Recovering semiclassical Einstein’s equation using generalized entropy

In this letter, we show that the Semiclassical Einstein’s Field Equation can be recovered using the generalized entropy Sgen\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{gen}$$\end{document}. This approach is reminiscent of non-equilibrium thermodynamics. Furthermore, contrary to the entanglement equilibrium approach of deriving the semiclassical Einstein’s equation, this approach does not require any such assumptions and still recovers its correct form. Therefore, in a sense, we also show the validity of the semiclassical approximation, a crucial approach for establishing a number of important ideas such as the Hawking effect.


Introduction
The connection between gravity and thermodynamics has long been explored.The first hints came from black hole thermodynamics when Hawking [1] showed that the area of a horizon(A) is a non-decreasing function of time.
dA dt ≥ 0 Later, Bekenstein [2] proposed the equality of horizon area and entropy as where γ is a constant of proportionality.Hawking [3] fixed this constant of proportionality by deriving the black hole temperature as where G is the universal gravitational constant and ℏ is the reduced Planck's constant.In 1995, Jacobson [4] derived Einstein's field equation from this proportionality of horizon area and entropy and the fundamental thermodynamic result dQ = T dS under equilibrium conditions.Some works under the non-equilibrium conditions were also presented later [5,6].Jacobson took Q as the heat flux across a causal horizon while temperature T was given by Unruh temperature [7] as observed by a local Rindler observer.Using these ingredients, he was able to derive Einstein's field equation in the form where the constant η = (4Gℏ) −1 and the constant Λ was identified as the cosmological constant.
The connection between gravity and thermodynamics was further explored by Padmanabhan (see [8] for a review) with some closely related follow-up articles from his collaborators [9,10,11,12].Recently, Verlinde [13] combined thermodynamic arguments and the holographic principle [14,15] and argued that gravity is an entropic force arising from the underlying microscopic theory to maximize its entropy.Very recently in 2016, Bousso et al. [16] conjectured the Quantum Focusing(QFC) which conjectures that the quantum expansion(Θ) cannot increase along any congruence, even in quantum states that would violate the classical focussing theorem: The quantum expansion allows to generalize the classical focusing theorem to the semiclassical case.The QFC has a wide variety of applications.It implies a Quantum Bousso Bound which has already gathered a considerable amount of evidence [17,18,19,20,21,16,22,23]. It also implies the quantum singularity theorems [24,25], the generalized second law of causal horizons and holographic screens [26] and new property of nongravitational theories, the Quantum Null Energy Condition(QNEC) [16,27,28,29].Arvin [30] presented a restricted quantum focusing which he argued is sufficient to derive all known essential implications of the quantum focusing and also proved it on brane-world semiclassical gravity theories.The Semiclassical Einstein's Field equation was derived using the entanglement equilibrium approach [31,32] which was essentially based on two assumptions: first, the vacuum reduced to a small ball is maximal when we vary it keeping the volume fixed and second, the variation of the entanglement entropy coming from the variation of the matter fields takes the following particular form where X is an operator in the QFT and the remaining symbols have the usual meaning.This second assumption was critically analyzed and was found to be somewhat problematic to an extent such that the entanglement equilibrium approach can only reveal linearized Einstein's equation or the full non-linear equation by restricting to some special choices of states(see [33] for details).
In this letter, we show that the generalized entropy approach can bypass all such assumptions and we can recover the full non-linear Semiclassical Einstein's Field equation which reflects on the naturalness and robustness of our approach.In the next section, we present our idea elaborately.
2 Recovering the Semiclassical Einstein's Field Equation Hawking [3] showed that black holes could evaporate, thus their horizon area can decrease via Hawking radiation.One of the key assumptions he made is the validity of semiclassical Einstein's field equation in the semiclassical regime given by where G µν is the Einstein tensor and ⟨T µν ⟩ is the expectation value of the energy-momentum tensor.We show how to recover this critical equation, thus establishing the validity of the semiclassical approximation under the assumption that the fluctuations in T ab are negligible.We consider a causal diamond in the large limit in the de-Sitter static patch (see Fig. (1)).In this limit, the boundary of the diamond coincides with the cosmological horizon of the dS space2 , that is, r = L.The conformal killing vector ζ a then becomes the timelike killing vector and H the killing horizon [34].The generalized entropy S gen is defined as where S out [H] is the entropy restricted to one side of H (here, the matter entropy outside the horizon H of the causal diamond).The Quantum Expansion (Θ) is defined similarly to the classical expansion θ as the response of S gen to deformations of H along the generator and the amount of deformation is measured by an affine parameter λ along the generator.This translates to the definition of Θ as Θ = lim Therefore, the variation of the generalized entropy is given by where η = (4Gℏ) −1 .The cosmological horizon H has a temperature given by where κ is the surface gravity.Since the interior of the causal diamond is unobservable as it is causally disconnected from the exterior, the integrated energy of the energy current of the matter given by ⟨T ab ⟩ζ a , is identified as heat [4].The energy flux across H can therefore be written as where k a is the (null) tangent vector to the horizon generators for an affine parameter λ so that ζ a = −2κλk a .Note the extra factor of 2 in the equation of heat (13).We recall that the relation between affine parameter λ and killing parameter v on a killing horizon is generally given as λ = −e −κv .But as we show here, in the semiclassical case, we should use instead λ = −e −2κv so as to recover the semiclassical equation.This can be understood as the vanishing of quantum expansion Θ to zeroth order in λ must occur at twice the rate to correctly impose the condition on spacetime curvature.Therefore, we get ζ a = −2κλk a , and δQ takes the form as given in (13).In the semiclassical case, the natural generalization to work with the generalized entropy is quantum expansion.The infinitesimal evolution of Θ is given by a linear expansion around its equilibrium value at p, up to the first order in λ, as it moves away from the equilibrium surface at λ = 0 along the generator To find Θ in terms of known quantities, we use its definition as [16] where θ is the classical expansion and S out the outside matter entropy.The evolution of Quantum Expansion Θ is governed by Therefore (10) becomes Since at zero order 3 in λ, Θ p = 0, we have 3 As can be noted here.In contrast with the classical case, the semiclassical case requires the vanishing of quantum expansion Θ to zeroth order and not the classical expansion θ for the semiclassical Einstein's equation to hold.This can be understood as follows: The quantum fields violate the classical focusing while even in this the quantum focusing (dΘ/dλ ≤ 0) holds.Therefore, focusing to the past of p must bring the quantum expansion to zero so that the increase in generalized entropy Sgen is proportional to the killing energy across it.This imposes a condition on the spacetime curvature that is governed by the evolution equation (17).The same cannot be said for the classical focusing since it does not hold in the semiclassical case.
We therefore get Before we use the Clausius relation δQ = T δS, let us write δS gen as δS gen = d i S + d e S, where and d i S is identified as the internal entropy production rate due to some dissipation [6,35] and it must be positive as required by the second law.Interestingly, this is twice that of the classical one and has contributions from both scalar and tensorial degrees of freedom.To see why d e S and d i S take these expressions can be understood by expressing (21) in terms of the Killing parameter v as as required by the second law (since the affine parameter λ and Killing parameter v on the horizon are related by λ = −e −2κv ).d e S is then identified as the reversible entropy change dS rev .Therefore, by subtracting d i S from δS gen and using the Clausius relation δQ = T dS rev = T d e S, we obtain This equation implies 2π for some function f .Local conservation implies ⟨T ab ⟩ is divergenceless, therefore, using the Bianchi identity, we recover the semiclassical Einstein's equation where Λ is identified as the cosmological constant of the de-Sitter space and η = (4Gℏ) −1 .Therefore, the constant Λ obtained in this case also carries a natural meaning.

Conclusion
In this work, we showed that the semiclassical Einstein's field equation can be recovered in the semiclassical case using the generalized entropy S gen .Since S gen is a cutoff independent quantity, it reveals information about the full quantum gravity theory and shows that the semiclassical approximation can be trusted as long as the fluctuations in T ab are negligible.Furthermore, we showed that this approach is reminiscent of non-equilibrium thermodynamics.The validity of the semiclassical equation has been previously established using the entanglement equilibrium approach.However, this approach applied to causal diamonds makes a number of assumptions and some of which turned out to be problematic.On the contrary, we need no such assumptions and still, we were successful in recovering the semiclassical equation making this approach natural and robust.

Figure 1 :
Figure 1: A causal diamond in the de-Sitter static patch for a ball-shaped spacelike surface Σ with dΣ as its boundary.p and p ′ are the future and past vertices of the diamond.ζ is a timelike killing vector in the large limit of the diamond when the boundary of the diamond coincides with the cosmological horizon.The dashed curves are the flow lines of the killing vector ζ.H is the killing horizon.