General relativity versus dark matter for rotating galaxies

A very general class of axially symmetric metrics in general relativity (GR) that includes rotations is used to discuss the dynamics of rotationally supported galaxies. The exact vacuum solutions of the Einstein equations for this extended Weyl class of metrics allow us to rigorously deduce the following: (i) GR rotational velocity always exceeds the Newtonian velocity (thanks to Lenz's law in GR). (ii) A non-vanishing intrinsic angular momentum (J) for a galaxy demands the asymptotic constancy of the Weyl (vectorial) length parameter (a)-a behaviour identical to that found for the Kerr metric. (iii) Asymptotic constancy of the same parameter a also demands a plateau in the rotational velocity. Unlike the Kerr metric, the extended Weyl metric can and has been continued within the galaxy, and it has been shown under what conditions Gauss and Ampere laws emerge along with Ludwig's extended gravito-electromagnetism (GEM) theory with its attendant non-linear rate equations for the velocity field. Better estimates (than that from the Newtonian theory) for the escape velocity of the Sun have been presented.

Automated summary

This article discusses a general class of axially symmetric metrics in general relativity that includes rotations, which are used to study the dynamics of rotationally supported galaxies. The exact vacuum solutions of the Einstein equations for this extended Weyl class of metrics lead to the following rigorous deductions: (i) The rotational velocity in general relativity always exceeds the Newtonian velocity due to Lenz's law. (ii) A non-zero intrinsic angular momentum for a galaxy requires the asymptotic constancy of the Weyl length parameter, similar to the Kerr metric. (iii) The asymptotic constancy of the parameter also implies a plateau in the rotational velocity. Unlike the Kerr metric, the extended Weyl metric can be extended within the galaxy, and it has been shown that Gauss and Ampere laws emerge along with Ludwig's extended gravito-electromagnetism (GEM) theory and non-linear rate equations for the velocity field. More accurate estimates for the Sun's escape velocity than those obtained from Newtonian theory have been presented.