The anomalous rotation of galaxies

When Newton's or Einstein's gravitational theory is applied on galactic and cosmological scales, various anomalies are found: most famously, the orbital speed of stars far from the centre of a galaxy is roughly constant, where the theory predicts that it should fall off with radius r as 1/√r (which, furthermore, appears to happen for at least one galaxy); this is called the anomalous rotation of galaxies. This has traditionally been accounted for by postulating the existence of dark matter, which we are unable to identify other than by its effect, via gravity, at galactic scales.

According to Newton's theory, which approximates general relativity fairly well in moderate environments like the outer galaxy, the square of the orbital speed of a (tiny thing like a) star orbiting the galactic centre is proportional to the mass closer to the centre than it divided by its distance from the centre. The observed constancy of speed thus implies that the amount of mass within any given distance of the galactic centre grows in proportion to the distance from the centre. The visible matter fails to account for this, so dark matter is postulated to provide the missing mass. If this extra mass falls within the galactic plane – i.e. is in the same place as the matter – then its density drops of as the inverse of the distance from the galactic centre (it's about 0.44 kg/m/m at our distance from the centre, where the galactic disk's thickness is of order 10 k ly, so the implied density would add a few tonnes in the volume of the Earth; the sides of a cube containing the mass of our solar system would be about one light year long); if it's spherically distributed, instead, then its density drops off as the inverse of the square of distance (and the density at our distance from the galactic centre comes out at about a kilogramme in the volume of the Earth; the sides of a cube containing the mass of our solar system would be about 14 light years long).

Anomalies have also been seen in the motions of space probes: most famously, the Pioneer spacecraft are escaping the solar system minutely slower than predicted by theory; but various probes flying by Earth on their way to other parts of the solar system have also shown anomalies, apparently linked to their trajectories' relationship to the Earth's spin.

Revising gravitational theory

The expansion of the universe is also observed to be speeding up, which doesn't fit with the standard theory unless we postulate some dark energy to cause it. Various folk have preferred to postulate modifications to the theory of gravity in preference to these postulated but invisible forms of matter and energy. If our theory of gravitation (whose definitive constant, G = 66.72e-12 N.m.m/kg/kg, is the least precisely measured of the fundamental constants of physics) were just minutely wrong, even the tiniest of corrections to it might provide an alternative explanation for the apparent expansion of the universe and the anomalous rotation of galaxies. Any experiment to measure such a correction would be extremely difficult on the scale of terrestrial laboratories; the effects involved are only seen on the galactic and cosmological scale.

An early MOdified Newtonian Dynamics theory, MOND, postulated that gravitation changes form at low field strengths – field strength of order c.H/6, where H is Hubble's constant and c is the speed of light (the product c.H is an accelleration: 0.69 nm/s/s or about 70 millionths of a millionth of the gravitational field strength on the surface of the Earth). MOND has been built on by various others which modify the Hamiltonian of General Relativity (so as to ensure invariances missing in the original MOND); notably Bekenstein's TeVeS (tensor vector scalar) theory; and John Moffat's STVG (scalar tensor vector gravity). HongSheng Zhao works on refining such theories. Philip Mannheim has developed a theory of conformal gravity based on the Weyl tensor (which has ten degrees of freedom in 4-dimensional space).

Further study

It occurs to me that I should check what the Kerr-Newman solution would imply, for bodies orbiting a spinning black hole at various distances – does the black hole's spin affect the orbital speeds in a way similar to what we observe for our galaxy's stars ? It seems unlikely (given that orbital speeds match expectation in the inner galaxy, where I'd expect the effect of the central black hole's spin to be most noticeable) that this can explain away the orbital speed anomaly, but it should at least be checked, as should the kindred effects on outer parts of the galaxy of inner parts' angular momenta.

There are good reasons to believe there are several black holes near the galactic centre, not just the super-massinve central one. It would thus be interesting to work out what scale of gravitational waves they should be producing.

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