Massive gravity with Lorentz symmetry breaking: black holes as heat engines
NNKU-2018-SF1
Massive gravity with Lorentz symmetry breaking:black holes as heat engines
Sharmanthie Fernando Department of Physics, Geology & Engineering TechnologyNorthern Kentucky UniversityHighland HeightsKentucky 41099U.S.A.
Abstract
In extended phase space, a static black hole in massive gravity is studied as aholographic heat engine. In the massive gravity theory considered, the graviton gaina mass due to Lorentz symmetry breaking. Exact efficiency formula is obtained fora rectangle engine cycle for the black hole considered. The efficiency is computed byvarying two parameters in the theory, the scalar charge Q and λ . The efficiency iscompared with the Carnot efficiency for the heat engine. It is observed that whenQ and λ are increased that the efficiency for the rectangle cycle increases. Whencompared to the Schwarzschild AdS black hole, the efficiency for the rectangle cycleis larger for the Massive gravity black hole. Key words : static, massive gravity, black hole, heat engine, anti-de Sitter space,efficiency
Black holes in anti-de Sitter space as a thermodynamical system has attracted lotof attention in a variety of contexts: when the negative cosmological constant istaken as the thermodynamical pressure of the black hole with the relation P = − Λ8 π ,the resulting thermodynamics lead to interesting features. In this extended phasespace, the first law of thermodynamics is modified by a term V dP and the mass M of the black hole is treated as the enthalpy rather than the internal energy E ofthe black hole [1] [2]. Many black holes in the context of extended phase space hasdemonstrated Van der Waals type phase transitions between small and large blackholes. Due to the large number of work published related to this topic we will mention [email protected] a r X i v : . [ g r- q c ] A ug ew here: [3] [4] [5] [6] [7][8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18][19]. Thereis a nice review on Black hole chemistry written by Kubiznak et.al. which gives acomprehensive summary of the interesting thermodynamical features of black holeswith a cosmological constant [20].In classical thermodynamics, there are four basic thermodynamical processes.They are isothermal, adiabatic, isobaric, and isochoric processes. In each of theseprocesses the thermodynamical quantities temperature, entropy, pressure, and, vol-ume are kept constant respectively. In a heat engine, a thermodynamical cycle ischosen which consist of aforementioned processes. For example, the Carnot cycle,which has the highest efficiency, have two isothermal and two adiabatic processesin the cycle. Brayton cycle has two isobaric and two adiabatic processes. In thispaper, the goal is to study a black hole in massive gravity as a heat engine. Theidea that black holes could be used as a working substance in a heat engine was firstpresented by Johnson [21]. In that paper, charged black hole in AdS space in D = 4was presented as an example. In an extension of that work, Johnson presented theBorn-Infeld AdS black hole as a heat engine in [22]. In [23], heat engines from dila-tonic Born-Infeld black holes were analyzed. Black holes in conformal gravity as heatengine was presented by Xu et.al. in [24]. Charged BTZ black hole in 2+1 dimen-sions as a heat engine was studied by Mo et.al. in [25]. Effects of dark energy on theefficiency of heat engines of AdS black holes were analyzed in detail by Liu and Mengin [26]. Class of black holes in massive gravity as heat engines were discussed byHendi et.al. [27]. The first study of rotating black holes as holographic heat engineswere done by Henniger et.al. [28]. Heat engines are defined for space-times that arenot black holes as well: Johnson [29] studied Taub-Bolt space-time as an example ofa heat engine and compared with the analogous Schwarzschild black hole as a heatengine.Massive gravity theories have become popular as a mean of explaining acceleratedexpansion of the universe without having to introduce a component of “dark energy.”There are many theories of massive gravity in the literature. There are large volumeof work related to theories in massive gravity [30] [31] [32] [33]: here we will mentiontwo nice reviews on the subject by de Rham [34] and Hinterbichler [35].In this paper we will consider a massive gravity theory where the graviton acquirea mass due to the Lorenz symmetry breaking. Here, Higgs mechanism for gravity isintroduced with space-time depending scalar fields that are coupled to gravity [36].The resulting theory will exhibit modified gravitational interactions at large scale.These models are free from ghosts and tachyonic instabilities around curved space aswell as in flat space [37] [38]. A review of Lorentz violating massive gravity theorycan be found in [39] [40]. The details of the theory is discussed in section 2.The paper is organized as follows: in section 2, the black hole in massive gravityis introduced. Thermodynamics and phase transitions of the massive gravity blackhole are discussed in section 3. Black hole as a heat engine with a rectangle cycleis introduced in section 4 and in section 5 the efficiencies are computed. Finally the2onclusion is given in section 6. The action for the massive gravity theory considered in this paper given by, S = (cid:90) d x √− g (cid:20) − π R + Ω F ( X, W ij ) (cid:21) (1)Here F is a function of our scalar fields φ µ . φ µ are minimally coupled to gravity bycovariant derivatives. F depends on two combinations of the scalar fields given by X and W defined in terms of the scalar fields as, X = ∂ µ φ ∂ µ φ Ω (2) W ij = ∂ µ φ i ∂ µ φ j Ω − ∂ µ φ i ∂ µ φ ∂ ν φ j ∂ ν φ Ω X (3)The constant Ω has dimensions of mass: it is in the order of (cid:113) m g M pl where m g isthe graviton mass and M pl the Plank mass [39] [36] [37] [38]. The scalar fields φ , φ i are responsible for breaking Lorentz symmetry spontaneously when they acquire avacuum expectation value. More details on the theory can be found in [39] [36] [37][38] [40]. A class of black hole solutions to the above action was derived in [41] and[42]. ds = − f ( r ) dt + dr f ( r ) + r ( dθ + sin θdφ ) (4)where, f ( r ) = 1 − Mr − γ Q r λ − Λ r Q represents a scalar charge related to massive gravity and γ = ±
1. The constant λ in is an integration constant and is positive. Λ is the cos-mological constant. A detailed description of this black hole is given by Fernando in[43]. Since for λ <
1, the ADM mass become divergent, such solutions will not beconsidered. When λ >
1, for r → ∞ the metric approaches the Schwarzschild-AdSblack hole with a finite mass M ; hence we will choose λ > γ = 1, the space-time of the black hole is very similar to the Schwarzschild-AdSblack hole and for γ = −
1, the geometry is similar to the Reissner-Nordstrom-AdScharged black hole. 3here are several works related to the black holes given above. P-V criticality andphase transitions of masive gravity black holes with a negative cosmological constantwere presented by Fernando in [43] [44]. Thermodynamics for Λ = 0 case were studiedin [45] [46]. Stability and quasi normal modes were studied in [47][48] [49] [50].
In this section we will derive the thermodynamical quantities and present the phasetransitions that could occur in massive gravity black holes.
The temperature of the black hole could be obtained by using the definition of Hawk-ing temperature: it is based on the surface gravity κ at the outer horizon, r h . It isgiven by, T H = κ π = 14 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) df ( r ) dr (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = r h = 14 π (cid:32) Mr h + γQ λr λ +1 h − r h (cid:33) (6)Since f ( r h ) = 0, the mass of the black hole could be written as, M = r h (cid:32) − γQ r ( λ ) h − r h Λ6 (cid:33) (7)The value of the mass M could be substituted to eq(6) and rewrite temperature interms of r h , Q, and λ as, T = 14 π (cid:32) r h − r h Λ + γ ( λ − Q r λ +1 h (cid:33) (8)The conjugate quantity for temperature, the entropy S is given by the area law, S = πr h for this black hole. As discussed in the introduction, in the extended phasespace, the pressure P is given by, P = − Λ8 π (9)The conjugate quantity for P, the volume V , is given by, πr h . The scalar potentialwhich is conjugated to the scalar charge Q is given by,Φ = − γQr λ − h (10)One could rewrite the temperature in terms of S and P as follows: T H = 14 √ πS (cid:32) P S + γ ( λ − Q (cid:18) πS (cid:19) λ/ (cid:33) (11)4he temperature is plotted against the entropy S in Fig.(1) and Fig.(2). For γ = 1,the temperature has a minimum: hence black holes cannot exist below this minimumfor a given Λ value. For γ = −
1, the temperature could have an inflection point. γ = Figure 1: The figure shows T vs S for γ = 1. Here λ = 2 . , P = 0 . Q = 0 . - γ = - Figure 2: The figure shows T vs S for γ = −
1. Here λ = 2 . P = 0 .
05. Thetop graph is at Q = 0 and the rest have Q = 0 . , . , . In the extended phase space, the mass of the black hole M is not considered as theinternal energy as it is usually done. Instead, it is considered as the enthalpy H.5ence, M = H = U + P V . Hence, the first law of the given black hole is given by, dM = T dS + Φ dQ + V dP (12)It is possible to combine the thermodynamical quantities,
M, P, V, S, T,
Φ, and Q toobtain the Smarr formula as, M = 2 T S + λ Q − P V (13)which also could be obtained using the scaling argument presented by Kastor et.al [1].When λ = 2, the Smarr formula simplifies to the one for the Reissner-Nordstrom-AdSblack hole obtained by Kubiznak and Mann in [3]. In eq.(6), one could substitute Λ = − πP and rearrange the equation to obtain P asa function of r h and T as, P = − πr h + T r h + Q γ (1 − λ )8 πr λh (14)Since the black hole radius r h is given by, r h = (cid:18) V π (cid:19) / (15)one could rewrite P in terms of V as, P = − π (cid:18) π V (cid:19) / + T (cid:18) π V (cid:19) / + Q γ (1 − λ )8 π (cid:18) π V (cid:19) ( λ +2) / (16)The pressure is plotted vs V as in Fig.(3) and Fig.(4). For γ = 1, there is a maximumpressure given by, P max = 4 πr h T (1 + λ ) − λ πr h (2 + λ ) (17)Since r h and V are related, P max also could be rewritten in terms of V m at which thepressure is maximum as, P max = (cid:16) (4 π ) / (3 V m ) / T (1 + λ ) − λ (cid:17) λ )(4 π ) / (3 V m ) / (18)Hence when the P > P max black holes dose not exist. This implies that there is amaximum value of the cosmological constant that the black holes could exist for agiven horizon radius (or volume). 6or γ = −
1, the behavior is quite different from what is of γ = 1. There arecritical points as demonstrated from Fig(4). γ = Figure 3: The figure shows P vs r h for γ = 1 for varying temperature. Here λ = 2 . Q = 0 . .00 0.05 0.10 0.15 0.20 0.25 - - T < T0 T = T0 T < Tc T = TcT > TcT > Tc Figure 4: The figure shows P vs V for γ = − λ = 1 . Q = 0 . For γ = −
1, there are phase transitions between small and large black holes. Thesephase transitions are first order and are similar to Van der Waals phase transitionsbetween gas and liquid under constant temperature. A thorough analysis of the phasetransitions were discussed in the paper by the current author in [43].There is a critical temperature T c at which the temperature which P vs V curvehas an inflection point. At that point, ∂P∂V = ∂ P∂V = 0 (19)8he inflection point occur at the volume V c given by, V c = 4 πr hc r hc = 12 (cid:16) Q ( λ − λ + 2)2 λ − (cid:17) /λ (21)The corresponding T c and P c are given by, T c = λπ ( λ + 1) (cid:16) Q ( λ − λ + 2)2 λ − (cid:17) − /λ (22) P c = λ π ( λ + 2) (cid:16) Q ( λ − λ + 2)2 λ − (cid:17) − /λ (23) There are two different heat capacities for a thermodynamical system: heat capacityat constant pressure, C P , and heat capacity at constant volume, C V . They are givenby, C P = T ∂S∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P = 2 S (cid:16) P S λ + S λ + π λ Q γ ( − λ ) (cid:17)(cid:16) P S λ − S λ − π λ Q γ ( − λ ) (cid:17) (24) C V = T ∂S∂T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V = 0 (25) C V is zero because the entropy S is proportional to V . Since the black hole is considered as a thermodynamical system with a
P dV term,one could extract mechanical work from the black hole. The given black hole has anequation of state which clearly articulate the relation between P and V . Net workcan be extracted from a given cycle in state space. If the net heat input is Q H , netheat out put is Q C and the net work out put from the system is W net , the relationbetween them are, Q H = W net + Q C . If the given thermodynamical cycle is given by two isothermal and two isentropicpaths, then the efficiency of the heat engine is given by, η c = W net Q H = (cid:18) − T C T H (cid:19) (26)9 .2 Efficiency of a cycle with two isochoric and two isobaricpaths In this paper, we focus on a thermodynamical path consisting of two constant pressureand two constant volume paths as shown in Fig.(5). The heat supplied along theisobaric path 1 → Q → = Q H = (cid:90) dQ = (cid:90) T dS (27)Since S = πr h , eq.(27) can be rewritten as, Q H = (cid:90) T H d ( πr h ) = 2 π (cid:90) r r T H ( r h ) r h dr h (28)Since the pressure P is constant along the path 1 →
2, and
Q, λ are also keptconstant, T H becomes a function of r h along 1 →
2. Hence the integral in eq.(28)could be computed to be, Q H = (cid:32) r h − γQ r ( λ − h − πr h P (cid:33) r r = M ( r ) − M ( r ) (29)Similarly the net heat out put from the system could be calculated along the path4 → Q → = Q C = M ( r ) − M ( r ) (30)Therefore, the efficiency of the the heat engine for this particular cycle is, η = W net Q H = 1 − ( M ( r ) − M ( r ))( M ( r ) − M ( r )) (31)The above formula was obtained by Johnson [22] by using enthalpy and the first lawof thermodynamics.It is also noted that since volume is a function of entropy in this black hole,isochoric path also represents adiabatic path.10 .00 0.05 0.10 0.15 0.20 0.250.00.20.40.60.81.0 VP γ = - Figure 5: The figure shows P vs V for γ = − λ = 1 . Q = 0 . P = P = 0 .
925 and minimum pressure, P = P = 0 . The goal of this section is to compute the efficiency of the black hole heat engine andcompare to the Carnot efficiency η C . We will restrict the black holes with γ = − Q for three values and perform the calculations, and second we will change λ andperform the computations. In both cases, V (or r h ) and the P for the coordinates 1and 4 are kept constant. Then V (or r h ) for coordinates 2 and 3 are varied to observehow the efficiency varies.Before we proceed, some cautionary remarks are in order: for a given mass, the11orizon radius r h is possible only if the parameters of the black hole are chosenappropriately. To clarify this further we have shown an example in the Fig.(6) of thefunction f ( r ) whose roots lead to the horizon radius r h . In this example, when themass is low, there are no horizons. Only for a mass greater than a critical value thatthe horizon radius exists.On the other hand, we also would like to make sure that for all the points on thechosen thermodynamical cycle that there is a real mass value. In Fig.(7), M is plottedvs r h for various values of P . The values of the pressures are chosen so that they arebetween the maximum and the minimum pressure of the cycle considered. It is clearthat the mass M is positive for all values of P chosen. However, it is interesting tonote that the mass has a minimum value for a given P . We did a spot check to makesure that for the range of r chosen, that M ( r ) , M ( r ) > - - r h f ( r ) M = = = Figure 6: The figure shows f ( r ) vs r for γ = − λ = 4 , Λ = − .
2, and Q = 0 .
3. 12 .0 0.1 0.2 0.3 0.4 0.50.00.20.40.60.81.0 r h M P = = = Figure 7: The figure shows M vs r h for γ = − λ = 1 . Q = 0 . P = P = 0 .
925 and P = P =0 .
503 respectively. Q In this section, the scalar charge Q will be varied for three values. Here, P , P , r = r will be kept constant. In the thermodynamical cycle in Fig.(5), the smallest temper-ature is at T . The cycle is chosen such that it does not include the region where thephase transition occur. Due to this reason, the critical temperature T c needs to besmaller than T for all chosen Q values. T c is computed by eq.(22). T is computedas, T = 14 π (cid:32) r + 8 πP r + γ ( λ − Q r λ +14 (cid:33) (32) T c and T are plotted together in the same graph in Fig.(8) for varying Q . For large Q , T c > T . Hence one need to choose the correct range of Q values to make surethat T c < T . 13 .10 0.15 0.20 0.25 0.300.00.10.20.30.40.50.6 Q T TcT4
Figure 8: The figure shows T c , T vs Q for γ = −
1. Here λ = 1 .
98, and P = 0 . Q is fixed, M ( r ) and M ( r ) will depend only on r = r , P and P . Now, η, η C are calculated using eq.(31) and plotted for 3 different values of Q in Fig.(9). r η c Q = = = r η Q = = = Figure 9: The figure shows η, η c vs r for γ = − Q . Here λ = 1 .
98 14 .26 0.28 0.30 0.32 0.340.800.850.900.95 r ηη c Q = = = Figure 10: The figure shows ηη c vs r for γ = − Q . Here λ = 1 . λ In this section, the value of λ is varied for three values. Once again, P , P , r = r will be kept constant. It is also important to make sure that T c < T for chosen valuesof λ . In Fig.(11), T c , T are plotted against λ . It is clear that there is a value λ that T c > T . Hence values of λ are chosen such that T c < T . λ T TcT4
Figure 11: The figure shows T c , T vs λ for γ = −
1. Here Q = 0 . P = 0 . .26 0.28 0.30 0.32 0.340.250.300.350.400.45 r η c λ = λ = λ = r η λ = λ = λ = Figure 12: The figure shows η, η c vs r for γ = − λ . Here Q = 0 . r ηη c λ = λ = λ = Figure 13: The figure shows ηη c vs r for γ = − λ . Here Q = 0 . In this paper, we have studied black holes in massive gravity with a negative cosmo-logical constant. in the extended phase space. Here the black hole has a pressuregiven by, P = − Λ8 π . There are two types of black holes for the values of γ in thetheory. Thermodynamical behavior differ significantly for γ = 1 and γ = −
1. For γ = 1, the black hole behaves similar to the Schwarzschild-anti-de Sitter black hole:the pressure has a maximum and the temperature has a minimum. For γ = −
1, theblack hole exhibits phase transitions for certain range of temperatures: for highertemperatures, the black holes behave like an ideal gas. Phase transitions are between16arge black holes and small black holes. A detailed analysis of the black holes in thiscontext is published by the current author in [43].The main goal of this paper is to study the massive gravity black hole as a heatengine. The thermodynamical cycle for the heat engine considered here is a rectanglein
P, V space with two isobaric and two isochoric processes. The efficiency of theheat engine taking black hole as the working substance is computed for the rectanglecycle as well as for the Carnot cycle by varying
Q, λ, r . When Q is increased, theefficiency for the rectangle cycle increases, but, the Carnot efficiency decreases. Theratio ηη c decreases when Q is increased. Hence to achieve better efficiency, a higher Q is appropriate. Larger the volume V (or r ), higher the η of the rectangle cycle.When Q = 0, we get the Schwarzschild AdS black hole. Hence from the graphs it canbe concluded that the Schwarzschild AdS black hole has a smaller efficiency for therectangle cycle compared with the massive gravity black hole.When λ is increased, the efficiency of the rectangle cycle as the Carnot cycleincreases. Both efficiencies increase with volume V (or r ). The ratio η/η c decreaseswith λ .The Reissner-Nordstrom AdS (RNAdS) black hole has the metric with f ( r ) = 1 − Mr + Q e r − Λ r . When Q e = Q and λ = 2, the massive gravity black holeand the RNAdS black hole are the same. Hence one can conclude that for λ > Q e = Q , the massive gravity black hole will have higher efficiency. When 1 ≤ λ < References [1] D. Kastor, S. Ray, & J. Traschen,
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