Rainbow gravity corrections to the entropic force

The entropic force attracts a lot of interest for its multifunctional properties. For instance, Einstein's field equation, Newton's law of gravitation and the Friedmann equation can be derived from the entropic force. In this paper, utilizing a new kind of rainbow gravity model that was proposed by Magueijo and Smolin, we explore the quantum gravity corrections to the entropic force. First, we derive the modified thermodynamics of a rainbow black hole via its surface gravity. Then, according to Verlinde's theory, the quantum corrections to the entropic force are obtained. The result shows that the modified entropic force is related not only to the properties of the black hole but also the Planck length $\ell_p$, and the rainbow parameter $\gamma$. Furthermore, based on the rainbow gravity corrected entropic force, the modified Einstein's field equation and the modified Friedmann equation are also derived.

1 the energy of a test particle. Therefore, the background of this spacetime can be described by a family of metrics (rainbow metrics) parameterized by Eℓ p . In Refs. [ [32,33] to study the modified entropy-area law [34]. His work showed that the modifications 50 are related to the rainbow functions. As a matter of fact, there are many works demonstrating that the RG 51 model is not unique. According to different phenomenological motivations, a series of RG models can be 52 obtained. Based on the varying speed of light theory, which is a model that can explain the cosmological 53 horizon, flatness, and cosmological constant problems of Big-Bang cosmology, Magueijo where γ, ℓ p and m represent the rainbow parameter, the Planck length and the mass of the test particle, 56 respectively. Obviously, Eq. (1) indicates that spacetime has an energy-dependent velocity c (E) = dE/dp = 57 1 − Eℓ p . Comparing it with the general form of MDR, namely, E 2 f 2 (Eℓ p ) − p 2 g 2 (Eℓ p ) = m 2 , one can easily 58 obtain the a new model of RG, whose rainbow functions are expressed as follows [36, 37] The organization of the paper is as follows. In Section 2, we calculate the modified Hawking temperature Generally, in order to obtain the modified metric of black holes in rainbow gravity, one can simply make the replacements dt → dt/f (Eℓ p ) for time coordinates and dx i → dx i g (Eℓ p ) for all spatial coordinates. Therefore, the metric of a static spherically symmetric black hole without charge in the framework of RG is where M , G, and dΩ 2 are the mass of a static spherically symmetric black hole, Newton's gravitational constant, and line elements of hypersurfaces, respectively [25]. The vanishing of [1 − (2GM /r)] f 2 (Eℓ p ) at point r H = 2GM indicates the presence of an event horizon. Eq. (4) reproduces the standard static spherically symmetric black hole if Eq. (3) holds. According to Eq. (4), surface gravity on the event horizon is given by where g 00 ′ = ∂ t g 00 . The above equation indicates that the surface gravity is modified by RG. It is well known that the Hawking temperature is defined in terms of surface gravity, that is, T H = κ/2π. Using the Hawking temperature-surface gravity relation, the RG-corrected Hawking temperature can be expressed as According to [40][41][42][43], in the vicinity of the Schwarzschild surface, there is a relationship between the minimum value of position uncertainty ∆x of emitted particles and the Schwarzschild radius r H , namely, ∆x ≃ r H . Then, by considering that the Heisenberg uncertainty principle ∆p∆x ≥ 1 still holds in RG, the energy uncertainty can be expressed as ∆E = 1/∆x. Therefore, one can obtain the following relation: where E is a lower bound on the energy of emitted particles. With the help of Eq. (7), the RG-corrected Hawking temperature can be rewritten as where T 0 = 1/8πGM is the original Hawking temperature. Using the first law of black hole thermodynamics, the modified entropy is given by In the above expression, we use the area of the horizon A = 16πG 2 M 2 . It is clear that the modified entropy 76 is related to the area of the horizon A, Planck length ℓ p , and rainbow parameter γ. Furthermore, there is 77 a logarithmic correction term in Eq. (9), which is coincident with those results achieved in previous works 78 (e.g., [44,45]). When γ = 0, Eq. (9) reduces to the original entropy, namely, S = A 4ℓ 2 p .

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3 The modified Newton's law of gravitation due to the rainbow 80 gravity 81 In the framework of quantum gravity, it is believed that information can be stored on a volume of space. This viewpoint is known as the holographic principle, which is supported by black hole physics and the anti-de Sitter/conformal field theory correspondence (AdS/CFT) correspondence. Now, following the idea of Verlinder, one can assume there is a screen in space and it can separate points. When the particles move from one side of the screen to the other side, the information is stored on the screen. In Ref.
[11], this screen is called a holographic screen. Therefore, according to the holographic principle and the second law of thermodynamics, when a test particle approaches a holographic screen, the entropic force of a gravitational system obeys the following relation: where F , T , ∆S, and ∆r represent the entropic force, the temperature, the change of entropy on holographic screen, and the displacement of the particle from the holographic screen, respectively. Eq. (10) means that a nonzero force leads to a nonzero temperature. Motivated by an argument in Ref. [1], which posits that the change in entropy is related to the information on the boundary, that is ∆S = 2πm∆r with the mass of particle m. Hence, a linear relation between the ∆S and ∆r is given by where ∆r = 1/m with the mass of the elementary component. In Ref. [46], Hooft demonstrated that the horizon of black holes can be considered as a storage device for information. Assuming that the amount of information is N bits, and combining the equipartition law of energy with the holographic principle, the number of bits N and area of horizon A obey the following relation: Substituting the entropy-area law S = A/4G and modified entropy Eq. (9) into Eq. (12), the relation between N and A can be rewritten as In the above equation, we utilize the natural units, which says G = ℓ 2 p . Next, by adopting the equipartition law of energy, the total energy of the holographic system can be written as Now, substituting the relations E = M and A = 4πr 2 , and Eq. (10)-Eq. (13) into Eq. (14), the entropic force becomes In previous work, people found that Einstein's field equation can be derived from the entropic force approach. Therefore, in this section, we derive the modified Einstein's field equation via the RG-corrected entropic force. Taking into account Eq. (13), one can generalize the entropy-corrected relation to the following form: which is the bit density on a holographic screen. Next, consider a certain static mass configuration with total mass M enclosed in the holographic screen S, and take the energy associated with it divided over N bits. It is easy to find that each bit carries a mass equal to T /2. Hence, the total mass can be expressed as and the local temperature on the screen is where e φ is the redshift factor as the local temperature T is measured by an observer at infinity [47]. Inserting the identifications for dN and T , Eq. (17) can be rewritten as where the right hand side (RHS) of Eq. (19) is the modified Komar mass while the original Komar mass is . Now, combining the Stokes theorem with the killing equation ∇ a ∇ a = −R b a ξ a , one can express the original Komar mass in terms of the Ricci tensor R ab and the Killing vector ξ a [13, 14] Equating Eq. (19) and Eq. (20), one obtains In the above equation, Σ represents the three-dimensional volume bounded by the holographic screen and n a is the volume's normal. On the other hand, following the viewpoint that was proposed by Seiedgar, the total mass M can be expressed as the stress-energy T µν [34], that is, Putting Eq. (22) into Eq. (21), one can obtain the following expression The RHS of Eq. (23) is a correction term, which is caused by quantum gravity. In order to obtain Einstein's field equation, the RHS should be rewritten as Substituting Eq. (22) and Eq. (24) into Eq. (23), one has Finally, the deformed Einstein's field equation is It is obvious that this field equation is affected by the geometry of spacetime, the energy-momentum tensor In this section, according to the methods which were proposed by [13-17, 24, 30, 31], the RG corrected Friedmann equation is derived via the modified entropic force approach. First, let us consider a 4-dimensional rainbow FRW universe, whose linear element is given by where dΩ 2 = dθ 2 + sin 2 θdϕ 2 represents the metric of a two-dimensional unit sphere, a is scale factor of the universe, and k is the spatial curvature constant. According to Refs. [50,51], it is suitable to use the notioñ r = f (Eℓ p ) a (t) r/g (Eℓ p ) and relation h µν ∂ µr ∂ νr = 0; the modified dynamical apparent horizon of FRW spacetime becomesr where H =ȧ/a is the Hubble parameter withȧ = ∂ t a. Significantly, Eq. (28) recovers the original apparent horizon of FRW spacetime when f (Eℓ p ) = 1 and g (Eℓ p ) = 1. If one assumes that the matter source in the FRW universe is a perfect fluid, the tress-energy tensor of this fluid can be expressed as Hence, the following continuity equation can be derived from the conservation law of energy-momentuṁ For obtaining the modified Friedmann equation, we need to consider a compact spatial region with volume V = (4/3) πr 3 . By incorporating the Newton's second law with the gravitational force, one has where χ (ℓ p , γ) = 1 − 2γℓ p /r − 2γ 2 ln (1/4r − γℓ p ) ℓ 2 p r 2 . Substituting Eq. (31) into Eq. (30), one can obtain the following acceleration equation with some manipulations: a a = − 4 3 πGρχ (ℓ p , γ) .
Furthermore, the total physical M can be defined as Next, using the active gravitational mass instead of the total mass in Eq. (33), the modified acceleration equation for the dynamical evolution of the FRW universe becomes a a = − 4 3 πG (ρ + 3p) χ (ℓ p , γ) .
It should be noted that the effect of correction terms in Eq. (35) is too small to detect because the apparent Finally, we derive the modified Friedmann equation from the RG corrected entropic force. Eq. (44) shows that the deformed Friedmann equation is related not only to the spatial curvature k, Hubble parameter H, 99 scale factor of the universe a, and equation of state parameter ω but also to the rainbow parameter γ and