Rotation curves of spiral galaxies: A general-relativistic model

Spiral galaxies are considered as static, spherically symmetric Dark Matter Configurations (DMC)in which non-zero rest-mass particles (NZRPs) move along appropriate trajectories. Using general relativity (GR), we show that a mass of dark matter about 127 - 212 $\times 10^{11} M_{\odot}$ is required inside the sphere of size $a \sim 10$ kpc for agreement with the observed typical orbital velocity ranging from 150 to 250 km sec$^{-1}$. In our model, it is possible to obtain flat, slightly rising, and even declining rotation curves. The model self-evidently explains the reason why most of the spirals show almost flat or slightly rising rotation curves and very few show declining curves. The NZRPs follow trajectories, and are finally trapped in circles of minimum radii $\sim 0.03a - 0.005a$ with rotation velocities reaching $\sim 6544-8445$ km sec $^{-1}$. The velocity values are consistent with observations of central regions of various galaxies. This scenario may also lead to conditions suitable for evolution of relativistic supermassive stars and supermassive black holes at the centres of various galaxies. Based on this study, we obtain the density parameter $\Omega \sim 0.127 - 0.212$, leading to an open model of the Universe with an age of $\sim 16.8 - 17.6$ Gyr, which is significantly higher than the presently estimated age of globular clusters $\sim 13 - 15$ Gyr.