# A scan of f(R) models admitting Rindler-type acceleration

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## Abstract

As a manifestation of a large distance effect Grumiller modified Schwarzschild metric with an extraneous term reminiscent of Rindler acceleration. Such a term has the potential to explain the observed flat rotation curves in general relativity. The same idea has been extended herein to the larger arena of \(f(R)\) theory. With particular emphasis on weak energy conditions (WECs) for a fluid we present various classes of \(f(R)\) theories admitting a Rindler-type acceleration in the metric.

### Keywords

Black Hole Dark Matter Ghost Gravity Model Einstein Field Equation## 1 Introduction

Flat rotation curves around galaxies constitute one of the most stunning astrophysical findings since 1930s. The cases can simply be attributed to the unobservable dark matter which still lacks a satisfactory candidate. On the general relativity side which reigns in the large universe an interesting approach is to develop appropriate models of constant centrifugal force. One such attempt was formulated by Grumiller in [1, 2], in which the centrifugal force was given by \(F=-\left( \frac{m}{r^{2}} +a\right) \). Here \(m\) represents the mass (both normal and dark), while the parameter “\(a\)” is a positive constant—called Rindler acceleration [3]—which gives rise to a constant attractive force. The Newtonian potential involved is \(\Phi \left( r\right) \sim -\frac{m}{r}+ar\), so that for \(r\rightarrow \infty \) the term \(\Phi \left( r\right) \sim ar\) becomes dominant. Since in Newtonian circular motion \(F=\frac{mv^{2}}{r}\), for a mass \(m,\) tangential speed \(v\left( r\right) \) and radius \(r\) are related by \(v\left( r\right) \sim r^{\frac{1}{2}}\) for large \(r\), which overall amounts to come slightly closer to the concept of flat rotation curves. No doubt, the details and exact flat rotation curves must be much more complicated than the toy model depicted here. Physically the parameter “\(a\)” becomes meaningful when one refers to an accelerated frame in a flat space, known as Rindler frame, and accordingly the terminology Rindler acceleration is adopted.

In [4] the impact of a Rindler-type acceleration is studied on the Oort cloud, and in [5, 6] the solar system constraints on Rindler acceleration are investigated, while in [7] bending of light in the model of gravity at large distances proposed by Grumiller [1, 2] is considered.

Let us add also that to tackle the flat rotation curves, Modified Newtonian Dynamics (MOND) in space was proposed [8]. Identifying a physical source for the Rindler acceleration term in the spacetime metric has been a challenge in recent years. An anisotropic fluid field was considered originally by Grumiller [1, 2], whereas nonlinear electromagnetism was proposed as an alternative source [8]. A fluid model with an energy-momentum tensor of the form \(T_{\mu }^{\nu }=diag[-\rho ,p,q,q]\) was proposed recently in the popular \(f(R)\) gravity [9]. For a review of the latter we propose [10, 11, 12]. By a similar strategy we wish to employ the vast richness of \(f(R)\) gravity models to identify possible candidates that may admit a Rindler-type acceleration. Our approach in this study beside the Rindler acceleration is to elaborate on the energy conditions in \(f(R)\) gravity. Although violation of the energy conditions is not necessarily a problem (for instance, any quantum field theory violates all energy conditions), it is still interesting to investigate the non-violation of the energy conditions. Note that energy conditions within the context of dark matter in \(f(R)\) gravity have been considered by various authors [13, 14]. This at least will filter the viable models that satisfy the energy conditions. In brief, for our choice of energy-momentum the weak energy conditions (WECs) can be stated as follows: (1) WEC1 says that the energy density \(\rho \geqslant 0\). (2) WEC2 says that \(\rho +p\geqslant 0\), and (3) WEC3 states that \(\rho +q\geqslant 0\). Among the more stringent energy conditions, the strong energy conditions (SECs) amounts further to \(\rho +p+2q\geqslant 0\), which will not be our concern in this paper. However, some of our models satisfy SECs as well. Our technical method can be summarized as follows. Upon obtaining \(\rho ,\) \(p\), and \(q\) as functions of \(r\) we shall search numerically for the geometrical regions in which the WECs are satisfied. (A detailed treatment of the energy condition in \(f(R)\) gravity was given by J. Santos et al. in [15].)

From the outset our strategy is to assume the validity of the Rindler modified Schwarzschild metric a priori and search for the types of \(f(R)\) models which are capable to yield such a metric. Overall we test ten different models of \(f(R)\) gravity models and observe that in most cases it is possible to tune the free parameters in rendering the WECs satisfied. In doing this we entirely rely on numerical plots and we admit that our list is not an exhaustive one in the \(f(R)\) arena.

The organization of the paper goes as follows. Section 2 introduces the formalism with derivation of density and pressure components. Section 3 presents 11 types of \(f(R)\) models relevant to the Mannheim metric. The paper ends with our conclusion in Sect. 4.

## 2 The formalism

## 3 \(f(R)\) Models covering the Rindler acceleration

### 3.1 The Models

*this model naturally unifies two expansion phases of the Universe: inflation at early times and cosmic acceleration at the current epoch”.*This model of \(f(R)\) is given by

## 4 Conclusion

In Einstein’s general relativity, which corresponds to \(f(R)=R\), the Rindler modification of the Schwarzschild metric faces the problem that the energy conditions are violated. For a resolution to this problem we invoke the large class of \(f(R)\) theories. From a cosmological standpoint the main reason that we should insist on the Rindler acceleration term can be described as follows: at large distances such a term may explain the flat rotation curves as well as the dark matter problem. Our physical source beside the gravitational curvature is taken to be a fluid with equal angular components. Being negative the radial pressure is repulsive in accordance with the expectations of the dark energy. Our scan covered ten different \(f(R)\) models and in most cases by tuning of the free parameters we show that the WECs are satisfied. In ten different models we searched primarily for the validity of WECs as well as for \(\frac{d^{2}f}{ dR^{2}}>0\), i.e. the stability. With some effort thermodynamic stability can also be checked through the specific heat. With equal ease \(\frac{df}{dR}>0,\) i.e., the absence of ghosts can be traced. Figure 1, for instance, depicts the model with \(f(R)=\sqrt{R^{2}+b^{2}}\), (\(b=\)constant) in which WECs and stability, even the thermodynamic stability, are all satisfied, however, it hosts ghosts since \(\frac{df}{dR}<0\) for \(R<0\). Finally, among all models considered herein, we note that Fig. 7 satisfies WECs, stability conditions, as well as the ghost-free condition for \(r>r_{\min }\) in which \(r_{\min }\ge r_{h}\) depends on the other parameters.

Finally we comment that the abundance of parameters in the \(f(R)\) theories is one of its weak aspects. This weakness, however, may be used to obtain various limits and for this reason particular tuning of parameters is crucial. Our requirements have been weak energy conditions (WECs), Rindler acceleration, stability, and the absence of ghosts. Naturally further restrictions will add further constraints, which may lead us to dismiss some cases that are considered as viable in this study.

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