If time travels slower nearer gravity wells, why can't the galaxy rotation speeds being faster on the outer edges than the inner areas be explained by relativity? What necessitates dark matter?
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Never mind. Problem solved. Special Relativity has no effect at all on the gravitation of a rotating spiral galaxy, Lorentz contracted on the opposite rim of the galaxy or not. – danshawen Jun 17 '17 at 00:09
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The necessity of GR to solving a particular problem can be assessed by calculating $GM/Rc^2$. Here, $M$ is the mass involved in producing a gravitational field at some separation $R$.
The rule of thumb is that if $(GM/Rc^2)\ll 1$ (i.e. is close to zero) then GR effects (time dilation) can be neglected roughly at that order of precision.
So if we take an object in a galaxy orbiting at $R$ of 10 kpc with a mass interior to its orbit of say $10^{11} M_{\odot}$, then $(GM/Rc^2) \sim 5\times 10^{-7}$. The redshift implied by this is of order 150 m/s, which is small compared with galactic speeds.
An interesting comparison would be the orbit of Mercury, where very small GR (i.e. non-Newtonian) effects are noticed with extremely careful measurements over many years (many orbits). Here, $(GM/Rc^2) \sim 3\times 10^{-8}$. So GR effects certainly can be detected at this level, but whereas Mercury is nearby and can be observed over many orbits, galaxies are far away and orbital periods are hundreds of millions of years. And in any case if you wanted to estimate the mass of the Sun using the orbit of Mercury and Newtonian gravity, your answer would be correct to several decimal places.
ProfRob
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See Why isn't the center of the galaxy "younger" than the outer parts? and Galactic Rotation Speeds - Ehrenfest Paradox, Gravitational time dilation, Dark Matter - all of the above?. The time dilation effects are tiny and far too small to explain the observed rotation curves.
John Rennie
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