Perhaps to open the floor on this question, I have added a write-up for a talk I gave at the IVS2014 meeting in Shanghai, China, to be published as a conference proceeding, it is entitled "Space geodesy, VLBI and the fourth pillar of geodesy; spacetime curvature". Some of the issues are mentioned in the write-up, without much detail though.

For convenience I add the file right here.

Dear Ludwig, I cannot help in these real microscopew and telescopes! However I Can recommend some theoretical microscopes and telescopes, as they are used in Robinsonian Nonsdardard Analysis. See e.g Keisler's Caalculus book http://www.math.wisc.edu/~keisler/calc.html

Dear Ludwig,

to be honest, your work is beyond my scientific horizon, but thank you for sharing. I´m going to try to understand.

Regards

Dear Ludwig

Wow this is way above my head. It looks very interesting and at the same time challenging. Now I am trying to read a few things about it. You may have to educate us about these concepts. Very interesting.

Asada, H. (2011). Gravitational Microlensing in Modified Gravity Theories Inverse-Square Theorem. Progress of theoretical physics, 125(2), 403-410.

Webb, S. (2012). A problem of some gravity. In New Eyes on the Universe (pp. 79-104). Springer New York.

Ohanian, H. C., & Ruffini, R. (2013). Gravitation and spacetime. Cambridge University Press.

Ludwig. I only grasped an small part of your text: the geophysical distortions of images for earth observers with the most advanced technology. I have observed at our earth level that each time a body moves it creates a vacuum behind it that affects other bodies, For example cyclists use to stay behind the first competitors to "save" energy and to take advantage from created vacuum, they say. Propellers create cavitation in turbines due to vacuum that generates bubbles of water that in some moment colapse. My question is if something similar occurs in cosmos motions: as the sun moves, besides gravity change of focus, does it create a kind of "vacuum" that effects the planets and even the light that travels in the space around? Is this already studied in your field, or is it a foolish question that I figure from "intuition"? Excuse me for these speculations about your interesting and complex topic. Thanks, emilio

Dear @Ludwig, As a Mathematician, the first question I ask is that: Is your problem well-defined? This includes the form and norm used; thereafter, Does there exist a solution to your problem? and, Is this solution unique?

There are other associated issues; but first I have one remark: you have to define your directions; I did not get this from your paper.

Would you mind defining "space-time curvature"?.

Hi Emilio, yes in a sense there is such an effect, so it is not a foolish question and your intuition is correct. A rotating mass 'drags' spacetime with it, this is predicted within the General Theory of Relativity, it is called frame-dragging or the Lense-Thirring effect. It has been measured to some level of accuracy using satellite laser ranging on the LAGEOS satellites and by the Gravity Probe B experiment. It is a very small effect in the relatively weak field of the Earth,. e.g. the argument of perigee of the LAGEOS satellite will advance by ~1.8 m per year. As the orbits are nearly circular, the advance in the perigee point is difficult to measure, in addition the classical orbital effects on the perigee advance is much larger, and the frame dragging measurement is also adversely affected by mismodelling of the even zonal harmonic coefficients of global gravity models.

Dear @Abedallah, a simple definition of spacetime curvature is that within the definitions of the General Theory of Relativity, which is a geometric theory of gravity, a clock will run slower in curved spacetime (time is curved) and light will be bent around a massive object such as the Sun, due to spacetime curvature (space is curved). If spacetime is not curved, it is assumed to be asymptotically flat and becomes similar to Minkowski spacetime, which can basically be defined as having a single time dimension combined with ordinary dimensions of space (e.g. based on an orthogonal x, y, z reference frame) which results in a four-dimensional manifold representation of spacetime.

Typically this asymptotically flat space would be assumed in the case of the Solar System, where measurements of say the light from a star as it bends around the Moon or planets is not measureable even though it exists, but close to the Sun it becomes measureable as it is at the 1 arcsecond level. So time delay and apparent directional displacement becomes an issue close to the Sun. Similarly, if you were to measure the position of the star by line-of-sight over the galactic bulge, its apparent position would be displaced from its true position due to spacetime curvature and you may want to consider it. With the rapid increase in measurement accuracies, spacetime curvature positional displacements become a dynamic property of space, so that extragalactic reference points exhibit positional displacement at the microarcsecond level. The VLBI technique measurement accuracy is at the 50 microarcsecond level, and Gaia is expected to deliver 10-20 microarcsecond for magnitude

Dear Ludwig,

It is a difficult and complex question for me. I'll keep learning about from this thread. Curiosity : what is the definition of dark matter and what are its components, and why the dark like color is used for it?

Dear Fairouz, dark matter is called dark as it does not interact with the electromagnetic force, so it is 'dark'...it does not radiate or reflect electromagnetic energy, so cannot be detected directly. It's existence is inferred from the rotational behaviour of galaxies, which rotate too fast to remain intact, unless additional (but invisible) matter is present. What it is made up of is an open question, there are just speculations at this time.

Thanks Ludwig for your detailed and kind answers to all our questions. Another question: At the level of earth and moon, has the Lense-Thirring effect been measured; if so, what was concluded so far for earth-moon system? emilio

The first issue that has to be dealt with (I hope as soon as the next IAU) is the galactic aberration, which is already above the VLBI noise. The acceleration of the Solar System by the galaxy causes a global poloidal vector spherical harmonic proper motion to all extragalactic source positions, and is not included in the ICRF. (Yes, this means that the ICRF will have to go to an "of date" model, with a static part appropriate for that data and a time varying part. This will be common in what I call "astrometric cosmology" - global effects will be at least as significant as the source jitter caused by (say) microlensing. See this for a little more : http://arxiv.org/abs/1310.2723

Thanks Marshall, yes the secular aberration drift is now a reality, seem to remember you predicted it somewhere. Well, I suppose most reference frames need an epoch, so the ICRF will have to get one. This is the 'problem' with science, the more you delve into it, and the better your equipment and measurements become, 'noise' gives way to signals, ad infinitum...of course it is very exciting and interesting, but the level of complexity increases quite sharply. I suppose at some stage we will have to add the de Sitter precession of the solar system as we rotate around the Galaxy centre...

Emilio, there have been attempts to measure its effect around Earth, but not around the Moon, although this can be calculated. Satellite Laser Ranging (SLR) data were used to evaluate the frame dragging effect (using the LAGEOS satellites) and some results have been published. See for instance work done by Lucchesi and Peron, http://journals.aps.org/prd/abstract/10.1103/PhysRevD.89.082002

To question raised by Fairouz about dark matter, Ludwig gave the exact answer in the order of the gravitational theory, i.e. " dark matter does not radiate or reflect electromagnetic energy, so cannot be detected directly".

Nevertheless I woud want to give a further non-conventional answer in the order of the relativistic theory in which dark matter is composed also of "dark bodies" that instead radiate electromagnetic energy but it cannot be detected by observer. Anyway the term "dark" is always referred to the fact that measuring instruments detect electromagnetic energy for no positive real frequency.

Back some time ago, I calculated a number of effects in astrometric cosmology, and they started entering in at the level of 1/10 of the rotation aberration. These include

galactic epicycles and the acceleration towards the Great Attractor, both of which are in the 1 - 10% range. The cosmological acceleration implied by our 600 km/sec motion with respect to the CMB is ~2 x 10^-3 H_0, which is a little smaller, about 0.3% of the galactic rotation term. I would actually expect other cosmological terms to start entering in at about the same level (nano arc sec/ year), and if we can get down to pico arc sec/year, a host of cosmological effects will come into play.

I dealt in the past with linear astronomical aberration and the Marshall Eubanks studies on the rotation aberration are very interesting. If I understood well, his theoretical and experimental studies prove the cyclic galactic rotation aberration produces greater effects than the linear aberration due to the cosmological acceleration. I agree little instead with considering the CMB like a sort of absolute reference frame.

As far as I could understand the problem, dark matter is a complicating factor which is difficult to account for. But inversely, will it be possible (in principle) to study distribution of dark matter using methods under consideration?

Absolutely, positional variations will have to be explained somehow, so unless baryonic matter can explain it, dark matter can be inferred from the observed dynamics. Finally, if accuracy of measurements can be brought to the

I agree suitable experimental devices should make dark matter "visible". I am reliant on the new version of the Doppler effect.

A second more promising way, from my wiewpoint, could be the survey of unstable elementary particles coming from dark bodies. In fact elementary particles have more flexible physical properties than energy quanta and photons. Photons and energy quanta have constant physical speed and only the relativistic speed can change. Instead for elementary particles whether the physical speed or the relativistic speed can change. Besides unstable elementary particles have very interesting physical properties that might allow their survey. Specifically I am looking at unstable cosmic electrons.

Fairouz and others,

DM is observed by its gravitational effects in about 10 different systems on cosmic scales, spiral galaxies is just one. The most important information comes from gravitational lensing and from the CMB anisotropies. There is very little DM in our Galaxy, even less within the Solar System.

Daniele, unstable elementary particles do not play any role in cosmological contexts.

I don't think that in standard DM it is possible to conclude that there is "very little DM in our Galaxy." From http://arxiv.org/pdf/1406.4130v1.pdf (which Ludwig Combrinck alerted me to)

"If the halo is spherical, 45% of the radial force on the Sun is provided by baryons, and only 4.3% of the Galaxy’s mass is baryonic. If the halo is flattened, the baryons contribute even less strongly to the local radial force and to the Galaxy’s mass."

This paper is brand new, but previous such analyses for the Milky Way come up with broadly similar conclusions.

For the Solar System, that I grant you. I don't see how DM can be more than ~ 10^-5 of the local mass, and it could well be less.

Yes Marshall, your numbers ar right, I did not express myself clearly..The rotation curve needs no DM. And there is no radiation from the galactic plane that can be associated with DM. So the DM is far outside the visible Galaxy.in an extended halo.

The question of course is what would the geometric distribution of DM in the solar system be...as ~ 10^-5 is large enough to have some effect on asteroid and planetary orbits, the Moon's orbit as well. There are still unexplained deviations e.g. the Moon's anomalous eccentricity rate of ~ 1.6 x 10 ^-11/yr, which is larger than that explained by the eccentricity rate contribution from tides on the Earth and Moon.

Marshall

It is not true that "only 4.3% of the Galaxy’s mass is baryonic". That figure refers to the mass of the Universe, Rather one must look at the ratio of mass-t-luminisity, M/L All of L is due to baryonic and electromagnetic radiation, on the other hand M is due to baryons and DM. If M/L=1 there is essentially no DM.

Typically stellar populations exhibit values M/L = 1-10 in solar units, in the solar neighborhood M/L = 2.5- 7, in the Galactic disk M/L = 1.0 -1:7 , thus very little DM.

Matts, what would be a reasonable guess then for the solar system? M/L = 2.5 - 7 is quite significant for the 'solar neighbourhood'.

Ludwig

I don't know on what information one has come to 2.5 - 7. Better not guess, just note that the value is very uncertain.

Ludwig - two definitions of "local"

First, near our location in the gaalxy - after Moni Bidin et al claimed to find a lack of dark matter near the solar system, http://adsabs.harvard.edu/abs/2012ApJ...751...30M, Bovy and Tremaine shot it down and found 0.008 +/- 0.003 Msun/pc^3

http://arxiv.org/abs/1205.4033

Since Chabrier, 2001, found a total stellar density of 0.045 +- 0.003 Msun/pc^3, this implies that only about 18% of the mass in the galactic plane is dark. A brand new review is http://arxiv.org/abs/1404.1938 - I haven't read it, just seeing in it preparing this note, but if you look at his Figure 2 (a century of dark matter), you will see that the estimate of 0.008 is pretty standard.

Second - inside the Solar System. These constraints depend _crucially_ on the type of dark matter. The solar system should capture dark matter, both primordially (during its formation) and in three-body interactions since formation. The three body interactions would be dominated by "gravity de-assists" from Jupiter, and are actually not very efficient (a newly captured DM particle needs another interaction to leave its Jupiter crossing orbit, or it is likely to be subsequently ejected from the system by Jupiter). Now, truly non-interacting particles (or, rather, only gravitationally interacting particles) would be left in elliptical obits and form some sort of more or less spherical, more or less exponentially decaying, cloud around the Sun. The Cassini-Saturn data have been used to set very tight constraints on _this_ sort of DM distribution, with the best (IMO) being the Pitjev & Pitjeva results http://arxiv.org/abs/1306.5534

"The dark matter mass in the sphere within Saturn's orbit should be less than 1.7⋅10−10M⊙". (These constraints are really based on Kepler's law, as a DM cloud would make an exterior planet orbit faster than predicted by Kepler's law.) I actually think that this constraint is an argument against WIMPs, as I think a higher fraction of the solar system mass would be WIMPs if they formed the total DM. (I get that 10^-5 to 10^-8 of the Solar System should be primordially captured DM, which even in the lower limit is > the Kepler's law limit.)

There are two types of DM (at least) that could evade this tight constraint. One is axions, which should form large (>> 1 ly) Bose Einstein condensates and thus resist being captured gravitationally. I don't know if anyone has ever considered actually modeling axion captures in the solar system, but someone should. The other is the condensed DM (basically, anything that evades the mass/cross section limits on DM by having a large mass instead of a small cross-section). Condensed DM generally does interact with normal matter, and so primordially captured condensed DM should be found at the center of ordinary matter bodies. The Kepler's law limits such as by Pitjev & Pitjeva don't apply here as the mass of the DM would mostly just be included in the masses of the Sun and planets, which has to be estimated from the data. (There is no Solar System body where we could predict the bulk density, independently of direct gravity measurements, to anything like a fractional error of 10^-5.)

Thank you Ludwig and Matts for your explanations. I have learned a little bit from this subject really fascinating but so far from my competencies and research experimental tools in my country. However i would like share experiments and calculus from an ultrasonic structural noise signal analysis captured from a steel material; The calculus data seems to be homogenous may be with a certain linearity, but yet I couldn't put interpretation or correlation with established knowledge from physics/material science. May be could it be discussed here if useful for the subject

Thank you

Fairouz....that seems to be a bit off topic...but thank you for your interest!

Fairouz,

Your problem has nothing to do with this thread. Please open a new thread with your question.

Hi Marshall, thanks for the links, very interesting! Matts also has an interesting review 'Astrophysical and cosmological probes of dark matter'. The Read paper has an interesting conclusion "The above makes hunting for the gravitational effect of dark matter near the Sun rather like looking for the proverbial needle in the haystack", which is probably an understatement. I see from the paper by Pitjev & Pitjeva quoting Kenneth Nordvedt that laser observations of the Moon gives an upper limit for the possible acceleration in the presence of dark matter of 3 · 10 ^−14 cm s −2, which is quite small. I am working on LLR data processing software and will have to figure out how I can make reliable estimates.

Thank you Ludwig and Matts for your thoughts. Matts you are right i'll open a special thread on this question.

I'd like just clarify my previous post. My idea follows the behavior of noisy signals, and as any signal, noise could bring in itself deep components from the analyzed milieu. For example in the context of the above study, material micro structural analysis could be highlighted.

So as any matter chemistry is from earth and atmospheric contents and when calculus carried a matter shape deeper than crystallographic state of the studied material, i have thought that may be the matter could bring in it self a memory from its cosmic formation

In order to be clear I will call MP the main four theories, for me obsolete, of Modern Physics: Special Relativity, General Relativity, Quantum Mechanics, Standard Model.

I will call instead CP the three theories of Contemporary Physics: Theory of Reference Frames (TR), Deterministic Quantum Physics, Non-Standard Model.

In MP, DM has an exclusively gravitational nature and explanation. In CP instead a substantial part of DM has relativistic nature and explanation. It is manifest then we must search great cosmological distances for the relativistic DM.

Truth to tell I have little or no interest in the gravitational DM, generated by mathematical singularities, particularly in DM that is inside the solar system. For this last Fairouz's proposal could have some physical meaning, but I see no physical possibility for that existence.

I know I have a very different viewpoint from almost all other researchers, because others argue inside the MP while I argue inside the CP.

Matts, you say "unstable elementary particles do not play any role in cosmological contexts". I would add to your statement the small but very important word "as yet".

In actuality what you say is true only in part, in fact the behavior of cosmic muons is considered in MP a very important trial in support of both the time dilation and Special Relativity. I think it goes without saying that the behavior of cosmic muons, and the same their nature, is explaned otherwise inside the TR and CP.

Daniele,

Cosmic muons offer good tests of relativistic time dilatation, but they play no role in the formation of structures, of the Universe.

For studying the structure of the universe cosmic photons, neutrinos and in general energy quanta have been used largely and it has been possible to deduce informations above all as per the Doppler effect and shifts of frequency and wavelength. I think we could have further informations making use of the new Doppler effect that doesn't present singularities like instead it happens relative to the Doppler effect in SR.

I am sure then the experimental research is able to have greater informations on the structure of the universe studying and taking the physical behavior of unstable elementary particles: for the present I am thinking rather about leptons than baryons.

Cosmic muons and other particles are unstable leptonic particles that must come from some celestial body and consequently they are able to provide us with useful informations about those celestial bodies.

In TR and in the Non-Standard Model (NSM) unstable leptons have very interesting physical properties and can provide us with important suggestions about the existence of relativistic DM and "dark bodies".

Daniele

The muons entering our atmosphere are not of cosmological origin. Wikipedia has a good review. Look in particular on the subtitle Secondary cosmic rays.

There were muons in the hot plasma during the early radiation era, the last ones being produced in electron-positron collisions at a temperature of about 50 Mev, when the age of the Universe was less than a millisecond. Also those muons decayed with a mean life of 2.2 microseconds. The structures of the Universe date back to the time when the Universe became matter-dominated at about 60000 years.

Dear Matts, thank you for your useful clarifications that confirm what I think and wrote in my previous comments. You are right, cosmic muons quoted by me in the previous comment are secondary and derive from other charged primary cosmic particles. The astronomy in the twentieth century considered justly first of all neutral cosmic radiations (photons, neutrinos, etc..). Now we know the astronomy for the future will consider always more charged cosmic rays (leptonic or baryonic) that allow to have always more precise knowledges on the structure of the Universe. Thechnologies and methodologies

for this type of astronomy are today more complex but it represents certainly the way for the future astronomy.