James

Just to give you a flavour, a Sept AGN paper abstract is here, discussing the high density jets of re-ionized matter. And one thing I forgot the Milky Way certainly DOES 'still have' and AGN. (iro Sagattarius A) Not (back) up to any great speed yet, so the 14 hypervelocity stars it has spat out recently are still in one piece, but there's also plenty of ions and gas.

Some are certainly worth paying for. The 'bipolar structures' referred in this one is a bit complex but can include 'kinetically decoupled' haloes and cores (rotating the other axis) which the recycling model explains.:

Claude-Andr茅 Faucher-Gigu猫re, Eliot Quataert The physics of galactic winds driven by active galactic nuclei MNRAS Volume 425 Issue 1, pages 605-622, 1 September 2012 http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2966.2012.21512.x/abstract

Active galactic nuclei (AGN) drive fast winds in the interstellar medium of their host galaxies. It is commonly assumed that the high ambient densities and intense radiation fields in galactic nuclei imply short cooling times, thus making the outflows momentum conserving. We show that cooling of high-velocity shocked winds in AGN is in fact inefficient in a wide range of circumstances, including conditions relevant to ultraluminous infrared galaxies (ULIRGs), resulting in energy-conserving outflows. We further show that fast energy-conserving outflows can tolerate a large amount of mixing with cooler gas before radiative losses become important. For winds with initial velocity聽vin聽≳ 10 000 km s−1, as observed in ultraviolet and X-ray absorption, the shocked wind develops a two-temperature structure. While most of the thermal pressure support is provided by the protons, the cooling processes operate directly only on the electrons. This significantly slows down inverse Compton cooling, while free-free cooling is negligible. Slower winds with聽vin聽∼ 1000 km s−1, such as may be driven by radiation pressure on dust, can also experience energy-conserving phases but under more restrictive conditions. During the energy-conserving phase, the momentum flux of an outflow is boosted by a factor ∼vin/2vs聽by work done by the hot post-shock gas, where聽vs聽is the velocity of the swept-up material. Energy-conserving outflows driven by fast AGN winds (vin聽∼ 0.1c) may therefore explain the momentum fluxes聽聽of galaxy-scale outflows recently measured in luminous quasars and ULIRGs. Shocked wind bubbles expanding normal to galactic discs may also explain the large-scale bipolar structures observed in some systems, including around the Galactic Centre, and can produce significant radio, X-ray and γ-ray emission. The analytic solutions presented here will inform implementations of AGN feedback in numerical simulations, which typically do not include all the important physics.

There are scores of recent ones more relevant to halo density and morphology.

Peter

Gurcharn

Thanks for your kind words. I'm no conspiracy theorist but you may have a point. There is room for 'old pals act' back scratching. Yet I also see some downsides, in limiting fair evaluation of many essays.

The big problem I think is finding time to read and absorb them. Mine for instance will not expose it's riches to a cursory skim over. But as I said on your blog I'm also very impressed with yours and it will certainly get a good rating from me. I think you may also like Regazas and Kingley Nixey's if you've not read it yet.

Best of luck.

Peter,

Thanks again for your reply, which was to some extent helpful in identifying the source of confusion. However, I must point out that you've not addressed the specific reference to the image caption referring to "Visible 'Dark Matter'. Galaxy Cluster CL 0024+17." As demonstrated by the NASA & Wikipedia references, this is definitely a misinterpretation.

Your last response stated:

"Just to give you a flavour, a Sept AGN paper abstract is here, discussing the high density jets of re-ionized matter. And one thing I forgot the Milky Way certainly DOES 'still have' and AGN. (iro Sagattarius A) Not (back) up to any great speed yet, so the 14 hypervelocity stars it has spat out recently are still in one piece, but there's also plenty of ions and gas."

The Milky Way certainly does NOT exhibit the relativistic jets characteristic of an AGN. As I understand, the enormous gamma-ray emitting polar bubbles of plasma are thought to be the remnants of past galaxy nucleus activity. Please see the NASA press release, NASA'S Fermi Telescope Discovers Giant Structure In Our Galaxy.

Your Claude-André Faucher-Giguère, Eliot Quataert reference abstract makes no mention of dark matter. I could not find a free version of that paper, however, I think much more to the point, an FQXi essay entry attributes the observed rotational velocity of spiral galaxies to those very same AGN outflows producing spiral arms moving to the galaxy periphery. This proposal also does not require dark matter to explain the observed rotational characteristics of spiral galaxies. Please see A New Model Without Dark Matter for the Rotation of Spiral Galaxies: The Connections Among Shape, Kinematics and Evolution by Mario Everaldo de Souza.

Still trying to understand the source of confusion regarding visible dark matter, I could not find any accessable Claude-André Faucher-Giguère papers in arXiv, but I did find some with contributor Eliot Quataert. Specifically, I found a very curious report, On the Structure of Hot Gas in Halos: Implications for the Lx-Tx Relation & Missing Baryons. As the title implies this model evaluation of, as I understand, hot gases in the intraclustr medium of galaxy clusters is also applied to the composition of galaxy dark matter halos. It also suggests that galaxies were formed from primordial dark matter.

What I think may be occurring is that researchers studying the characteristics of hot gaseous intracluster media (ICM) (perhaps produced by AGN outflows) are applying their results to supposedly galactic dark matter halos thought to have been accreted from the hot gas/presumptive dark matter ICM, depleted of baryonic gases.

I can understand how confusing this is from the varying perspective of those studying hot gases in the ICM and the presumedly colocated galaxy cluster dark matter, and those that specified the presence of dark matter to explain the non-Keplerian rotational characteristics of spiral galaxies.

Keep in mind that the visible ICM matter is thought to be twice as massive as the visible galactic matter of all galaxies within a galaxy cluster. It's difficult to understand how the more massive visible ICM could have been ejected from the relativistic jets of less massive AGN.

Moreover, galactic dark matter halos are thought to be about 10x the visible mass of spiral galaxies - this mass seeming to be necessary. It's also very difficult to believe that AGNs produce much more massive dark matter galactic halos.

At any rate, I have yet to see any reference to "visible 'dark matter'" in galactic halos. I don't think there's any support for this interpretation in any of the literature.

I hope this helps, Jim

Jim

Our 'dark matter' Halo is 10^12 solar masses, far more than visible baryonic matter. In my caption I put 'dark matter' in inverted commas as I'm clearly saying I'm not proposing the common interpretation. That density is consistent with kinetic effects and CMB data as well as gravity, but I also accept your own proposal as worthy of consideration.

Mine is simply one of the alternative valid views; That we live in a 'dark energy/ matter universe' where visibly baryonic matter in only

    • [deleted]

    Dear Peter,

    Thanks for your valuable support.

    In my opinion, at this stage of the contest we should not get bogged down in minor points of differences in our viewpoints. We should pickup about 40 to 50 essays that are broadly to our liking, generally represent our viewpoints in different styles and are novel contributions in our opinion. We should rate them 'high' to see them in the list of finalists.

    I definitely want to see your essay in the list of finalists!

    Best of Luck

    G S Sandhu

    Peter,

    Thanks very much for your consideration. I must say, though that the image caption "Visible 'dark matter'" certainly does not even clearly convey your assertion that it represents (as stated in the Figure 1 caption) "... a diffuse plasma medium of ions, CO and molecular gas." Neither interpretation is correct, as the 'visible' 'clouds' actually illustrates a gravity map inferred by the researchers. No plasma medium has been detected within any inferred galactic dark matter halo - no 'clouds' or anything else is visible there.

    Sorry to be so obstinate - I do see value in your emphasis on the importance of plasma media in the universe. I'm just compelled to clarify points of confusion.

    Best wishes - you're doing very well in the competition!

    • [deleted]

    James

    Thanks. The 'map' is overlaid on the image on the left, also showing REAL X ray results. The caption is, correctly, on the visible band HST image. That was precisely my point; That 'dark matter' need NOT be some mysterious unknown substance. The Ostriker quote gives the 'Concordance' view which I vary from. That fact, or that it's a different modification to yours, cannot make it 'incorrect.'

    Your following comment "No plasma medium has been detected within any inferred galactic dark matter halo - no 'clouds' or anything else is visible there." is indeed incorrect.

    Firstly you have two propositions; 1. "No plasma medium has been detected", which is quite wrong. 'detection' covers all forms of detection not just the limited visible wavelengths. It is detected everywhere, and generally at increasing densities nearer matter. Its normally considered as 'electron density'. Gravitation is only one indicator of a number used. In fact I disagree with gravitational mass estimation level from lensing as it ignores the kinetic diffraction well known from optics etc. (and kSZ).

    2.; "no 'clouds' or anything else is visible there." Agreed, plasma scattering is not detectable in the optical range, but is otherwise. CO and molecular gas can only come from bound ions (from pair production). Wherever we find gas being formed we find the ions that form it (less in long established undisturbed gas clouds). Also check the sums; the gravitational effects can and do include the gas itself (or the plasma density is even greater!)

    I'm sorry if my thesis appears to disagrees with yours, but that does not make mine wrong. I make mine very clear in the text, there are no ghost-like dark characters, just Eddie and the electrons, and Penny and the protons, who might just couple one day!

    I've now read Evaraldo's essay as you suggested, and commented. It's original and I can see why you like it, as it offers your idea support. But beware of being unscientific. I suggest the best science is about 'falsifying' a model, so zeroing in and honestly analysing the conflicting, not supporting evidence. Evaraldo's does seem to conflict with much evidence, i.e. he has ring galaxies expanding not accreting! (just look at the 'spokes of the Cartwheel galaxy) so it should be an interesting process.

    You'll have seen from my papers (I hope) that I do however agree with his note ref formation from Quasars. Do comment on those.

    Best wishes.

    Peter,

    "The 'map' is overlaid on the image on the left..."

    This is incorrect - the gravity map overlay I've repeated referred to is included only in the image on the right. The process using weak gravitational lensing to generate the gravity map used to produce the artificial dark matter 'cloud' overlay image is clearly and undeniably explained in the NASA announcement: Hubble Finds Ghostly Ring of Dark Matter. Both the overlaid and unmodified images are included in that document.

    If you will refer to the above NASA document I'll be happy to discuss any issues you might have.

    • [deleted]

    James,

    You're right. If we considered 'visible' as only the optical em range the right hand image is also not strictly 'visible', same as as the left one. However, in astronomy we're almost never dealing with the optical range (which is a tiny part of observation), so 'visible' is shorthand for visible 'at' 'in' or 'by'...' i.e. it's synonymous with 'detectable'. It is a very anthropocentric view to think the 'visibly' range is more important than any other. Astronomy is by it's very nature NOT anthropocentric! The ring shown was 'detectable' by the standard lensing based mass estimation techniques that have been used for decades. The technique is not that complex, based largely on radius of curvature. Remember, our eyes use lensing, so even our visible image of the universe depends on it! Present astronomy is equivalent to 'bionic eyes', including taking lensed images and inferring both original image and lens characteristics.

    The matter is indeed then rendered 'visible', and that sense it is better than just 'optical range' detection, but, I agree it is also open to as much or even more misinterpretation as other observation (certainly including optical!

    I also agree that estimation of galaxy and cluster mass from lensing is faulty. It has always been to high, and I consider 'caustics' and 'gravity wells' are used too liberally as 'patches' to cover the cracks. The 'Curved space-time' basis is in any case still unexplained (however many claim it is!).

    But consider this carefully; It is not just lensing that tells up of the IGM, and whether from 'space-time curved by gravity' or from simple diffraction by the (mainly) free electrons found, (with protons, positrons, CO and basic bound molecular gas) the lensing exists. (Look carefully at the optical image and you can see the many curved 'smears' around the centre which they used).

    Now we know plasma DOES refract light, and very effectively, but has a refractive index very close to 1. That's why it normally isn't detectable optically. That does not mean it is not visible by other means (including first hand by Voyager 2, and other closer probes including Cluster).

    Now I say the plasma etc. we find out there moves and diffracts just like plasma anywhere else, so acts like a giant lens, and propose that the fact that lensing can be quantitatively explained in this way, along with unexplained effects such as kSZ, Faraday Rotation, ellipticity, aberration, Chirality etc, is no accident. Nothing we have found contradicts this hypothesis, only current interpretation does so.

    You have a different view, which I agree should be studied considered equally with all others, as the de Souza proposal. So I should then ask questions, such as do you agree there is diffuse plasma CO and molecular gas out there, and why that might not be expected to behave with the plasma we find locally. The question is one I have asked proponents of the mainstream interpretation, with, as yet, no credible answer.

    I hope that fully answers your points.

    Best wishes

    Peter

      • [deleted]

      Peter,

      Thanks for you consideration and understanding. However, I object to the use of the term 'visible' in regard to the dark matter illustrated in your Fig. 1 image of galaxy cluster Cl 0024+17. That image has effectively been 'photoshopped' by NASA to illustrate the presence of dark matter that has been inferred through a complex series of analyses including the statistical analysis of minute optical distortions imparted to thousands of background galaxies - a very complex process subject to error. Again, the process is described in Hubble Finds Ghostly Ring of Dark Matter. As you say, I don't think the optical effects produced by the ICM is adequately considered in galaxy cluster weak gravitational lensing analyses used to infer dark matter.

      No, this is not an issue of semantics, the term 'inferred' is applied to describe how the presence of dark matter is hypothetically determined. The term 'visible' indicates that an object can be seen through the direct detection of photons, usually but not necessarily those within the visible spectrum. Many objects, including the intracluster medium (ICM) of galaxy clusters, are visible using X-ray telescopes, for example. The 'clouds' illustrating the inferred presence of dark matter in the referenced NASA document do not represent any photons. NASA also did not intend those 'clouds' to illustrate the presence of any enormous clouds of plasma encircling the galaxy cluster. As such, I insist that your Fig. 1 image caption is misleading, as the statement: "The 'clouds' represent a diffuse plasma medium..." is completely false.

      In the case of galaxy clusters, I agree with you that identified weak gravitational lensing results from the combined effects of ICM diffusion and the gravitational curvature of spacetime distorted by the combined mass of the cluster's galaxies and the more massive ICM.

      Where your conception differs from both my essay and Mario Everaldo de Souza's A New Model Without Dark Matter for the Rotation of Spiral Galaxies: The Connections Among Shape, Kinematics and Evolution is that you seem accept the analyses that compensatory mass provided by some form of undetected matter is necessary to account for identified gravitational effects. My essay and de Souza's (along with several galactic models referenced in my essay that also do not require dark matter to account for observed spiral galaxy rotation) are complementary: I argue that galactic dark matter is erroneously inferred, while de Souza's model explains spiral galaxy evolution without dark matter.

      Regarding "So I should then ask questions, such as do you agree there is diffuse plasma CO and molecular gas out there, and why that might not be expected to behave with the plasma we find locally:"

      The presence of massive, hot plasma comprising the ICM of galaxy clusters is confirmed by detection of x-ray emissions. I don't think there is any compelling evidence that enormous massive halos of plasma envelope galaxies, contributing to their discrete rotational characteristics. I think that only proper representations of ordinary galactic mass and its gravitational effects is necessary to account for galactic rotational characteristics.

      At this point in our discussion I cannot expect you to acknowledge the inappropriateness of your "Visible 'dark matter'" caption. I also understand that I'm being intractable in my arguing this point, but I assure you there is nothing personal intended.

      Best wishes, Jim

      Jim

      I agree for non astronomer readers the word 'detected' would have been better here. Working in 3 sciences I have to regularly switch 'languages' subconsciously. Discussing an astronomical image with my 'FRAS' hat on, the term 'visible' is appropriate, but I agree, human eyes nor the HST can 'see' it, only it's effects. You'll recall I don't subscribe to long range 'photons' but your point is also valid for waves. However, the detection certainly isn't 'completely false', just based on as questionable an interpretation as most other theories!

      Where I think you are mislead, affecting your view, is that what is termed 'weak lensing' does NOT mean and is not 'weak' in the way you suggest. All characteristics, including the shape (a disc to a thin arc), deflection (thousands of parsecs), and temporal delta, are very sold and substantial. Did you know that the delay to light from lensing can be over three years!! (One side of a lensing body to another, confirmed by precise spectroscopic pattern matching emissions). I'm saying you certainly WILL find compelling evidence if you look harder in the right places. It is a crime that the best places, the PR journal pages, are 'pay per view'. But you can't refuse to pay then refuse to recognise the existence of the compelling evidence there! (Also; have you read the links I gave you yet. It seems not). I've campaigned for better access; http://independent.academia.edu/JacksonPeter/Papers/1920871/SUBJUGATION_OF_SCEPTICISM_IN_SCIENCE and it's slowly coming, at least in the UK.

      What is far more important is the main thrust of my essay, that there is NO OTHER CURVATURE OF SPACE TIME than refraction. The matter doing the refracting itself has 'gravitational potential' so the whole thing can be resolved with no mystery or strange non-detectable dark matter. Did you look a the Rick Kingsley-Nixey essay Fig 2? you're arguing over relative trivia while missing the big picture!

      Have a look, and lets get back to science.

      Best wishes

      Peter

      • [deleted]

      I am thinking to the detection idea, or collapse in quantum mechanics, reading your Act 1-Scene 1.

      I am thinking on the double slit experiment: the screen is a quantum macroscopic object; so is the photon that make interference, or the multiple screen not-correlated electron (and proton) to make quantum effect? Is it possible to distinguish the two effects?

      In general a measure is a collapse of a quantum function using a macroscopic object, this is the reason of my problem.

      I think that the Act 1-Scene 2 is very interesting: what happen in a half transparent(index n)-half vacuum tube in movement with velocity v, with two mirror that reflect a photon? What is the law n(v) in the rest system, and in the movement system? What happen for a neutrino in the same system? If there exist an hypothetical transparent observer (like very intelligent jellyfish), then the light velocity is the maximum velocity in the medium (for example glycerol)? The neutrino is more quick!

      I must make some calculus, but your article give me some suggestions.

      Amusing event: the same post has been write before (for error) in the Matthew Peter Jackson blog!

      The netiquette require the erasing of this post.

      Saluti

      Domenico

      Peter,

      Thinking more about the additional effects contributing to galaxy cluster lensing, as you mentioned, I recalled seeing the Nature News article back in June, Galaxy clusters caught in motion, clearly describing the recent identification of the kSZ effect. That had been my first exposure to either the tSZ of kSZ effect.

      Interestingly, I commented back then, speculating that the SZ effects should apply not only to CMB photons but all others as well, likely contributing to weak lensing effects produced by galaxy clusters that are now attributed only to gravitational effects. I further speculated that the result may be the overestimation of the total galaxy cluster mass necessary to produce the identified weak lensing effects. This then would at a minimum result in the overestimation of dark matter present within galaxy clusters.

      My assertion is that the hypothesized enormous amounts of galactic dark matter have been misconceived, and that huge galactic DM halos do not actually exist.

      That would not necessarily preclude the existence of dark matter to explain observed galaxy cluster lensing, or its inference (however that's accomplished) as the structural backbone of the 'cosmic web'.

      I strongly suspect that, at large scales, no enormous dark matter halos envelop galaxies. I'm also beginning to suspect that, at very large scales, much of the effects attributed to dark matter in and among galaxy clusters may be the product of ICM and amorphous plasmas, as I think you suggest.

      Best wishes on your work (although I suggest you focus on galaxy clusters & above :-) Jim

      Domenico,

      The mirror problem resolves with c in the medium frame both on approach and after reflection, because the 'transition zone' must of course work both ways. It is equivalent to a 'fluid dynamic coupling' of inertial frames. At the boundary there are electrons at rest in BOTH media frames, with turbulence between, but all re-emitting at their own c. This applies to ALL media. Only the extinction distance changes (with 'birefringence' apparent during the change).

      I think this has proved too difficult dynamic visualisation of most readers, so introducing theoretical speculative neutrino's would just confuse more. We need to assimilate the concept that there is BOTH real and APPARENT speed, so apparent superluminal speed is allowed with no violation of c.

      At the (twin-slit) back screen, there is no co-motion, so the (probably part re-blended) arriving 'photon' or wave energy charges the particles, and as each reaches threshold energy it re-emits, forming the small 'dot' pattern. The intensity of combined interacting waves from the 'slit edge' emissions changes across the screen according to Huygens construction to form the interference pattern. This can simply be shown in simple experiments I've done myself. By moving the back screen position during the experiment. A 3D interference picture can then be built up in the space behind the slits. You may have read this last year; http://vixra.org/abs/1007.0022

      Does this accord with your thoughts? What it does seem to do is unify all physics and derive curved space time from QM by changing the optical axis of re-emissions ('refraction'). Is that emerging for you, or can you see any shortcomings?

      Also the little maths I give has been challenged. Do you have a view on that?

      Many thanks, and Best wishes

      Peter

        • [deleted]

        Peter,

        I recognize that you are well prepared to defend your view. I have read about it over and over:

        "I think this has proved too difficult dynamic visualisation of most readers, so introducing theoretical speculative neutrino's would just confuse more. We need to assimilate the concept that there is BOTH real and APPARENT speed, so apparent superluminal speed is allowed with no violation of c."

        I don't think that I have having a problem visualizing it. But there remains something that I have asked in past and feel uncertain about of your answer:

        Does relativity's length contraction consist only of remote appearances due to effects which occur upon the photons, doing the reporting, during their travels including their arrivals? In other words, does the original object suffer length contraction as a real local physical effect before the reporting photons begin their travels and undergo their own changes? Is length contraction only a remote visual effect?

        Time and talk may have blurred my memory about your position. But, I do find myself reading your messages and today at least am uncertain about your answer. Thank you.

        James

        James

        Lengths contract by an intuitive Doppler shift process due to the non zero distance between components of matter. It does NOT then apply at all to 'idealised rigid bodies', either real or apparently. Luckily there is no such thing as a completely rigid body, but there are degrees of compressibility.

        The non-zero time of interaction during media co-motion is the key. Imagine a compressible body on collision course with a medium. When they interact the spaces between the particles close up = length contraction. Now back to SR; Say a light pulse or string of photons lasts 1/10th sec (or is 30,000km 'long'). If it enters a co-moving medium of n=~1 approaching at v= 3,000km/sec it will compress, or 'contract' by 10%, agreed? As the propagation speed is still ~c, to an observer at rest in the new medium the length of TIME of the pulse in his frame will also have reduced by 10%.

        This is purely a Doppler effect, but on wavelength lambda. Frequency is only a derivative of lambda and speed, so follows inversely due to conserved c in all frames. (I identify the quantum mechanism = all re-emission at c).

        Now a different case; An observer staying at rest in the first frame watching two light pulses distance D apart enter the other frame (the co-moving medium). This is again intuitive. He will see the same thing; the distance between the pulses contracted. No tricks, fully intuitive and real, except of the course the SPEED of the pulses in the other medium is then only apparent, not real.

        Finally the case of the light reaching an observer. Consider the observers lens a 'medium', which it is, and this is then precisely the same as the first case. Lambda changes subject to observer medium motion. So all real physical changes.

        This simple reality implements CSL via the quantum mechanism and axiom of absorption (at any closing speed) but re-emission at local c. Ergo, physics is unified with causal Local Reality, and neither QM or SR have any paradoxes.

        I'm a little dismayed so few seem to be able to assimilate that from the essential components in the essay . Was there anything above exceptionally difficult to understand, or that you're not comfortable with?

        If so do point them out, if not, any ideas how it can better be communicated?

        Best wishes

        Peter

          • [deleted]

          Peter,

          Ok, I think what I have been missing in my question is that I am not asking about special relativity. I am interested in what happens in the real world. If there is a long straight closed tube with a vacuum inside it, and, that tube is lying stationary on the surface of the Earth, and, an object inside that tube is moving through it at very high speed relative to the tube and Earth, will that object experience length contraction as a real physical effect due to its velocity relative to the tube and Earth? What happens to photons afterwards is not a point of confusion. I picture the object as a thin rod with its length parallel to the tube.

          James

          dear Peter

          this is just to say that I enjoyed reading your essay

          as you expected (your post on the blog for my own essay) I found in it some intriguing physical intuitions and a little theatre, all combined in enjoyable reading material

          best wishes for the competition

          Giovanni

          James

          No. It would not contract. Either in reality or apparently.

          I think you misunderstood my previous reply. I was also certainly not discussing special relativity but only the real world. The real mechanisms I describe produce all the effects we observe.

          Someone once drew up theory to try to try to explain the apparent paradoxes in those observations (i.e. CSL). The real mechanistic solution bears little resemblance to that theory, and shows there were b=never any real paradoxes, just m=limited comprehension.

          This is important as it also makes sense of QM, meaning that all classical observed effects are fully explained by (known but better interpreted) quantum mechanisms. Is this is too big for physics to assimilate?

          Peter

          PS Back to your rod. There are two varying cases. If the rod is accelerated it will contract during acceleration (not much if it's rigid). And if the observer is at rest as the rod moves further into the distance, then there will be an 'apparent' contraction, which we call 'perspective'. Both these, like the primary cases in my last post, should be intuitive. Are you in agreement?

          • [deleted]

          Peter,

          Ok I am clear about your view. Thank you.

          James