Role of Mann Counterterm in Gravitational Energy
aa r X i v : . [ g r- q c ] J u l Role of Mann Counterterm in Gravitational Energy
Shoichiro
Miyashita ∗ Department of Physics, Waseda University, Okubo 3-4-1, Shinjuku, Tokyo 169-8555, Japan (Dated: July 28, 2020)
Abstract
In 1999, R. B. Mann proposed a counterterm that is some sort of generalization of the well-knownHolographic counterterm and that can eliminate the divergence of the gravitational action of asymp-totically AdS and flat spacetimes (Phys. Rev. D (1999) 104047 [1]). I show it is not only foreliminating the divergence of such spacetimes but also for setting the ground state energy to zerofor any d -dimensional spacetimes with an S d − × R boundary geometry, and speculate it is also truefor spacetimes with any (suitable) boundary geometry and topology. PACS numbers:
Proper Gravitational Action
For a field theory defined on a ( d −
1) dimensionalmanifold B with a fixed background pseudo-Riemanniangeometry γ , there is a ambiguity of adding a term Z B d d − y √− γf ( γ ) , (1)where f is a local function of geometric scalar quantitessuch as R , R ij R ij . For example, consider a scalar fieldtheory I scalar [ ϕ, γ ]= Z B d d − y √− γ (cid:20) − γ ij ∂ i ϕ∂ j ϕ − V ( ϕ ) (cid:21) (2)with a nonzero potential minimum V min = min Φ ∈ R V (Φ) =0. If we add the term I scalar,ct [ γ ] = Z B d d − y √− γV min , (3)the ground state energy computed from the total action I scalar,proper [ ϕ, γ ] = I scalar [ ϕ, γ ] + I scalar,ct [ γ ] (4)is now set to zero for any (suitable) B and γ . I callsuch a action the proper action .The objective of this latter is to raise and pursue thepossibility that adding Mann counterterm I Mann [ γ ] [1] (see eq. (7), (8), (9)) to Einstein-Hilbert-York-Gibbons-Hawking action [3, 4] leads to the proper gravitational ∗ Electronic address: s-miyashita”at”aoni.waseda.jp This simple subtraction works only classically. Quantum me-chanically, a slight modification of I scalar,ct is needed, that de-pends on more finer information of V , in addition to V min , andthat of γ . A closely related work was done by Lau at almost the same time[2]. action I GR,proper [ g ] = I EH [ g ] + I Y GH [ g ] + I Mann [ γ ] (5) I EH [ g ] = 116 πG Z M d d x √− g ( R ( d ) − I Y GH [ g ] = 18 πG Z B d d − y √− γ Θwhere B is time-like boundary(ies) of M , γ is a boundarymetric that admits a time-like Killing vector. I neglectother boundaries and joints in (5). If this is the case, theground state energy given by Brown-York tensor [5] ofthe action τ properij = − √− γ δI GR,proper [ g ] δγ ij (6)is zero for any suitable boundary geometry ( B , γ ), andfor any value of Λ, at least, at semi-classical level.The first thing I have to mention about Mann countert-erm is that it is equivalent to the well-known Holographiccounterterm I ct [ γ ] [6, 7], completely for even spacetimedimension (as stated in [1]), and up to a finite contribu-tion for odd dimension when Λ < γ is of infinitevolume. This means that Mann counterterm can subtractall IR divergences for asymptotically AdS spacetimes (or,in terms of AdS/CFT [8–10], all UV divergences of theholographic CFT [11, 12] ). Apparently, this is the prop-erty that a counterterm of the gravitational action shouldhave. The next is, contrary to I ct [ γ ], Mann countertermcan also be used to eliminate the divergence of asymp-totically flat spacetimes of 4- and 5-dimension [13].What is mentioned about Mann counterterm newly inthis letter is, as stated above, that it is not only for elim-inating divergences of the gravitational action or gravi-tational energy of spacetimes with a boundary of infinitespatial volume, but also for setting the ground state en-ergy to zero, at least for spacetimes with some restrictedclass of ( B , γ ), and hopefully, for those with any suitable( B , γ ), namely, the action (5) is proper. Explicitly, afterintroducing Mann counterterm in a little detail, I showthe ground state energy of spacetimes with an S d − × R boundary geometry of any radius and any value of Λ iszero (for 4 ≤ d ≤ . I end the letter withsome remarks on Mann counterterm and the proper grav-itational action. Mann Counterterm
Mann counterterm is constructed out purely fromgeometric quantities that refer to the information of theboundary metric γ , as is the Holographic counter term I ct [ γ ], I Mann [ γ ] = − πG Z B d d − y √− γ Θ Mann ( γ ) (7)Θ Mann ( γ ) = l d = 3[ C ( d ) R − C ( d )Λ] d = 4 , (cid:20) D ( d ) (cid:18) R ij R ij + d − d − R (cid:19) − D ( d )Λ R + D ( d )Λ (cid:21) d = 6 , C a ( d ) , D a ( d ) are C ( d ) = d − d − , C ( d ) = 2( d − d − ,D ( d ) = 2( d − ( d − d − , D ( d ) = 4( d − ( d − d − , (9) D ( d ) = 4( d − ( d − . Although it is useful for such a simple situation, the backgroundsubtraction method itself is somewhat conceptually unsatisfac-tory for setting the ground state energy to zero (also for justsubtracting the divergence) because we need to know the groundstate solution g gs (or a meaningful reference state solution g ref ) a priori . The advantage of the counterterms that depend onlyon γ , such as I ct [ γ ], is the counterterms themselves have theinformation of the ground states energy (or that of a meaningfulreference state) and we do not need to know g gs ( g ref ). Un-fortunately, I ct [ γ ] is not defined for Λ ≥ S d − × R boundary even for Λ <
0. Theground state energy computed by using I ct [ γ ] deviates from zeroand its deviation depends on the radius of S d − . Precisely, the original counterterm proposed in [1] is only for4-dimension case. The cases of 5 ≤ d ≤ d = 5 case was also shown in[13]). Note that l denotes the AdS radius and the d = 3 caseis special, works only for Λ <
0, and is completely sameas the Holographic counterterm [6]. Therefore, I will notconsider the d = 3 case here. We can easily check that · it coincides with I ct [ γ ], completely for even dimension,and up to finite contribution for odd dimension, when wetake large l or small curvature limit , · it well-behaves for Λ ∈ R , · the contribution is same as the background subtrac-tion method [4] for asymptotically flat spacetimes withan S d − × R boundary topology.The explicit form of the corresponding Brown-Yorktensor is τ properij = − πG (cid:20) Θ ij − γ ij (Θ − Θ Mann ) − ∂ Θ Mann ∂γ ij (cid:21) (10)2 ∂ Θ Mann ∂γ ij = d = 31Θ Mann C ( d ) R ij d = 4 , Mann (cid:20) D ( d ) (cid:18) R ki R kj + d − d − RR ij (cid:19) − D ( d )Λ R ij (cid:21) d = 6 , S d − × R Boundary Geometries
I show the utility of Mann counterterm for a class ofspacetimes with an S d − × R boundary geometry, whosemetric form is γ = − d ˜ t + r b d Ω d − , (11) r b is the radius of S d − . The ground state of spacetimeswith the boundary would be pure flat, dS, and AdS space- The reason why I show only up to 7-dimension case is just I ct [ γ ] is explicitly shown only up to 7-dimension in [7]. Butthe derivation of higher dimensional case is straightforward ifwe know the Holographic counterterm of the dimension as I willcomment later. For small x , (1+ ax ) ≃ ax − a x and (1+ ax + bx ) ≃ ax + (cid:0) b − a (cid:1) x + (cid:0) − ab + a (cid:1) x . time with a cut-off at r = r b ; g gs = − f ( r ) dt + 1 f ( r ) dr + r d Ω d − (12) f ( r ) = (cid:18) − d − d −
1) Λ r (cid:19) (13) r ∈ [0 , r b ]Only for the Λ > r b < q ( d − d − . The extrinsic curvature isΘ tt = 2Λ r b ( d − d − p f ( r b ) , (14)Θ ti = 0 , (15)Θ ab = p f ( r b ) r b σ ab , (16)Θ = 1 p f ( r b ) (cid:20) − r b ( d − d −
1) + ( d − r b f ( r b ) (cid:21) (17)where σ ij is the projection of γ to S d − and a, b, · · · isthe index of a coordinate on S d − , that is, σ it = σ it = 0.Since Ricci tensor R ij and Ricci scalar R of the boundarymetric (11) are R ij = ( d − σ ij r b , R = ( d − d − r b , (18)Θ Mann and 2 δ Θ Mann δγ ij are calculated asΘ Mann = ( d − r b p f ( r b ) , (19)2 δ Θ Mann δγ ij = 1 r b p f ( r b ) σ ij , (20)for 4 ≤ d ≤
7. From these quantities, we can easilyconfirm that τ properij = 0 , (21)for 4 ≤ d ≤
7, for any r b and for any Λ . Another wayto see this is to compute the free energy at zero tempera-ture through the canonical partition function defined byEuclidean path integral [4]. The Euclidean saddle corre-sponding to thermal equilibrium at low temperature 1 /β is obtained by Wick rotating t = − iτ the metric (12). What I mean by “ground state” here is, in a classical mechanicalsense, the most lowest energy solution with the boundary condi-tion (11) and, in a quantum mechanical sense, the coarse-grainedand decohered history of the ground state with a overwhelmingprobability (if exists) at some level of coarse-graining. Except r b > q ( d − d − for Λ > The on-shell value of each term of the Euclidean versionof (5) is I EEH [ g gs,E ] = − β Ω d − πG Λ( d − d − r d − b p f ( r b ) , (22) I EY GH [ g gs,E ] = − β Ω d − πG (cid:20) d − r b − d − (cid:21) r d − b p f ( r b ) , (23) I EMann [ γ gs,E ] = β Ω d − πG ( d − p f ( r b ) r d − , (24)where Ω n is the volume of unit n dimensional sphere.Then F = β I EGR,proper [ g gs,E ] = 0 . This subtraction isalways valid for any spacetimes with an S d − × R bound-ary geometry both of infinite and finite spatial volume . Therefore, when we analyze standard gravitationalthermodynamics [4, 14–16], all we need is to just addMann counterterm (7). Some Remarks
Higher dimensionThe extension to higher dimension is straightforward.For example, using Maclaurin expansion formula for(1 + x ) , we can construct Mann counterterm for d = 8 , d = 8 , d = 9). Wecan check that eq. (19), (20), (21), (24), and F = 0is also hold for d = 8 ,
9. For more higher dimension,Mann counterterm can be constructed by using theformula for (1 + x ) [ d − ] , would properly subtract theground state energy for spacetimes with an S d − × R boundary geometry, and can be used for gravitationalthermodynamics of higher dimension.Λ as the parameter of GRSomeone might feel strange about assigning zero en-ergy to dS/AdS spacetimes enclosed by an S d − × R For the d = 4 , For asymptotically AdS spacetimes of odd dimension, there isknown to exist the non-vanishing ground state energy if we com-pute it by using I ct , that is interpreted as the Casimir energyof the Holographic CFT [6]. The existence is related to the am-biguity of the Holographic counterterm of odd dimension, thatis, the freedom of adding curvature invariants of an appropri-ate mass dimension to I ct . For large l or small curvature limit, I Mann and I ct are differ by such invariants and the differenceexactly compensates the Casimir energy for asymptotically AdSspacetimes with an S d − × R boundary geometry. Explicitly, (cid:0) ax + bx + cx (cid:1) ≃ ax + (cid:0) b − a (cid:1) x + (cid:0) c − ab + a (cid:1) x + (cid:0) − ac − b + a b − a (cid:1) x . boundary (12) since the bulk can be thought as filled witha positive/negative energy if they regard Λ as matterand consider the corresponding energy-momentum tensor T Λ µν = − Λ8 πG g µν . Actually, from this point of view, theirintuition would be correct. On the other hand, from an-other point of view, one can regard Λ as just the parame-ter of the gravitational theory, and think (12) as “empty”dS/AdS spacetimes. In this letter, I took the latter pointof view and assigned zero energy to them by subtractingthe “fictitious energy” of Λ, like we do for asymptoticallyAdS spacetimes in the context of AdS/CFT. Eventually,this matter is nothing but just difference of viewpoint, orchoice of a reference of energy.Other boundary geometry and topologyAlthough Mann counterterm are shown to workfor some class of spacetimes, it is far from sayingthat Einstein-Hilbert-York-Gibbons-Hawking-Mann ac-tion (5) is the proper gravitational action. Pursuing thepossibility is left for future work. Whether it is or not,it would be interesting to check whether Mann countert-erm for d ≥ S d − × R bound-ary topology but also that with another boundary ge-ometry, such as S n × R d − − n (for d ≥ B , γ ) can be seen as addingit to the action of the Holographic QFT on ( B , γ ). Al-though the Holographic dual of gravity with a boundaryof finite spatial volume is less-known compared to thatof asymptotically AdS spacetimes, one promising thingmay be that it is non-local [20]. Suppose Mann coun-terterm leads to the proper gravitational action as wellas the proper Holographic non-local QFT action. Thedifferent point from usual local field theories, such as theHolographic CFT, is that, for subtracting the groundstate energy , Mann counterterm Lagrangian that isnon-polynomial of curvature invariants is needed for theHolographic non-local QFT whereas a counterterm La-grangian for local field theories, such as Mann countert-erm Lagrangian for the Holographic CFT, that is polyno-mial of them is sufficient. At the moment, the meaningof this difference is unclear to me. I hope it could shedsome light to general Holography and the Holographicnon-local QFT. Acknowledgement
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