Galactic gravitational field

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I was wondering whether someone has been able to find an exact solution of the Einstein field equations for the gravitational field of a galaxy (in which the galaxy could be modeled as a spinning disk for instance). Such a solution would clearly not be spherically symmetrical, though cylindrical symmetry could be assumed (as an approximation). Any references ?

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Chris Hillman
Hi, notknowing,

An exact solution modeling an isolated "rigidly rotating" disk of dust has been found, and this may turn out to be the most important exact solution since the Kerr solution. It is called the Neugebauer-Meinel dust and you can read all about it in various arXiv eprints http://arxiv.org/find/gr-qc/1/AND+AND+all:+dust+all:+Meinel+all:+Neugebauer/0/1/0/all/0/1. Note that related solutions have more recently been found (by similar methods) which represent nonrigidly rotating disks of dust.

You wrote "cylindrical symmetry could be assumed", but you mean "axial symmetry can be assumed". In Newtonian terms, cylindrical symmetry would mean that the equipotentials form nested cylinders (as for a line mass). Axial symmetry means that the equipotentials are axially symmetric (far from a Newtonian disk of dust, they should be slightly oblate spheroids).

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Hi, notknowing,

An exact solution modeling an isolated "rigidly rotating" disk of dust has been found, and this may turn out to be the most important exact solution since the Kerr solution. It is called the Neugebauer-Meinel dust and you can read all about it in various arXiv eprints http://arxiv.org/find/gr-qc/1/AND+AND+all:+dust+all:+Meinel+all:+Neugebauer/0/1/0/all/0/1. Note that related solutions have more recently been found (by similar methods) which represent nonrigidly rotating disks of dust.

You wrote "cylindrical symmetry could be assumed", but you mean "axial symmetry can be assumed". In Newtonian terms, cylindrical symmetry would mean that the equipotentials form nested cylinders (as for a line mass). Axial symmetry means that the equipotentials are axially symmetric (far from a Newtonian disk of dust, they should be slightly oblate spheroids).
Thanks for these interesting references. Indeed, I meant "axial symmetry".
The motivation for my question is related to the missing mass problem (in galaxies). Usually one compares the observed rotation curves (of stars) to the rotation curve based on a Newtonian potential and which reveals the well-known discrepancy. I was wondering whether there could be some extra GR-related terms (leading to an extra effective "force" towards the galaxy centre) which could mimic "extra matter". Has this been investigated in detail ? Would the corrections be so small that they can be ruled out to explain the galaxy rotation curves ? One could for instance use the derived metric to obtain the path of a small test body just outside of the disk.

Chris Hillman
Gtr needed for galactic gravitation? No.

The motivation for my question is related to the missing mass problem (in galaxies). Usually one compares the observed rotation curves (of stars) to the rotation curve based on a Newtonian potential and which reveals the well-known discrepancy. I was wondering whether there could be some extra GR-related terms (leading to an extra effective "force" towards the galaxy centre) which could mimic "extra matter". Has this been investigated in detail ? Would the corrections be so small that they can be ruled out to explain the galaxy rotation curves ?
I was too rushed to say this before, but at the averaged matter densities at galactic scales inferred from visible matter (shining stars), Newtonian gravitation should be quite adequate to determine the rotation curve. Thus, the observed rotation curves would be a huge problem for any theory of gravitation constructed to fit solar system scale data. Either gravitation behaves very differently at galactic scales (for various reasons, hardly anyone buys this as the explanation), or there's some mass-energy we don't yet know about associated with galaxies. The lensing observations discussed by Sean Carroll at his blog and in other places provides some very convincing if indirect evidence that there's nothing wrong with gtr (or other "close mimics") at galactic scales; rather, it seems that there is indeed some mass-energy that corresponds to something we haven't yet identified. Actually, quite a lot of mass-energy.

So the best short answer to your question of whether relativistic corrections to Newtonian gravitation have any relevance to the galactic rotation curve problem is probably "no".

As for a centrally directed "gravitational force" associated with a disk which is a bit stronger (in the equatorial plane) than the "gravitational force" for an equal mass spherical object, that's true in Newtonian gravitation, but doesn't help at all with the rotation curve problem. There is nothing mysterious about this; the equipotentials look like spheres at large radii but nearer the object they look more like the shape of the equal-mass surfaces in the matter distribution inside the object; in this case, they look more like a disk, i.e. they become flattened and look like nested oblate spheroids near the disk. These are "more closely packed" in the equatorial plane near the disk, so the force is a big greater. Something similar happens in gtr and other such theories.

One could for instance use the derived metric to obtain the path of a small test body just outside of the disk.
If you are back to the Neugebauer-Meinel exact solution modeling (in gtr) a rigidly rotating disk of dust, then, sure, in principle you can determine the geodesics of the vacuum exterior (everything outside a certain disk in the equatorial plane).

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I was too rushed to say this before, but at the averaged matter densities at galactic scales inferred from visible matter (shining stars), Newtonian gravitation should be quite adequate to determine the rotation curve. Thus, the observed rotation curves would be a huge problem for any theory of gravitation constructed to fit solar system scale data. Either gravitation behaves very differently at galactic scales (for various reasons, hardly anyone buys this as the explanation), or there's some mass-energy we don't yet know about associated with galaxies. The lensing observations discussed by Sean Carroll at his blog and in other places provides some very convincing if indirect evidence that there's nothing wrong with gtr (or other "close mimics") at galactic scales; rather, it seems that there is indeed some mass-energy that corresponds to something we haven't yet identified. Actually, quite a lot of mass-energy.

So the best short answer to your question of whether relativistic corrections to Newtonian gravitation have any relevance to the galactic rotation curve problem is probably "no".
It seems that the scientific community is divided in two distinct camps: those who believe in dark matter and those who do not (I belong to the latter). It would be quite interesting to know the actual proportions .
I have my own personal theory on this, part of which is published (see R. Van Nieuwenhove, Is the missing mass really missing ?, Astronomical and Astrophysical Transactions, 1996, Vol. 16, pp. 37-40).
The lensing observations you mentioned are indeed an indirect indication only and are not convincing to me (and probably also not convincing to a lot of other scientists).

In short, what I think is that one needs a quantum theory of gravitation to describe the strange things we see today (Pioneer anomaly, missing mass problem, ..). So, I think that the additional effects introduced by quantum gravity will not only have an impact on the things going on at a very small scale but also at cosmological scales (such as on the scale of a galaxy and larger). I could expand a lot more on this but since I am not allowed to introduce personal theories in this forum, I leave it hereby.

Chris Hillman