisher] for details.

[25] C. C. Dyerand E. Honig, “Conformal Killing horizons,” Journal

of Mathematical Physics 20, 409–412 (1979).

[26] Ted Jacobson and Gungwon Kang, “Conformal invariance of 

black hole temperature,”   Class.Quant.Grav.   10, L201–L206

(1993), arXiv:gr-qc/9307002 [gr-qc].

[27] Eanna E. Flanagan, “The Conformal frame freedom in theories

of gravitation,” Class. Quant. Grav.  21, 3817 (2004), arXiv:gr-qc/0403063 [gr-qc].

[28] G.W. Gibbons and S.W. Hawking, “Action Integrals and Par-

tition Functions in Quantum Gravity,”   Phys.Rev.   D15, 2752–

2756 (1977).

[29] G.W. Gibbons and S.W. Hawking, “Cosmological Event Hori-

zons, Thermodynamics, and Particle Creation,” Phys.Rev. D15,

2738–2751 (1977).

[30] James M. Bardeen, B. Carter, and S.W. Hawking, “The Four

laws of black hole mechanics,”  Commun.Math.Phys.  31, 161–

170 (1973).

http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1007/BF02345020http://dx.doi.org/10.1007/BF02345020http://dx.doi.org/10.1007/BF02345020http://dx.doi.org/10.1007/BF02345020http://dx.doi.org/10.1007/BF02345020http://arxiv.org/abs/1402.3589http://arxiv.org/abs/1402.3589http://dx.doi.org/%2010.1016/0550-3213(85)90418-3http://dx.doi.org/%2010.1016/0550-3213(85)90418-3http://dx.doi.org/%2010.1016/0550-3213(85)90418-3http://dx.doi.org/10.1103/PhysRevD.34.373http://dx.doi.org/10.1103/PhysRevD.34.373http://dx.doi.org/10.1103/PhysRevD.34.373http://dx.doi.org/10.1103/PhysRevD.34.373http://dx.doi.org/%2010.1103/PhysRevD.48.4545http://dx.doi.org/%2010.1103/PhysRevD.48.4545http://dx.doi.org/%2010.1103/PhysRevD.48.4545http://dx.doi.org/%2010.1103/PhysRevD.48.4545http://arxiv.org/abs/gr-qc/9309001http://arxiv.org/abs/gr-qc/9309001http://dx.doi.org/10.1103/PhysRevLett.71.666http://dx.doi.org/10.1103/PhysRevLett.71.666http://dx.doi.org/10.1103/PhysRevLett.71.666http://dx.doi.org/10.1103/PhysRevLett.71.666http://dx.doi.org/10.1103/PhysRevLett.71.666http://arxiv.org/abs/hep-th/9303048http://arxiv.org/abs/hep-th/9303048http://dx.doi.org/%2010.12942/lrr-2011-8http://dx.doi.org/%2010.12942/lrr-2011-8http://dx.doi.org/%2010.12942/lrr-2011-8http://dx.doi.org/%2010.12942/lrr-2011-8http://arxiv.org/abs/1104.3712http://arxiv.org/abs/1104.3712http://dx.doi.org/%2010.1142/S0218271812420060http://dx.doi.org/%2010.1142/S0218271812420060http://dx.doi.org/%2010.1142/S0218271812420060http://dx.doi.org/%2010.1142/S0218271812420060http://arxiv.org/abs/1204.6349http://arxiv.org/abs/1204.6349http://arxiv.org/abs/1204.6349http://dx.doi.org/10.1103/PhysRevLett.56.1885http://dx.doi.org/10.1103/PhysRevLett.56.1885http://dx.doi.org/10.1103/PhysRevLett.56.1885http://dx.doi.org/10.1103/PhysRevLett.56.1885http://arxiv.org/abs/gr-qc/9705006http://arxiv.org/abs/gr-qc/9705006http://dx.doi.org/10.1088/1126-6708/2009/06/021http://dx.doi.org/10.1088/1126-6708/2009/06/021http://dx.doi.org/10.1088/1126-6708/2009/06/021http://dx.doi.org/10.1088/1126-6708/2009/06/021http://arxiv.org/abs/0901.3865http://arxiv.org/abs/0901.3865http://dx.doi.org/10.1103/PhysRevD.82.124019http://dx.doi.org/10.1103/PhysRevD.82.124019http://dx.doi.org/10.1103/PhysRevD.82.124019http://dx.doi.org/10.1103/PhysRevD.82.124019http://dx.doi.org/10.1103/PhysRevD.82.124019http://arxiv.org/abs/1007.1493http://arxiv.org/abs/1007.1493http://dx.doi.org/10.1103/PhysRevD.87.069904,%2010.1103/PhysRevD.85.104049http://dx.doi.org/10.1103/PhysRevD.87.069904,%2010.1103/PhysRevD.85.104049http://dx.doi.org/10.1103/PhysRevD.87.069904,%2010.1103/PhysRevD.85.104049http://arxiv.org/abs/1105.3445http://arxiv.org/abs/1105.3445http://dx.doi.org/10.1103/PhysRevLett.75.1260http://dx.doi.org/10.1103/PhysRevLett.75.1260http://dx.doi.org/10.1103/PhysRevLett.75.1260http://dx.doi.org/10.1103/PhysRevLett.75.1260http://arxiv.org/abs/gr-qc/9504004http://arxiv.org/abs/gr-qc/9504004http://dx.doi.org/10.1007/JHEP04(2014)195http://dx.doi.org/10.1007/JHEP04(2014)195http://dx.doi.org/10.1007/JHEP04(2014)195http://dx.doi.org/10.1007/JHEP04(2014)195http://arxiv.org/abs/1308.3716http://arxiv.org/abs/1308.3716http://dx.doi.org/%2010.1007/JHEP03(2014)051http://dx.doi.org/%2010.1007/JHEP03(2014)051http://dx.doi.org/%2010.1007/JHEP03(2014)051http://dx.doi.org/%2010.1007/JHEP03(2014)051http://arxiv.org/abs/1312.7856http://arxiv.org/abs/1312.7856http://arxiv.org/abs/1405.2933http://arxiv.org/abs/1405.2933http://dx.doi.org/%2010.1103/PhysRevD.87.084047http://dx.doi.org/%2010.1103/PhysRevD.87.084047http://dx.doi.org/%2010.1103/PhysRevD.87.084047http://dx.doi.org/%2010.1103/PhysRevD.87.084047http://arxiv.org/abs/1212.6824http://arxiv.org/abs/1212.6824http://dx.doi.org/10.1007/JHEP12(2014)045http://dx.doi.org/10.1007/JHEP12(2014)045http://dx.doi.org/10.1007/JHEP12(2014)045http://dx.doi.org/10.1007/JHEP12(2014)045http://arxiv.org/abs/1302.1878http://arxiv.org/abs/1302.1878http://dx.doi.org/%2010.1103/PhysRevD.69.064006http://dx.doi.org/%2010.1103/PhysRevD.69.064006http://dx.doi.org/%2010.1103/PhysRevD.69.064006http://dx.doi.org/%2010.1103/PhysRevD.69.064006http://arxiv.org/abs/hep-th/0310022http://arxiv.org/abs/hep-th/0310022http://arxiv.org/abs/hep-th/0410168http://arxiv.org/abs/hep-th/0410168http://arxiv.org/abs/hep-th/0410168http://dx.doi.org/10.1063/1.524078http://dx.doi.org/10.1063/1.524078http://dx.doi.org/10.1063/1.524078http://dx.doi.org/10.1063/1.524078http://dx.doi.org/10.1063/1.524078http://dx.doi.org/%2010.1088/0264-9381/10/11/002http://dx.doi.org/%2010.1088/0264-9381/10/11/002http://dx.doi.org/%2010.1088/0264-9381/10/11/002http://dx.doi.org/%2010.1088/0264-9381/10/11/002http://dx.doi.org/%2010.1088/0264-9381/10/11/002http://arxiv.org/abs/gr-qc/9307002http://arxiv.org/abs/gr-qc/9307002http://dx.doi.org/10.1088/0264-9381/21/15/N02http://dx.doi.org/10.1088/0264-9381/21/15/N02http://dx.doi.org/10.1088/0264-9381/21/15/N02http://dx.doi.org/10.1088/0264-9381/21/15/N02http://arxiv.org/abs/gr-qc/0403063http://arxiv.org/abs/gr-qc/0403063http://arxiv.org/abs/gr-qc/0403063http://dx.doi.org/%2010.1103/PhysRevD.15.2752http://dx.doi.org/%2010.1103/PhysRevD.15.2752http://dx.doi.org/%2010.1103/PhysRevD.15.2752http://dx.doi.org/%2010.1103/PhysRevD.15.2752http://dx.doi.org/%2010.1103/PhysRevD.15.2752http://dx.doi.org/%2010.1103/PhysRevD.15.2738http://dx.doi.org/%2010.1103/PhysRevD.15.2738http://dx.doi.org/%2010.1103/PhysRevD.15.2738http://dx.doi.org/%2010.1103/PhysRevD.15.2738http://dx.doi.org/%2010.1103/PhysRevD.15.2738http://dx.doi.org/10.1007/BF01645742http://dx.doi.org/10.1007/BF01645742http://dx.doi.org/10.1007/BF01645742http://dx.doi.org/10.1007/BF01645742http://dx.doi.org/10.1007/BF01645742http://dx.doi.org/10.1007/BF01645742http://dx.doi.org/10.1007/BF01645742http://dx.doi.org/%2010.1103/PhysRevD.15.2738http://dx.doi.org/%2010.1103/PhysRevD.15.2738http://dx.doi.org/%2010.1103/PhysRevD.15.2752http://dx.doi.org/%2010.1103/PhysRevD.15.2752http://arxiv.org/abs/gr-qc/0403063http://arxiv.org/abs/gr-qc/0403063http://dx.doi.org/10.1088/0264-9381/21/15/N02http://arxiv.org/abs/gr-qc/9307002http://dx.doi.org/%2010.1088/0264-9381/10/11/002http://dx.doi.org/%2010.1088/0264-9381/10/11/002http://dx.doi.org/10.1063/1.524078http://dx.doi.org/10.1063/1.524078http://arxiv.org/abs/hep-th/0410168http://arxiv.org/abs/hep-th/0410168http://arxiv.org/abs/hep-th/0310022http://arxiv.org/abs/hep-th/0310022http://dx.doi.org/%2010.1103/PhysRevD.69.064006http://arxiv.org/abs/1302.1878http://dx.doi.org/10.1007/JHEP12(2014)045http://arxiv.org/abs/1212.6824http://dx.doi.org/%2010.1103/PhysRevD.87.084047http://dx.doi.org/%2010.1103/PhysRevD.87.084047http://arxiv.org/abs/1405.2933http://arxiv.org/abs/1405.2933http://arxiv.org/abs/1312.7856http://dx.doi.org/%2010.1007/JHEP03(2014)051http://arxiv.org/abs/1308.3716http://dx.doi.org/10.1007/JHEP04(2014)195http://arxiv.org/abs/gr-qc/9504004http://dx.doi.org/10.1103/PhysRevLett.75.1260http://arxiv.org/abs/1105.3445http://dx.doi.org/10.1103/PhysRevD.87.069904,%2010.1103/PhysRevD.85.104049http://dx.doi.org/10.1103/PhysRevD.87.069904,%2010.1103/PhysRevD.85.104049http://arxiv.org/abs/1007.1493http://dx.doi.org/10.1103/PhysRevD.82.124019http://dx.doi.org/10.1103/PhysRevD.82.124019http://arxiv.org/abs/0901.3865http://dx.doi.org/10.1088/1126-6708/2009/06/021http://arxiv.org/abs/gr-qc/9705006http://dx.doi.org/10.1103/PhysRevLett.56.1885http://dx.doi.org/10.1103/PhysRevLett.56.1885http://arxiv.org/abs/1204.6349http://arxiv.org/abs/1204.6349http://dx.doi.org/%2010.1142/S0218271812420060http://arxiv.org/abs/1104.3712http://dx.doi.org/%2010.12942/lrr-2011-8http://arxiv.org/abs/hep-th/9303048http://dx.doi.org/10.1103/PhysRevLett.71.666http://dx.doi.org/10.1103/PhysRevLett.71.666http://arxiv.org/abs/gr-qc/9309001http://dx.doi.org/%2010.1103/PhysRevD.48.4545http://dx.doi.org/10.1103/PhysRevD.34.373http://dx.doi.org/%2010.1016/0550-3213(85)90418-3http://arxiv.org/abs/1402.3589http://dx.doi.org/10.1007/BF02345020http://dx.doi.org/10.1007/BF02345020http://dx.doi.org/10.1103/PhysRevD.7.2333http://dx.doi.org/10.1103/PhysRevD.7.2333

8/18/2019 1505.04753v3

8/8

8

[31] W.G. Unruh, “Notes on black hole evaporation,”   Phys.Rev.

D14, 870 (1976).

[32] Geoffrey L. Sewell, “Quantum fields on manifolds: PCT and

gravitationally induced thermal states,” Annals Phys. 141, 201–

224 (1982).

[33] Peter D. Hislop and Roberto Longo, “Modular Structure of the

Local Algebras Associated With the Free Massless Scalar Field

Theory,” Commun.Math.Phys. 84, 71 (1982).

[34] J.J Bisognano and E.H. Wichmann, “On the Duality Condition

for Quantum Fields,” J.Math.Phys. 17, 303–321 (1976).[35] Horacio Casini, Damián A. Galante, and Robert C. Myers, in

preparation.

[36] Antony J. Speranza, in preparation.

[37] This was pointed out to me by Madhavan Varadarajan.

[38] Thomas M. Fiola, John Preskill, Andrew Strominger, and

Sandip P. Trivedi, “Black hole thermodynamics and infor-

mation loss in two-dimensions,”   Phys. Rev.   D50, 3987–4014

(1994), arXiv:hep-th/9403137 [hep-th].

[39] Eric Benedict and So-Young Pi, “Entanglement entropy of non-

trivial states,”  Annals Phys.   245, 209–224 (1996),   arXiv:hep-

th/9505121 [hep-th].

[40] Seth Lloyd, “The quantum geometric limit,” (2012),

arXiv:1206.6559 [gr-qc].

[41] W. Pauli, Theory of Relativity (Pergamon Press, 1958).

[42] R.P. Feynman, Lectures on Gravitation (Westview Press, 2002).

[43] R.P. Feynman, “The Feynman Lectures on Physics, Vol. 2,”

http://www.feynmanlectures.caltech.edu/.

[44] Vivek Iyer and Robert M. Wald, “Some properties of Noether

charge and a proposal for dynamical black hole entropy,”

Phys.Rev. D50, 846–864 (1994), arXiv:gr-qc/9403028 [gr-qc].[45] Vivek Iyer, “Lagrangian perfect fluids and black hole mechan-

ics,”  Phys.Rev.  D55, 3411–3426 (1997),  arXiv:gr-qc/9610025

[gr-qc].

[46] Stephen R. Green, Joshua S. Schiffrin, and Robert M.

Wald, “Dynamic and Thermodynamic Stability of Relativis-

tic, Perfect Fluid Stars,”  Class.Quant.Grav. 31, 035023 (2014),

arXiv:1309.0177 [gr-qc].

[47] Stefan Hollands and Robert M. Wald, “Stability of Black Holes

and Black Branes,” Commun.Math.Phys. 321, 629–680 (2013),

arXiv:1201.0463 [gr-qc].

http://dx.doi.org/10.1103/PhysRevD.14.870http://dx.doi.org/10.1103/PhysRevD.14.870http://dx.doi.org/10.1103/PhysRevD.14.870http://dx.doi.org/10.1103/PhysRevD.14.870http://dx.doi.org/10.1016/0003-4916(82)90285-8http://dx.doi.org/10.1016/0003-4916(82)90285-8http://dx.doi.org/10.1016/0003-4916(82)90285-8http://dx.doi.org/10.1016/0003-4916(82)90285-8http://dx.doi.org/10.1016/0003-4916(82)90285-8http://dx.doi.org/10.1007/BF01208372http://dx.doi.org/10.1007/BF01208372http://dx.doi.org/10.1007/BF01208372http://dx.doi.org/10.1063/1.522898http://dx.doi.org/10.1063/1.522898http://dx.doi.org/10.1063/1.522898http://dx.doi.org/10.1063/1.522898http://dx.doi.org/10.1103/PhysRevD.50.3987http://dx.doi.org/10.1103/PhysRevD.50.3987http://dx.doi.org/10.1103/PhysRevD.50.3987http://dx.doi.org/10.1103/PhysRevD.50.3987http://dx.doi.org/10.1103/PhysRevD.50.3987http://arxiv.org/abs/hep-th/9403137http://dx.doi.org/10.1006/aphy.1996.0007http://dx.doi.org/10.1006/aphy.1996.0007http://dx.doi.org/10.1006/aphy.1996.0007http://arxiv.org/abs/hep-th/9505121http://arxiv.org/abs/hep-th/9505121http://arxiv.org/abs/1206.6559http://arxiv.org/abs/1206.6559http://www.feynmanlectures.caltech.edu/http://www.feynmanlectures.caltech.edu/http://dx.doi.org/10.1103/PhysRevD.50.846http://dx.doi.org/10.1103/PhysRevD.50.846http://dx.doi.org/10.1103/PhysRevD.50.846http://dx.doi.org/10.1103/PhysRevD.50.846http://arxiv.org/abs/gr-qc/9403028http://dx.doi.org/%2010.1103/PhysRevD.55.3411http://dx.doi.org/%2010.1103/PhysRevD.55.3411http://dx.doi.org/%2010.1103/PhysRevD.55.3411http://dx.doi.org/%2010.1103/PhysRevD.55.3411http://arxiv.org/abs/gr-qc/9610025http://arxiv.org/abs/gr-qc/9610025http://arxiv.org/abs/gr-qc/9610025http://dx.doi.org/10.1088/0264-9381/31/3/035023http://dx.doi.org/10.1088/0264-9381/31/3/035023http://dx.doi.org/10.1088/0264-9381/31/3/035023http://dx.doi.org/10.1088/0264-9381/31/3/035023http://arxiv.org/abs/1309.0177http://arxiv.org/abs/1309.0177http://dx.doi.org/10.1007/s00220-012-1638-1http://dx.doi.org/10.1007/s00220-012-1638-1http://dx.doi.org/10.1007/s00220-012-1638-1http://dx.doi.org/10.1007/s00220-012-1638-1http://arxiv.org/abs/1201.0463http://arxiv.org/abs/1201.0463http://arxiv.org/abs/1201.0463http://dx.doi.org/10.1007/s00220-012-1638-1http://arxiv.org/abs/1309.0177http://dx.doi.org/10.1088/0264-9381/31/3/035023http://arxiv.org/abs/gr-qc/9610025http://arxiv.org/abs/gr-qc/9610025http://dx.doi.org/%2010.1103/PhysRevD.55.3411http://arxiv.org/abs/gr-qc/9403028http://dx.doi.org/10.1103/PhysRevD.50.846http://www.feynmanlectures.caltech.edu/http://arxiv.org/abs/1206.6559http://arxiv.org/abs/hep-th/9505121http://arxiv.org/abs/hep-th/9505121http://dx.doi.org/10.1006/aphy.1996.0007http://arxiv.org/abs/hep-th/9403137http://dx.doi.org/10.1103/PhysRevD.50.3987http://dx.doi.org/10.1103/PhysRevD.50.3987http://dx.doi.org/10.1063/1.522898http://dx.doi.org/10.1007/BF01208372http://dx.doi.org/10.1016/0003-4916(82)90285-8http://dx.doi.org/10.1016/0003-4916(82)90285-8http://dx.doi.org/10.1103/PhysRevD.14.870http://dx.doi.org/10.1103/PhysRevD.14.870