In Verlinde’s Entropic Gravity (EG), it was conjectured that ordinary surfaces are holographic screens that obey the First Law of Thermodynamics (tFLT) [
1]. The theory primarily used tFLT equation,
, where
is the distance of the test particle from the holographic screen,
T is the temperature in the screen and
is the change in entropy
S. It was argued that as
, i.e., as the test particle touches the screen and increases the entropy, it induces gravitational force as a kind of entropic force [
2]. Thus in EG, a test particle that enters a gravitational field is likened to a polymer molecule that enters a region immersed in a thermal bath. Such a condition gives rise to an entropic force at molecular and atomic levels. According to EG, a test particle that enters a gravitational field also undergoes a similar condition where the entropic force is the gravitational force itself. This analogy was heavily criticized [
3,
4]. In the paper of Wang and Braunstein [
5], it was shown that for surfaces away from horizons, tFLT fails and therefore undermines the key thermodynamic assumption of the EG program.
In this paper, it will be shown that there is no need to use tFLT but instead, one can fully develop an Entropic Gravity model that purely uses the Second Law of Thermodynamics alone as it was originally used in the seminal works in Black Hole Physics. In retrospect, the basis of Entropic Gravity is the Holographic Principle based on the conjecture of Bekenstein [
6] in the 1970s that the entropy of a black hole is proportional to the surface area of the black hole, i.e.,
, where
S is the entropy within the black hole,
is the surface (horizon) area of the black hole with radius
and
is a constant of proportionality. Subsequently, the value for
was established to be the Planck Length [
7] such that,
where
is the Planck length and
is the Planck area. This result was interpreted by many as the quantity
or the number of cells in the holographic screen is a measure of entropy within the volume of space enclosed by the holographic screen. The question that Verlinde wanted to answer in his version of EG is how can this be applicable to describe gravity in a non-black hole setting. However, the relation
seems to be the only central argument in Verlinde’s work and by using it, separates the role of other quantities like the mass or energy content in a holistic way. In Verlinde’s approach to the derivation of Newton’s gravity law equation, the role of the mass enters via the use of tFLT and the Equipartition Theorem. In this paper, although tFLT and the Equipartition Theorem will not be used, a modified Newton’s law of gravity will be derived that is similar to a Modified Newtonian Dynamics (MOND) theory [
8]. Our motivation is derived from the fact that the changes in the strength of gravity are proportional to the matter density. In Poisson Equation,
, the strength of gravity is given by the scalar potential
and proportional to the density
. In Gauss’ Theorem of gravity,
, the strength of gravity given by
F is proportional to the changes in matter content or density for a constant volume and area. It gives us,
Hence, we suggest here that this fundamental result in the description of gravity should be integrated into the EG approach by not just using the quantity
, but via the quantity,
as a measure not only of the number of the smallest possible units of density but also as a measure of the density of information or entropy of the system. It is defined by the Planck (linear) density
, where
is the Planck Mass and
is the Planck Length. Notice that
can also be written in terms of quantities
and
. This will be shown to be aligned with the earlier work of Vopson [
9] with his extended Landauer’s Principle that relates mass and energy with the entropy of a system,
where the information corresponds in observing
N set of event
X such that
is the information entropy function
,
T is the temperature at which the bit of information is stored and
is the Boltzmann constant. Thus, the main difference of our work with Verlinde’s and others [
10] that also derived a model of MOND via the EG approach, is that the Energy Equipartition Principle,
, will not be used here but to be replaced by the expression,
, as a quantized representation of energy in terms of the Planck energy
. This is consistent with the fact that everything is expressed in terms of the Planck scale units rather than with quantities that are emergent or within the macroscopic and quantum scale. Moreover, the use of
is consistent also with what had been suggested above that the magnitude of gravity is proportional to the density and not just to mass or to (properties of) space alone. In a way, the intrinsic relation of matter and space in the description of gravity is still maintained by using a simple mathematical approach that measures the mass and information density of the system at the fundamental level. Lastly, the consequence of the Area Theorem which shows the relation of entropy to the mass, via the equation,
, will be emphasized here. It is considered here as a key result that relates the mass to the information that can reside inside a black hole. This, of course, is a simplification since other quantities such as the electric charge and the spin can also increase the entropy of a black hole. Hence, if only the mass-entropy relation is to be considered, the questions we set to answer here are: “Can we still use this result in a non-black hole setting?" and “Can the Area Theorem in Black Hole Physics be applied to a galactic setting in which a relatively large magnitude of gravity is also involved?". In the end, the main objective of this paper is not to derive Newtonian Gravity as Verlinde had done in his original paper on EG but to derive a new model of gravity similar to MOND or to a Chameleon Theory as a possible alternative to the Dark Matter hypothesis. The paper is organized as follows: In
Section 1, a modified Newtonian gravity equation was derived, and in
Section 2 Vopson’s Energy-Mass-Information Equivalence Principle was discussed. Then in
Section 3, 4, 5, and 6, we showed that the model predicts the MOND equation, the Tully-Fisher Relation, the External Field Effect, and the observation on Wide Binary Stars, respectively. In
Section 7, a relativistic version of the model, consistent with Special Relativity, is discussed leading to a modified Kepler’s Third Law. This is different from the TeVeS model of Bekenstein which extensively uses Tensor and Lagrangian Formalism within the framework of General Relativity (GR). In
Section 8, 9,10, and 11, topics such as the Horizon Mass, Mercury’s Perihelion Shift, Light Deflection, and Equivalence Principle, were all derived from the model.