Probable Entropic Nature of Gravity in Ultraviolet and Infrared Limits, Part I. An Ultraviolet Case

This work presents a study of the possibility for extending the well-known results of E.Verlinde concerning the entropic nature of gravity to the ultraviolet region (Planck's energies) and also the derivation of quantum corrections to Einstein Equations


Introduction
In the last 15-20 years new very interesting approaches to gravity studies have been proposed, which may be divided into "thermodynamical" and "theoretical-informational" approaches. The approach suggested in the pioneer work by T. Jacobson [1] has been considerably extended in a series of remarkable papers by T.Padmanabhan [2]- [13]. The paper by E.Verlinde [14] stating a secondary character of the gravitational interaction and its entropic nature was published in 2010 after the appearance in the ArXiv. The paper [14] introducing such specific terms as "Entropic Force" has been followed by numerous studies (e.g. [15]- [26] and others). In this work the author studies the possibility for extension of the results given in [14] to the ultraviolet region (Planck's energies) and presents the derivation of quantum corrections to Einstein Equations using the dimensionless small parameter α introduced by the author in his previous works [27] - [39].

Fundamental Quantities and Their High-Energy Deformation
In this section the author uses the "ideology", terms, and notation introduced in [14] to extend the corresponding results to the ultraviolet and infrared gravity regions. It may be stated that the results in [14] have been obtained for the "medium" energies, i.e. for the range of well-known energies, where the General Relativity (GR) is valid. But owing to modern knowledge, in the ultraviolet and infrared limits gravity may be modified. As regards the ultraviolet (Planck) scale, this idea has been proposed long ago [40] - [45] and in this situation the word "may" we have to replace by the word "must". As for low energies, there are many recent publications considering the infrared (for great distances) modification of gravity (e.g. [46], [47]). And the modification can have a solid experimental status in the nearest future [48]. Naturally, when we are concerned with extension of some results to higher or lower energies, the principle of conformity must be executed undeviatingly: on going to the known energy scales the known results must be reproduced. In this Section it is shown that the fundamental quantities A,N,T , defined in [14] and associated with the holographic screen (where A -its surface area, N -number of data bits "existing" on A, T -its temperature), may be supplementary defined for the region of high energies so that at normal energies they be coincident with values given in [14]. In the process we take the corresponding quantities for the stationary Schwarzshild black hole as most natural holographic object. The idea is as follows: formulae for the correction of the fundamental quantities within the Generalized Uncertainty Principle (GUP) for the holographic screen S from [14] are similar to those for the black hole. Let us consider Section 5.2 (Derivation of Einstein Equations) in [14]. In this Section formula (5.32) for a "bit density" on the holographic screen is given as However, when the holographic principle [49]- [53] is valid, N is actually the entropy S up to the factor S ∼ N and hence from (1) it follows directly that dS ∼ dA G . ( What are the changes in S on going to high (Planck) energies? The answer to this question is already known owing to the fact that at these energies the Heisenberg Uncertainty Principle (HUP) is replaced by GUP [54]- [61]: where ℓ 2 = α ′ l 2 p and α ′ is the dimensionless numerical factor. The wellknown Bekenstein-Hawking formula for the black hole entropy in the semiclassical approximation [62], [63] is modified by the corresponding quantum corrections on going from HUP to GUP [64]- [67]. In particular, [65]: where the expansion coefficients c n ∝ α ′2(n+1) can always be computed to any desired order of accuracy. The general form of quantum corrections for the black hole entropy derived in (5) remains valid for any horizon spaces and, in particular, for the holographic screen S from [14]. Specifically, in [68], [69] the logarithmic correction was obtained in the following form: omitting the Boltzmann constant k B as a factor and assuming it to be equal to unity in what follows. Higher-order corrections may be derived using the Taylor-series expansion in terms of the small parameter l 2 p /A in a similar way to the Taylor-series expansion of the right-hand side in (9) in terms of the small parameter 4l 2 p /A. This is valid as GUP gives the ultraviolet cutoff at the level of l min ∼ l p . In this way at high energies we have S → S GU P and hence N → N GU P . Assuming in the notation of [14] that we directly obtain Now, coming back to [14], in terms of N GU P we can define the holographic screen area, as measured at high energies, by where G and -gravitational and Planck constants, respectively, and N GU P is given by (9). Considering that we, similar to [14], assume that the speed of light c = 1, then, according to l 2 p = G , from (9) and (10) we have In this case an exact value of the constant in the right-hand side of (11) is of no great importance as further we need the relation (1), being primarily interested in dA GU P rather than in A GU P , i.e. the constant in the righthand side of (11) is insignificant. So, (1) has a fairly definite analog at high energies that on going to the known low energies gives (1). There is a single considerable difference, in [14] the quantity N was defined in terms of A and dN was defined in terms of dA but in the case under study the situation is opposite: A GU P is defined in terms of N GU P and dA GU P in terms of dN GU P . The logic series is here as follows: The GUP-correction problem of the temperature T for the arbitrary holographic screen S has been studied in [70]. Actually, this case is identical to the case of the Schwarzschild black hole and T GU P was derived as a series (formulas (1),(7) from [70]) where the factors in the right-hand side (14) may be computed in the explicit form and at low energies T GU P → 0. Thus, we can have a GUP -analog of Komar's mass in ((5.33) from [14]) that in the low-energy limit gives the well-known Komar formula [71],( [72], p.289). It is clear that the "GUP-deformed Komar's mass" M GU P in the first term (15) as a summand has the known Komar's mass [71],((11.2.9 -11.2.10), [72]) ((5.34), [14]) 3 N GUP ,A GUP and M GUP in Terms of Unified Small Parameter If feasible, it is desirable to express all the above-derived fundamental quantities in terms of a unified parameter. As shown by the author in [39], [73], this is possible for black holes within the scope of GUP and a role of the unified small parameter is played by the parameter introduced previously in [27]- [36] as follows: where x is the measuring scale, l min ∼ l p by virtue of GUP (3), and 0 < α ≤ 1/4. Obviously, the principal results obtained in [39], [73] remain in force for an arbitrary screen S and may be applied to the quantities N GU P ,A GU P , M GU P defined in the preceding Section. Substituting from GUP (3) l min = 2 √ α ′ l p and using the formula where R is the radius of the screen S, we get N GU P (9) of the following form: Here α R is a value of α parameter at the point R and σ is a equal to 16α ′ π . It is convenient to refer to the form N GU P derived in (18) as to the α -representation.
Using (10) and (11), we can easily obtain α -representation for A GU P Also, it is clear that M GU P (15) may be derived in terms of α R .
Here α x is considered as a deformation parameter for the Heisenberg algebra on going from HUP to GUP. Generally speaking, initially the construction of such a deformation was realized with other parameters (e.g. [60], [61]). But it is easily shown that QFT parameter of the deformations associated with GUP may be expressed in terms of the parameter α that has been introduced in the approach to the density matrix deformation [38], [39]. Here the notation of [74] is used. Then ( [39], p. 943) and In this case from (20), (21) it follows that β ∼ 1/p 2 , and for x min ∼ l p , β corresponding to x min is nothing else but where P pl is Planck's momentum: P pl = /l p . In this manner β is varying over the following interval: where λ is a numerical factor and the second term in (20) is accurately reproduced in the momentum representation (up to the numerical factor) by α In the case under study convenience of using α x stems from its smallness, its dimensionless character, and ability to test changes in the radius R of the holographic screen S.

Quantum Corrections to the Principal Result and Ultraviolet Limit
Based on the aforesaid, we can proceed to generalization of the results from Section 5.2 of [14] and to derivation of equations for a gravitational field within the scope of GUP. We must consider two absolutely different cases.

Quantum Corrections to the Principal Result
It is assumed that the screen radius is given by S In terms of the deformation parameter α x introduced in the previous Section, we have So far we are not concerned with redefinition of the lower limit for α R . This is interesting when going to the infrared limit.
Then the principal result from the final part of Section 5.2 in [14] remains valid owing to the replacement of M (formula (5.33) from [14]) by to M will be simply a (small) quantum correction for the principal result.
In this way, because in the case of GUP-corrections the left side of formula (5.33) and hence the left side in formulas (5.34), (5.35) from [14] is dependent on α R , the right sides of the corresponding formulas are also dependent on α R ,in particular the quantities T ab , R ab , g ab from (5.37). But in fact this relation at low energies ((25) or (26)) is insignificant since α R at low energies (as distinct from high Planck's energies)is varying very slowly, practically showing continuity though being discrete in character. Indeed, as long as there is the minimal length l min ∼ l p , all the lengths measured are its multiples and hence α R is a discrete non-uniformly varying quantity. Then, due to ((25) or (26)), the difference between two successive values of α R is as follows: for R ≫ l p or, that is the same, for R ≫ l min giving a value close to zero. And assuming in this case that α R is continuously varying from R and all the quantities in Section 5.2 of [14] are also continuously dependent on α R (26), we can write down the ("α -analog" of formula (5.37) in [14])as where the dependence of T ab [α] and R ab [α] on α = α R is completely determined, in accordance with [72], [14], by the integral M GU P [α] (15). Besides, it is assumed that n a and ξ b are dependent on α, though the dependence is dropped.
Next, similar to [14], from (28)we can derive the α-deformed Einstein Equations using the method from [1]. Note that both this method and its minor modification given in ( [14], end of Section 5.2) in this case are valid because α R is small and continuous, the whole system being continuously dependent on it.
Solutions of the α-deformed Einstein Equations represent a series in α R , and for α R → 0 or for α ′ = 0 from formula (3) become the corresponding solutions of (Section 5.2 in [14]).
Using the result obtained in [75], we can easily extend the above result to the case with a nonzero cosmological term Λ = 0. In [75] Komar's formula was generalized to the case of a nonzero Λ. All the arguments from (Section 5.2 of [14]) in this case remain valid and formula (5.37) takes the following form: We can easily obtain the α -analog of the last formula with the dynamic cosmological term Λ(α) as a corresponding complement to the right-hand side (28). Analysis of the relationship between Λ and α, applicable in this case as well, will be given in Section 4.2.

Ultraviolet Limit
In the case in question we suggest that the screen S has a radius on the order of several Planck's lengths where ξ is a number on the order of 1 or The problem is which object puts the limit for such a screen S. It may be assumed that if T ab = 0 then the object may be represented only by Planck's black hole or by a micro-black hole with a radius on the order of several Planck's lengths. Clearly, the methods of [14] and [1] are not in force for such screen S. Specifically, there are no classical analogs of N, T , and M for the screen. Moreover, it is impossible to use the result of [1] as "a very small region the space-time" is no longer "an approximate Minkowski space" [14].
Also, such micro-black hole is a horizon space, jet at high energies (Planck scales). As is known, for horizon spaces, black holes in particular, at low energies (semiclassical approximation) the results of [12] are valid.
At the horizon (and we are interested in this case only) Einstein's field Equations may be written as a thermodynamic identity ( [12] formula (119)) where R = a is a radius of a black hole (i.e. of the screen S), P = T R R is the trace of the momentum-energy tensor and radial pressure, and the horizon location will be given by simple zero of the function f (R), at R = a. The main ingredients of (32) may be written in terms of the deformation parameter α with the coefficients containing only the numerical factors and fundamental constants [73]. Also, the work [73] presents two possible variants of high-energy (Planck) α -deformation α → 1/4 (32). Hereinafter, we assume that the energy -momentum tensor of matter fields is not traceless T a a = 0, In this case it is assumed that in the high-energy (ultraviolet (UV))limit the thermodynamic identity (32) is retained but now all the quantities involved in this identity become α-deformed (α → 1/4). All the quantities Υ in (32) are replaced by the corresponding quantities Υ GU P with the subscript GUP. Then the high-energy α-deformation of equation (32) takes the form Substituting into (34) the corresponding quantities T GU P (α), S GU P (α), E GU P (α), V GU P (α), P (α) and expanding them into a Lau-rent series in terms of α, close to high values of α, specifically close to α = 1/4, we can derive a solution for the high energy α-deformation of the general relativity (34) as a function of P (α). Provided at high energies the generalization of (32) to (34)is possible, we can have the high-energy α-deformation of the metric. It is noteworthy that in (34) T GU P this time is calculated from ( [64], formula (10)) with subsequent replacement of l p by √ G for c = 1. It is especially interesting to consider the following case.

Case of nonequilibrium thermodynamics ([73], section (6.2))
In this case the α -dependent dynamic cosmological term Λ(α) = 0 appears in the right-hand side of (34). Then, with the addition of Λ(α) = 0 , the α -representation (34)(for = 1) is given as follows(( [73], formula (53)): where α = α R ≈ 1/4, f ′ (α) = 4πk B T GU P (α) (37) and the derivative in the left-hand side of (37) is taken with respect to α. Λ(α) in the right-hand side of (37) may be subjected to a series expansion in terms of α, in compliance with the holographic principle [49]- [53] as applied to the Universe [76]. In [77], [38], [39], [73] in the leading order this expansion results in the first power, i.e. we have where Λ p -initial value of Λ ≈ Λ 1/4 derived using the well-known procedure of "summation over all zero modes " and the Planck momentum cutoff [78], [79]. Actually, (38)is in a good agreement with the observable Λ = Λ observ . Because a radius of the visible part of the Universe is given as R = R U niv ≈ 10 28 cm, it is clear that α R ≈ 10 −122 and (38)is completely consistent with the experiment [79]. Note that, proceeding directly from a quantum field theory but without the use of the holographic principle, we can have only a rough estimate of Λ that, on the whole, is at variance with Λ observ . Such an estimate may be obtained in different ways: by simulation [80]; using the cutoff [78] but now in the infrared limit; with the use of the Generalized Uncertainty Principle for the pair (Λ, V ), where V -four-dimensional volume [38], [39]. In the αrepresentation in this case the expansion in terms of α results in the second leading order that, obviously, is at variance with the accepted facts.

Conclusion
I. A very interesting case of the zero energy-momentum tensor for matter fields T ab = 0 and , specifically the case of P (α) = 0 in the right-hand side of (36) has remained beyond the scope of the final Section. We can state the problem more specifically: for which conditions in this case we can derive a solution in the form of the de Sitter space with large values of α? This problem is also important when we try to find whether it is possible to derive the initial inflation conditions [81], [82] for T ab = 0 on the basis of the foregoing analysis.
Note that the dynamic cosmological term Λ(α) correlates well with inflation models [81], [82] as the latter require a very high Λ at the early stages of the Universe, and this is distinct from Λ = Λ exper in the modern period. Of great interest is the recent work [83], where a mechanism of the vacuum energy decay in the de Sitter space is established to support a dynamic nature of Λ. II. The deformation parameter α R has a double meaning.
As κ = 1/R -curvature with the radius R, α R = κ 2 l 2 min is nothing else but the squared curvature multiplied by the squared minimal area and is explicitly dependent on the energy E. On the other hand, it is seen that, at least at the known energies, from the definition of the bit number N in ( [14], formula (3.10)) we get α R ∼ 1/N. But, because α R = α R [E], this suggests that N = N[E], as demonstrated in the text on going to higher energies N ⇒ N GU P .
Nevertheless, on going to lower energies, i.e. in the infrared limit, the same should be true: the bit number must be a function of energy. This problem and the relevant questions touched upon in this paper will be further considered in subsequent works of the author.