The entropic formulation of the inertia and the gravity relies on quantum, geometrical and informational arguments. The fact that the results are completly classical is missleading. In this paper we argue that the entropic formulation provides new insights into the quantum nature of the inertia and the gravity. We use the entropic postulate to determine the quantum uncertainty in the law of inertia and in the law of gravity in the Newtonian Mechanics, the Special Relativity and in the General Relativity. These results are obtained by considering the most general quantum property of the matter represented by the Uncertainty Principle and by postulating an expression for the uncertainty of the entropy such that: i) it is the simplest quantum generalization of the postulate of the variation of the entropy and ii) it reduces to the variation of the entropy in the absence of the uncertainty.
Deep Dive into Entropic Law of Force, Emergent Gravity and the Uncertainty Principle.
The entropic formulation of the inertia and the gravity relies on quantum, geometrical and informational arguments. The fact that the results are completly classical is missleading. In this paper we argue that the entropic formulation provides new insights into the quantum nature of the inertia and the gravity. We use the entropic postulate to determine the quantum uncertainty in the law of inertia and in the law of gravity in the Newtonian Mechanics, the Special Relativity and in the General Relativity. These results are obtained by considering the most general quantum property of the matter represented by the Uncertainty Principle and by postulating an expression for the uncertainty of the entropy such that: i) it is the simplest quantum generalization of the postulate of the variation of the entropy and ii) it reduces to the variation of the entropy in the absence of the uncertainty.
In a recent paper, Verlinde has put forward a very interesting proposal for the origin of the inertia, the law of force, the law of gravity and the General Relativity as emergent phenomena [1]. Acording to this theory, the spacetime can be described as an information device made of holographic surfaces (screens) on which the information about the physical systems can be stored. The relevant information about the physical dynamics can be recovered by analysing the variation of the information on the screens and it is independent of the details of the particular theory used to describe the physical system. The screens behave as streched horizons in the black hole physics and define emergent holographic directions in which the spacetime grow. The information on the screens is described by the information entropy and, as the black hole entropy, it is encoded in a number of bits proportional to the area of the screen [3]. The total energy of the degrees of freedom satisfy the equipartition theorem. Also, it is postulated that the information entropy is maximized by the entropic forces that act along the holographic directions and are defined by gradients of the entropy. If one associates noninertial frames to these forces, the corresponding accelerated observers measure a redshifted information at the Unruh temperature [6]
where k B is the Boltzmann’s constant, c is the speed of light and a is the acceleration. For a particle of mass m, it is postulated in [1] that the variation of the entropy along the holographic direction be linear in the separation ∆x between the particle and the screen
where l c = /mc is the Compton length. Thus, the Compton length defines the units of the entropy change. The variation of the entropy ∆S corresponds to an arbitrary variation of the energy ∆W at the thermodynamical equilibrium (with T = T U ) which, at its turn, can be interpreted as the result of a macroscopic force F acting on the particle over the distance ∆x
The above assumptions lead to the law of force for m and when applied to a spherical screen around a mass M they deliver the Newton’s law of gravity. Also, when properly generalized, the postulates allow one to obtain the Einstein’s equations [1] 1 . Several features and consequences of the entropic postulate have already been explored. The compatibility between the entropic gravity and the loop quantum gravity was proved in [9]. Applications to cosmology, Friedmann’s equations and D-brane cosmology were developed in [10,11,12,13,14]. The connection between the entropic gravity and the black holes was explored in [15,16,17,18,19]. A modification of the entropic force to include the deviation from the equipartition theory was proposed in [20]. The derivation of a dark energy term from the entropic gravity was given in [21]. Further relations with the thermodynamics were discussed in [22,23,24]. A speculation about the possible interpretation of the Coulomb force as an entropic force was done in [25]. The realisation of the information entropy in terms of light was investigated in [26]. Other gometrical, field theoretical and informational aspects of the theory are presented in [27,28,29,30,31].
In the above construction, the holographic principle [7,8] and the information entropy [3] play fundamental roles in establishing the entropic nature of the inertia and of the law of force. On the other hand, since the Planck constant cancells out in the law of force and the gravitational force, it seems that the quantum effects are irrelevant to the inertia and gravity despite the quantum nature of the key concepts from the relations (1) and ( 2). Thus, one may ask the important question whether there are quantum corrections to the laws of Mechanics derivable from the entropic postulate. The aim of this paper is to show that this questions has a positive answer if one takes into account the quantum structure of matter in the most general form of the Uncertainty Principle.
The paper is organized as follows. In Section 2 we briefly review the Verlinde’s postulate. Then we use it to derive the uncertainty in the entropy of the holographic screen if the particle is subjected to the Uncertainty Principle. We also argue that the uncertainty in the entropy should imply an uncertainty in the adiabatic force which can be interpreted as a quantum correction to it. In Section 3 we discuss the same situation in the Special Relativity as well as in the General Relativity. The last section is devoted to some discussions.
2 Non-Relativistic Entropic Force and the Uncertainty Principle Consider a system composed by a holographic screen S and a quantum test particle of mass m. Due to the quantum nature of the particle, the separation between S and m is determined only up to the uncertainty δx that satisfies the Heiseberg’s relation
where the uncertainty in the momentum along the transverse direction to S is δp. If the particle were classical and δx were zero, the variation of the entropy
…(Full text truncated)…
This content is AI-processed based on ArXiv data.
