BLACK HOLES IN STRING THEORY

Transcription

1 Black holes in sting theoy N Sadikaj & A Duka Pape pesented in 1 -st Intenational Scientific Confeence on Pofessional Sciences, Alexande Moisiu Univesity, Dues Novembe 016 BLACK OLES IN STRING TEORY NDRIÇIM SADIKAJ 1, ANILA DUKA 1 1 Faculty of Technical Sciences, Univesity Ismail Qemali Vloё, Albania Coesponding autho ndsadikaj@gmailcom Abstact: A black hole is a egion of space time exhibiting such stong gavitational effects that nothing not even paticles and electomagnetic adiation such as light can escape fom inside it Fom the theoetical point of view, black holes povide an intiguing aena in which to exploe the challenges posed by the econciliation of geneal elativity and quantum mechanics Since sting theoy pupots to povide a consistent quantum theoy of gavity, it should be able to addess these challenges In fact, some of the most fascinating developments in sting theoy concen quantum-mechanical aspects of black hole physics These ae the subject of this pape This pape is devoted to tying to find a micoscopic quantum desciption of black holes One of the most impotant achievements of sting theoy in ecent times is the constuction of examples that povide an affimative the themodynamic desciption of black holes The themodynamics of stings is govened lagely by the exponential gowth of the numbe of quantum states accessible to a sting, as a function of its enegy The behavio of the entopy indicates that at high enegies the tempeatue appoaches a finite constant, the agedon tempeatue We explain how the counting of sting states can be used to give a statistical mechanics deivation of the entopy of black holes The calculations give esults in qualitative ageement with the entopy of Schwazschild black holes and in quantitative ageement with the entopy of cetain chaged black holes Key wods: Sting Theoy, Black ole, Entopy, D-banes 1 Intoduction In 197, Bekenstein was the fist to suggest that black holes should have a well-defined entopy e wote that a black hole s entopy was popotional to its (the black hole s) event hoizon Bekenstein also fomulated the genealized second law of themodynamics, black hole themodynamics, fo systems including black holes Both contibutions wee affimed when Stephen awking poposed the existence of awking adiation two yeas late In 197, awking pefomed a lengthy calculation that convinced him that paticles do indeed emit fom black holes Today this is known as Bekenstein- awking adiation In most physical systems the themodynamic entopy has a statistical intepetation in tems of counting micoscopic configuations with the same macoscopic popeties, and in most cases this counting equies an undestanding of the quantum degees of feedom of the system Sting theoy, being a theoy of quantum gavity, should be able to descibe black holes As we shall see, although sting theoy is usually well appoximated by local quantum field theoy, in the neighbohood of a black hole hoizon the diffeences become exteme The analysis of these diffeences suggests a esolution of the black hole dilemma and a completely new view of the elations between space, time, matte, and infomation 1 Black holes in geneal elativity In ode to intoduce the eade to some basic notions of black-hole physics, let us begin with the simplest black-hole solutions of geneal elativity in fou dimensions, which ae the Schwazschild and Reissne -Nodstöm black holes The latte black hole is a genealization of the Schwazschild solution that is electically chaged Anothe genealization, known as 5

2 Intedisplinay Jounal of Reseach and Development Alexande Moisiu Univesity, Duës, Albania Vol (IV), No, 017 the Ke black hole, is a black hole with angula momentum 11 Schwazschild black hole In Schwazschild coodinates, the Schwazschild geomety is manifestly spheically symmetic and static In Schwazschild coodinates (t,, θ, φ) the metic is given by: 1 D D = + + ΩD ds 1 ( ) dt 1 ( ) d d, with D 16π = ( D ) Ω D D whee: 1 µ ν ds = (1 ) dt (1 ) d dω = gµνdx dx d d sin d Ω = θ + θ φ The coodinate t is called Schwazschild time, and it epesents the time ecoded by a standad clock at est at spatial infinity, GM is known as the Schwazschild adius and denoted, G is Newton s constant and the coodinate is called the Schwazschild adial coodinate The Schwazschild metic only depends on the total mass M and it educes to the Minkowski metic as M 0 The suface =, called the event hoizon, sepaates the pevious two egions This metic is stationay in the sense that the metic components ae independent of the Schwazschild time coodinate t, so that / t is a Killing vecto This Killing vecto is time-like outside the hoizon, null on the hoizon, and space-like inside the hoizon It becomes clea that M has the intepetation of a mass by consideing the weak field limit, that is, the asymptotic In this limit we should ecove Newtonian gavity The Newtonian potential φ in these stationay coodinates can be ead off fom the tt component of the metic: gtt : (1+ φ) As a esult, in the case of the Schwazschild black hole, φ = The paamete M is the black-hole mass The fou-dimensional Schwazschild metic can be genealized to D dimensions, whee it takes the fom: ee Ω n is the volume of a unit n-sphee Fo lage, this again detemines the Newton potential and theefoe the black-hole mass M 1 Schwazschild solution in Kuskal- Szekees coodinates Finally we can bing the black hole metic to the fom: h / = h ν + + Ω ds e ( d du ) d, t whee u = e cosh 1 / ( 1) t ν = e sinh( ) 1 / ( 1) ( ), The aea of the event hoizon is: A= π = 16 π( ) 1 Reissne-Nodstöm black hole The genealization of the Schwazschild black hole to one with electic chage Q, but no angula momentum, is called the Reissne-Nodstöm black hole In fou dimensions the metic of a Reissne-Nodstöm black hole can be witten in the fom: ds dt d d 1 = Δ +Δ + Ω, whee: Δ= Q G

3 Black holes in sting theoy N Sadikaj & A Duka This metic is a solution to Einstein s equations in the pesence of an electic field G µν = R µν 1/Rg µν = 8πGT µν 1 Extemal Reissne-Nodstöm black hole fo D = In the limiting case = o M G = Q the black hole is called extemal, and it has the maximal chage that is allowed given its mass The metic of an extemal Reissne-Nodstöm black hole takes the fom: whee: 0 ds = (1 ) dt + (1 ) + dω 0 0 = 15 Extemal Reissne-Nodstöm black hole fo D = 5 An extemal Reissne-Nodstöm black hole in D = 5 is of inteest in connection with the micoscopic deivation of the black-hole entopy Its metic can be witten in a fom simila to 0 0 = + ) + Ω ds (1 ) dt (1 d Using this expession, it is easy to see that the hoizon at = 0 has adius 0, and theefoe and its aea is: A=Ω = π 0 0 The mass and chage of this black hole ae: Q π M = = G G 5 Black-hole themodynamics Classical black holes behave like themodynamical objects chaacteized by a tempeatue and an entopy The entopy S of the system is defined in tems of the numbe of states as: 0 5 S (E) = k ln Ω (E), whee: k is Boltzmann s constant The tempeatue T of the system is defined in tems of the deivative of the entopy with espect to the enegy: 1 S = T E The tempeatue of the Schwazschild black hole is: 1 T = 8π Fo a Schwazschild black hole the entopy is: S = π M G Black holes in sting theoy This section consides supesymmetic (and hence extemal) black holes that have finite entopy in the supegavity appoximation These include theechage black holes in five dimensions and fou-chage black holes in fou dimensions, which can be intepeted as appoximations to solutions of tooidally compactified sting theoy Fo this class of compactifications, finite-hoizon-aea black-hole solutions that ae asymptotically flat only exist in the supegavity appoximation in fou and five dimensions 1 Extemal thee-chage black holes fo D = 5 The simplest nontivial example fo which the entopy can be calculated involves supesymmetic black holes in five dimensions that cay thee diffeent kinds of chages These can be studied in the context of compactifications of the type IIB supesting theoy on a five-tous T 5 The analysis is caied out in the appoximation that five of the ten dimensions of the IIB theoy ae sufficiently small and the black holes ae sufficiently lage so that a five-dimensional supegavity analysis can be used N=8 supegavity fo D=5 The supegavity theoy in question is N = 8 supegavity in five dimensions This contains a numbe of one-fom and two-fom gauge fields 56

4 Intedisplinay Jounal of Reseach and Development Alexande Moisiu Univesity, Duës, Albania Vol (IV), No, 017 Thee-chage black holes in five dimensions can be appoximation, and this equies intoducing thee obtained by taking Q 1 D1-banes wapped on an S 1 of diffeent kinds of excitations The mass of the black adius R inside the T 5, Q 5 D5-banes wapped on the hole M to be T 5 = T S 1, and n units of Kaluza-Klein momentum π along the same cicle Each of these objects beaks M = M 1 + M + M whee, M = i half of the supesymmety, so altogethe 7/8 of the G5 supesymmety is boken, and one is left with solutions that have fou conseved supechages The fact that the masses ae additive in this way is a Othe equivalent sting-theoetic constuctions of consequence of the fom of the metic oweve, this these black-hole solutions ae elated to the one had to be the case, because the BPS condition is consideed hee by U-duality tansfomations Thee satisfied, and the chages ae additive The entopy is: ae a vaiety of ways to analyze this system One of them is in tems of a five-dimensional gauge theoy A π gl s s Since the Q 1 D1-banes ae embedded inside the Q 5 S = = M1MM D5-banes, this configuation can be descibed G5 RV entiely in tems of the U(Q 5 ) wold-volume gauge theoy of the D5-banes In this desciption a D-sting In tems of the chages, one obtains the esult: wound on a cicle is descibed by a U(Q 5 ) instanton that is localized in the othe fou diections So, altogethe, thee ae Q 1 such instantons The Kaluza- S = π QQ 1 5n Klein momentum can also be descibed as excitations in this gauge theoy The five-dimensional metic Nonextemal thee-chage black holes fo descibing this black-hole can be obtained fom the D = 5 ten-dimensional type IIB theoy by wapping the The extemal thee-chage black-hole solutions in five coesponding banes as descibed above, o it can be dimensions given above have non extemal constucted diectly In eithe case, the esulting genealizations, which descibe non supe symmetic metic can be witten in Einstein fame in the fom: black holes with finite tempeatue These black holes ae descibed by the metic whee: ds = -λ dt + λ (d + d Ω ) -/ 1/ λ = + i 1 1 whee: -/ 1/ d ds = -h λ dt + λ + d Ω, h h=1-0 and i λ = This solution descibes an extemal thee-chage black hole with a vanishing tempeatue T = 0 The hoizon of the black hole is located at = 0, and its aea is: A = π 1 This vanishes when any of the thee chages vanishes, which is the eason that thee chages have been consideed in the fist place Put diffeently, one needs to beak 7/8 of the supe symmety in ode to fom a hoizon that has finite aea in the supe gavity A S= =16 π G M1MMM G with i = 0 sinh α i, i = 1,, The mass of this black hole can be ead off using the same ules as befoe esulting in: π 0 M = (cosh α1 + cosh α+ cosh α ) 8G 5 57

5 Black holes in sting theoy N Sadikaj & A Duka The aea of the hoizon is: ae not pesent in the usual Einstein theoy of gavity It does not tell us what they ae Sting theoy does A =π = π 0cosh α1cosh αcosh α povide a micoscopic famewok fo the use of statistical mechanics In all cases the entopy of the appopiate sting system agees with the Bekenstein- The entopy is then given by: awking entopy This, if nothing else, povides an existence poof fo a consistent micoscopic theoy of A π V 0 6 black hole entopy S= = cosh α 6 1cosh αcosh α G5 lp 5 Refeences [1] B Zwiebach (009) A Fist Couse in Sting This also allows the entopy to be ewitten in fom: Theoy Second Edition [] K Becke, M Becke, J Schwaz (007) A Sting Theoy and M-theoy, a moden intoduction S= = π ( Qi + Qi) G [] L Susskind, J Lindesay (005) Black holes, 5 1 infomation and Sting Theoy evolution The ologaphic Univese Extemal fou-chage black holes fo D = [] R C Myes (001) Black holes and Sting The constuction of supe symmetic black holes in Theoy [5] CG Callan, RC Myes and MJ Pey (1989) fou dimensions is quite simila to the fivedimensional case Black oles In Sting Theoy [6] R M Wald (001) The themodynamics of black holes -1/ 1/ ds = -λ dt + λ ( d + d Ω ), [7] S Calip (015) Black ole Themodynamics whee axiv:110186v i λ = The mass of the black hole is M = M, 1 i whee M i i G = The aea of the hoizon, which is located at = 0, is A = 1 Putting these facts togethe, the esulting entopy is: A S= =16 π G M1MMM G Conclusions The existence of black hole entopy indicates the existence of micoscopic degees of feedom which 58

6 Intedisplinay Jounal of Reseach and Development Alexande Moisiu Univesity, Duës, Albania Vol (IV), No,