On the dark matter as a geometric effect in $f (\mathcal {R})$ gravity

Abstract

A mysterious type of matter is supposed to exist because the observed rotational velocity curves of particles moving around the galactic center and the expected rotational velocity curves do not match. This type of matter is called dark matter. There are also a number of proposals in the modified gravity which are alternatives to the dark matter. In this contrast, in 2008, Christian G. Böhmer, Tiberiu Harko and Francisco S.N. Lobo presented an interesting idea in Böhmer et al. (Astropart Phys 29(6):386–392, 2008) where they showed that a $f (\mathcal {R})$ gravity model could actually explain dark matter to be a geometric effect only. They solved the gravitational field equations in vacuum using generic $f (\mathcal {R})$ gravity model for constant velocity regions (i.e. dark matter regions around the galaxy). They found that the resulting modifications in the Einstein Hilbert Lagrangian is of the form $\mathcal {R}^{1+m}$, where $m=V_{tg}^2/c^2$; $V_{tg}$ being the tangential velocity of the test particle moving around the galaxy in the dark matter regions and c being the speed of light. From observations it is known that $m\approx \mathcal {O}(10^{-6})$ (Böhmer et al. 2008; Salucci et al. in Mon Not R Astron Soc 378(1):41–47, 2007; Persic et al. in Mon Not R Astron Soc 281:27–47, 1996; Borriello and Salucci in Mon Not R Astron Soc 323(2):285–292, 2001). In this article, we perform two things (1) We show that the form of $f (\mathcal {R})$ they claimed is not correct. In doing the calculations, we found that when the radial component of the metric for constant velocity regions is a constant then the exact solutions for $f (\mathcal {R})$ obtained is of the form of $\mathcal {R}^{1-\alpha }$ which corresponds to a negative correction rather than positive claimed by the authors of Böhmer et al. (2008), where $\alpha$ is the function of m. (2) We also show that we can not have an analytic solution of $f(\mathcal {R})$ for all values of tangential velocity including the observed value of tangential velocity 200–300 km/s (Salucci et al. 2007; Persic et al. 1996; Borriello and Salucci 2001) if the radial coefficient of the metric which describes the dark matter regions is not a constant. Thus, we have to rely on the numerical solutions to get an approximate model for dark matter in $f (\mathcal {R})$ gravity.

Appendices

Appendix 1: Derivation of the equations

As discussed, from Eq. (5), we see that $A_\mu \equiv (F\mathcal {R}_{\mu \mu }-\nabla _\mu \nabla _\mu F)/ g _{\mu \mu }$ is independent of the index $\mu$, thus $A_\mu -A_\nu =0$. The proof is as follows: Eq. (5) can be written as

$$\begin{aligned} F(\mathcal {R})\mathcal {R}_{\mu \nu }-\nabla _\mu \nabla _\nu F(\mathcal {R})= & {} \dfrac{1}{4}g_{\mu \nu }\left( F(\mathcal {R})g^{\alpha \beta }\mathcal {R}_{\alpha \beta } -g^{\alpha \beta }\nabla _\alpha \nabla _\beta F(\mathcal {R}) \right) \end{aligned}$$

(40)

$$\begin{aligned} F(\mathcal {R})\mathcal {R}_{\mu \nu }-\nabla _\mu \nabla _\nu F(\mathcal {R})= & {} \dfrac{1}{4}g_{\mu \nu }g^{\alpha \beta }\Big ( F(\mathcal {R})\mathcal {R}_{\alpha \beta } -\nabla _\alpha \nabla _\beta F(\mathcal {R}) \Big ) \end{aligned}$$

(41)

Since the metric considered is static and spherically symmetric (1) with diagonal entries thus $\mu =\nu$ and $\alpha =\beta$. Equation (41) now becomes

$$\begin{aligned} F(\mathcal {R})\mathcal {R}_{\mu \mu }-\nabla _\mu \nabla _\mu F(\mathcal {R})= & {} \dfrac{1}{4}g_{\mu \mu }g^{\alpha \alpha }\Big ( F(\mathcal {R})\mathcal {R}_{\alpha \alpha } -\nabla _\alpha \nabla _\alpha F(\mathcal {R}) \Big ) \qquad \end{aligned}$$

(42)

$$\begin{aligned} \dfrac{1}{g_{\mu \mu }}\Big (F(\mathcal {R})\mathcal {R}_{\mu \mu } -\nabla _\mu \nabla _\mu F(\mathcal {R})\Big )= & {} \dfrac{1}{4}g^{\alpha \alpha }\Big ( F(\mathcal {R})\mathcal {R}_{\alpha \alpha } -\nabla _\alpha \nabla _\alpha F(\mathcal {R}) \Big ) \end{aligned}$$

(43)

$$\begin{aligned} \dfrac{1}{g_{\mu \mu }}\Big (F(\mathcal {R})\mathcal {R}_{\mu \mu } -\nabla _\mu \nabla _\mu F(\mathcal {R})\Big )= & {} \dfrac{1}{4 g_{\alpha \alpha }} g_{\alpha \alpha }g^{\alpha \alpha }\Big ( F(\mathcal {R})\mathcal {R}_{\alpha \alpha } -\nabla _\alpha \nabla _\alpha F(\mathcal {R}) \Big )\nonumber \\ \end{aligned}$$

(44)

Since, $g^{\alpha \alpha }g_{\alpha \alpha }=4$. Thus, above equation now becomes

$$\begin{aligned} \dfrac{1}{g_{\mu \mu }}\Big (F(\mathcal {R})\mathcal {R}_{\mu \mu } -\nabla _\mu \nabla _\mu F(\mathcal {R})\Big )=\dfrac{1}{g_{\alpha \alpha }} \Big ( F(\mathcal {R})\mathcal {R}_{\alpha \alpha } -\nabla _\alpha \nabla _\alpha F(\mathcal {R}) \Big ) \end{aligned}$$

(45)

Since $\alpha$ is free index, we can replace it by $\nu$. Thus,

$$\begin{aligned}&\dfrac{1}{g_{\mu \mu }}\Big (F(\mathcal {R})\mathcal {R}_{\mu \mu } -\nabla _\mu \nabla _\mu F(\mathcal {R})\Big )=\dfrac{1}{g_{\nu \nu }}\Big ( F(\mathcal {R}) \mathcal {R}_{\nu \nu }-\nabla _\nu \nabla _\nu F(\mathcal {R}) \Big ) \end{aligned}$$

(46)

$$\begin{aligned}&A_\mu =A_\nu \quad \text {for all} \, \mu \,\,\text {and} \, \nu , \end{aligned}$$

(47)

where $A_\mu =\dfrac{1}{g_{\mu \mu }}\Big (F(\mathcal {R})\mathcal {R}_{\mu \mu }-\nabla _\mu \nabla _\mu F(\mathcal {R})\Big )$ and $A_\nu =\dfrac{1}{g_{\nu \nu }}\Big (F(\mathcal {R})\mathcal {R}_{\nu \nu }-\nabla _\nu \nabla _\nu F(\mathcal {R})\Big )$. Here, $\nabla _i\nabla _j=\partial _i\partial _j+\Gamma ^{\sigma }_{ij}$.

From the above relation $A_\mu -A_\nu =0$ we can write three equations ($A_0-A_1=0,~ A_1-A_2=0\text { and }A_0-A_2=0$; ‘0’ means ‘t’, ‘1’ means ‘r’, ‘2’ means ‘$\theta$’ and ‘3’ means ‘$\phi$’).³

The equation $A_0-A_1=0$ can be written as

$$\begin{aligned} \dfrac{1}{g_{00}}\Big (F(\mathcal {R})\mathcal {R}_{00}-\nabla _0\nabla _0 F(\mathcal {R})\Big )=\dfrac{1}{g_{11}}\Big (F(\mathcal {R})\mathcal {R}_{11}-\nabla _1\nabla _1 F(\mathcal {R})\Big ) \end{aligned}$$

(48)

As the metric in consideration is static, Eq. (48) now becomes

$$\begin{aligned} \dfrac{F,_{11}}{g_{11}}+\left( \dfrac{\mathcal {R}_{00}}{g_{00}} -\dfrac{\mathcal {R}_{11}}{g_{11}} \right) F-\left( -\dfrac{\Gamma ^{1}_{00}}{g_{00}} +\dfrac{\Gamma ^{1}_{11}}{g_{11}} \right) F,_{1}=0 \end{aligned}$$

(49)

This leads to Eq. (7). The equation $A_1-A_2=0$ can be written as

$$\begin{aligned} \dfrac{1}{g_{11}}\Big (F(\mathcal {R})\mathcal {R}_{11} -\nabla _1\nabla _1 F(\mathcal {R})\Big )=\dfrac{1}{g_{22}}\Big (F(\mathcal {R}) \mathcal {R}_{22}-\nabla _2\nabla _2 F(\mathcal {R})\Big ) \end{aligned}$$

(50)

This leads to Eq. (8). The rr component of the field equations (3) leads to Eq. (9). Equation (4) leads to Eq. (10)

Appendix 2: Comparison of equations and results with [1]

In this section, we perform a comparison of our equations and results with the Böhmer et al. [1] (Table 1).⁴

Table 1 Comparison of equations

1.1 Comparison of results

Böhmer et al. in [1] gave only one $f(\mathcal {R})$ gravity function, $f(\mathcal {R})=f_0 \mathcal {R}^{(1+m)}$, as being the replacement of the dark matter when in fact there should have been two function as Eq. (18) in [1] is second order differential equation. On the other hand, we found two different $f(\mathcal {R})$ gravity functions given by Eqs. (26) and (30).