Seifert -Solutions of Einstein's equations give flat galactic rotation curves

[timmdeeg]

The Importance of Being Symmetric: Flat Rotation Curves from Exact Axisymmetric Static Vacuum Spacetimes

... Analyzing the low-velocity limitcorresponding to the Newtonian approximation of the Schwarzschild metric, we find an effective logarithmic potential. Thisyields flat rotation curves for test particles undergoing rotational motion within the spacetime described by the line elements,in contrast to Newtonian rotation curves. This analysis highlights how important the symmetry assumptions are for deriving general relativistic solutions.
One example of physical objects that are generally described in the static vacuum low-velocity limit (reducing to Newtonian gravity in the spherically symmetric case) and exhibit axial symmetry are disk galaxies. We show that symmetries and appropriate line elements that respect them are crucial to consider in such settings. In particular, the solutions presented here result in flatrotation curves without any need for dark matter. While these exact solutions are limited to static vacuum spacetimes, their application to physical galaxies relies on appropriate approximations. Nonetheless, they offer valuable insights into explanations for flat rotation curves in galaxies and their implications for dark matter.

Antonia Seifert is a master's student of Prof. Bartelmann, Heidelberg.

 

[gentzen]

What is the relation of these axisymmetric static solutions of Einstein's equations to BSM physics?

 

[timmdeeg]

I think you are right, "Special and General Relativity" would fit better, if supposed to be correct.

 

[ohwilleke]

It belongs in BSM. The prevailing understanding is that GR does not produce flat rotation curves.

 

[timmdeeg]

But prevailing understanding can be expanded, in this case by new solutions of the EFE.

I wonder if these have the potential to falsify the field self-interaction proposed by Deur, as Seifert states in the abstract "In particular, the solutions presented here result in flat rotation curves without any need for dark matter."

If true are there any reasons to still stick to Dark Matter?

 

[renormalize]

But prevailing understanding can be expanded, in this case by new solutions of the EFE.
If true are there any reasons to still stick to Dark Matter?

Evidence for Dark Matter goes well beyond just flat galaxy rotation curves. Its existence helps to explain, for example, the following:

• Cosmic microwave background angular power spectrum
• Velocity dispersions in elliptical galaxies
• Gravitational lensing observations
• Dynamics of the Bullet cluster
• Etcetera

(See https://en.wikipedia.org/wiki/Dark_matter)
Any purely gravitational explanation (whether MOND, novel solutions of General Relativity, or something else) of the galaxy rotation curves would have to be shown to also explain these other phenomena in order to banish Dark Matter.

 

[timmdeeg]

Ok, thanks, got it.

 

[ohwilleke]

Any purely gravitational explanation (whether MOND, novel solutions of General Relativity, or something else) of the galaxy rotation curves would have to be shown to also explain these other phenomena in order to banish Dark Matter.

Mostly, they do that. Often pretty trivially. The Bullet Cluster does rule out one specific class of modified gravity theories, but does not generically rule them out. Both Moffat's MOG theories and Deur's theories, for example, can explain the Bullet Cluster.

 

[renormalize]

Mostly, they do that. Often pretty trivially. The Bullet Cluster does rule out one specific class of modified gravity theories, but does not generically rule them out. Both Moffat's MOG theories and Deur's theories, for example, can explain the Bullet Cluster.

Is there a nice review article that summarizes the alternative gravitational explanations for the experimental evidence that motivates the dark matter hypothesis?

 

[timmdeeg]

A question to this part of the paper:

4.2 Importance of Nonlinearities

Apart from the unknown mass configuration in the singularities asa source of spacetime curvature, the nonlinear nature of general relativity also causes curvature to enhance itself, although constrainedby the imposed symmetry conditions. This suggests an analogy ofthe vacuum solution to the pure field case in Quantum Chromodynamics (QCD), as this theory is nonlinear as well. Investigating theLagrangian as done for the path integral in QCD, we find that for thegravitational field 𝜓𝜇𝜈 = 𝑔𝜇𝜈 − 𝜂𝜇𝜈, the general relativistic purefield Lagrangian can be expressed in the polynomial form (see e.g.,Zee 2013)

L =∑︁∞𝑛=0(16𝜋𝐺𝑀)𝑛2 𝜓𝑛(𝜕𝜓𝜕𝜓), (73)

where the 𝑛 = 1 term is of order √𝐺𝑀, which supports the findingof √︁𝜇𝐺𝑀 as the coefficient of the effective potential in Section 3.4.Furthermore, the nonlinearities in general relativity cause selfinteraction, which will be relevant beyond a certain mass scale. Forexample, Newtonian gravity is not able to explain Mercury’s perihelion shift, but the Schwarzschild solution is needed for this. Inthis paper, we have shown that the form of the metric and the trajectories of test particles are crucially influenced by the symmetryconditions imposed to derive them. This also affects the mass scalesat which general relativistic effects become relevant in systems withcylindrical symmetry, opposed to those with spherical symmetry. For example, galaxies are generally assumed not to be massive enough to exhibit general relativistic effects known for spherically symmetricsettings such as black holes. However, considering their axisymmetric configuration without spherical symmetry but differing lengthscales characterising the 𝜌- and 𝑧-directions, this assumption has to be re-evaluated and general relativistic effects can become relevant.

Could this be interpreted as a step towards Deur's field-selfinteraction?

It seems, there are two possibilities to describe flat galactic rotation curves:

1. a dark matter halo around galaxies, the present model
2. GR assuming cylindrical symmetry wihout dark matter

I am curious how these fit together.

 

[renormalize]

It seems, there are two possibilities to describe flat galactic rotation curves:
1. a dark matter halo around galaxies, the present model
2. GR assuming cylindrical symmetry wihout dark matter

Don't forget MOND and its variants.

 

[Vanadium 50]

I have a really, really hard time believing rotation curves are caused by some hitherto unknown GR effect. Dark Matter gravity is 5x luminous matter gravity. Newtonian gravity is (approximately) the T₀₀ term in the stress-energt tensor.

The largest other terms are down by one factor of β, which is around 1/1000. So any effect is ~5000x too small. "Non-linearity" simply means I can throw higher powers of β in, which doesn't help unless I have a leading coefficient of not 1/2, not 8π, but thousands.

As I said in another post, "numbers matter".

 

[gentzen]

I have a really, really hard time believing rotation curves are caused by some hitherto unknown GR effect. Dark Matter gravity is 5x luminous matter gravity. Newtonian gravity is (approximately) the T00 term in the stress-energt tensor.

A moderator in a German speaking quantum physics forum was also skeptical, and even ventured a concrete guess how unjustified conclusions could have arisen from correct formulas (http://www.quanten.de/forum/showthread.php5?p=105770#post105770):

In einem logarithmischen Potential hast du natürlich flache Rotationskurven. Das hast du aber auch bei Newton. Und ein (näherungsweise) logarithmisches Potential findest du in der Umgebung einer linienförmigen Masse.

Viel Rauch um Nichts. Das ganze ART-Gedöns könnte man sich sparen und mit Newton rechnen. Dann ist es aber so unkompliziert, dass man sich nicht einbilden kann, eine Lösung für DM gefunden zu haben.

Das ist zumindest meine Einschätzung.

 

[strangerep]

Sheesh! I really wish more people would actually study this type of paper before posting opinions.

OK, here goes...

Seifert solves the vacuum(!) Einstein field equations in cylindrical coordinates. She has in mind the case of "cylindrically distributed" matter. (She seems unaware that many disk-like matter distributions have already been solved analytically in Newtonian gravity.) In her words:

A prime example [...] is the visible matter in a galaxy. Assume an observer within the disk to which the main visible mass content appears to be located at the centre of the galaxy and the mass distributed across the surrounding disk is negligible compared to this central mass. If we do not consider any dark matter halo but only visible baryonic matter, the cylinder line element applies.

If the main mass is near the center and we neglect the rest, then we don't have a galaxy-like scenario. Rather, we have an approximately spherically symmetric situation. But she persists with cylindrical coordinates, obtaining a radially-logarithmic solution (and a weird $z$ solution that looks rather unphysical to me). Her $g_{00}$ is of the form $-C\, \ln(r/R)$, where $R$ and $C$ are constants of integration.

Now, if one simply applies Newtonian gravity to such a situation (with the potential $\Phi$ having only $r$-dependence), we have an simple Laplace equation $$0 ~=~ \Delta \Phi ~=~
\frac{1}{r} \; \partial_r \left( r\, \partial_r \Phi \right) ~,$$with an elementary solution of the form $\Phi = \kappa \ln r + C$ where $\kappa$ and $C$ are constants. If there is $z$ dependence, one gets extra terms (arising from the usual techniques in separation of variables), but the $z$ dependence looks unphysical to me. (I leave the details as an exercise for the reader.)

The key point is: if one blithely uses cylindrical coordinates naively, one can get a logarithmic potential, but there is no transition region between low and high acceleration regimes that are observed in real galactic rotation curves. You don't need full GR to see this -- it happens quite plainly for the same case with Newtonian gravity.

Moreover, Seifert uses the known Tully-Fisher relationship between orbital velocity and total mass to choose the constants in her solution. (See right-hand column on p5). But this is backwards. One is expected to derive a universal Tully-Fisher relationship, involving a universal acceleration scale -- not something which is fudged from an integration constant (hence could be different for every galaxy).

Sorry, but I hereby call "rubbish" on this paper.

 

[ohwilleke]

A moderator in a German speaking quantum physics forum was also skeptical, and even ventured a concrete guess how unjustified conclusions could have arisen from correct formulas (http://www.quanten.de/forum/showthread.php5?p=105770#post105770):

The English translation (per Google) of the quoted material in German is as follows:

Of course, in a logarithmic potential you have flat rotation curves. But you also have that with Newton. And you can find an (approximately) logarithmic potential in the vicinity of a linear mass.

Much ado about nothing. You could save yourself all the ART nonsense and calculate with Newton. But then it's so uncomplicated that you can't imagine that you've found a solution for DM.

At least that is my assessment.