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Under natural assumptions on the thermodynamic properties of space and time with the holo-graphic principle, we reproduce a MOND-like behaviour of gravity on particular scales of mass and length, where Newtonian gravity requires a modification or extension if no dark matter component is introduced in the description of gravitational phenomena. The result is directly obtained with the assumption that a fundamental constant of nature with dimensions of acceleration needs to be introduced into gravitational interactions. This in turn allows for modifications or extensions of the equipartion law and/or the holographic principle. In other words, MOND-like phenomenology can be reproduced when appropriate generalised concepts at the thermodynamical level of space and/or at the holographic principle are introduced. Thermodynamical modifications are reflected in extensions to the equipartition law which occur when the temperature of the system drops below a critical value, equals to Unruh’s temperature evaluated at the acceleration constant scale introduced for the description of the gravitational phenomena. Our calculations extend the ones by [1] in which Newtonian gravity is shown to be an emergent phenomenon, and together with it reinforces the idea that gravity at all scales is emergent.

The laws for black hole mechanics have suggested a remarkable similarity with the three laws of thermodynamics, in such a way that quantities associated to black hole properties have their corresponding thermodynamic equivalent interpretation [

All the above suggest the possibility for a deep relation between thermodynamics and gravity. This has been studied mainly in the relativistic regime under the concept of emergent gravity, considering thermodynamics as a more fundamental theory from which, general relativity can be derived (see e.g. [

In recent years, a growing number of independent observations have suggested that gravity requires modification [

where

where n is a constant that must be fixed via astronomical observations. A large value of n means that the function

For the case of spherically symmetric mass distribution, this extended Newtonian gravity approach proposal reproduces a MOND-like phenomenology [

In another attempt to obtain the MOND-like force formula, [

thus interpreting it as a cut off temperature below which, modifications to the equipartition law must occur. This approach takes into account the fact that the dynamical sector needs to be modified and so, the validity of Newton’s law of gravitation remains unaffected. As explained by the dimensional analysis of [

In this work, we show how, using arguments about thermodynamics and information, it is possible to derive in several ways an equation for the gravitational force in an extended modified gravity regime, which supports the idea that gravity can be understood as an emergent force, i.e. a consequence of deeper fundamental principles. The article is organised as follows: in Section 2 we review the main hypothesis made by [

We begin this section reviewing briefly some of the main ideas and hypothesis made in Verlinde’s work [

being A the area of the spherical screen and

The main motivation by [

for a constant volume. Let us now find the expression for the gravitational force by considering gravity as an entropic force. For this, we follow the approach by [

when a displacement

To do so, note that the dimensional relevant quantities of the problem are the energy E associated to the screen, it’s temperature T―or more important for a dimensional analysis treatment its energetic temperature

where F is an unknown function of four dimensionless parameters, x was defined in Equation (1),

The ratio

where

In the remaining of this section, we study three separate cases associated with the previous relation:

Case (A)

Let us consider

The physics behind this choice of parameters can be understood under the basis of the case studied by [

Equating relation (9) to

Substituting this into (5), and employing (6), as made by Verlinde, the resultant entropic force is:

Taking

In other words, the equipartition energy must satisfy the following condition:

Case (B)

Let us now consider the case when

Once again, using the equivalence between mass and energy, and the explicit form of Planck’s area, we can write:

which with the aid of Equations (5) and (6), gives following expression for the entropic force:

The choice

The acceleration exerted on the test mass m is then given by:

It is convenient to express it in this form, since we can easily compare it with (1). It can be observed that given the form proposed for the energy E, in the Newtonian regime, the ratio

Case (C)

Finally, we study the case for which

The choice

To summarise this section, note that the inclusion of Milgrom’s acceleration constant as a fundamental quantity of nature related to gravitational phenomena, is capable of generalising the equipartition law in such a way that either Newton or MOND force formulae can be obtained. This result reinforces the idea that, at all scales, gravity is an emergent force with a thermodynamic nature.

Let us now search for a MOND-like force formula assuming that the equipartition law has its usual form and allowing for modifications related to the way in which information is stored on the holographic screens (cf. Equation (4)). The only additional constant that needs to be taken into consideration is

where

With this in mind, we can follow Verlinde’s analysis. Assuming the validity of the equipartition law of energy, and the equivalence between mass and energy inside the screen, along with Equation (20), it follows that:

Direct substitution of the previous equation on relation (6), with the aid of the first law of thermodynamics (5) yields the following expression for the acceleration caused by this entropic force:

The choice

Finally, a possible alternative way to modify Verlinde’s result it to combine these two approaches, i.e. assume that both the equipartition law of energy and the holographic principle get modified when MONDian effects are introduced. Based on the previous analysis, this can be studied if we introduce the parameter x as follows:

where

and so,

according to Equation (1) with

As explained by [

In this article we have introduced this extra fundamental constant of nature

1) In the first approach, the equipartition law is modified and the holographic principle keeps its standard form, resulting on a temperature scale

2) In a second approach, modifications to the holographic principle―with the equipartition law unchanged― have showed to be able to explain in a natural way how gravity transits from a Newtonian regime to a modified one.

The important point about these two different ways is that they are consistent with the general formula for acceleration experienced by a particle under a gravitational field given in (1). In this sense, and in the context of emergent gravity, it suggests that the transition observed across different astrophysical systems could be a consequence of a modification at a deeper level in the equipartition law and/or in the holographic principle, i.e. the observed effects at large scales in gravitational systems reflect the behaviour of physical laws at a deeper thermodynamical level. More generally, it has been also considered the possibility of modifications of both the holographic principle and the equipartition law in such regimes.

As pointed by [

A full non-relativistic theory of gravity can be constructed assuming a modification of inertia as described by [

With a few natural assumptions about space and information, the main result of this article is to show that gravity can be considered an emergent phenomenon also in the MONDian regime. This suggests that the force of gravity on this extended regime is not a fundamental force of nature, but a consequence of the inherent properties of space and time. Since [

We thank an anonymous referee for his fruitful comments on the first version of this article. This work was supported by a DGAPA-UNAM grant (PAPIIT IN111513-3) and a CONACyT grant (240512). DAC and SM thank support granted by CONACyT 480147 and 26344. The authors gratefully acknowledge the comments made by Ehoud Pazy and Hristu Culetu for the valuable comments made of an earlier version of this article.