Flat galactic rotation curves from geometry in Weyl gravity

Abstract

We searched for a resolution of the flat galactic rotation curve problem from geometry instead of assuming the existence of dark matter. We observed that the scale independence of the rotational velocity in the outer region of galaxies could point out to a possible existence of local scale symmetry and therefore the gravitational phenomena inside such regions should be described by the unique local scale symmetric theory, namely Weyl’s theory of gravity. We solved field equations of Weyl gravity and determined the special geometry in the outer region of galaxies. In order to understand the effective description of gravitational phenomena, we compared individual terms of so called Einstein–Weyl theory and concluded that while the outer region of galaxies are described by the Weyl term, the inner region of galaxies are described by the Einstein-Hilbert term.

Appendices

Appendix A: Values of $R$, $C_{\mu \nu \rho \sigma } C^{\mu \nu \rho \sigma }$ and $\alpha$

In this appendix we compare values of $R$ and $\alpha C_{\mu \nu \rho \sigma } C^{\mu \nu \rho \sigma }$ at various regions in the galaxy and inside the Solar system. We first calculate them in the outer region of the galaxy both at the onset of the flat rotation curves region and at the edge of the galaxy to show that in the outer region of the galaxy the Weyl term dominates over the Einstein-Hilbert term. The calculation in the outer region also determines an order of magnitude value for $\alpha$ in a typical galaxy. With the same value for $\alpha$, we also show that in the inner region of the galaxy the Einstein-Hilbert term dominates over the Weyl term. The value of $\alpha$ in the inner region could be less than its value in the outer region of galaxy (perhaps due to less amount of integrated curvature (Donoghue 2017)), in which case the dominance of the Einstein-Hilbert term would be even more pronounced. Inside the Solar system the value of $\alpha$ was calculated some time ago in (Mannheim 2007) and it was shown that the Solar system tests of gravity are not affected by the existence of the Weyl term.

We use the realistic mean values for a typical galaxy as given in (Sofue 2016). For the bulge we assume exponential spheroid model with volume density given by $\rho (r) = \rho _{ob} \exp (-r/r_{ob})$. Mass of the bulge is given by $M_{b} = 8\pi r_{ob}^{3} \rho _{ob}$. We take onset of outer region to be $r_{b} = 2.2 r_{ob}$ (Mannheim 2006) as explained in the introduction. Given the mean values for the bulge mass, $M_{b} = 2.3 \pm 0.4 \times 10^{10}\ M_{\odot }$, and the bulge scale radius, $r_{ob} = 1.5 \pm 0.2~\mbox{kpc}$ of a typical galaxy (see Table 2 of (Sofue 2016)), we calculate $R$ and $C_{\mu \nu \rho \sigma } C^{\mu \nu \rho \sigma } = C^{2}$ at $r = r_{b}$ from the relations (61) to be

$$\begin{aligned} R (r_{b}) =& \frac{G}{c^{2}} \frac{M_{b}}{r_{ob}^{3}}e^{-r_{b}/r_{ob}} \, , \\ C^{2} (r_{b}) =& 48 \frac{G^{2}}{c^{4}} \frac{M_{b}^{2}}{r_{ob}^{6}} \left (\left (\frac{r_{ob}}{r_{b}}\right )^{3} - \frac{1}{6} e^{-r_{b}/r_{ob}} \right )^{2} \, . \end{aligned}$$

(62)

Thus the ratio of $\alpha C^{2}$ to $R/2\kappa$ at $r = 2.2 r_{ob}$ is about 50 for $\alpha = 4.74 \times 10^{80}~\mbox{kg}\cdot \mbox{m}^{2} / \mbox{sec}$.

We now repeat this calculation at the edge of galaxy with the mean radius of a typical galaxy given by $r_{g} = 16 r_{od}$ (Mannheim 2006) with $r_{od}$ being the disk scale radius. For the disk we assume exponential disk model with volume density given by $\rho (r,z) = \rho _{od} \exp (-r/r_{od}-|z|/z_{od})$ in cylindrical coordinates. Vertical profile scale radius is typically $z_{od} \approx 0.1 r_{od}$. Mass of the disk is then given by $M_{d} = 4\pi z_{od}r_{od}^{2} \rho _{od}$. Given the mean values for the disk mass, $M_{d} = 5.7 \pm 1.1 \times 10^{10}\ M_{\odot }$, the total baryonic mass, $M_{g} = M_{b} + M_{d} = 7.9 \pm 1.2 \times 10^{10}\ M_{\odot }$, and the disk scale radius, $r_{od} = 3.3 \pm 0.3~\mbox{kpc}$ of a typical galaxy (see Table 2 of (Sofue 2016)), we calculate $R$ and $C_{\mu \nu \rho \sigma } C^{\mu \nu \rho \sigma } = C^{2}$ at $r = r_{g}$ from the relations (61) to be

$$\begin{aligned} R (r_{g}) =& 20\frac{G}{c^{2}} \frac{M_{d}}{r_{od}^{3}}e^{-r_{g}/r_{od}} \, , \\ C^{2} (r_{g}) \approx & 48 \frac{G^{2}}{c^{4}} \frac{M_{g}^{2}}{r_{g}^{6}} \, . \end{aligned}$$

(63)

Thus the ratio of $\alpha C^{2}$ to $R/2\kappa$ at $r = r_{g}$ is about 11.3 with the same value of $\alpha$.

In the inner region of galaxy, however, we need to show just the opposite, that the Einstein-Hilbert term dominates the Weyl term and therefore the geometry should be described by general relativity. We take the value of $\alpha$ in the inner region of galaxy to be equal to its value in the outer region of galaxy. We calculate $R$ and $C_{\mu \nu \rho \sigma } C^{\mu \nu \rho \sigma } = C^{2}$ at $r_{i} = 0.2 r_{ob}$ from the relations (61) to be

$$\begin{aligned} R (r_{i}) =& \frac{G}{c^{2}} \frac{M_{b}}{r_{ob}^{3}}e^{-r_{i}/r_{ob}} \, , \\ C^{2} (r_{i}) \approx & \frac{48 G^{2}}{c^{4}} \frac{M_{b}^{2}}{r_{ob}^{6}} e^{-r_{i}/r_{ob}} \left ( \frac{1}{6!} \frac{r_{i}}{r_{ob}} \right )^{2} \, . \end{aligned}$$

(64)

Thus the ratio of $R/2\kappa$ to $\alpha C^{2}$ at $r = r_{i}$ is about 33.2 with the same value of $\alpha$. A smaller value of $\alpha$, as expected from its running behavior, would give even more pronounced dominance of the Einstein-Hilbert term in the inner region.

Inside the Solar system the value of $\alpha$ was calculated some time ago in (Mannheim 2007) with a different approach and it was shown that the Solar system tests of gravity are not affected by the existence of the Weyl term. In (Mannheim 2007), it is found that $\alpha = 3.29 \times 10^{75}~\mbox{kg}\cdot \mbox{m}^{2} /\mbox{sec}$. This value is smaller than the value obtained for the required phenomenological behavior in the galaxy. Thus, unconventional running behavior of $\alpha$ as suggested in (Donoghue 2017) that might depend on some physical property of the system besides the energy is evidently seen here. It is imperative that more rigorous analysis should be done to understand the true nature of the coupling $\alpha$.

Appendix B: Interrelations of Bach tensor components

The form of the metric solution is given by (17) with $d\Omega _{k}^{2} \equiv \frac{1}{1-kx^{2}}dx^{2}+(1-kx^{2})dy^{2}$ corresponding to two dimensional hyperbola, torus and sphere geometries of the solid angle, with $k$ taking values −1, 0, 1, respectively.

Since Bach tensor has trace and divergence free properties, field equations are not independent. We first note that “xx” and “yy” components of Bach tensor are the same, $B^{x}_{x}=B^{y}_{y}$. Then the trace of equations (10) is given by

$$ B^{r}_{r} + B^{t}_{t} +2B^{x}_{x}=0. $$

(65)

Furthermore the divergence of the Bach tensor $\nabla _{\mu }{B^{\mu }_{\nu }}=0$ is calculated to be

$$ \left (\frac{d}{dr}+\frac{2+w}{r}\right )B^{r}_{r}-\frac{w}{r} B^{t}_{t}- \frac{2}{r} B^{x}_{x}=0. $$

(66)

Solving these two equations, we find that $B^{x}_{x}$ and $B^{t}_{t}$ are given in terms of $B^{r}_{r}$ as

$$\begin{aligned} B^{t}_{t} =& -\frac{1}{1-w}\left ( r\frac{d}{dr}+3+w\right )B^{r}_{r} \ , \end{aligned}$$

(67)

$$\begin{aligned} B^{x}_{x} =& \frac{1}{2(1-w)}\left ( r\frac{d}{dr}+2(1+w)\right )B^{r}_{r} \ . \end{aligned}$$

(68)

If a solution satisfies the equation $B^{r}_{r} = 0$, then all the field equations (10) are satisfied by that solution due to above relations.