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Thread: Hyperspherical Universe without Singularities.

  1. #31
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    Thank you for that Shaula.

    You say " I am not going to your site to get the paper - rules are you present it here"

    Very well, I attach the relevant equation at the bottom. The equation writing facility on this site doesn't seem very extensive, and it seems to leave distracting brackets all over the result. (or am I using it wrongly?)

    I do hope that not everyone here has so steadfastly refused to look at my original paper.

    I would dearly like to perform the calculation you challenge me to, can anyone point me towards datasets on the expected rotation of the baryonic stellar matter at various radii, plus also the observed rotation figures, for Andromeda and Triangulum. I do not have academic levels of access.

    "Not obviously wrong" sounds encouraging, thanks for that.

    Regards, Pete.
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  2. #32
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    Quote Originally Posted by Peter J Carroll View Post
    4) We perhaps need to run some calculations and Reality Check has implied something we could do here to provide more hard evidence either way.

    5) May I suggest that we create rotation curves for the nearby galaxies of Andromeda and Triangulum which have been recently shown to have vast gas halos, using Equation 9 from the main paper here https://www.specularium.org/hypersphere-cosmology
    For this we would just need datasets on the expected rotation of the baryonic stellar matter at various radii, plus also the observed rotation figures.
    As it seems a four figure IQ is required to use excel equation syntax and my mathematician friend is stuck in Australia, would someone care to run the calculation?
    We? This is not a collaborative environment. If anything needs to be worked out or created in the presentation of your theory, you are responsible for getting it done.
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  3. #33
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    Thanks for that Peters Creek.

    I would have thought people here might have jumped at this chance to fatally falsify HC.

    However, if I’m to attempt to perform the calculations in the hope of providing supporting evidence for HC then perhaps we had better agree on acceptable evidence. I only have access to what lies in the public domain.

    Do the following sources of rotation curve data appear acceptable?


    Rotation curves for M31 and M33 Galaxies.
    https://iopscience.iop.org/article/1...881/aa79f3/pdf
    https://commons.wikimedia.org/wiki/F...riangulum).png
    https://www.researchgate.net/publica...er_Halo_of_M33
    https://vixra.org/pdf/1810.0427v1.pdf
    file:///C:/Users/pete/Downloads/1810.0427v3.pdf
    https://www.rmg.co.uk/sites/default/...f_M31_(HL).pdf
    https://arxiv.org/pdf/astro-ph/0603143.pdf

    When it comes to the newly discovered giant gas halos around M31 and M33 the estimates of their sizes and densities seems to remain a matter of some debate. I could try for a fit with some of the estimates below or I could try back calculating to an expected size and density and see if that falls close to current estimates.

    Does the following list provide acceptable estimates?

    Gas Halos around M3 and M3
    https://www.aanda.org/articles/aa/pd...aa26395-15.pdf (lower limit)
    https://iopscience.iop.org/article/1...38-4357/aba49c (can anyone get an institutional login for this please?)
    https://iopscience.iop.org/article/1...-637X/804/2/79
    https://arxiv.org/pdf/1605.01628.pdf
    If at the same distance as M33 it has a diameter of 18 kpc and a HI mass of 1.22 x 10^7M solar.
    https://www.researchgate.net/publica...angulum_Galaxy
    Paper requested.

    Fortunately, I just remembered my eldest acquired expertise in Excel doing her Biology PhD and she has volunteered to crunch the numbers.

    Regards, Pete.

  4. #34
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    Quote Originally Posted by Peter J Carroll View Post
    I would have thought people here might have jumped at this chance to fatally falsify HC.
    It is your job to provide evidence for it, not everyone elses' job to try to dig through your equations and prove you wrong (I tried that once and look where that got me - "Oh well even if what I have done is mathematically nonsense it is still right").

    Quote Originally Posted by Peter J Carroll View Post
    I do hope that not everyone here has so steadfastly refused to look at my original paper.
    If you had provided evidence or something that sparked my interest then I would have read it. You are really just repeating what you said int he last thread and throwing in a few more claims for good measure. Now you are asking us to spend time and effort picking through your paper and posts to do the work you should have done before you claimed that your ideas did away with dark matter and fixed galactic rotation curves.

  5. #35
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    Quote Originally Posted by Peter J Carroll View Post
    IF03: Show how you derive GR in HC so that you can then use gravitational redshift inside a mass, Peter J Carroll. ...
    My questions make the point that you are using GR when you do not have any evidence that GR exists in your HC universe .
    Irrelevant stuff about Frequently Asked Questions in Cosmology: What is the evidence for the Big Bang?. Unsupported assertions about what you want your HC universe to be is irrelevant.

    An ignorant citation of How Mathematical ‘Hocus-Pocus’ Saved Particle Physics is an example of your misunderstanding of how science works or a lack of reading comprehension. Ideas in science start as doubted. Evidence is gathered to support or debunk the the idea until it is accepted or rejected. That is why renormalization was accepted. Argument from incredibility does not make this mathematical procedure wrong.

    The "Gödel-Gomel-Zimmerman MOND effect" remain a fantasy because the papers you cite are not about MOND, neither is about your HC, one is not even about galaxy rotation curves, and one is not even a paper !
    An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation by Kurt Gödel, (1 July 1949).
    The Discrepancy in Galaxy Rotation Curves by Gomel and Zimmerman (5 August 2019).

    A "Gödel’s insight seems particularly adaptable to HC and MOND" fantasy as the title of his paper shows (it is a GR solution and your universe has no GR) and there is nothing applicable to MOND as reading the paper itself shows. A particle will have an intrinsic angular momentum. Apply that to a galaxy and it will have that momentum.
    "fields he wasn’t aware of when he wrote it" gibberish or more ignorance. Kurt Gödel was an educated mathematician who certainly knew about hyperspheres.
    A "makes predictions consistent with the rest of the HC hypothesis" fantasy.
    You emphasize that Gödel's paper is irrelevant to your HC!
    IFO4b Answer. He calculated w = 2sqrt pi G d where w = angular velocity, you can apply that w as acting on anything that rotates about anything chosen as a centre. This makes only an infinitesimal difference at planetary scales but a significant difference on the scale of galaxies, and a big difference on the scale of the entire universe.
    This is a constant angular velocity for a particle or an entire galaxy. Just imagining that it explains galaxy rotation curves which are changing angular velocities for stars inside a galaxy is not science.

    A bit of "ferocious guardians of orthodoxy " paranoia. MOND is an explanation for galaxy rotation curves that is even better than dark matter! Many MOND papers have been published. Gomel and Zimmerman is orthodox physics. We have been using the physics in their paper for about 300 years!

    IF05: Show how adding Gödel's angular momentum produces galaxy rotation curves that match observations, Peter J Carroll.
    A "IFO5 Answer. https://www.specularium.org/componen...lum-m33-galaxy" lie. That is an addition to the overall rotation of the M33 (Triangulum) galaxy from Gödel's paper which is mainstream cosmology. A galaxy rotation curve is not a single rotation and you know this, Peter J Carroll. Getting close to a value from an unpublished preprint that has nothing to do with your idea does not mean anything. This is really simple. This unpublished, not peer reviewed, preprint can derive galaxy rotation curves from orthodox physics and get a match to measured galaxy rotation curves without any of your HC! That preprint does not cite Gödel's paper. Rumors that the next preprint will cite Gödel's paper does not meant anything.

    Some "energy equality between the temperature of space" ignorance. Eddington's Temperature of Space has an effective temperature of starlight as 3.18 K which is not the temperature of the CMB.

    I wrote "Misconception about type 1a supernovae ("presumed to have the same intrinsic brightness"). Type 1a supernovae are known to have same peak magnitude! Textbook astrophysics shows the white dwarfs will explode at about the same mass. Closer supernovae can have their distances measured by other means and we see that the distances match." and you do not understand the point! The point is that the peak intensity being similar is a real measured feature of this universe.

    "visualise the geometry inside a glome style hypersphere" nonsense. There is no geometry inside a hypersphere because it has nothing insides it. This is a hypersphere
    In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its centre. It is a manifold of codimension one—that is, with one dimension less than that of the ambient space.
    A 1D hypersphere is the set of 1D points in a 2D space that are a distance r from a center. This is 2 points on a line.
    A 2D hypersphere is the set of 2D points in a 3D space that are a distance r from a center. This is a circle.
    A 3D hypersphere is the set of 3D points in a 4D space that are a distance r from a center. This is the surface of a sphere.
    A 4D hypersphere is the set of 4D points in a 5D space that are a distance r from a center. This may be the surface of a 4-sphere?
    Last edited by Reality Check; 2021-Feb-08 at 09:12 PM.

  6. #36
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    Also note some textbook cosmology, Peter J Carroll. By assuming that GR is valid in your cosmology, you are suggesting that your universe is either expanding or contracting! These are the only 2 stable GR solutions for "a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected". It may be static but unstable where any random density fluctuations turn it into an expanding or contracting universe.
    The classical Gödel metric has no Hubble expansion and thus is not applicable to this universe.
    Following Gödel, we can interpret the dust particles as galaxies, so that the Gödel solution becomes a cosmological model of a rotating universe. Besides rotating, this model exhibits no Hubble expansion, so it is not a realistic model of the universe in which we live, but can be taken as illustrating an alternative universe, which would in principle be allowed by general relativity (if one admits the legitimacy of a nonzero cosmological constant). Less well known solutions of Gödel's exhibit both rotation and Hubble expansion and have other qualities of his first model, but travelling into the past is not possible. According to S. W. Hawking, these models could well be a reasonable description of the universe that we observe, however observational data are compatible only with a very low rate of rotation.[4] The quality of these observations improved continually up until Gödel's death, and he would always ask "is the universe rotating yet?" and be told "no, it isn't".[5]
    ETA: Who knows what adding an imaginary "hyperspherical lensing" to the Gödel metric would do?
    Last edited by Reality Check; 2021-Feb-09 at 01:33 AM.

  7. #37
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    Thank you for this Reality Check.

    I would have preferred to have had something from you about the acceptability of the datasets I mentioned for the forthcoming calculation of galactic rotation curves. There seems an element of filibuster or broken record technique to your debating style, but I will nevertheless attempt to answer those of your questions that I have not already answered.

    My questions make the point that you are using GR when you do not have any evidence that GR exists in your HC universe .

    Answer. I have used both the Schwarzschild metric for a static black hole and the Gödel metric for a rotating dust solution. Both of these derive directly from GR do they not?

    Irrelevant stuff about Frequently Asked Questions in Cosmology: What is the evidence for the Big Bang?. Unsupported assertions about what you want your HC universe to be is irrelevant.

    An ignorant citation of How Mathematical ‘Hocus-Pocus’ Saved Particle Physics is an example of your misunderstanding of how science works or a lack of reading comprehension. Ideas in science start as doubted. Evidence is gathered to support or debunk the the idea until it is accepted or rejected. That is why renormalization was accepted. Argument from incredibility does not make this mathematical procedure wrong.

    The "Gödel-Gomel-Zimmerman MOND effect" remain a fantasy because the papers you cite are not about MOND, neither is about your HC, one is not even about galaxy rotation curves, and one is not even a paper !
    An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation by Kurt Gödel, (1 July 1949).
    The Discrepancy in Galaxy Rotation Curves by Gomel and Zimmerman (5 August 2019).

    A "Gödel’s insight seems particularly adaptable to HC and MOND" fantasy as the title of his paper shows (it is a GR solution and your universe has no GR) and there is nothing applicable to MOND as reading the paper itself shows. A particle will have an intrinsic angular momentum. Apply that to a galaxy and it will have that momentum.
    "fields he wasn’t aware of when he wrote it" gibberish or more ignorance. Kurt Gödel was an educated mathematician who certainly knew about hyperspheres.
    A "makes predictions consistent with the rest of the HC hypothesis" fantasy.
    You emphasize that Gödel's paper is irrelevant to your HC!
    IFO4b Answer. He calculated w = 2sqrt pi G d where w = angular velocity, you can apply that w as acting on anything that rotates about anything chosen as a centre. This makes only an infinitesimal difference at planetary scales but a significant difference on the scale of galaxies, and a big difference on the scale of the entire universe.
    This is a constant angular velocity for a particle or an entire galaxy. Just imagining that it explains galaxy rotation curves which are changing angular velocities for stars inside a galaxy is not science.

    Answer. Yes, a system be it a galaxy or a universe, will have an intrinsic angular velocity expressed in radians. You have to multiply that by the radius at which you wish to derive the rotational or orbital velocity.

    A bit of "ferocious guardians of orthodoxy " paranoia. MOND is an explanation for galaxy rotation curves that is even better than dark matter! Many MOND papers have been published. Gomel and Zimmerman is orthodox physics. We have been using the physics in their paper for about 300 years!

    Answer. Yes, and I hope to show that if you add in the Gödelian component you will get the observed rotation curve, so long as you take the gas halo into account.

    IF05: Show how adding Gödel's angular momentum produces galaxy rotation curves that match observations, Peter J Carroll.
    A "IFO5 Answer. https://www.specularium.org/componen...lum-m33-galaxy" lie. That is an addition to the overall rotation of the M33 (Triangulum) galaxy from Gödel's paper which is mainstream cosmology. A galaxy rotation curve is not a single rotation and you know this, Peter J Carroll. Getting close to a value from an unpublished preprint that has nothing to do with your idea does not mean anything. This is really simple. This unpublished, not peer reviewed, preprint can derive galaxy rotation curves from orthodox physics and get a match to measured galaxy rotation curves without any of your HC! That preprint does not cite Gödel's paper. Rumors that the next preprint will cite Gödel's paper does not meant anything.

    Answer, okay, so I’m trying to establish agreement on which datasets are acceptable to you. This will either beat Gomel and Zimmerman to the post or my idea will crash and burn.

    Some "energy equality between the temperature of space" ignorance. Eddington's Temperature of Space has an effective temperature of starlight as 3.18 K which is not the temperature of the CMB.

    Answer. No but the energy equality does look like an interesting coincidence, I claim no more at this stage.

    I wrote "Misconception about type 1a supernovae ("presumed to have the same intrinsic brightness"). Type 1a supernovae are known to have same peak magnitude! Textbook astrophysics shows the white dwarfs will explode at about the same mass. Closer supernovae can have their distances measured by other means and we see that the distances match." and you do not understand the point! The point is that the peak intensity being similar is a real measured feature of this universe.

    Answer. Yes and at closer distances the redshifts do have a fairly linear relationship to distances measured by these other means. The whole point of the Perlmutter study was to explore that relationship at higher redshifts, and they turned out to be progressively ls linear, leading to the idea of an accelerating expansion.

    See https://www.specularium.org/hypersph...here-cosmology

    The blue points show Perlmutter’s data, redshift against distance calculated from apparent magnitude. Plainly the distances will become very vast at redshifts in double figures and rise towards infinity at redshifts corresponding to near big bang phenomena. The orange points show the effect of correcting for hyperspherical lensing and indicate that the curve will flatten as redshifts become very high.

    "visualise the geometry inside a glome style hypersphere" nonsense. There is no geometry inside a hypersphere because it has nothing insides it. This is a hypersphere
    In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its centre. It is a manifold of codimension one—that is, with one dimension less than that of the ambient space.
    A 1D hypersphere is the set of 1D points in a 2D space that are a distance r from a center. This is 2 points on a line.
    A 2D hypersphere is the set of 2D points in a 3D space that are a distance r from a center. This is a circle.
    A 3D hypersphere is the set of 3D points in a 4D space that are a distance r from a center. This is the surface of a sphere.
    A 4D hypersphere is the set of 4D points in a 5D space that are a distance r from a center. This may be the surface of a 4-sphere?

    Answer. Yes, and as I hope I have already made clear, I’m referring solely to the 3D ‘surface’ of the 3-Sphere which is called a Glome. This lies ‘embedded’ in a fourth dimension which lies at right angles to all the 3 dimensions of the 3D space. We cannot ‘see’ this dimension except through the small positive spacetime curvature it creates.

    Flatlanders on the surface of an ordinary sphere will perceive a curvature if they look on a big enough scale. I claim that we do not live in a flat universe. GR describes local curvature due to mass. HC describes overall curvature due to mass.

    Formal Request. I would be pleased to hear if the suggested datasets for the rotation curves calculation meet your approval.

    Regards, Pete.

  8. #38
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    Peter J Carroll,

    Can I request that you please use the "reply with quote" function, as it's very difficult for me to follow where you're quoting other posts and where you're giving your own statements.
    "I'm planning to live forever. So far, that's working perfectly." Steven Wright

  9. #39
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    Quote Originally Posted by Peter J Carroll View Post
    Thank you for this Reality Check.

    I would have preferred to have had something from you about the acceptability of the datasets I mentioned for the forthcoming calculation of galactic rotation curves. ...
    Nothing to do with the fundamental error you made. A galaxy rotation curve is still not a single speed. You have not calculated any galaxy rotation curves yet.
    IF03: Show how you derive GR in HC so that you can then use gravitational redshift inside a mass [or any other result from GR], Peter J Carroll.
    Gravitation redshift, the Schwarzschild metric and the Gödel metric derive directly from GR which you still have not shown to exist in HC.
    IF05: Show how adding Gödel's angular momentum produces galaxy rotation curves that match observations, Peter J Carroll.

    More irrelevant stuff about the points in my post. You are still relying on an an irrelevant GR paper and an irrelevant unpublished, not peer reviewed preprint.
    Quote Originally Posted by Reality Check View Post
    The "Gödel-Gomel-Zimmerman MOND effect" remain a fantasy because the papers you cite are not about MOND, neither is about your HC, one is not even about galaxy rotation curves, and one is not even a paper !
    An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation by Kurt Gödel, (1 July 1949).
    The Discrepancy in Galaxy Rotation Curves by Gomel and Zimmerman (5 August 2019).
    You do not understand that 3.18 K is not 2.72548±0.00057 K (Eddington's temperature is not the CMB temperature), starlight is not space, effective does not mean actual and repeat "energy equality between the temperature of space" ignorance. Eddington's Temperature of Space is nothing to do with the CMB.
    Measurements of Omega and Lambda from 42 High-Redshift Supernovae by Perlmutter et.al. published 1999 supports a non-zero lambda cosmology. A web page abusing it for an obvously invalid universe does not mean anything.
    Yes a 3-sphere is called a glome. A pity that we exist in a universe with 4 dimensions, 3 of space and another of time ! You need a 4-sphere.
    A nonsensical "Formal Request". Of course measurements of real galaxy rotation curves meet everyone's approval. Questions IF03 and IF05 is you firstly showing that GR exists in your universe and then using Godel's results to predict galaxy rotation curves that fit the datasets. Repeating a calculation for a overall rotation for 1 galaxy (M33) is not a galaxy rotation curve or an answer to either question. Not answering IF03 first makes any answer to IF05 a waste of everyone's time.
    Last edited by Reality Check; 2021-Feb-09 at 01:34 AM.

  10. #40
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    Okay, I see this thread is going nowhere.

    Peter J Carol, you had better post something significantly about your universe, something that your "referees" can work with, like math and derivisions instead of walls of text. Just dumping an equation does NOT suffice, and neither does linking to your pdf, you will have to present all here on the board. Those are the rules.

    Also, to make Noclevername's request official, please when answering use the "quote" function, which you can easily "cut up" in smaller chunks by copypasting the quote tags around parts of the text.
    All comments made in red are moderator comments. Please, read the rules of the forum here, the special rules for the ATM section here and conspiracy theories. If you think a post is inappropriate, don't comment on it in thread but report it using the /!\ button in the lower left corner of each message. But most of all, have fun!

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  11. #41
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    That's fair enough Tusenfem.

    I will go quiet for a few days whilst we perform the lengthy calculation.

    Results to follow.

    Regards, Pete.

  12. #42
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    Quote Originally Posted by Peter J Carroll View Post
    I will go quiet for a few days whilst we perform the lengthy calculation.

    Results to follow.
    Can I make a plea that you detail your working as well as the results when you present them here?

    And good luck - hopefully a detailed, worked example will help to clarify some of the bits of this idea that I don't think have been made clear here. Look forwards to seeing what you come up with.

  13. #43
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    Galactic Rotation Curves

    Galactic Rotation Curves for M31 and M33 Galaxies, and all Disc Galaxies.

    Whilst preparing this piece I was dismayed to find that the overwhelming number of papers on galactic structure and behaviour do not append their raw data, just the resulting graphs.
    Estimates of the baryonic masses of the M31 and M33 galaxies vary surprisingly widely.
    In the absence of datasets with which to construct fresh expected and observed rotation curves for M31 and M33 I have used the graphs provided by Gomel and Zimmerman

    https://www.preprints.org/manuscript/201908.0046/v1

    These I append below.

    Both these graphs show what Gomel and Zimmerman call a ‘Non-Inertial System Contribution’. This acts as an angular velocity w, which by wr adds an extra rotational/orbital velocity at any radius r.

    I submit that Gödel’s old formula w = 2(sqrt pi G density) could supply this angular velocity.

    The calculation uses order of magnitude figures that encompass both galaxies. As they have similar sizes. I hope that the following calculations show that the addition of a Gödelian rotation COULD resolve the galactic rotation anomaly without recourse to the hypothesis of dark matter. I have dropped the consideration of the gas halos beyond the disc radius for the time being.

    Both galaxies have a missing angular velocity component of order 1 e-16 radians/second.

    The calculation treats the galaxies as spheres with a radius equal to that of their stellar discs.

    Data: Msolar ~ 2 e30 kg. G = 6.674 e-11. 1 light year ~1 e16 metres
    If w = 2sqrt (pi G m/volume)
    Then w = 2 sqrt (5 e-11 x m/r^3)

    Order of magnitude calculation: -

    Additional angular velocities of the Galaxies M31 and M33
    Baryonic mass ~ 1 e11 solar masses ~ 2 e41 kg.
    Radius ~ 1 e5 light years ~ e21 metres.
    Radius cubed = 1 e63 metres cubed
    5 e-11 x 2 e41 / 1 e63 = e-32
    w = has the order of 1 e-16 radians/second.

    The above very scrappy demonstration depends on using mass approximations and size approximations for M31 and M33, and the result lies open to endless debate and argument for the time being. Gomel and Zimmerman’s fits seem based on using convenient approximations derived from flexible estimates.
    However strong evidence exists that For All Disc Galaxies w = 2 e-16 radians/second.

    https://astronomy.com/news/2018/03/a...-billion-years

    In a study published March 9 in The Monthly Notices of the Royal Astronomical Society, astronomers announced the discovery that all disk galaxies rotate about once every billion years, no matter their size or mass.

    “It’s not Swiss watch precision,” said Gerhardt Meurer, an astronomer from the International Centre for Radio Astronomy Research (ICRAR), in a press release. “But regardless of whether a galaxy is very big or very small, if you could sit on the extreme edge of its disk as it spins, it would take you about a billion years to go all the way round.”

    The rotation velocity at the edge of a galactic disc depends almost entirely on the wr component, that is why the rotation curves go flat.

    Calculation

    distance/velocity = time
    2 pi r / wr = t
    2 pi / 2 e-16 = t
    pi e16 = t = 3.142 e16 seconds.

    A billion years equals 3.155 e16 seconds!

    Thus, if a Gödelian mechanism can account for the rotation curves of disc galaxies then every disc galaxy has a w component of almost exactly 2 e-16 radians/second.

    Rather than encouraging further inconclusive arguments about the material already gone over exhaustively here, I would like to use the remaining time available to debate other of the controversial assertions of HC. Please see my next post. Regards, Pete.
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  14. #44
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    Entropy

    Entropy! They do say that if your theory breaks the second law of thermodynamics then there is no hope for it.

    All around us in our kitchens and in the solar system we see the relentless rise of entropy. The dissipative processes of life remain driven by the gradual degradation of the material in the sun. According to many theorists, the big bang represents a moment of exceptionally low entropy in the universe and it has increased steadily ever since.

    HC asserts that the entropy of the universe remains constant overall.
    See equations 11 and 12 appended.

    Thermodynamics asserts that the entropy of a closed system can never decrease, it can only increase, or remain constant.

    HC posits a closed system. Yet we seem to observe mainly increasing entropy everywhere.

    HC thus implies some large-scale entropy decreasing processes going on somewhere.

    Professor Ian Stewart has opined, at least in his popular books, that he regards material accreting under gravity as an effective reversal of entropy.

    The weak force neutron-proton (or d quark - u quark) interaction underlies the build up and breakdown of heavier elements beyond hydrogen. (joke – hydrogen – a colourless odourless gas that slowly turns into people). Protons turn into neutrons inside stars. Free neutrons turn into hydrogen in space.

    Both n to p reactions and p to n reactions create neutrinos or antineutrinos and few things seem to absorb neutrinos, so in a temporally unlimited universe we might run into a sort of Olber’s paradox of neutrino fluxes. However, we do not yet know if neutrinos exist as Dirac or Weyl or Majorana fermions and as their own antiparticles which can annihilate to appear as energy in the form of photons.

    https://arxiv.org/abs/1404.6540

    this paper on the ‘new’ giant gas halo around Andromeda indicates that it could well act as a huge recycling yard for the disc. It contains not only a lot of hydrogen for use as fuel for new stars but also a lot of ‘metals’ in the astronomical sense, in the form of dust sloughed off by old exploding stars.

    Exploding stars return material to space, Remnant free supernovae may occur.

    In addition to the Hawking radiation predicted to make small black holes evaporate quickly, HC predicts that huge ones will tend to eject mass and energy because the Gödelian spin of the entire universe will tend to overspin them.

    See equation 10 appended.

    Thus, I assert that the universe, if finite but unbounded in space and time, can maintain a constant entropy. Regards, Pete.
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  15. #45
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    Quote Originally Posted by Peter J Carroll View Post
    Galactic Rotation Curves for M31 and M33 Galaxies, and all Disc Galaxies. ...
    Astronomy datasets tend to be very large and thus papers usually do not include the raw data. They give references to the datasets that are used.
    The graphs in Gomel and Zimmerman will be correct for the measured galaxy rotation curves.
    Ignorance that a result from GR (Gödel’s solution) will be the same as a result from classical physics.
    A repeat of no galaxy rotation curves makes this post irrelevant.
    All disk galaxies rotate once every billion years is interesting.

  16. #46
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    Quote Originally Posted by Peter J Carroll View Post
    Entropy! They do say that if your theory breaks the second law of thermodynamics then there is no hope for it. ...
    You imagine that there are entropy decreasing processes in your universe that somehow balance out the observed increases in entropy.

    Someone's opinion is not science. Some "weak force neutron-proton" and neutrino nonsense. An irrelevant citation of Evidence for a Massive, Extended Circumgalactic Medium Around the Andromeda Galaxy A "Thus, I assert" non sequitur when nothing you wrote before even implied that there were entropy decreasing processes in your universe.

    A fantasy that "huge" black holes "eject mass and energy" because Gödelian spin will "overspin" them. A non-charged, rotating black hole is described by the Kerr metric has the usual event horizon. Rotating black holes never eject mass or energy.

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    Quote Originally Posted by Reality Check View Post
    Astronomy datasets tend to be very large and thus papers usually do not include the raw data. They give references to the datasets that are used.
    The graphs in Gomel and Zimmerman will be correct for the measured galaxy rotation curves.
    Ignorance that a result from GR (Gödel’s solution) will be the same as a result from classical physics.
    A repeat of no galaxy rotation curves makes this post irrelevant.
    All disk galaxies rotate once every billion years is interesting.
    I do not doubt that Gomel and Zimmerman have used reliable figures for the observable rotation curves themselves which we can seemingly measure accurately. However I do wonder about the other lines drawn under the rotation curves for the contributions from gas mass and disc mass, particularly as estimates of total galactic baryonic masses do seem rather flexible.

    Yes indeed, the Billion Year Rotation of the edges of ALL disc galaxies does seem highly indicative of some unrecognised physics going on, it surprises me that this surprise hasn't attracted more attention. A Gödel-Gomel-Zimmerman type mechanism could account for it much more simply by setting w = 2 e-16 radians/second for all disc galaxies rather than asserting that dark matter must somehow always arrange itself in precisely the right distribution and quantity relative to ordinary matter to account for the effect.
    Last edited by Peter J Carroll; 2021-Feb-13 at 12:01 AM. Reason: numerical typo

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    Quote Originally Posted by Reality Check View Post
    You imagine that there are entropy decreasing processes in your universe that somehow balance out the observed increases in entropy.

    Someone's opinion is not science. Some "weak force neutron-proton" and neutrino nonsense. An irrelevant citation of Evidence for a Massive, Extended Circumgalactic Medium Around the Andromeda Galaxy A "Thus, I assert" non sequitur when nothing you wrote before even implied that there were entropy decreasing processes in your universe.

    A fantasy that "huge" black holes "eject mass and energy" because Gödelian spin will "overspin" them. A non-charged, rotating black hole is described by the Kerr metric has the usual event horizon. Rotating black holes never eject mass or energy.
    I dispute that the neutron-proton weak force reversible reaction has no relevance to cosmology, it underlies the Big Bang nucleosynthesis model which accounts for the Helium abundance but the Lithium abundance rather poorly. LCDM cosmology asserts a one way evolution and implies the continuous build up of 'metals' with iron as the ultimate nuclear ash.

    No, I haven't so far introduced anything here about entropy, it appears in the original brief paper that we are not supposed to refer to here, however as part of this presentation I'm introducing it now, with equations, for criticism. Global entropy reversing processes in closed systems obviously do not occur, although local ones do not seem entirely forbidden. I argue that this universe can accommodate both local increases and local decreases in entropy and thus maintain a constant global entropy (which the second law of thermodynamics does not forbid.)

    Kerr rotating black holes exist on paper but I have seen no reports of their observation. The massive black holes in galactic cores often prove difficult to observe because of the densely packed stars around them and the massive fluxes of radiation from their accretion discs and infalling matter. A black hole will certainly have a spin of some sort because of the infalling matter. However I submit that the spacetime geometry inside any black hole must become hyperspherical if it creates a gravitationally closed space and thus it will spin as a 3-sphere rather than as an ordinary 2-sphere. 3-sphere spin proves difficult to visualise but imagine all the particles on randomly orientated circular paths as in the Hopf Fibration.

    Regards, Pete.
    Last edited by Peter J Carroll; 2021-Feb-12 at 05:59 PM.

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    Quote Originally Posted by Peter J Carroll View Post
    Yes indeed, the Billion Year Rotation of the edges of ALL disc galaxies does seem highly indicative of some unrecognised physics going on, it surprises me that this surprise hasn't attracted more attention. A Gödel-Gomel-Zimmerman type mechanism could account for it much more simply by setting w = 2 e-16 radians/second for all disc galaxies rather than asserting that dark matter must somehow always arrange itself in precisely the right distribution and quantity relative to ordinary matter to account for the effect.
    No, Peter J Carroll. This will be solved with recognized physics. This has nothing to do with Gödel or a fantasy of a ""Gödel-Gomel-Zimmerman type mechanism". This is not dark matter making galaxies rotate. Galaxies rotate overall from the conservation of angular momentum during the collapse of their original gas clouds. That would occur even without dark matter. Dark matter modifies the rotation of stars and gas of the galaxies.

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    Quote Originally Posted by Peter J Carroll View Post
    I dispute that the neutron-proton weak force reversible reaction has no relevance to cosmology...
    Nothing to do with what I wrote, Peter J Carroll. You wrote a fantasy about entropy. You wrote ignorance about black holes. You wrote some "weak force neutron-proton" and neutrino nonsense in a post about entropy.

    It is textbook physics that there is "one way evolution". Big Bang nucleosynthesis produces the original mix of elements up to Li. Stellar nucleosynthesis turns hydrogen into heavier elements up to Fe. Explosive (supernova) nucleosynthesis produces heavier elements. This is one way for the simple reason that non-radioactive elements do not decay!

    A post about entropy is an introduction to what you imagine about entropy in your universe. The criticism is that all you have is a story. You imagine that entropy is constant in your universe. You must have a source of decreasing entropy to balance the increasing entropy we observe. You have no idea what that is. Detailing into the weak force and neutrinos is not that source.

    Every black hole we have observed would be a Kerr black hole! Bodies formed from collapsing gas clouds rotate. Stars rotate. The black bodies they form rotate. Supermassive black holes either formed from collapsing gas clouds or more likely as stellar black holes that ate stars and gas (and black holes). They will rotate. Maybe somewhere en the universe a supermassive black hole ate material that cancelled its rotation. The event horizon will still stop anything from escaping.

    A black holes and "hyperspherical" space doing whatever you imagine fantasy. It s fantasy until you have a theory of gravity that is comparable to GR. Then you can say what happens to black holes in your universe.
    Last edited by Reality Check; 2021-Feb-14 at 08:24 PM.

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    Could you give us your working definition of "finite but unbounded"? This phrase has always seemed a bit oxymoronic to me. Or mathematical trickery. I appreciate what you are trying to do as regards cosmological redshift, but can you really do the spherical cow thing with the entire universe?

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    Quote Originally Posted by Lucretius View Post
    Could you give us your working definition of "finite but unbounded"? This phrase has always seemed a bit oxymoronic to me. Or mathematical trickery. I appreciate what you are trying to do as regards cosmological redshift, but can you really do the spherical cow thing with the entire universe?
    It's always seemed a bit like trickery to me too. But I think it generally means something like this: a universe that is (for example) only ten kilometers in any direction, but once you travel in that direction you get back to where you started from. So you can keep going forever, even though the space itself is not infinite. It would happen if say your universe was like the inner tube of a bicycle, but in all directions.
    As above, so below

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    Quote Originally Posted by Jens View Post
    It's always seemed a bit like trickery to me too. But I think it generally means something like this: a universe that is (for example) only ten kilometers in any direction, but once you travel in that direction you get back to where you started from. So you can keep going forever, even though the space itself is not infinite. It would happen if say your universe was like the inner tube of a bicycle, but in all directions.
    Greetings Lucretius and Jens. Yes, the surface of an ordinary sphere forms a finite but unbounded two-dimensional space, you can walk as far as you like without encountering an edge but it has a finite size. A few posts back on Page 1 Feb 5th 05.51 pm I posted a fairly extensive piece about visualising a three-dimensional finite and unbounded space. (its come out a bit tightly spaced, perhaps the formatting here has compressed its size, but it often makes the whole idea suddenly click in people's heads when they read it)

    Yes a bicycle inner tube does as a torus, have a similar topology to a hypersphere. The Hopf Fibration shows that a hypersphere can be represented in 3D as a torus.

    Finite and unbounded in time might seem an odd idea but in the closed HC universe you can only 'see' or get signals from any event up to 13 billion years ago. Redshift and lensing obliterate any signal originating from before that. Similarly you can only send a signal the same distance into the future for the same reasons, your signal will fade to nothing after that much time.

    I don't think finite but unbounded time leads to visiting the exact past again. The universe can do something slightly different on each of its awesomely long 'days', much as we can whistle different tunes whilst endlessly circling the globe.

    Regards, Pete.
    Last edited by Peter J Carroll; 2021-Feb-15 at 02:30 PM. Reason: grammar

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    Quote Originally Posted by Reality Check View Post
    Nothing to do with what I wrote, Peter J Carroll. You wrote a fantasy about entropy. You wrote ignorance about black holes. You wrote some "weak force neutron-proton" and neutrino nonsense in a post about entropy.

    It is textbook physics that there is "one way evolution". Big Bang nucleosynthesis produces the original mix of elements up to Li. Stellar nucleosynthesis turns hydrogen into heavier elements up to Fe. Explosive (supernova) nucleosynthesis produces heavier elements. This is one way for the simple reason that non-radioactive elements do not decay!

    A post about entropy is an introduction to what you imagine about entropy in your universe. The criticism is that all you have is a story. You imagine that entropy is constant in your universe. You must have a source of decreasing entropy to balance the increasing entropy we observe. You have no idea what that is. Detailing into the weak force and neutrinos is not that source.

    Every black hole we have observed would be a Kerr black hole! Bodies formed from collapsing gas clouds rotate. Stars rotate. The black bodies they form rotate. Supermassive black holes either formed from collapsing gas clouds or more likely as stellar black holes that ate stars and gas (and black holes). They will rotate. Maybe somewhere en the universe a supermassive black hole ate material that cancelled its rotation. The event horizon will still stop anything from escaping.

    A black holes and "hyperspherical" space doing whatever you imagine fantasy. It s fantasy until you have a theory of gravity that is comparable to GR. Then you can say what happens to black holes in your universe.
    My apologies for presuming that you had detailed knowledge of particle physics. The proton-neutron transmutation mediated by the weak force plays a critical role in nucleosynthesis whether at a big bang or in stars. In the big bang scenario a suitable selection of parameters can account for the helium abundance in the universe (~25%) and the stars cannot have made this much in the time available since a big bang. However this model runs into problems accounting for the lithium abundance https://en.wikipedia.org/wiki/Cosmol...ithium_problem

    In the HC model the universe effectively has unlimited time and the hydrogen-helium ratio exists as a natural equilibrium. Indeed the overall proportions of all elements remain roughly constant. Now you say 'non radioactive elements do not decay', well they certainly do if you treat them roughly enough, not only in particle accelerators and probably in the hottest stars but also in neutron stars where elements breakdown into neutronium - neutron plasma, and neutron stars do explode.
    Free neutrons in space decay to form protons and electrons (and anti-neutrinos). I mention all this because HC posits a universe which does not evolve towards some end state with all the hydrogen transmuted into heavier elements and all the matter stuck in black holes. As you said, the current model of galactic evolution remains questionable, and embarrassingly tight for the timescale. The recently discovered vast gas halos around nearby galaxies could well act as the recycling yards for galaxies which have had something similar to their current form for far longer than we think.

    Agreed, black holes must have a rotation, and if the material inside hasn't collapsed to a singularity it should have an orbital velocity of lightspeed.

    Only a small number of exact solutions to the field equations of General Relativity have been found. HC uses the Schwarzschild solution and the Gödel solution. HC also uses the GR principle of gravitational redshift, albeit adapted to space curved in all three spatial dimensions, but it offers this as a testable prediction. In general, GR will do nicely for HC.

    Regards, Pete.
    Last edited by Peter J Carroll; 2021-Feb-15 at 03:53 PM. Reason: grammar & afterthought

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    Quote Originally Posted by Peter J Carroll View Post
    My apologies for presuming that you had detailed knowledge of particle physics. ...
    Anyone with any knowledge of physics knows that the weak force is irrelevant to your entropy fantasy. I happen to have a detailed knowledge of particle physics (an MSc gained some decades ago).

    Once again: Any problems with the Big Bang does not mean that your idea is correct. Writing the word hypersphere does nt explain the abundance of Li in this universe using your idea!

    You keep on making the HC "model" even worse, Peter J Carroll.
    • A HC universe "which does not evolve towards some end state with all the hydrogen transmuted into heavier elements..." .
      In this universe stars do fuse hydrogen transmuted into heavier elements. The metals in stars increases and this is a way to measure their age. That is a fatal flaw with any cosmology with a static universe - they have explain why we do not see stars whose age is trillions of years.
      IF06: Where are the stars that are enormously older than 13.8 billion years, Peter J Carroll?
      Stars will stop forming partially because most (not all) H has been fused into metals. Fusion in stars needs a certain density of H. Star formation ceases in this universe after ~1014 years. See A Dying Universe: The Long Term Fate and Evolution of Astrophysical Objects by Fred C. Adams, Gregory Laughlin (1997) ("Thus, by the cosmological decade η ≈ 14, essentially all normal star formation in galaxies will have ceased").
      IF07: Why are stars still being formed in your "unbounded in both space and time" universe, Peter J Carroll
    • A fantasy that gas haloes around galaxies are "recycling yards" when they are just clouds of gas!
    • Emphasizing the ignorance of using GR which so far does not exist in your universe!
    • Emphasizing ignorance of GR.
      There are many exact solutions of GR.
      The Schwarzschild solution is for the field outside of a massive object.
      Gravitational redshift is not a principle. It is one of many prediction of GR which have all been tested and passed those tests. This universe is a GR universe as far as we can tell.
      GR has curved spacetime, i.e. "space curved in all three spatial dimensions". This is not a hyperphere which has no curvature defined at all ! This is one of the fundamental concepts of mathematical physics that you need to learn. A space useful in physics is not not an arbitrary set of points. It is a set of points with at least a structure and a definition of differentiation. That gives little things like velocity, acceleration, etc.
    • Emphasizing the ignorance of cherry picking an aspect or 2 of GR!
      If you use GR then you are stating that GR is correct. You know that an exact solution of GR states that this universe must be expanding or contracting (or unstably static). You cannot use GR in your static universe.

    A "As you said, the current model of galactic evolution remains questionable" fantasy when I have not written that. What I wrote was: Even if the models of galaxy formation and evolution are wrong, this does not make your idea correct.

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    I humbly refer to HC as a ‘model’ out of deference to the principle that ‘cosmologists are seldom right but never in doubt’ (Ladau).


    Over the course of taking a detailed interest in the subject for 25 years I've seen the Hubble time change wildly. The value recently seemed to stabilise around 13.8 bnyr after making complicated adjustments for the Perlmutter observation that distance from apparent magnitude seemingly contradicts Hubble's law that redshift rises in direct proportion to distance. But now cosmology has another crisis; The Hubble Tension https://arxiv.org/abs/2010.12164

    The age of stars remains a matter of debate. Methuselah star https://www.space.com/how-can-a-star...-universe.html


    https://en.wikipedia.org/wiki/Stella...the%20universe.

    How can we possibly tell the age of stellar remnants such as white dwarfs, black dwarfs, and black holes?

    Plus things like this throw doubt on the whole galactic evolution idea https://phys.org/news/2021-02-portra...formation.html

    I think that if you look at that paper on gas halos you will see evidence for heavy element dust remnants in the halos.

    Stellar evolution seems largely predicated upon no further gas in-fall. In quiet parts of galaxies that probably holds, but Andromeda is due to pass right through the Milky way in a few billion years time and such galactic collisions seem the long-term norm in the universe.

    The Gödel metric applies to both the entire universe and to structures within it. According to this the Schwarzschild metric can apply similarly, depending on the observers frame of reference.

    https://en.wikipedia.org/wiki/Schwarzschild_metric

    "The Schwarzschild metric has a singularity for r = 0 which is an intrinsic curvature singularity. It also seems to have a singularity on the event horizon r = rs. Depending on the point of view, the metric is therefore defined only on the exterior region r > rs ,only on the interior region r < rs or their disjoint union."

    Yes the universe must either contract or expand if only the simplest metric, the Schwarzschild metric, applies. However in the Gödel metric a rotation can stabilise it at constant size.

    In the attached equations below I have shown an equality between those metrics. I have done this by effectively multiplying r both sides by pi, as pi r = L.

    Any gravitationally closed space will have a radius r apparent from the outside but an antipode length L where L = pi r on the inside.

    Regards, Pete.
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  27. #57
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    Quote Originally Posted by Peter J Carroll View Post
    I humbly refer to HC as a ‘model’ out of deference to the principle that ‘cosmologists are seldom right but never in doubt’ (Ladau).


    Over the course of taking a detailed interest in the subject for 25 years I've seen the Hubble time change wildly. The value recently seemed to stabilise around 13.8 bnyr after making complicated adjustments for the Perlmutter observation that distance from apparent magnitude seemingly contradicts Hubble's law that redshift rises in direct proportion to distance. But now cosmology has another crisis; The Hubble Tension https://arxiv.org/abs/2010.12164

    The age of stars remains a matter of debate. Methuselah star https://www.space.com/how-can-a-star...-universe.html


    https://en.wikipedia.org/wiki/Stella...the%20universe.

    How can we possibly tell the age of stellar remnants such as white dwarfs, black dwarfs, and black holes?

    Plus things like this throw doubt on the whole galactic evolution idea https://phys.org/news/2021-02-portra...formation.html

    I think that if you look at that paper on gas halos you will see evidence for heavy element dust remnants in the halos.

    Stellar evolution seems largely predicated upon no further gas in-fall. In quiet parts of galaxies that probably holds, but Andromeda is due to pass right through the Milky way in a few billion years time and such galactic collisions seem the long-term norm in the universe.

    The Gödel metric applies to both the entire universe and to structures within it. According to this the Schwarzschild metric can apply similarly, depending on the observers frame of reference.

    https://en.wikipedia.org/wiki/Schwarzschild_metric

    "The Schwarzschild metric has a singularity for r = 0 which is an intrinsic curvature singularity. It also seems to have a singularity on the event horizon r = rs. Depending on the point of view, the metric is therefore defined only on the exterior region r > rs ,only on the interior region r < rs or their disjoint union."

    Yes the universe must either contract or expand if only the simplest metric, the Schwarzschild metric, applies. However in the Gödel metric a rotation can stabilise it at constant size.

    In the attached equations below I have shown an equality between those metrics. I have done this by effectively multiplying r both sides by pi, as pi r = L.

    Any gravitationally closed space will have a radius r apparent from the outside but an antipode length L where L = pi r on the inside.

    Regards, Pete.
    Where is the beef. I never see any math here!
    The moment an instant lasted forever, we were destined for the leading edge of eternity.

  28. #58
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    Quote Originally Posted by Peter J Carroll View Post
    ...More logical error of false dichotomy snipped....
    The age of stars remains a matter of debate. Methuselah star https://www.space.com/how-can-a-star...-universe.html
    More evidence that you are arguing from a stance of ignorance of astronomy and some science, Peter J Carroll!
    The age of stars is a matter of science. We have found 1 (one) star that may be older than the universe. The Methuselah star has a measured age whose errors includes the age of the universe. In short, there is a 20% chance that the current age of the star is less than the age of the universe.
    We can tell the age of white dwarfs by applying science. They start at a certain temnperure and cool down predictably. And the temperature of the coolest white dwarfs is one observational limit on the age of the universe.
    Black dwarfs are evidence that the universe is too young for white dwarfs to cool down to be black.
    You are persisting in your logical error of false dichotomy with Portrait of young galaxy throws theory of galaxy formation on its head.
    One more time: There are many internet physics cranks some with ideas about galaxy formation. There are many scientists, some with competing scientific models of galaxy formation. You have no idea or model of galaxy formation! If one model is wrong than you are not correct - one of the others is correct.
    You do not fully understand what you cite. This one observation is that the cold gas in ALESS 073.1 is rotating such to need the existence of a bulge and imply spiral arms. We do not expect galaxies at 1.2 billion years to have bulge or spiral arms. Science treats a single observation as tentative. The lead author of the study Dr. Federico Lelli makes this clear in another article
    “The main message is that galaxies can form extremely fast,” says Lelli. He says they may go through an early stage of rapid growth, then mature more slowly. But he says it is hard to make generalisations about this based on a single image, so more are needed to see if this is common for other galaxies.
    Gas halos having dust still leaves you with a fantasy that gas haloes around galaxies are "recycling yards" when they are just clouds of gas [and dust]!
    More

    You add more ignorance and repeat HC fantasies
    • The Gödel metric is not the Schwarzschild metric.
      The Gödel metric can be applied to dust, etc. because it has no restriction on its coordinates.
      The Schwarzschild metric has a explicit restriction that it applies outside of a massive object.
      In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild vacuum or Schwarzschild solution) is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero.
      ...
      The Schwarzschild metric is a solution of Einstein's field equations in empty space, meaning that it is valid only outside the gravitating body. That is, for a spherical body of radius R the solution is valid for r>R.
    • The Schwarzschild metric is not used in cosmology.
      The Friedmann–Lemaître–Robertson–Walker metric is the "exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected.[1][2][3]" used in the Big Bang.
    Last edited by Reality Check; 2021-Feb-16 at 09:00 PM.

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    Firstly, greetings Copernicus, I will address your post towards the end of this one.

    In Answer to your points Reality Check,

    If Black Dwarfs exist now, we will have great difficulty observing them except perhaps by lensing or occlusion effects. If we do manage to observe one, we will find it as impossible to assign an age to it, as with any Black Hole.

    If you now accept that the giant gas halos contain heavy element dust, then it presumably originates from previously exploded stars.

    The galaxy evolution article states that lighter stars can survive for vastly longer than the apparent age of the universe. Astronomers assign ages to stars based on heavy element build up, however they also assign ‘second generation’ status to stars which do exhibit significant amounts of ‘metals’. Our sun is often described as a second-generation star containing material from previous supernovae. Most of the stars (at least in this vicinity) seem to have planets. If most stars in the galaxy already have planets, then that may lead to a modification of galactic evolution ideas.

    Recent observations, including the gas halos, suggest that far more recycling may occur within galaxies than previously factored in. I raise these points, along with the material about fully formed galaxies at extreme redshifts, merely to suggest that fitting galactic and stellar evolution hypotheses to a 13.8 bnyr timescale may not be necessary.

    Thus, I cannot venture a quantitative answer to your IFOs 6 & 7.

    How very odd that you and I both found conflicting quotes about the Schwarzschild metric in the same article, and no apparent resolution of it within the article.

    The FLWR metric is AN exact solution to the field equations of GR, it is not THE exact solution to the field equations of GR, not merely a point of grammar - you said there were many.

    HC uses the Schwarzschild and Gödel solutions and hopefully equations 1-5 previously attached, show how they can work together.

    Copernicus, I’m not sure to whom your comment “Where is the beef. I never see any math here!” is addressed.

    I have just eight more equations with supporting arguments about physical principles to present for falsification before we run out of time here.

    I present equations 11-15 which follow from each other, as PDFs for consideration below.

    These provide an interpretation of the Dirac Large Number Hypothesis that does not involve either the idea that fundamental constants evolve with time or that the large number correlations arise purely as a coincidence at this epoch. Rather, they suggest that the numbers, or at least the orders of magnitude, e20, e40, e60, e120 reflect something fundamental about the universe at any point in time.

    https://en.wikipedia.org/wiki/Dirac_...ers_hypothesis

    https://arxiv.org/ftp/physics/papers/0502/0502049.pdf

    Regards, Pete.
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