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WalterP

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Everything posted by WalterP

  1. For good measure Disillusioned, here are links to sites and articles I thought would be relevant to our examination of WLC's claims about H & P's 'proof'. Given what's gone between us here in this thread, I think it's now right for me to share this info with you, for your consideration. https://www.physicsforums.com/insights/proofs-in-mathematics/ https://www.nsta.org/publications/news/story.aspx?id=52402 https://oregonstate.edu/instruction/bb317/scientifictheories.html https://en.wikipedia.org/wiki/Mathematical_proof https://undsci.berkeley.edu/teaching/misconceptions.php https://www.birmingham.ac.uk/Documents/college-eps/college/stem/Student-Summer-Education-Internships/Proof-and-Reasoning.pdf Thank you. Walter.
  2. Disillusioned, Here are two excerpts from Julian Baggini's book, A Short History of Truth: Consolations for the Post-Truth World. Chapter 5, Empirical Truths. Pages 54 / 56. Etymologically, empirical means ‘from experience’, and experience seems to be telling us that an empirical approach leaves us with uncertainty, rather than knowledge. Far from being a weakness, however, the open-endedness of empirical inquiry is actually its strength. David Hume made this point wonderfully when he observed that ‘all the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact.’ Relations of ideas concern truths of maths, geometry and pure logic. As we have seen, such truths are, in effect, true by definition, but they tell us nothing about the real world. Matters of fact, in contrast, cannot be established by pure logic. That also means they cannot be established with 100 per cent certainty. ‘The contrary of every matter of fact is still possible,’ warned Hume. ‘That the sun will not rise tomorrow is no less intelligible a proposition, and implies no more contradiction than the affirmation, that it will rise.’ Indeed, we can easily imagine circumstances in which we would have to accept that the sun is unlikely to rise tomorrow, such as if a massive asteroid were about to hit the Earth. A lack of certainty is therefore part of the deal with empirical truth. We need to give up on it in order to take up the possibility of knowledge of the world. Absolute certainties can only be obtained about purely conceptual matters, such as axioms of mathematics and laws of logic. If we want to know about the world then there is potentially no end of discoveries – for ourselves or the entire human race – that might force us to alter our opinions. (That’s why despite my best efforts, it is almost inevitable that I have unintentionally stated at least one falsehood in this book.) What we hold to be true is constantly open to being tested, which makes the truths that pass the test more reliable. The strength of empirical truth resides in the fact that it is always open to scrutiny, revision and rejection. Chapter 4, Reasoned Truths. P 44. In modern philosophy, the two great representatives of this optimistic ‘rationalist’ tradition were Rene Descartes and Baruch Spinoza. Both were impressed by the ways in which proofs were generated in mathematics and geometry. They thought that with due care and diligence, it would be possible to replicate this precision in all areas of human knowledge. ‘Those long chains composed of very simple and easy reasonings, which geometers customarily use to arrive at their most difficult demonstrations,’ wrote Descartes, ‘had given me occasion to suppose that all the things which come within the scope of human knowledge are interconnected in the same way.’ David Hume worked out the basic flaw in rationalism back in the eighteenth century. All pure reason could analyse was the relationship between concepts. But this tells us nothing about the relationship between the things in the world those concepts relate to. ‘1 + 1 = 2’ is a truth about numbers, but says nothing about what happens when you put two physical things together, where they might annihilate each other, merge into one, or multiply. I lighted upon these Disillusioned because they seemed to be telling me something important about the way WLC was presenting H & P's entirely mathematical 'proof' of singularity theory as a Matter of Fact. As a brute fact of reality. As an ontological truth. This cannot be so! Singularity theory is a construct of human endeavour and so MUST BE an epistemic truth. Thank you. Walter.
  3. Hello Disillusioned. Its taken me a bit longer to get back to you on this, but here I am. Agreed. Also, what you say here intersects with almost uncanny accuracy with what I had planned to delve into about David Hume's thoughts on this, in the Failed Cosmology thread. And this agrees with the uncertainty of knowledge, your very point above. Agreed. If ontological truths are absolute, then the inherent uncertainty of epistemic truths would seem to indicate that things epistemic cannot establish things ontological. The ephemeral cannot generate the permanent. Yes, yes and yes again. Now for a confession, Disillusioned. I much prefer the neatness and cleanness of hard science data to the slipperiness of these philosophical concepts. But, I know that these concepts are of great importance. So, like a teenager who tries to get out of doing the chores, I've found myself putting off dealing with them. You can even see evidence of my reluctance in the way I've suggested that you and Josh and I put off dealing with the philosophical aspects of WLC's apologetics until after we've tackled the hard science. Mea culpa! Agreed. Bohr can be a bitter pill for some to digest because there's still a widely prevailing, but incorrect, view that science tells us exactly how nature is. No. Science provides a tentative, incomplete and provisional working description of how we think nature is. Agreed. I might add that particle physics relies just as heavily upon assumption and inference, if not more so. The Higgs boson was not actually detected, per se. Instead, a number of events and interactions were detected by the LHC and when these were matched against simulations of how the Higgs was expected to behave, there was a high degree of correspondence. When the scientists were satisfied that the level of confidence was at 5 sigma, they announced the detection. It's also sobering to note that the complete range of decay products from the interactions couldn't be fully detected. This was known before the machine was even built. The ingenious solution to that problem was to precisely calculate the amount of energy going in to a collision and then subtract the amount being detected leaving the event. The energies of the detected muons, photons, W and Z particles, etc. was known and when all of these had been accounted for, the shortfall equaled the predicted energy of the undetectable decay products. Perform that procedure often enough, eliminate as many variables as possible, account for human error and if what you have left matches what you were looking for, then (even though you couldn't detect 100% of the resulting particles) you can say that you've detected the Higgs. The above agrees with what Hume's had to say about the difference between Relations of Ideas and Matters of Fact. (More about this to follow, I think. The time seems right.) I can't fault your logic here, Disillusioned. I find myself agreeing with you all the way. Mostly because I've had to cover similar ground in the five years I spent researching, checking and testing WLC's claims about singularity theory. All I would say about that period is that I felt much more certain about the science aspects I covered than the epistemic/philosophical aspects. But now might be the very time for me to engage more fully with the latter. That being so, I'll follow this post with another that contains some material that i believe is very relevant. Thank you. Walter.
  4. Evidence that totally excludes the use of faith, Edgarcito. Posted October 28 Then faith destroys the meaning of words. The meaning of words is agreed upon by reason, not by faith. When someone (in this case, you) chooses to assign their own individual meanings to words, then communication breaks down. Hence, this thread. It becomes impossible to reason with a devout Christian (or Edgarcito) because they use their faith to destroy the meaning of words. That's an example of you in your faith-driven mode, Edgarcito. So, to reaffirm my answer to your question, in this thread I will only accept evidence (and dialogue) from you if you agree to totally exclude faith from the equation. With faith gone it will interesting to see if its possible to reason with you. Thank you. Walter.
  5. No offence taken, Justus. If you check my two posts for your attention I clearly stated that the interior volume, that is, the inside of the sphere, did not figure in any way in the geometry of the universe in modern cosmology. I also said that only the surface of the sphere that is of relevance, because that surface has no edge or boundary. Therefore, if the contents of the surface are compressed by the shrinkage of the sphere, there is no need of a hard boundary to compress them. The shrinkage (or expansion) of the space itself does the compression. An ordinary, 3-dimensional sphere has no depth to it's surface and so isn't used in cosmology. But a 4-dimensional hypersphere does have the needed extra dimension. https://en.wikipedia.org/wiki/Hypersphere Please remember that just because something isn't understood, that doesn't make it BS. I don't understand the technical details of optometry, but I know that it works and that it's not BS. An infinity is not what you think it is, Justus. https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel Infinities are not fixed entities. They can be added to and they can grow. Again, because you and I don't necessarily understand something, that doesn't make it BS. Thank you. Walter.
  6. Hello Josh. I appreciate your confusion and I agree that these are not easy concepts to visualize. As it happens, in the next step in the Failed Cosmology of WLC thread I was going to explain the apparent contradiction between what appear to be two very different things. The observed flatness of space and accelerating 'opening out' of space by the cosmological constant (CC). If you recall, early on in the thread I said that we would be covering what the CC is and how it is measured. 1. What is the Cosmological Constant? 2. How is it measured? 3. Why did Hawking and Penrose make these four assumptions in the first place? So far we've haven't tackled these three points out of their given order. We've covered 3, with the opacity of the CMB forcing Hawking and Penrose to make assumptions. I posted info on the CC on Tuesday, so that's the start of us covering point 1. Once you and Disillusioned are happy to move on to point 2, then we can cover how the CC is measured. When we get there I can more readily explain some of the things you mentioned above. Is that ok? One thing I can do here and now is help with this. "But a universe is merely some observable region within the context of space." https://en.wikipedia.org/wiki/Observable_universe The term 'observable universe' is used to apply to any region of space that any observer can see. https://en.wikipedia.org/wiki/Universe The term 'universe' is used to apply to everything that exists, regardless of what any observers can see. Josh, the former is knowable by us because it is simply the limit of how far we can see. Whereas, the latter is unknown to us. As I'm sure you're aware, almost all cosmology proceeds by assumption and inference and not by direct observation. The speed of light sees to that. Therefore, we assume and infer that space extends beyond the limit of our visual horizon. The name we give to the observable universe and the space beyond that limit is 'the universe'. The name we give the space within that limit is the 'observable universe'. The two should never really be conflated or swapped over. Doing that causes misunderstanding and confusion. Space within the observable universe has been measured to be flat. But, as I've shown in this thread, the wider universe beyond our horizon might be Flat, Closed or Open. We just don't know for sure. Science papers like the ones cited in this thread are all trying to discover if clues within the observable universe can tell us about the flatness or curvature of the entire universe. My stated position is that we need much better data before we are in a position to find an answer. If you look again at the sequence of graphics I posted you can see that an entire universe can start off as a quantum-sized hypersphere that inflates to become so vast that it looks flat to any observer within it. Those last words deserve repeating. A sufficiently-large hyperspherical universe looks FLAT TO ANY OBSERVER WITHIN IT. That's all my graphics can show. No more. But the graphic you posted shows an ensemble of separate hyperspherical universes from GOD'S point of view. From a position 'outside' (?) of 'anywhere' (?) in space and time. None of us can ever aspire to see reality from such a viewpoint. You see how I'm even struggling to find the right words to use? Such images, even if they are based upon peer-reviewed science papers, are at their best, no more than educated guesses and at their worst, unbridled speculation. I can make an educated guess if you'd like me to, but I wont indulge in unbridled speculation about things we can never observe. To recap, I'll explain about the measured flatness of the observable universe and how that relates to the entire universe in the Failed Cosmology thread. I hope this post has been of some help. Thank you. Walter.
  7. That's a kind offer Pantheory, but with all due respect, I must decline. My take on the current 'issues' in cosmology is that we will soon(ish) be in possession of data that will resolve them or perhaps overturn the current paradigm. As you know, I'm not wed to any particular theory, model or paradigm and so it's not a big deal for me to wait and see what pans out. It's my view that the current cosmic distance ladder has too many rungs that are poorly constrained by the data. A good reappraisal of the lowest rungs will come some time after 2021, when the last Gaia data is released. https://www.cosmos.esa.int/web/gaia/release I'm of the opinion that, together with such advances as the JWST, the EELT and the LSST, the Gaia data will bring a sea change in our understanding of the cosmos. To be honest, I can't deny that I'm eagerly awaiting the mid 2020's. Thanks again. Walter.
  8. On Tuesday, I wrote this. The simple answer to the question, Is the universe Closed, Flat or Open? is that it's too soon to say for sure. Our observable universe is simply too small a volume of space for us to say for sure, as of today. It's possible that future observations may pin the answer down, but as far as I'm concerned, the jury's still out. I tend to take what the NewsScientist says with a pinch of salt. They have something of a reputation for sensationalizing issues and over-simplifying complex topics. Today, after skimming over articles on various science forums I noticed this. https://arxiv.org/pdf/1903.12097.pdf It's extremely technical and I can't claim to understand much of it, but this section is relevant to this thread. Our measurements support a value of H0 that is closer to that one found by the BAO methodology rather than the higher value from the Cepheids. Interestingly, the 8 H0−Ωm contours from γ-ray attenuation are roughly orthogonal to results from other techniques, which makes our results nicely complementary to those from other probes. In order to improve the H0 measurement we need to measure optical depths up to the largest possible energies. This is difficult with LAT because of the limited photon statistics. However, it may be possible with the future Cherenkov Telescope Array (CTA, Hinton et al. 2019). In a nutshell, this paper seems to be supporting the Flat universe of the LCDM Concordance Model. Please note that I do not necessarily support any particular cosmological model, so all I am pointing out here is what I said on Tuesday - the jury's still out. If different measurements appear to be coming to different conclusions about the geometry of the universe, then the issue is still in a state of flux and further, better measurements are needed to settle things. Thank you. Walter.
  9. Josh, As far as I understand, Pantheory has it right. These are 2D depictions that are drawn to look 3D, but I think they're representing 4D hyperspheres and not ordinary, three dimensional spheres. I've tried to explain this to Justus, who favors a 3D, spherical universe with a hard boundary. But that model violates the Copernican principle, which requires that no location be central or special in any way. https://www.ex-christian.net/topic/82672-robin-collins-and-fine-tuning-for-discoverability/ Thank you. Walter.
  10. Hello Disillusioned. I intend to do your reply justice over the weekend. Thank you. Walter.
  11. Thank you, Forrest. I appreciate what you say. Walter.
  12. Hello Josh. I think we may have got our wires crossed. I did answer your question, but from the only reasonable scientific viewpoint that we have - our own. I showed from the diagrams that it's impossible for us to tell if our own universe is closed, flat or open. To us, there is no appreciable curvature of space. I also qualified this by adding that our particular universe could be curved closed or open, but on scales that are simply far too large for me to show on a computer screen. The fallout of this is that it's just an exercise in speculation for me to go way beyond the confines of our universe. If there's insufficient information for us to determine the geometry of our own universe, then it logically follows that that there's NO information to guide us about what lies beyond it. Re-reading your question in the context of the graphic you posted I now see that you wanted me to talk about the geometries of bubble universes, as seen from a viewpoint somewhere out in the multiverse itself. Is that right? If so, then I'm sorry, but I really don't think that I can do more than politely decline your request. However, I can give you an answer (of sorts) to this. The flat appearance in the above, is just an appearance if all universes are bubbles that never actually flatten out. It looks like approaching the spherical earth from space until you get close enough that the sphere starts looking flatter. So bubbles, where a bubble can look flat from one perspective. But for all bubbles to ever get to literally flat, like I said, it seems to follow that they'd all merge into one instead of remaining as individual. I'm curious what the cosmological (standard model) answer to this inquiry can entail. If you look back, I posted a diagram explaining the origin of the Concordance Model, in response to something mentioned by Pantheory. The lowest yellow box (Lambda Cold Dark Matter) is today's currently accepted model for mainstream cosmology. If that's what you meant by the standard model, than I have to inform you that in terms of evidence and data, it says nothing about a multiverse. An examination of the function of the diagram shows that each stage in the development of our cosmological theories was prompted by Observations (purple boxes, on the right) and Theories (blue boxes, on the left) that attempt to explain what was observed. So, starting with Observations, everything from Expansion of the Universe, down to Accelerating Expansion and Flat Universe deals with things within our observable universe. How could it be otherwise? Theories, from General Relativity and the Copernican Principle, down to Inflation are the theories that are used to explain the right-hand column, what is observed within our universe. And this could be the point where you're getting confused. By definition, it impossible for us to say whether other universes are expanding, what their elemental abundances are, what their galaxy rotation curves are, whether or not they have their own CMB's, whether their expansions are accelerating or if they are flat universes or not. That covers the Observations column. That being so, it logically follows that the theories designed to explain these things in our universe cannot be used to say anything meaningful about other universes. That covers the Theories column, but with one important caveat. The last box, Inflation, does have the potential to predict that our universe is not the only one that there is. However, there is a very serious and immovable roadblock in the way of rigorous science, when it comes to an Inflationary multiverse. Scientific rigor demands that a theory be testable by experiment. Clearly, in the case of cosmology, that does not mean experimentation in a Earthbound laboratory. No, what is meant is that, to be properly scientific, a theory must make testable predictions that can be confirmed or ruled out on the basis of observed evidence. Where no observations are possible, no testing of the theory in question is possible. The theory is therefore impossible to verify or falsify. The merit of such a theory is, to say the least, very questionable. When it comes to what lies beyond the visual limits of our own universe, Inflation is just such an un-testable theory. As is Roger Penrose's Conformal Cyclic Cosmos theory or Paul Steinhardt's Ekpyrotic theory. None of these three theories can be falsified or verified by observations made of what lies outside of our universe. They stand or fall on the basis of what they predict about this universe. Josh, to reinforce the point I've just made, think back to our discussions about William Lane Craig's cosmology and why he cannot use Hawking and Penrose's singularity theory. If you recall, that theory was falsified in 1998 by supernova observations. That data came from within our universe, not from outside of it. That's the only standard that counts - if you want to talk rigorous cosmology. I hope that this does give you some kind of answer. Thank you. Walter.
  13. This might be of interest to you, Justus. This graphic shows the surface of the Earth displayed first as a flat plane. Like any ordinary map, the dimension of height is compressed down to zero. So, even Mount Everest has no height in the Earth on the left. In the next step the sides of the Earth are folded round and in the last one on the right they seamlessly meet up and the flat, square Earth is transformed into a hollow sphere. There's nothing inside the sphere, because the dimension of height (or depth) has instead been compressed into the surface. This is how scientists usually display the geometry of the universe, using the balloon analogy. They compress one dimension into a surface. Therefore, anything that exists inside the three dimensions of the sphere (but not in the interior volume of it) can never reach or find a boundary or edge. Such a universe is said to be 'unbounded' to anyone living in it. Which is why cosmology doesn't require a hard, physical edge to compress or decompress its contents. Thank you. Walter.
  14. That's fine, Pantheory. Just for the record, since my declared position on cosmology is one of wait-and-see, I hope you'll understand that I was asked by JoshPantera to comment on LogicalFallacy's opening post. By doing that I'm just explaining things, as best as I understand them and not necessarily claiming that this or that theory is the right one. Thank you. Walter.
  15. Pantheory, The word 'discordances' isn't really nebulous. It stems from the usage of the Lambda Cold Dark Matter model being referred to as the Concordance Model of Cosmology. Therefore, data which doesn't seem to fit this model is said to generate 'discordance'. References 17 and 18 show how the words are used. https://arxiv.org/abs/1806.04649 Concordance and Discordance in Cosmology https://arxiv.org/pdf/1703.05959.pdf Planck Confronts Large Scale Structure : Methods To Quantify Discordance Thank you. Walter. https://www.researchgate.net/figure/How-the-Concordance-Model-of-Cosmology-was-developed-Theories-and-observations-motivated_fig2_308806912
  16. https://www.nature.com/articles/s41550-019-0906-9.epdf?referrer_access_token=qwhMEJ0wzDHdfB97wJVustRgN0jAjWel9jnR3ZoTv0NqA1r1vCcOgNn4LDpENPnwoeHNKdiccffHhZYAzAfWirButjQZIlE_kiDrG2iFXHU26guoLHtcK-8Bd16B0iBJXq0caiLWtJzstN0I7uk0LQxGEBosJTSceW-apXV_94PvsFmpVcGaw28tdjMSFwnpofsTl3o-XTORTfdjxoPw_sOoF3uj6pyMdfiy1SOFQKs%3D&tracking_referrer=www.livescience.com Above is a link to the paper that this thread is all about. I can't seem to copy and paste from the body of the text, so I'd like to draw attention to a sentence beneath paragraphs beneath Figure 8, on page 6. "Fortunately, future measurements will fully confirm or falsify current tensions and the PL18 evidence for curvature." This agrees with the closing comments of my last post, yesterday. 'The issue of Closed, Flat or Open is far from settled.' In a nutshell, the jury is still out and we need more and better measurements to discover what the universe is telling us. Thank you. Walter.
  17. Josh, Agreed. There's a lot to go through in the stuff you've linked to and I don't have time right now to do it all justice. Also, if you don't mind, I'd like to save this until later on, because we still have other aspects of cosmology to cover and then the epistemic and philosophical issues of how Craig uses and misuses the science. So, thanks for this and we will get back to it, I promise you. As to Craig making serious errors, if you recall I said that he'd shot himself in the foot by using a theory (H & P's 1970 singularity paper) that had been falsified before he went on online with his arguments at Reasonablefaith.com. Well, in the body of what I posted yesterday is material that I can use to show how Craig has shot himself in the other foot! (Ouch!) Before I can actually say where and how, it's best if you and Disillusioned read and digest what I've written. Thank you. Walter.
  18. Hello again Disillusioned. I've been giving some thought to one aspect of this thread, which you've outlined for us over several posts. Specifically, the differences between ontological and epistemic truths. To be honest, while I've only taken a brief look at the links you supplied and haven't gone into their content deeply. So, the thinking I've been doing is really only related to what's been said here, rather than stemming from the material contained in those links. Ok, confession over. If I were to sum up the extent of my thinking, I reckon it would come to this. Given what I know about cosmology and science in general, it seems to me that almost everything we think we know about physical reality must fall within the realm of epistemic truth. And conversely, the entire body of ontological truth that is available to us must therefore be really rather small. I say this because almost everything to do with astronomy and cosmology is based upon inference and assumption and not not direct observation or measurement. In a similar way, a great deal of geology must follow the same pattern. We have penetrated only a few miles into the Earth with our drilling and our mines. All of the other information we've gleaned about the interior of the Earth is either gained forensically or by seismological probing. This suggests to me that ontological truths must occupy a realm that is very closely linked to our place of existence - currently, the land, the sea, the air, low earth orbit and (arguably) the Moon. However, would I be right in thinking that these ontological truths cannot be directly linked to our sense experience? I ask this because, if were unaware that a grand piano was falling on to me, the existence of it would still constitute a brutal ontological truth, regardless of my lack of sense experience of it. So, when it comes to ontological truths, what do you think about my two ideas, Disillusioned? Thank you. Walter. p.s. Ooops! Just re-thought my argument! A black hole destroying the Earth would be a VERY brutal ontological truth. So, please factor that into the mix.
  19. This graph illustrates the conundrum faced by cosmologists who are trying to find out if the entire universe is Closed, Flat or Open. Even though four different possibilities are shown, please look closely at what we see can of each of them, Now. Yes, they all look the same to us. By looking at the curvature of space inside the observable universe, we simply cannot tell which type of curvature entire universe possesses. The size of the red ellipse (or circle) in the second and third diagrams I posted is far, far too small to see even the slightest degree of curvature. Even though the grid of squares I used to represent the wider universe looks flat, it could curve closed or open on scales far larger than I can show on any computer screen. Think of an ant trying to find out if the planet Earth is flat or curved by measuring what it can see from the top of a blade of grass. It's just impossible! So, when it comes to trying to find answers, we are forced to rely upon what we can see within the observable universe. To do that we have to use measurements of things like the Cosmic Microwave Background Radiation. Sure enough, the NewScientist article is about the 2018 data release from the Planck satellites' measurements of the CMBR. Catch is, for all the precision of that data, there's still enough leeway for different interpretations and there are still many assumptions involved in making these interpretations. The issue of Closed, Flat or Open is far from settled. Thank you. Walter.
  20. The simple answer to the question, Is the universe Closed, Flat or Open? is that it's too soon to say for sure. Our observable universe is simply too small a volume of space for us to say for sure, as of today. It's possible that future observations may pin the answer down, but as far as I'm concerned, the jury's still out. I tend to take what the NewsScientist says with a pinch of salt. They have something of a reputation for sensationalizing issues and over-simplifying complex topics. Anyway, why is the observable universe too small? Well, since Inflation is the currently-accepted paradigm of mainstream cosmology, I'll use that to illustrate why we don't measure up.
  21. I was posting something about the Cosmological Constant in the 'Failed Cosmology' thread, when you posted this Josh, so I'm in a good place to answer your question. I'll do it with a series of diagrams. Cheers. Walter.
  22. Ok guys. Here's something about the Cosmological Constant that not only explains something about it but also pulls it together with other stuff you may have heard of. Closed, Flat and Open universes. The Big Crunch, Big Freeze and Big Rip. https://en.wikipedia.org/wiki/Cosmological_constant The Wiki article does a reasonable job of explaining what the cosmological constant (hereafter, CC) is, but I think it helps to understand a bit more about the historical context of Einstein’s dissatisfaction with the field equations of General Relativity (hereafter, GR). These days most people understand that the Milky Way galaxy is just one of many galaxies and that they are all in motion, moving with respect to each other. But this was not the case in first twenty years of the 20th century. For centuries before that, men of science believed that the universe was static, unchanging and eternal. New stars were born to replace those that died, but the overall shape and behaviour of the universe didn’t change on the largest scales. These ‘largest scales’ were also different from what we understand today. Up until 1923 it was generally believed that what we call the Milky way galaxy was the entire universe. The very concept of an individual galaxy wasn’t even part of mainstream cosmology. Various spiral-shaped smudges of light could be seen, but the telescopes of that time couldn’t tell whether these were small, nearby gas clouds spinning around a central core or hugely distant, spiral systems of billions of stars. https://en.wikipedia.org/wiki/Galaxy In 1917, Heber Curtis observed nova S Andromedae within the "Great Andromeda Nebula" (as the Andromeda Galaxy, Messier object M31, was then known). Searching the photographic record, he found 11 more novae. Curtis noticed that these novae were, on average, 10 magnitudes fainter than those that occurred within our galaxy. As a result, he was able to come up with a distance estimate of 150,000 parsecs. He became a proponent of the so-called "island universes" hypothesis, which holds that spiral nebulae are actually independent galaxies. Einstein was writing his theory of GR before it was generally accepted that our galaxy is not the whole universe. He held to the idea that there was no large-scale motion to be seen in the universe (i.e., only the Milky Way) and was therefore troubled when his GR equations didn’t successfully describe the static universe, he believed existed. Because gravity is exclusively attractive in GR, his calculations lead to an unstable universe that collapsed inwards upon itself. This was not the result he wished to see. So, he postulated something new. He added an ‘anti-gravitational’ term to his equations, trying to exactly balance the rate of the universe’s collapse, with an opposing, repulsive force of equal strength. This was the cosmological constant. In the mid 1920’s, when Edwin Hubble found evidence that ours was not the only galaxy and that they were all moving away from each other Einstein dropped the CC from his GR equations. There was no longer any need to postulate the CC to achieve a static universe. Observations clearly showed that the universe was not static on the largest scales. It wasn’t until the 1980’s and the advent of Inflationary theory that cosmology entertained the idea that gravity could be reversed and made to ‘push’ instead of ‘pull’. But that’s another story. In the mid 1960’s Stephen Hawking realized that Roger Penrose’s collapsing star (black hole) singularity theory had an interesting feature that he could use in a novel way. Because Einstein’s GR equations do not have a fixed and absolute frame of reference in either time or space, the direction of time could be reversed in the calculations, without disturbing the accuracy of the results. So, if a collapsing star of high enough mass inevitably leads to a black hole and a singularity, then the time-reversed version of that scenario is equally valid. Just as a black hole pulls in space-time, matter and energy, so a time-reversed version would ‘push out’ space-time, matter and energy. In GR, both scenarios are equally valid. In 1969 Hawking and Penrose worked together to write their 1970 paper, ‘The Singularities of Gravitational Collapse and Cosmology’, concluding that according to only GR, the universe must have had a definite beginning, with an initial singularity as the ultimate source of all the entire universe. One of the advantages of using a time-reversed singularity was that Hawking and Penrose could partially explain why the universe was expanding, with galaxies moving away from each other. The following questions highlight the problem. If gravity is always attractive in GR, why is it that we observe the universe expanding? What was it that caused the galaxies to begin moving apart in the first place? Surely there’s nothing in GR that can cause gravity to ‘push’ and not ‘pull’? Einstein dropped the CC from his GR field equations and it played no part in cosmology up until the 1980’s. So how did Hawking and Penrose get GR to ‘push’ and describe an expanding universe, back in 1970, without using the CC? Here’s their solution. They realized that if in a normal, forward-moving time frame a black hole singularity always ‘pulls’, then in a reversed and backward-moving time frame, the opposite kind of singularity must always ‘push’. Their time-reversed singularity provides the necessary ‘push’ that causes the universe to expand and causes the galaxies to begin moving away from each other. After the initial push of the Big Bang, they expected the universe to do one of two things. The mutual gravitational attraction of all the matter in the universe might be enough to eventually overcome the inertia of the initial push, leading to the expansion slowing down, stopping and then reversing into a contraction. This is called the Closed Universe scenario. The shrinking universe becomes hotter and hotter as its contents are squeezed more and more tightly together. The popular name for this is the Big Crunch. https://en.wikipedia.org/wiki/Big_Crunch Or, the gravity of the universe’s matter might exactly equal the force of the initial push, leading to a Flat universe. Here, the galaxies coast slowly apart from each other forever. The universe doesn’t collapse, but expands and cools, eventually becoming a cold and sterile void. This is called the Big Freeze. https://en.wikipedia.org/wiki/Future_of_an_expanding_universe As we now know, neither Hawking nor Penrose were expecting a third option – a scenario where the universe’s expansion would be accelerated by some kind of ‘pushing’ force. Enter the Cosmological Constant in 1998, after an absence of 75 years. Their singularity theory was configured to work only in a Closed or Flat universe, where GR is only ‘pulling’ and gravity is only attractive. But the CC is an anti-gravitational force, that ‘pushes’, speeding up the expansion of the universe. It throws a monkey wrench into the gears of GR and makes the H – P singularity theory unworkable in our accelerating universe. Any universe with a positive CC is known as an Open universe and may well lead to what is commonly known as the Big Rip. https://en.wikipedia.org/wiki/Big_Rip Thank you. Walter.
  23. As you will see Justus, the size of the dots, stars and galaxies on the surface of each balloon doesn't change. They get no bigger or smaller. But the area of the balloon's surface between them does expand. Now, reverse the expansion and whatever material (hot gas in the very early universe) is in any square of the balloon's surface will be compressed and will condense. It's not being compressed by any kind of edge or boundary, because the surface of a sphere has no edge or boundary. As each square gets smaller, the material within it compresses further and heats up. Eventually, if you reverse the expansion enough, all the material in the universe will be so compressed that it will become an ultra-dense, super-heated plasma. This is the hot, dense state that corresponds to the earliest instants after the Big Bang. So, your point about condensing was relevant, but you were thinking in terms of what happens inside the balloon. As I've just explained, compression or expansion happens within the balloon's surface and not its interior volume. Therefore, cosmology has no need to posit a hard 'edge' to the universe. Also, every square on the balloon's surface is exactly the same as every other. The view from each one is exactly the same as every other. No particular square is more important or more central than any other. They all experience the same amount of compression or expansion as each other. This equality satisfies the Copernican Principle. Thank you. Walter.
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