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The black hole, predicted by general relativity, is an object of long-standing interest to physicists. It was Bekenstein who first proposed that black holes actually possess entropy [1]. In sharp contrast to standard thermodynamic notions wherein entropy is supposed to be a function of volume, he suggested that the entropy of a black hole is proportional to the horizon area. Since then a variety of different theoretical methods have been used to calculate the Bekenstein-Hawking entropy, such as using the quantum fields near the horizon [2], quantum field theory in a fixed background [3], the entanglement entropy [4], string theory [5-10], loop quantum gravity [11, 12], Noether charge [13, 14], induced gravity [15], the causal set theory [16], the symmetry near the horizon [17, 18], and using the inherently global characteristics of a black hole spacetime [19]. It has been shown that the classical Bekenstein-Hawking entropy depends not only on the black hole parameter, but also on the coupling, which reduces the Lorentz violation [20]. Considering the complexity involved in these methods, finding a simpler way to calculate the entropy of a black hole is an important task.
Energy is another important issue besides entropy in black hole physics. In higher-order gravitational theories, the energy of a black hole is still an open problem. Several efforts to find a satisfactory answer to this issue have been carried out [21-25]. It was shown that the entropy and energy of a black hole can be simultaneously obtained in Einstein's gravity using the horizon first law [26], but it cannot work in higher-order gravitational theories. Recently a new horizon first law, in which both the entropy and the free energy are derived concepts, was suggested in Einstein's gravity and Lovelock's gravity; the standard horizon first law can be recovered by a Legendre projection [27]. In [28], it was found that the new horizon first law still works in
f(R) theories by introducing the effective curvature fluid: it can give not only the energy but also the entropy of black holes, which are in agreement with the known results in the literature. Here we will consider the new horizon first law and the entropy and the energy issues inf(R,RμνRμν) gravity.The paper is organized as follows. In section 2, we briefly review the new horizon first law. In section 3, we discuss the entropy and the energy of black holes in
f(R,RμνRμν) theory. In section 4, applications are considered. Finally, in section 5 we briefly summarize our results. -
According to the suggestion proposed in [29], that the source of thermodynamic system is also that of gravity, the radial component of the stress-energy tensor can act as the thermodynamic pressure,
P=Trr|r+ , then at the horizon of Schwarzschild black hole the radial Einstein equation can be written asP=T2r+−18πr2+,
(1) which can be rewritten as a horizon first law after a imaginary displacement of the horizon,
δE=TδS−PδV , with E as the quasilocal energy and S as the horizon entropy of the black hole [26]. As the temperature T in the Eq. (1) is identified from the thermal quantum field theory, independent of any gravitational field equations [27], while the pressure P in (1), according to the conjecture proposed in [29], is identified as the radial component of the matter stress-energy, it is reasonable to assume that the radial field equation of a gravitational theory under consideration takes the form [27]P=D(r+)+C(r+)T,
(2) where C and D are analytic functions of the radius of the black hole,
r+ ; in general, they depend on the gravitational theory under consideration. Varying the Eq. (1) and multiplying the geometric volumeV(r+) , it is straightforward to have a new horizon first law [27]δG=−SδT+VδP,
(3) with the Gibbs free energy as
G=∫r+V(r)D′(r)dr+T∫r+V(r)C′(r)dr=PV−ST−∫r+V′(r)D(r)dr,
(4) and the entropy as [27]
S=∫r+V′(r)C(r)dr.
(5) Under the degenerate Legendre transformation
E=G+TS−PV , yields the energy as [28]E=−∫r+V′(r)D(r)dr.
(6) This procedure was first discussed in Einstein's gravity and Lovelock's gravity, which only give rise to a second-order field equation [27]. It was generalized to
f(R) gravity with a static spherically symmetric black hole [28] or with a general spherically symmetric black hole [30] and was also applied to the D-dimensionalf(R) theory [31].In the next section, we will investigate whether this procedure can be applied to more complicated cases, such as
f(R,RμνRμν) theory, and whether Eqs. (5) and (6) can still be used to obtain the entropy and the energy in the theory we consider. -
As shown in section 2, the new horizon first law works well in Einstein's theory, Lovelock gravity [27], and
f(R) theory [28, 30, 31]. Does it still works in other gravitational theories such asf(R,RμνRμν) theory? We consider this question in this section. In four-dimensional spacetime, the general action off(R,RμνRμν) theory with source is given byI=∫d4x√−g[f(R,RμνRμν)16π+Lm],
(7) where
Lm is the matter Lagrangian andf(R,RμνRμν) is a general function of the Ricci scalar R and the square of the Ricci tensorRμν . We take the unitsG=c=ℏ=1 . Varying the action (7) with respect to metricgμν yields the gravitational field equations asGμν≡Rμν−12Rgμν=8π[TμνfR+18πΩμν],
(8) where
fR≡∂f∂R andTμν=2√−gδLmδgμν is the energy-momentum tensor of matter.Ωμν is the tress-energy tensor of the effective curvature fluid and is given byΩμν=1fR[12gμν(f−RfR)+∇μ∇νfR−gμν◻fR−2fXRαμRαν−◻(fXRμν)−gμν∇α∇β(fXRαβ)+2∇α∇(μ(Rαν)fX)],
(9) where
A(μν)=12(Aμν+Aνμ ),X≡RμνRμν , andfX≡∂f∂X . Inserting the following two derivational relations∇α∇β(fXRαβ)=Rαβ∇α∇βfX+(∇βR)(∇βfX)+12fX◻R,
(10) and
∇α∇μ(fXRαν)+∇α∇ν(fXRαμ)=Rαν∇α∇μfX+Rαμ∇α∇νfX+12(∇μfX)(∇νR)+12(∇νfX)(∇μR)+(∇αfX)(∇μRαν)+(∇αfX)(∇νRαμ)+fX∇μ∇νR+2fXRαμνλRαλ+2fXRμλRλν,
(11) into Eq. (9), the tress-energy tensor of the effective curvature fluid
Ωμν is simplified asΩμν=1fR[12gμν(f−RfR)+∇μ∇νfR−gμν◻fR−fX◻Rμν−Rμν◻fX−gμνRαβ∇α∇βfX−12fXgμν◻R+Rαν∇α∇μfX+Rαμ∇α∇νfX−fX∇μ∇νR−2fXRαμνλRαλ].
(12) For a static spherically symmetric black hole whose geometry is given by
ds2=−B(r)dt2+dr2B(r)+r2dΩ2,
(13) where the event horizon is located at
r=r+ the largest positive root ofB(r+)=0 withB′(r+)≠0 , the(11) components of the Einstein tensor isG11=1r2(−1+rB′+B),
(14) with the primes denoting the derivative with respect to r. At the horizon, since
B(r+)=0 , it reduces toG11=1r2(−1+rB′).
(15) While the radial components of the tress-energy tensor of the effective curvature fluid
Ωμν at the horizon takes the following formΩ11=1fR[12(f−RfR)−12B′f′R−B′(fXR11)′−2fXB′R11r++2fXB′R33r+−2fX(R11)2].
(16) Substituting Eqs. (15) and (16) into Eq. (8), and considering
P=Trr|r+ , we derive8πP=−fRr2++fRB′r+−12(f−RfR)+12B′f′R+B′(fXR11)′+2fXB′R11r+−2fXB′R33r++2fX(R11)2.
(17) This equation is very complicated; therefore, how can we determine the function
C(r+) in Eq. (2)? Even worse, this equation depends on higher derivatives of B, and hence we can no longer follow the same approach as the one we followed for Einstein's theory in which we obtained the entropy and energy from Eqs. (5) and (6) directly. In higher-derivative gravity, to use the horizon first law,δE=TδS−PδV , usually one should reduce the higher-derivative field equations to lower-derivative field equations via a Legendre transformation [32, 33]. Here we try a new method. If we obtain the entropy by using other methods, then using the new horizon first law (3) and the degenerate Legendre transformationE=G+TS−PV , we can derive the energy. Asf(R,RμνRμν) is a diffeomorphism invariance of the gravitational theory, the entropy can be obtained using the Wald method, which is presented in the Appendix. Taking into account the volume of the black holeV(r+)=4πr3+/3 , the pressure in Eq. (17), the Hawking temperatureT=B′(r+)/4π , and the entropy given in the Appendix, the new horizon first law (3) can be rewritten asδG=−13πr2+(fR+2fXR11)δT+4πr3+T3δ(fR2r+)+4πr3+T3δ(fXR11r+)+13πr3+δ(Tf′R)+23πr3+δ[T(fXR11)′]−112r3+δ(f−RfR)−16r3+δ(fRr2+)+4πr3+3δ(4πfXT2r2+)−4πr3+3δ(TfXr3+)+13r3+δ[fX(R11)2],
(18) and
TS−PV is given byTS−PV=13πr2+fRT+23πr2+fxR11T−23πr3+T(fxR11)′−13πr3+Tf′R+16fRr++112r3+(f−RfR)−13r+fXB′2+13fXB′−13r3+fX(R11)2.
(19) According to the degenerate Legendre transformation
E=G+TS−PV , we haveδE=δ(G+TS−PV)=[12fRδr++14r2+(f−RfR)δr++r+fXB′R33δr+−r2+fX(R11)2]δr+,
(20) or equivalently
E=∫r+[12fR+14r2+(f−RfR)+r+fXB′R33−r2+fX(R11)2]dr+.
(21) When
fX=0 , Eq. (21) returns to the result obtained inf(R) theory [28]. Using Eqs. (21) and (29), we can calculate the energy and the entropy of the black hole inf(R,RμνRμν) theory. -
For application, we consider a simple but important example: the most general quadratic-curvature gravity theory with a cosmological constant in four dimensions; its Lagrangian density is given by an arbitrary combination of scalar curvature-squared and Ricci-squared terms, namely,
f(R,RμνRμν)=R+αRμνRμν+λR2−2Λ,
(22) where
α andλ are constants, andΛ is the cosmological constant. For this theory, we havefR=1+2λR andfX=α . We find from (29) that in spacetime with metric (13), the entropy isS=πr2+(1+2λR+2αR11).
(23) Substituting Eq. (22) into Eq. (21), the energy of the black hole is given by
E=12∫r2+[1r2++2λRr2++12αRμνRμν−12λR2+2αB′R33r+−2αR112−Λ]dr+=−14∫[(6αR112+2R11αB″+2αR222+2Λ+λR2)r2++4R11αr+B′−2(2αR22+2λR+1)]dr+.
(24) In the case of a Schwarzschild-(A)ds black hole, for example, whose metric is given by
B(r)=1−2Mr−Λr23 , the entropy (23) and the energy (29), respectively, return toS=πr2+(1+2λR+2αR11)=πr2+[1+2(α+4λ)Λ],
(25) and
E=[1+2(α+4λ)Λ]M,
(26) where we have used
B(r+) = 0. Eqs. (25) and (26) are consistent with the results obtained in [34, 35]. The energy in [35] was computed using the Abbott-Deser-Tekin method and a qualitatively different way of regularizing the Iyer-Wald charges. On the other hand, in [34] the energy was calculated using the horizon first law after taking a Legendre transformation. Whenα=0 , we obtain the results inR+λR2−2Λ theory. While forα=0 andλ=0 , we get the results in Einstein's gravity with the cosmological constant. -
We have discussed whether the new horizons first law is still valid in
f(R,RμνRμν) theory. Unlike the approach taken in Einstein's gravity, we cannot directly derive the entropy and energy via Eqs. (5) and (6). We must first obtain the entropy using other methods, such as Wald formula, then we can use the new horizon first law, the degenerate Legendre transformation, and the gravitational field equations to derive the energy of the black hole inf(R,RμνRμν) theory. For application, we have considered quadratic-curvature gravity and have presented the entropy and the energy for a static spherically symmetric black hole, especially for a Schwarzschild-(A)ds black hole where the results are consistent with those obtained in the literature. Whether this procedure can be applied to other complicated cases, such as Einstein-Horndeski-Maxwell theory [36], is worth studying in the future. In future works, we also plan to compare our results with those obtained by using other methods such as the Misner-Sharp and Abbott-Deser-Tekin methods.We thank J. Zhai for the helpful advice.
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In the bulk of the present paper, we studied the entropy and energy of a black hole (13) in
f(R,RμνRμν) theory. To confirm whether our results are reasonable, here we calculate the entropy by using the Wald formula, which takes the following formS=−2π∮δLδRabcdϵabϵcddV2,
(27) where L is Lagrangian density of the gravitational field,
dV2 is the volume element on the bifurcation surfaceΣ , andϵab is the binormal vector toΣ normalized asϵabϵab=−2 . For the metric (13) the binormal vectors can easily be found asϵ01=1 andϵ10=−1 . ForL=f(R,RμνRμν)16π , we can getδLδRabcd=116π(fRδRδRabcd+fXδRμνRμνδRabcd)=116π(gc[agb]dfR+2Rμνgσρδa[μδbσ]δcνδdρfX).
(28) For the metric (13), we have
gc[agb]dϵabϵcd=−2 and2Rμνgσρδa[μδbσ]δcνδdρξabξcd=4R00g11=−4R11 . Since the integral (27) is to be evaluated on shell, finally we have the entropy asS=A(r+)4(fR+2fXR11)
(29) in
f(R,RμνRμν) theory for the black hole (13).
Horizon thermodynamics in f(R,RμνRμν) theory
- Received Date: 2020-04-29
- Available Online: 2020-11-01
Abstract: We investigate whether the new horizon first law still holds in