# Cardy-Verlinde formula in Taub-NUT/Bolt-(A)dS space

Chong Oh Lee Department of Physics, Chonbuk National University, Jeonju 561-756, Republic of Korea
August 20, 2023
###### Abstract

We consider a finite action for a higher dimensional Taub-NUT/Bolt-(A)dS space via the so-called counter term subtraction method. In the limit of high temperature, we show that the Cardy-Verlinde formula holds for the Taub-Bolt-AdS metric and for the specific dimensional Taub-NUT-(A)dS metric, except for the Taub-Bolt-dS metric.

###### pacs:
11.25.Hf, 11.25.Tq

## I Introduction

The AdS/CFT duality was first conjectured by 't Hooft:1973jz in his search for relationship between gauge theories and strings. The AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj ; Aharony:1999ti ; D'Hoker:2002aw asserts there is an equivalence between a gravitational theory in the bulk and a conformal field theory in the boundary. According to AdS/CFT, a -dimensional S-(A)dS action is given by

 A=AB+A∂B+Act (1)

where the bulk action , action boundary , and counterterm action are given as

 AB = 116πGd+1∫Mdd+1x√−g(R−2Λ), A∂B = −18πGd+1∫∂Mddx√−γΘ, Act = −18πGd+1∫∂Mddx√−γ{−d−1l (2) −lR2(d−2)F(d−3) −l32(d−2)2(d−4) ×(RabRab−d4(d−1)R2)F(d−5) +l5(d−2)3(d−4)(d−6) ×(3d+24(d−1)RRabRab−d(d+2)16(d−1)2R3 +d−22(d−1)Rab∇a∇bR−Rab□Rab +12(d−1)R□R)F(d−7)+⋯},

where a negative cosmological constant is , is the trace of extrinsic curvature. Here, is a step function, 1 when , 0 otherwise. The boundary action is added to the action to obtain equations of motion well behaved at the boundary. Then the boundary energy-momentum tensor is expressed in Brown:1992br

 2√−γδA∂Bδγab=Θab−γabΘ. (3)

The counterterm action is added to the action to remove the divergence appearing as the boundary goes to infinity Balasubramanian:1999re . For low dimensional S-AdS, a few terms in the counterterm action were explicitly evaluated in Balasubramanian:1999re ; Emparan:1999pm . Using the universality of the structure of divergences, the counterterm action for arbitrary dimension is suggested in Kraus:1999di . This action (1) leads to the entropy via the Gibbs-Duhem relation

 S=ET−A (4)

where denotes the temperature and is the total energy.

The entropy of the (1+1)-dimensional CFT is expressed in terms of the Virasoro operator and the central charge , the so-called the Cardy formula Cardy:1986ie . Using conformal invariance, the generalized Cardy formula in arbitrary dimension is shown to be given universal form as Verlinde:2000wg (for the review articles of the issue, see, e.g., Nojiri:2001fa ; Nojiri:2002hz ; Lidsey:2002ah )

 SCFT=2πR√ab√Ec(2E−Ec), (5)

where and are certain constants. denotes the radius of the universe at a given time and is the Casimir energy defined by

 Ec=dE−(d−1)TS. (6)

Employing AdS/CFT dual picture, is fixed to exactly, in particular, for a -dimensional CFT on  Verlinde:2000wg . Then, the entropy is given as

 SCFT=2πRd−1√Ec(2E−Ec), (7)

which is shown to hold for Schwarzschild (A)dS (S-(A)dS) Verlinde:2000wg ; Cai:2001sn , charged (A)dS Cai:2001jc ; Cai:2001tv , Kerr-(A)dS Klemm:2001db ; Cai:2001tv , and Taub-Bolt-AdS Birmingham:2001vd . There are many other relevant papers on the subject Wang:2001bf ; Wang:2001bv ; Setare:2002ss ; Setare:2002qa ; Setare:2003fg . Thus, one may naively expect that the entropy of all CFTs that have an AdS-dual description is given as the form (7). However, AdS black holes do not always satisfy the Cardy-Verlinde formula (see, e.g., Cai:2001jc ; Gibbons:2005vp ). Therefore, one intriguing question is whether this formula is valid for higher dimensional Taub-NUT-(A)dS at high temperature. In this Letter, we will endeavor to do this.

When the total number of dimension of the spacetime is even, , for some integer , the Euclidean section of the arbitrary (d+1)-dimensional-Taub-NUT-AdS metric, for a fibration over a series of the space as the base space , is given by Chamblin:1998pz ; Hawking:1998ct ; Mann:1999bt ; Awad:2000gg ; Clarkson:2002uj ; Astefanesei:2004ji ; Astefanesei:2004kn (for the generalized versions of the issue, see, e.g., Mann:2003zh ; Mann:2005ra )

where represents a NUT charge for the Euclidean section, and the metric function has the general form

 f(r) = r(r2−N2)u∫r[(p2−N2)up2 (9) + (2u+1)(p2−N2)u+1l2p2]dp−2mr(r2−N2)u,

with a cosmological parameter and a geometric mass .

Requiring the NUT solution occurs. Then for AdS spacetime the inverse of the temperature arises from imposed condition in order to ensure regularity in the Euclidean time and radial coordinate Chamblin:1998pz ; Hawking:1998ct ; Mann:1999bt ; Awad:2000gg ; Clarkson:2002uj ; Astefanesei:2004ji ; Astefanesei:2004kn

 β=4πf′(r)∣∣∣r=N=2(d+1)πNq, (10)

where is the period of . Here is a positive integer, which originates from removing Misner string singularities. Using counter term subtraction method the regularized action is given as Chamblin:1998pz ; Hawking:1998ct ; Mann:1999bt ; Awad:2000gg ; Clarkson:2002uj ; Astefanesei:2004ji ; Astefanesei:2004kn

 INUT = (11) ×Γ(2−d2)Γ(d+12)β.

Employing the thermal relation the total energy can also be written by

 E = (4π)d2(d−1)Nd−2((d+1)N2−l2)32π2ql2 (12) ×Γ(2−d2)Γ(d+12),

and the entropy is given as Chamblin:1998pz ; Hawking:1998ct ; Mann:1999bt ; Awad:2000gg ; Clarkson:2002uj ; Astefanesei:2004ji ; Astefanesei:2004kn

by the Gibbs-Duhem relation where denotes the conserved mass

 M=(d−1)(4π)d216π32m. (14)

Substituting (10), (12), and (13) into (6), one gets the Casimir energy Verlinde:2000wg

 Ec = (15) ×Γ(2−d2)Γ(d+12).

From now on, for convenience we use instead of the universe radius in (7) since the AdS metric is always asymptotically taken to be Fefferman:1984aa

where the is put to , and the roman indexes and refer to boundary coordinates. When in the formula (7) is taken to be , the CFT entropy is given as

 SCFT = 4πl√|Ec(2E−Ec)|(d+1)(d−1)(d−2), (17) = (4π)d2|dN2−l2|(−1)[d2]Γ(d+12)Γ(2−d2)4π(d+1)(d−2)q,

where is the Gauss number (greatest integer less than or equal to x). Here it seems that the difference from the standard Cardy-Verlinde formula (7) is due to the distinctive nature of NUT solution in AdS space like asymptotically locally AdS (ALAdS) metric. In the limit of high temperature, , leading term in the entropy of CFT can be expressed as

 SCFT = (4π)d2(d+1)(d−2)Nd−116πq (18) ×(−1)[d2]Γ(d+12)Γ(2−d2) = (−1)[d2]SNUT,AdS.

This result shows that the entropy of the Taub-NUT-AdS space suffices to be the generalized Cardy-Verlinde formula (5) for all even . This is reasonable because the Taub-NUT-AdS metric has the thermodynamically stable range depending on the magnitude of the NUT charge i.e. any NUT solution in AdS space for all odd is thermodynamically unstable in the lime .

Requiring and , the Bolt solution occurs. In Taub-Bolt-AdS metric, the inverse of the temperature, the total energy, and the entropy are respectively

 β=4πf′(r)∣∣∣r=rB=4πl2rBl2+(2u+1)(r2B−N2), (19)
 E = (4π)uu8π(u∑k=0([]luk)(−1)kN2kr2u−2k−1B2u−2k−1 (20) +u+1∑k=0(u+1k)(−1)kN2kr2u−2k−1B2u−2k−1),
 SBolt,AdS = (4π)uβ16πl2[(2u−1)(2u+1)(−1)uN2u+2rB +u∑k=0(uk)(−1)kN2kr2u−2kB ×((2u−1)l2(2u−2k−1)rB +(2u+1)(2u2+3u−2k+1)rB(2u−2k+1)(u−k+1))],

where . The CFT entropy is written as

 SCFT=2πl√Ec(2E−Ec)2u√2u−1, (22)

where is fixed to . In the high temperature limit, the CFT entropy well suffices to be Cardy-Verlinde formula as the following

Note that the higher dimensional Taub-Bolt-AdS space follows the generalized Cardy-Verlinde formula (5) even if Taub-Bolt-AdS space exactly satisfies the Cardy-Verlinde formula (7Birmingham:2001vd .

## Iii Taub-NUT/Bolt-dS black hole

Taub-NUT-dS metric is obtained from the Taub-NUT-AdS metric by replacing , and one has Clarkson:2003wa ; Clarkson:2003kt ; Mann:2004mi

 ds2dS = − dr2g(r)+(r2−N2)u∑i=1[dθ2i+sin2(θi)dϕ2i],

where is given as

 g(r) = −r(r2−N2)u∫r[(p2−N2)up2 (25) − (2u+1)(p2−N2)u+1l2p2]dp+2mr(r2−N2)u.

Using parallel way as in the previous case, the inverse of the temperature, the total energy, the entropy, and the Casimir energy are obtained

 β=4πg′(r)∣∣∣r=N=2(d+1)π|N|q, (26)
 E = (4π)d2(d−1)Nd−2((d+1)N2+l2)32π2ql2 (27) ×Γ(2−d2)Γ(d+12),
 Ec = (29) ×Γ(2−d2)Γ(d+12).

In this case, in (7) is fixed to . Then, the entropy in boundary CFT is expressed as

 SCFT = 4πl√Ec(2E−Ec)(d+1)(d−1)(d−2), (30) = (4π)d2(dN2+l2)(−1)[d2]Γ(d+12)Γ(2−d2)4π(d+1)(d−2)q.

In high temperature limit, leading term in the entropy of the CFT for all even is precisely matched with that in the entropy of the Taub-NUT-dS space as the following

 SCFT = (4π)d2(d+1)(d−2)Nd−1Γ(4−d2)Γ(3+d2)16π32q (31) = SNUT,dS.

For the Bolt solution in dS space, the inverse of the temperature, the total energy, and the entropy are respectively

 β=4πf′(r)∣∣∣r=rB=−4πl2rBl2+(2u+1)(r2B−N2), (32)
 E = (33) +u+1∑k=0(u+1k)(−1)kN2kr2u−2k−1B2u−2k−1),
 SBolt,dS = (4π)uβ16πl2[−(2u−1)(2u+1)(−1)uN2u+2rB +u∑k=0(uk)(−1)kN2kr2u−2kB ×(−(2u−1)l2(2u−2k−1)rB +(2u+1)(2u2+3u−2k+1)rB(2u−2k+1)(u−k+1))],

where . The CFT entropy is given as

 2πl√|Ec|(2E−Ec)2u√2u−1, (35)

where is fixed to . As the NUT charge goes to 0, the CFT entropy becomes

 SCFT=(4π)u4N2u(ql22u2+3u+1)2u=−SBolt,dS, (36)

which shows that no entropy of the Taub-Bolt-dS metric satisfies the Cardy-Verlinde formula. This means that any Bolt solution in dS space is thermodynamically unstable at high temperature limit.

## Iv Conclusion

We have considered that the Taub-NUT/Bolt-(A)dS metric in general even dimension, and have checked that its metric suffices to be the the Cardy-Verlinde formula. In the limit of high temperature, we showed that the Taub-Bolt-AdS space well follows the generalized Cardy-Verlinde formula (7) rather than the Cardy-Verlinde formula (5). It seems that the modification of the standard Cardy-Verlinde formula (5) is due to the distinctive property of the Taub-NUT solution such as the ALAdS metric. It was proven that the leading term of the CFT entropy at the boundary for all even is exactly matched with that of the entropy in the Taub-NUT-(A)dS space by using the generalized Cardy-Verlinde formula at high temperature. Thermal stability of the Taub-NUT-(A)dS solution for all odd and Taub-Bolt-dS solution for all is determined by the magnitude of the NUT charge so that the negative entropy occurs as the NUT charge goes to 0. Finally, the the breaking of Cardy-Verlinde formula in the Taub-Bolt-dS metric seems to reflect the fact that there is no Bolt solution in dS space due to the absence of hyperbolic NUT in AdS space Mann:2004mi .

## Acknowledgements

We are grateful to Cristian Stelea for useful comments. This work was supported by the BK 21 project of the Ministry of Education and Human Resources Development, Korea (C.O.L.).
{"ft_lang_label":"__label__en","ft_lang_prob":0.72560203,"math_prob":0.9723875,"size":12478,"snap":"2023-40-2023-50","text_gpt3_token_len":4191,"char_repetition_ratio":0.15023248,"word_repetition_ratio":0.0985,"special_character_ratio":0.3459689,"punctuation_ratio":0.22825702,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9861106,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-03T15:10:01Z\",\"WARC-Record-ID\":\"<urn:uuid:b501c4ed-ee6b-4719-89ea-0df72f8942e9>\",\"Content-Length\":\"604980\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:90478746-1e83-4004-9f91-a7d72cdf137e>\",\"WARC-Concurrent-To\":\"<urn:uuid:dbc72773-c4f8-4cd5-bd2e-ee5bcb8d945f>\",\"WARC-IP-Address\":\"104.21.14.110\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/0807.2685/\",\"WARC-Payload-Digest\":\"sha1:RZGDHTVAEECHFTO3DPXE6Z74WETLQ7BO\",\"WARC-Block-Digest\":\"sha1:7MQYC5UGU3QKMPAVNJZ462D2D76X2RL6\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100508.23_warc_CC-MAIN-20231203125921-20231203155921-00378.warc.gz\"}"}