Thermodynamics of (1+1) dilatonic black holes in global flat embedding scheme

We study thermodynamics of (1+1) dimensional dilatonic black holes in global embedding Minkowski space scheme. Exploiting geometrical entropy correction we construct consistent entropy for the charged dilatonic black hole. Moreover, (1+1) dilatonic black holes with higher order terms are shown to possess (3+2) global flat embedding structures regardless of the details of the lapse function, and to yield a generic entropy.

Si nce (1+ 1) di m ensi onal bl ack hol es associ ated w i th stri ng theory was proposed [ 1] , there have been l ots of progresses such as di scovery of U -dual i ty between two di m ensi onaldi l atoni c bl ack hol es [ 2,3,4,5]and ve dim ensi onalone i n the stri ng theory. A therm alH aw ki ng e ect on a curved m ani fol d [6,7]can be l ooked at as an U nruh e ect [8]i n a gl obalem beddi ng M i nkow skispace (G EM S).T hi sG EM S approach [ 9,10,11]coul d suggesta uni ed deri vati on oftherm odynam i cs for vari ous curved m ani fol ds [ 9] and the (5+ 1) G EM S structure of (3+ 1) Schwarzschi l d bl ack hol e sol uti on [ 12]was obtai ned [ 9] .
In thi s paperwe study therm odynam i csof(1+ 1)di l atoni c bl ack hol es i n the G EM S schem e. U si ng geom etricalentropy correcti on we can obtai n consi stent entropy for a charged di l atoni c bl ack hol e. M ore general(1+ 1) di l atoni c bl ack hol es are show n to possess (3+ 2) G EM S structures regardl ess ofthe detai l s ofthe l apse functi on w i th hi gher order term s,and to yi el d a generi c entropy form ul a.
W e startw i th two-di m ensi onaldi l atoni c bl ack hol es[ 2, 3,4,5]associ ated w i th the type IIA stri ng theory and i ts com pacti cati on to ve di m ensi ons w hose m etri c i s the productofa three-sphereand an asym ptoti cal l y attwodi m ensi onal geom etry. T he ten-di m ensi onal type IIA superstri ng sol uti on consi sts of a sol i toni c N S 5-brane w rappi ng around the com pact coordi nates, say, x 5 , x i (i= 6;7;8;9)and a fundam entalstri ng w rappi ng around x 5 , and a gravi tati onalwave propagati ng al ong x 5 . In the stri ng fram e,the 10-m etri c,di l aton and 2-form el d B are gi ven as [ 13,14,15,16] [ 13,14] , and then perform i ng an T 5 ST 6789 ST 5 transform ati on [ 17]and an SL(2, R ) coordi nate transform ati on associ ated w i th the O (2, 2) T -dual i ty group,together w i th the sam e set of reverse S and T transform ati ons,one can obtai n the ve-di m ensi onalbl ack hol e m etri c (1) e 2 = r 2 r 2 0 + si nh 2 ; (2) w here H 5 = r 2 0 =r 2 . N ext,perform i ng di m ensi onalreducti on i n thex 5 ,x i (i= 6;7;8;9)di recti onsw i th = ,one can arri ve at the ve-di m ensi onalbl ack hol e m etri c [18] ds 2 = 1 r 2 0 r 2 1 + r 2 0 si nh 2 r 2 2 dt 2 + r 2 r 2 0 1 1 dr 2 + r 2 0 d 2 3 ; (3) and the di l aton w hi ch i s tri vi al l y i nvari antunder the dim ensi onalreducti on to yi el d the above resul t (2). H ere one notes that the m etri c (3) i s the product ofthe two com pl etel y decoupl ed parts,nam el y,a three-sphere and an asym ptoti cal l y at two-di m ensi onalgeom etry w hi ch descri bes the two-di m ensi onal charged di l atoni c bl ack hol e. Introduci ng a new vari abl e x w i th Q = 2=r 0 e Q x = 2 r 2 r 2 0 + si nh 2 (m 2 q 2 ) 1=2 ; w here m and q are the m ass and charge ofthe di l atoni c bl ack hol e,one can obtai n the wel l -know n (1+ 1)charged di l atoni c bl ack hol e [ 2,3,4] w here the l apse functi on i s gi ven as W e can then obtai n the hori zon x H and x i n term s of the m ass m and the charge q: B y usi ng these rel ati ons,we can rew ri te the l apse functi on as Fi rst,we consi der the uncharged di l atoni c bl ack hol e 2-m etri c from w hi ch we can construct (3+ 1) di m ensi onalG EM S w here the surface gravi ty i s gi ven by k H = Q =2. U si ng the G EM S (6) and the rel ati on G 4 = G 2 V 2 (detai l s of w hi ch w i l l be di scussed l ater), w here V 2 i s a com pact vol um eV 2 = 2=Q gi ven al ong z 2 onl y,wecan then obtai n the desi red entropy (7) w hi ch i s consi stent w i th the previ ous resul t i n [ 3,18] .
Second,for a charged di l atoni c bl ack hol e case associated w i th them etri c(4),wecan constructa (3+ 2)G EM S ds 2 = (dz 0 ) 2 + (dz 1 ) 2 + (dz 2 ) 2 + (dz 3 ) 2 (dz 4 ) 2 gi ven by the coordi nate transform ati ons, si nh k H t; w here the surface gravi ty i s gi ven by H ere one can al so check that, i n the uncharged l i m i t q ! 0,the above coordi nate transform ati onsare exactl y reduced to the previ ous one (6) for the uncharged di l atoni c bl ack hol e case. M oreover,one can easi l y obtai n the rel ati on between z 3 and z 4 as fol l ow s In the standard G EM S approach,al lthe i nform ati ons for the entropy com e from the areas them sel ves associ ated w i th the event hori zons. H ere the N ew ton constants G n i n the hi gher di m ensi onalem beddi ngs are i mpl i ci tl y treated to be the sam e as the ori gi nalG d ofthe d-di m ensi onalbl ack hol es [ 9] .H owever,i n the(1+ 1)di l atoni c bl ack hol e cases,we coul d not obtai n the areas i n term softhe eventhori zonsdue to the del ta-functi on-l i ke behavi orsatthe eventhori zons,w hi ch arecharacteri sti cs of the (1+ 1) di l atoni c bl ack hol es. A s i n (7), i n order to obtai n the entropi es, we thus expl oi t an al ternati ve schem e, w here the entropy i nform ati ons are extracted from the N ew ton constants G n w hi ch are now spl i tted i nto two factors: G n = G d V n d w i th the vol um es of the com pact m ani fol ds V n d . To be m ore speci c, i n order to cal cul ate the entropy for the charged di l atoni c bl ack hol e,we rstconsi dera detectoron the eventhorizon at x = x H w here the detector onl y sees a com pact m ani fol d V 3 al ong the z 3 and z 4 di recti ons,gi ven by T he N ew ton constant i s then gi ven by N ote that,even though we have used the x H (q)i n cal cul ati on ofthe above entropy S(x H ),the nalresul t does notcontai n any i nform ati on ofthe charge q and m ass m associ ated w i th the even hori zons x H and x , to yi el d the sam e entropy (7) ofthe uncharged case. D i erent from the uncharged case, we have another event hori zon x = x , w here we have another compact m ani fol d w i th vol um e V 3 (x ) = z 4 (x ) to yi el d the m odi ed N ew ton constantG 5 = G 2Ṽ3 w i thṼ 3 = V 3 (x H )+ V 3 (x ) = z 4 (x H )+ z 4 (x ),si nce the detector at the event hori zon x = x can see two com pact m ani fol ds at x = x and x = x H . M oreover, i t has been cl ai m ed i n [ 19]that the entropy ofa charged bl ack hol e shoul d decrease w i th the absol ute val ue ofthe bl ack hol e charge. W e can then obtai n the entropy l oss due to the exi stence ofthe com pact m ani fol d at to yi el d the totalentropy S = S(x H ) S ofthe di l atoni c charged bl ack hol e as fol l ow s w hi ch i s consi stent w i th the previ ous resul t i n [ 3,18] . N ote thati n the vani shi ng charge l i m i t q ! 0,the above entropy i s reduced to that ofuncharged case (7). M oreover, w i thout the U -dual i ty transform ati ons di scussed above,we can obtai n the consi stent entropy (11) vi a the G EM S em beddi ngs and thei r associ ated geom etri calentropy correcti ons. Fol l ow i ng the standard procedure i n generalrel ati vi ty, one can obtai n the 2-accel erati on,the H aw ki ng tem perature and the bl ack hol e tem perature c n e n Q x ; (12) w here c 1 = 2m ,c 2 = q 2 and c n (n 3) are coe ci ents ofhi gher order term s. N ote that the l apse functi on (12) can be rew ri tten i n term s ofthe event hori zons x n (n = 1;2;: : : ) w i th x 1 = x H and x n > x n + 1 , (1 e Q (x x n ) ): A si n theprevi ouscase,wecan obtai n thesurfacegravi ty, the 2-accel erati on and the H aw ki ng tem perature i n m ore generalform w hi ch are i ndependent of the di m ensi onal i ty of the G EM S structures. W e construct the G EM S em beddi ng sol uti ons for our general (1+ 1) di l atoni c bl ack hol e by m aki ng an ansatz ofthree coordi nates(z 0 ;z 1 ;z 2 )i n (15) to yi el d H ere we have used the fact that the term s i n the parenthesi s i n the second l i ne can be expressed i n term s of di erence oftwo posi ti ve de ni te term s w here F and G can be read o from (15). W e can thus obtai n the (3+ 2) di m ensi onalG EM S ds 2 = (dz 0 ) 2 + (dz 1 ) 2 + (dz 2 ) 2 + (dz 3 ) 2 (dz 4 ) 2 gi ven by the coordi nate transform ati ons, N ote that,as i n (9),z 4 can be expressed i n term s ofz 3 : N ow we com m ent on the di m ensi onal i ty ofthe G EM S em beddi ngs i n the general di l atoni c bl ack hol es. T he charge param eter c 2 = q 2 i ntroduces one m ore ti m e-l i ke di m ensi on to yi el d two ti m e di m ensi onal i ti es w i th (3+ 2) G EM S structures for the charged di l atoni c bl ack hol e. In the generaldi l atoni c bl ack hol esw i th c n (n = 1;2;: : : ), even though we have hori zonsx n m ore than two onesx H and x of the charged di l atoni c bl ack hol e, the G EM S structures are xed as (3+ 2) di m ensi ons w i th no m ore i ncreasi ng di m ensi onal i ty,si nce onl y two posi ti ve de ni te term sF 2 and G 2 areenough to descri betheterm sof (14) regardl ess ofw hatever the l apse functi on N 2 has hi gher order term s w i th c n (n = 1;2;: : : ).
N ext, we cal cul ate the entropy for the general(1+ 1) di l atoni c bl ack hol e w i th hi gher order term s. A s i n the charged di l atoni c bl ack hol e case,a detectoron the event hori zon atx = x H onl y seesa com pactm ani fol d V 3 al ong the z 3 and z 4 di recti ons to yi el d the entropy (10) at x = x H . H owever,we have othereventhori zonsx = x n (n = 2;3;: : : )associ ated w i th com pactm ani fol dsw i th vol um es V 3 (x n )= z 4 (x n ) to yi el d the N ew ton constant T he exi stences of the com pact m ani fol ds at x = x n (n = 2;3;: : : ) thus yi el d the geom etri calentropy correcti on ori gi nated fromG 5 : so that, together w i th S(x H ) w hi ch has the sam e form as(10),we can obtai n the totalentropy S = S(x H ) S ofthe general(1+ 1) di l atoni c bl ack hol e S = 1 4G 2 P n = 2 z 4 (x n ) P n = 1 z 4 (x n ) : In thecharged casew i th c 1 = 2m ,c 2 = q 2 and c n = 0 (n 3), by expl oi ti ng the expl i ci t expressi on for z 4 i n the G EM S (8) we obtai n for the hori zons x 1 = x H and ; A ftersom eal gebra w i th thei denti ti es(5),substi tuti on of z 4 (x 1 )and z 4 (x 2 )i n (17)i nto thegeneri centropy form ul a (16) reproduces the previ ous resul t (11). Si m i l arl y, for the uncharged case w i th c 1 = 2m and c n = 0 (n 2),we can easi l y check that the entropy (16) i s reduced to the previ ous one (7). For m ore general cases w i th c 1 = 2m ,c 2 = q 2 and nonvani shi ng c n (n 3),we can nd the expressi on for z 4 i n the G EM S (15), w hi ch i s gi ven by an i ntegralform .D i erentfrom thecharged case w i th z 4 (x n ) (n = 1;2) i n (17),for thi s generaldi l atoni c bl ack hol e we do not have expl i ci t anal yti c expressi ons for z 4 (x n ) at the m om ent so that we cannot proceed to eval uate the entropy vi a the form ul a (16). H owever,i f the coe ci entsc n aregi ven expl i ci tl y,we can nd x n and z 4 (x n ),w i th w hi ch the generi c entropy (16) i s supposed to yi el d to al lordera resul tconsi stentw i th thatgi ven i n [ 3] .
In concl usi on, we have i nvesti gated the hi gher dim ensi onalgl obal at em beddi ngs of (1+ 1) (un)charged and general di l atoni c bl ack hol es. T hese two di m ensi onaldi l atoni c bl ack hol es are show n to be em bedded i n the (3+ 1) and (3+ 2)-di m ensi ons for the uncharged and charged two-di m ensi onal di l atoni c bl ack hol es, respecti vel y.M oreover,i n the generaldi l atoni c bl ack hol es w i th hi gherorderterm s,even though wehavehori zonsx n m ore than two ones x H and x ofthe charged di l atoni c bl ack hol e,the G EM S structures have been show n to be xed as (3+ 2) w i th no m ore i ncreasi ng di m ensi onal i ty. D i erent from the uncharged case,i n order to obtai n theentropy ofthe(1+ 1)charged di l atoni cbl ack hol es,we have taken i nto accountal lthe com pactm ani fol d associated w i th the event hori zons to yi el d the m odi ed N ewton constant. Expl oi ti ng the geom etri cal entropy correcti on ori gi nated from the m odi ed N ew ton constant, we have obtai ned the entropy for the charged di l atoni c bl ack hol esand even forthe generaldi l atoni c bl ack hol es. It i s qui te si gni cant to obtai n the consi stent entropi es through the G EM S em beddi ngsand thei rassoci ated geom etri calentropy correcti ons,w i thoutgetti ng i nvol ved i n the U -dual i ty transform ati ons associ ated w i th the type IIA stri ng theory.
A cknow ledgm ents T he author woul d l i ke to thank Prof. C hi ara R .N appifor hel pfuldi scussi ons and worm hospi tal i ty at Pri nceton U ni versi ty,w here a part ofthi s work has been done. T hi s work was supported by the Ew ha W om ans U ni versi ty R esearch G rant of2004.