Probable Entropic Nature of Gravity in Ultraviolet and Infrared Limits—Part I: An Ultraviolet Case

Abstract

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.

1. Introduction

In the last 15–20 years, new and 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 Jacobson [1] has been considerably extended in a series of remarkable papers by Padmanabhan [2–13]. The paper by 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].

2. 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, that is, 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 to the ultraviolet (Planck) scale, this idea has been proposed long ago [40–45], and in this situation we have to replace the word “may” 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; ongoing to the known energy scales, the known results must be reproduced.

In this section, it is shown that the fundamental quantities 𝐴, 𝑁, and 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 Schwarzschild black hole as the 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 𝑑𝑁=𝑑𝐴.𝐺ℏ (1) 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.𝐺ℏ (2) What are the changes in 𝑆 ongoing 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]: ℏΔ𝑥≥Δ𝑝+ℓ²Δ𝑝ℏ, (3) where ℓ²=𝛼^′𝑙²_𝑝 and 𝛼^′ is the dimensionless numerical factor. The well-known Bekenstein-Hawking formula for the black hole entropy in the semiclassical approximation [62, 63], 𝑆^BH=𝐴4𝑙²_𝑝, (4) is modified by the corresponding quantum corrections ongoing from HUP to GUP [64–67].

In particular, [65], 𝑆^BH_GUP=𝐴4𝑙²_𝑝−𝜋𝛼^′24𝐴ln4𝑙²_𝑝+^∞_𝑛=1𝑐_𝑛𝐴4𝑙²_𝑝^−𝑛+const., (5) 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: 𝑆_ln=𝐴4𝑙²_𝑝+𝛼4𝐴ln𝑙²_𝑝, (6) 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 𝑙²_𝑝/𝐴: 𝑆_GUP=𝐴4𝑙²_𝑝+𝛼4𝐴ln𝑙²_𝑝+^∞_𝑛=1𝑐_𝑛̃𝐴𝑙²_𝑝^−𝑛+const. (7) in a similar way to the Taylor-series expansion of the right-hand side in (9) in terms of the small parameter 4𝑙²_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_GUP, and hence 𝑁→𝑁_GUP. Assuming in the notation of [14] that 1𝑆=4𝑁, (8) we directly obtain 𝑁_GUP=𝐴𝑙²_𝑝+𝐴𝛼ln𝑙²_𝑝+4^∞_𝑛=1𝑐_𝑛̃𝐴𝑙²_𝑝^−𝑛+const. (9) Now, coming back to [14], in terms of 𝑁_GUP we can define the holographic screen area, as measured at high energies, by 𝐴_GUP≡𝐺ℏ𝑁_GUP, (10) where 𝐺 and ℏ are gravitational and Planck constants, respectively, and 𝑁_GUP is given by (9). Considering that we, similar to [14], assume that the speed of light 𝑐=1, then, according to 𝑙²_𝑝=𝐺ℏ, from (9) and (10), we have 𝐴_GUP𝐴=𝐴+𝐺ℏ𝛼ln𝐺ℏ+4𝐺ℏ^∞_𝑛=1𝑐_𝑛̃𝐴𝐺ℏ^−𝑛+const. (11) 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_GUP rather than in 𝐴_GUP; that is, the constant in the right-hand side of (11) is insignificant. So, (1) has a fairly definite analog at high energies 𝑑𝑁_GUP=𝑑𝐴_GUP𝐺ℏ (12) that ongoing 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_GUP is defined in terms of 𝑁_GUP and 𝑑𝐴_GUP in terms of 𝑑𝑁_GUP. The logic series is here as follows: 𝐴⟹𝑁⟹𝑁_GUP⟹𝐴_GUP. (13) 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_GUP was derived as a series (formulas (1), (7) from [70]): 𝑇_GUP=𝑇1+Θ_𝑇𝑇²+⋯=𝑇+Θ_𝑇𝑇³+⋯=𝑇+𝑇_GUP, (14) where the factors in the right-hand side (14) may be computed in the explicit form and at low energies 𝑇_GUP→0.

Thus, we can have a GUP - analog of Komar’s mass in ((5.33) from [14]) 𝑀_GUP=12_𝒮𝑇_GUP𝑑𝑁_GUP=12_𝒮𝑇+𝑇_GUP𝑑𝑁_GUP=12𝐺ℏ_𝒮𝑇_GUP𝑑𝐴_GUP, (15) that the low-energy limit gives the well-known Komar formula [71], [72, page 289].

It is clear that the “GUP-deformed Komar’s mass” 𝑀_GUP in the first term (15) as a component has the known Komar’s mass [71], [72, (11.2.9–11.2.10)], [14, (5.34)] 1𝑀=4𝜋𝐺_𝒮𝑇𝑑𝐴. (16)

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: 𝛼_𝑥=𝑙²_min𝑥², (17) 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_GUP, 𝐴_GUP, and 𝑀_GUP defined in the preceding section.

Substituting from GUP (3) 𝑙_min=2√𝛼^′𝑙_𝑝 and using the formula 𝐴=4𝜋𝑅², where 𝑅 is the radius of the screen 𝒮, we get 𝑁_GUP (9) of the following form: 𝑁_GUP=𝑁+𝛼ln𝜎𝛼_𝑅⁻¹+4^∞_𝑛=1𝑐_𝑛̃𝜎^−𝑛𝛼^𝑛_𝑅+const. (18) Here, 𝛼_𝑅 is a value of 𝛼 parameter at the point 𝑅, and 𝜎 is equal to 16𝛼^′𝜋. It is convenient to refer to the form 𝑁_GUP derived in (18) as to the 𝛼—representation.

Using (10) and (11), we can easily obtain 𝛼—representation for 𝐴_GUP: 𝐴_GUP=𝐴+𝛼𝐺ℏln𝜎𝛼_𝑅⁻¹+4𝐺ℏ^∞_𝑛=1𝑐_𝑛̃𝜎^−𝑛𝛼^𝑛_𝑅+const. (19) Also, it is clear that 𝑀_GUP (15) may be derived in terms of 𝛼_𝑅.

Here, 𝛼_𝑥 is considered as a deformation parameter for the Heisenberg algebra ongoing 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, page 943], 𝑥,𝑝=𝑖ℏ1+𝛽²𝑝²+⋯,Δ𝑥_min≈ℏ𝛽∼𝑙_𝑝. (20)(21) In this case, from (20), (21), it follows that 𝛽∼𝟏/𝐩², and for 𝑥_min∼𝑙_𝑝, 𝛽 corresponding to 𝑥_min is nothing else but 1𝛽∼𝑃²_𝑝𝑙, (22) where 𝑃_𝑝𝑙 is Planck’s momentum: 𝑃_𝑝𝑙=ℏ/𝑙_𝑝.

In this manner, 𝛽 is varying over the following interval: 𝜆𝑃²_𝑝𝑙≤𝛽, (23) 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_𝑥=𝑙²_min/𝑥²∼𝑙²_𝑝/𝑥²=𝑝²/𝑃²_𝑝𝑙𝑥,𝑝=𝑖ℏ1+𝛽²𝑝²+⋯=𝑖ℏ1+𝑎₁𝛼_𝑥+𝑎₂𝛼²_𝑥+⋯. (24) 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.

4. Quantum Corrections to the Principal Result and Ultraviolet Limit

Based on the aforesaid, we can proceed the generalization of the results from Section 5.2 of [14] and the derivation of equations for a gravitational field within the scope of GUP.

We must consider two absolutely different cases.

4.1. Quantum Corrections to the Principal Result

It is assumed that the screen radius is given by 𝒮𝑅≫𝑙_𝑝. (25) In terms of the deformation parameter 𝛼_𝑥 introduced in the previous section, we have 𝛼_𝑅≪14. (26) So far we are not concerned with the 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 𝑀_GUP=𝑀_GUP[𝛼_𝑅] (15). The “𝛼_𝑅-complement” (i.e., the difference 𝑀[𝛼_𝑅]=𝑀_GUP[𝛼_𝑅]−𝑀) 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] are 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, and 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 nonuniformly varying quantity. Then, due to ((25) or (26)), the difference between two successive values of 𝛼_𝑅 is as follows: Δ_min𝛼_𝑅=𝛼_𝑅−𝛼_𝑅+𝑙min∼𝑙³_min𝑅³, (27) 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 2_Σ𝑇_𝑎𝑏[𝛼]−12𝑇[𝛼]𝑔_𝑎𝑏[𝛼]𝑛^𝑎𝜉^𝑏=1𝑑𝑉4𝜋𝐺_Σ𝑅_𝑎𝑏[𝛼]𝑛^𝑎𝜉^𝑏𝑑𝑉, (28) where the dependence of 𝑇_𝑎𝑏[𝛼] and 𝑅_𝑎𝑏[𝛼] on 𝛼=𝛼_𝑅 is completely determined, in accordance with [14, 72], by the integral 𝑀_GUP[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 ([14], end of Section 5.2) in this case are valid because 𝛼_𝑅 is small and continuous; the whole system is 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: 2_Σ𝑇_𝑎𝑏−12𝑇𝑔_𝑎𝑏𝑛^𝑎𝜉^𝑏=1𝑑𝑉4𝜋𝐺_Σ𝑅_𝑎𝑏+Λ𝑔_𝑎𝑏𝑛^𝑎𝜉^𝑏𝑑𝑉. (29) 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.

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 𝑅≈𝜉𝑙_min=2𝛼^′𝜉𝑙_𝑝, (30) where 𝜉 is a number on the order of 1 or 𝛼_𝑅≈14. (31) 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 microblack hole with a radius on the order of several Planck’s lengths.

Clearly, the methods of [1, 14] 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 microblack 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)): ℏ𝑐𝑓^′(𝑎)4𝜋_𝑘𝐵𝑇𝑐³𝑑1𝐺ℏ44𝜋𝑎²_𝑑𝑆−12𝑐⁴𝑑𝑎_−𝑑𝐸𝐺=𝑃𝑑4𝜋3³_𝑃𝑑𝑎_𝑉, (32) 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, (33) similar, in particular, to the case under study (32) 𝑃=𝑇^𝑅_𝑅≠0.

4.2.1. Case of Equilibrium Thermodynamics [73, Section  6.1]

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_GUP with the subscript GUP. Then, the high-energy 𝛼-deformation of (32) takes the form 𝑘_𝐵𝑇_GUP(𝛼)𝑑𝑆_GUP(𝛼)−𝑑𝐸_GUP(𝛼)=𝑃(𝛼)𝑑𝑉_GUP(𝛼). (34) Substituting into (34) the corresponding quantities 𝑇_GUP(𝛼), 𝑆_GUP(𝛼), 𝐸_GUP(𝛼), 𝑉_GUP(𝛼), and 𝑃(𝛼) and expanding them into a Laurent 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_GUP this time is calculated from [64, formula (10)] 𝑇^BH_GUP=14𝜋ℏ𝑅2𝛼^′2𝑙²_𝑝1−𝛼1−^′2𝑙²_𝑝𝑅²=ℏ𝛼_𝑅⁻¹4𝜋𝛼^′𝑙_𝑝1−1−𝛼_𝑅^1/2 (35) with subsequent replacement of 𝑙_𝑝 by √𝐺ℏ for 𝑐=1.

It is especially interesting to consider the following case.

4.2.2. 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 ℏ=1) is given as follows [73, formula (53)]: −𝛼²𝑓^′(𝛼)−12𝛼=16𝜋𝛼^′2𝑃(𝛼)𝐺²−𝐺Λ(𝛼), (36) where 𝛼=𝛼_𝑅≈1/4, 𝑓^′(𝛼)=4𝜋𝑘_𝐵𝑇_GUP(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 [38, 39, 73, 77] in the leading order, this expansion results in the first power, that is, we have Λ𝛼_𝑅∼𝛼_𝑅Λ_𝑝, (38) where Λ_𝑝 is the 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_Univ≈10²⁸cm, 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 is four-dimensional volume [38, 39]. In the 𝛼—representation in this case, the expansion in terms of 𝛼 results in the second leading order Λ𝛼_𝑅∼𝛼²_𝑅Λ_𝑝, (39) that, obviously, is at variance with the accepted facts.

5. 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=𝜅²𝑙²_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 ongoing to higher energies 𝑁⟹𝑁_GUP. (40) Nevertheless, ongoing to lower energies, that is, 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.