The Dark Matter Problem General Relativistic Galactic Rotation Curves in a Friedman Dust Universe with Einstein's Lambda
| Scientific Paper | |
|---|---|
| Title | The Dark Matter Problem
General Relativistic Galactic Rotation Curves in a Friedman Dust Universe with Einstein's Lambda |
| Read in full | Link to paper |
| Author(s) | James G Gilson |
| Keywords | Cosmology, Dust Universe, Dark Energy, Dark matter |
| Published | 2010 |
| No. of pages | 17 |
Read the full paper here
Abstract
In this paper, the general relativistic replacement for the Newtonian inverse square law of gravitation is obtained from the Friedman Cosmology equations. This version of the inverse square law is shown to contain information about the amount of dark energy mass contained in a specific region through a mass term MΛ− dependent on Einstein's Lambda and, importantly for this paper, it also contains information about the amount of dark matter mass in the same region through a term MP+. This work derives from the Dust Universe Model which gives a complete cosmological description of the movement and evolution of the astrophysical space substratum which as usual is represented by a spatially uniform or constant mass density distribution at zero pressure. Thus definite spatial regions of the substratum can only be regarded as holding regions for un clumped mass, as primitive galaxies might be described. Consequently, to describe actual galaxies that have condensed from such a region, the more general solution of Einstien's Field eqtions involving the pressure term is needed to explain clumping and the resultant galactic form. The general relativist version of the inverse square law is written in a form applicable to the case of bound circular orbiting about a spherically symmetric central gravitational spatially distributed source force. Thus the behaviour of masses cycling within or outside the source region can be analysed. The formula for the galactic rotation curves for stars rotating within or outside the source region is obtained. A very simple galactic model is used consisting of just two components, the halo and the bulge with all visible orbiting stars. The conclusion is that the pressure term from general relativity and in the consequent Friedman equations is adequate to explain the constancy of the function of rotational velocity as a function of orbital distance from the centre of gravity starting at the massive core of the galaxy. A simple and parameter adaptable computer program using Mathematica has been constructed to display diagrams of galactic rotation curves. This program is available for downloading.
Overview
James G. Gilson, of the School of Mathematical Sciences at Queen Mary University of London, argues that the Dark Matter problem is not evidence for an unseen species of matter but a consequence of using the wrong inverse square law. His claim is that the correct general relativistic replacement for Newton's law — read directly off the Friedman acceleration equation — carries two extra mass terms that Newtonian gravity does not have: a negatively gravitating term set by Einstein's Λ, and a positively gravitating term arising from the pressure P in the stress-energy tensor. Ordinary galactic dynamics discards the second of these by assuming zero pressure. Restore it, Gilson says, and flat rotation curves follow without any missing matter.
The paper is an application of Gilson's earlier A Dust Universe Solution to the Dark Energy Problem and its sequels. In that model the cosmic substratum is a spatially uniform zero-pressure dust, so uniform regions can only hold unclumped mass; to describe an actual galaxy that has condensed out of such a region, the pressure term must be reinstated. The departure from the mainstream account is therefore conservative in one sense and radical in another: Gilson insists that no modification of Einstein's field equations, and no modification of Newton at large distances of the MOND kind, is needed — only an honest reading of what general relativity already says. As he puts it in the conclusions, "there seems to have been little clear recognition of the actual form and structure taken by the Einstein general relativity replacement for the Newton inverse square law."
The argument
The generalised inverse square law
Gilson starts from the Friedman acceleration equation with Λ,
r̈(t)/r(t) = Λc2/3 − (4πG/3)(ρ(t) + 3P(t)/c2).
The pressure adds to the mass density to give an effective density ρP = ρ + 3P/c2. Defining a dark energy density ρΛ† = Λc2/4πG (twice Einstein's usual ρΛ = Λc2/8πG, a doubling he takes over from his earlier work), a sphere volume VP = 4πrP3/3, masses MP+ = ρPVP and MP− = ρΛ†VP, and coupling constants G+ = +G, G− = −G, he rewrites the same equation as
r̈P(t) = −G−MP−(t)/rP2(t) − G+MP+/rP2(t).
This, he says, is "the general relativity generalisation for Newton's inverse square law of gravitation that is implied by Einstein's field equations with Λ." It generalises Newton in three respects: the radius may depend on time; negatively gravitating material enters through MP−; and pressure adds positively gravitating mass through MP+. The derivation assumes spatially uniform densities, but Gilson then invokes the Newtonian shell result to argue that the mass inside radius r may afterwards be redistributed arbitrarily provided its total and centre of mass are unchanged.
A two-component galaxy
The galactic model is deliberately minimal: two concentric spheres of constant density, one of radius ri representing the bulge and its visible orbiting stars, and a larger one of radius ri′ > ri representing the halo, each cut off sharply at its boundary. Setting radial velocity and transverse acceleration to zero for a circular orbit gives
v22(r) = (G/r)(M+(r,ri) + MP(r,ri′)) − GMΛ(r)/r,
which Gilson compares term by term with the classical v22(r) = GM(r)/r. Outside a mass distribution the classical formula gives v2 ∝ r−1/2; inside a uniform sphere it gives v2 ∝ r. The observed galactic curves are neither: they are approximately flat, lying above the falling case and below the rising one. That, in Gilson's framing, is exactly the dark matter problem — how much extra mass, and where.
Fixing the equation of state
Cosmological pressure is written through an equation of state P(r,ri′) = c2ρ(r,ri′)ω(r,ri′). Gilson notes that for the Λ material ω = −1, and then says of the positively gravitating halo material that "a first reasonable shot at the value for the ω above is the value +1", giving
MGR+ = V(r)(ρ(r,ri) + 3ρ(r,ri′)).
The pressure term thus contributes three times the halo density on top of the ordinary density. Choosing ρ(r,ri′) = ρ(r,ri) with ri′ = ri gives a halo-to-visible mass ratio of exactly 4, which Gilson matches to the observational consensus that dark matter exceeds ordinary matter by a factor of four or five, and to the dust universe model's present-epoch split of 75% dark energy to 25% normally gravitating matter. He adds candidly that "it is clearly easy to find the values for the quantities concerned... to find any value that might be determined from experiment to be the correct value." A Mathematica notebook, grcs.nb, generates the rotation curves; the paper reproduces three diagrams — halo alone, bulge plus orbiting stars, and the sum in v2 and in v — showing the summed curve flattening beyond the core.
Conclusion
Gilson traces the problem to Fritz Zwicky's cluster mass measurements and observes that the dominant response has been to propose modifying Newtonian gravity at large distances, which would entail modifying general relativity since Newton is its limiting case. He rejects this route: the required content was in Einstein's equations from the start. He identifies the halo with the pressure-induced mass MP+ and repeats a remark from his earlier dark energy paper — that pressures are not usually visible, and that this may be why dark energy is not seen, adding that a density equivalent to about five hydrogen atoms per cubic metre would not be visible anyway.
Assessment
The attractive feature of the approach is its economy of hypothesis. Gilson introduces no new particle, no new field, no new force law and no adjustable interpolation function; he takes the Friedman acceleration equation, an equation every cosmologist accepts, and rearranges it into a form that displays the pressure and Λ contributions as masses inside a sphere. Writing the result as an inverse square law with two coupling constants G+ and G− is a genuinely clarifying piece of bookkeeping, and the paper is unusually honest about the crudeness of its own galactic model — Gilson says plainly that the sharp-edged uniform spheres are unrealistic, that real visible distributions tail off indefinitely and are often spiral, and that his curves therefore carry unphysical cusps.
The difficulties are structural rather than arithmetical. The decisive step is not derived but chosen: ω = +1 is offered as "a first reasonable shot", and no physical argument is given for why the halo material should have a stiff equation of state with P = ρc2 — the causal limit, matter as stiff as anything can be. The factor 3ρ and hence the ratio 4 follow entirely from that choice, and Gilson himself concedes that ri′ is a free parameter tunable to any desired ratio. The agreement with the observed four-to-one figure is therefore a fit, not a prediction, and the paper does not claim otherwise as clearly as it might.
More seriously, the Friedman equations from which the law is read off describe a homogeneous and isotropic universe. Applying them to a galaxy — a bound, virialised, decoupled object that is by construction not part of the expanding substratum — requires justification that the paper does not supply; the shell-theorem argument used to relax the uniformity assumption is a Newtonian result invoked inside a general relativistic derivation. Nor does a stiff-fluid halo sit easily with what is actually observed of dark matter, which is inferred to be dissipationless and pressureless precisely because it does not settle into a disc as ordinary gas does. And the model is confronted only with the qualitative fact of flatness. It is not tested against the Tully-Fisher relation, against the varying halo-to-light ratios across galaxy types, or against the bullet-cluster-type observations in which the Gravitational Lensing mass is spatially offset from the visible baryons — an offset that a pressure term tied to the local matter density cannot easily produce, since Gilson's MP is proportional to a density that sits where the matter sits.
Finally the doubled dark energy density ρΛ† = 2ρΛ is carried in from earlier papers without re-derivation here, so a reader who does not already accept the dust universe model has no way to check it from this text alone. The paper is best read as a proposal about where to look — at the pressure term rather than at new matter — rather than as an established solution.