
Welcome to the International Skeptics Forum, where we discuss skepticism, critical thinking, the paranormal and science in a friendly but lively way. You are currently viewing the forum as a guest, which means you are missing out on discussing matters that are of interest to you. Please consider registering so you can gain full use of the forum features and interact with other Members. Registration is simple, fast and free! Click here to register today. 
25th June 2023, 04:46 PM  #281 
Illuminator
Join Date: Nov 2020
Posts: 4,240


__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

25th June 2023, 04:57 PM  #282 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696

Can you explain exactly and mathematically how the FLRW metric is incompatible with general relativity?
I don't care in the slightest whether you trust me. You don't have to trust me  just read and understand (I realise the latter is impossible for you) the standard text books and stop cherry picking and/or misunderstanding your experts. 
__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

25th June 2023, 05:04 PM  #283 
Philosopher
Join Date: Oct 2009
Posts: 5,497

Those are the same authors who say And yet, in ΛCDM the timetime component of the metric, g_{tt}, is set at 1 at all times and stages of cosmic evolution, even under conditions where it may not be mathematically robust to do so.That's quite a silly thing for those authors to say, because g_{tt} = − c^{2} in this familiar metric form: ds^{2} = − c^{2} dt^{2} + dx^{2} + dy^{2} + dz^{2}What this shows is that those authors were speaking imprecisely. Everyone who knows what they're talking about is aware that the numerical value of g_{tt} depends on the coordinate system you prefer. In particular, as that metric form shows, its value depends on the units you prefer to use. The authors who said that silly thing would agree with what I just said, and would hasten to explain what they really meant when they wrote those silly words. It's safe to assume the author and sole proponent of Helland physics can't explain what those authors really meant when they wrote those silly words. 
25th June 2023, 05:10 PM  #284 
Illuminator
Join Date: Nov 2020
Posts: 4,240


__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

25th June 2023, 05:34 PM  #285 
Illuminator
Join Date: Nov 2020
Posts: 4,240


__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

25th June 2023, 05:42 PM  #286 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696


__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

25th June 2023, 05:43 PM  #287 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696

NT

__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

25th June 2023, 08:29 PM  #288 
Philosopher
Join Date: Oct 2009
Posts: 5,497

correction
Correcting the highlighted word:
That paper was written in French (not Dutch), but was published in an obscure Belgian (not Dutch) journal. In 1931, after Arthur Eddington had described it as a "brilliant solution", part of the paper was translated into English and Lemaître, building on that paper, proposed a "Primeval Atom", which is now more commonly known as the Big Bang. ETA: Not long before Lemaître's death, he learned that cosmic background radiation from that "Primeval Atom" had been detected. 
25th June 2023, 10:39 PM  #289 
Illuminator
Join Date: Nov 2020
Posts: 4,240


__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 12:56 AM  #290 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696

Suit yourself. We all know that you don’t understand what you are parroting.
Also the fact that you still don’t understand the difference between dimensions and units, and you still don’t understand that there was no contradiction between the two statements is because you haven’t a clue about any of this stuff. You can carry in talking to yourself now. 
__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

26th June 2023, 01:39 AM  #291 
Illuminator
Join Date: Nov 2020
Posts: 4,240

You're the one who said (in one post) that it had dimensions and units of inverse seconds.
Melia's argument that LCDM is incompatible with with the Einstein equivalence principle is based on the change of dominating ingredient throughout the history of the universe. In times when it's accelerating and times when its decelerating, I think he's suggesting that the nonfree fall and the free fall associated with each can't both be described g_{tt}=1 (or c^{2} if you prefer). That of course wouldn't apply to a pure cosmological constant FLRW model such as de Sitter space. However, Willem de Sitter said that g_{tt}=1 is fundamentally at odds with relativity itself. Here is a quote from one of his papers I already mentioned:
Quote:
https://ui.adsabs.harvard.edu/link_g....1217D/PUB_PDF On the relativity of inertia. Remarks concerning Einstein's latest hypothesis Also by de Sitter in 1917.
Quote:
Quote:

__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 03:55 AM  #292 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696

Indeed I did, and I have already said that that was a sloppy statement and corrected it. What I should have said was that, as defined by the relationship H=ȧ/a, H has dimensions of T^{1} and units of s^{1}. That is not inconsistent with the statement that as normally understood in cosmology H is given in units of km s^{1}Mpc^{1}, which is is an expansion rate or speed per unit distance. I have already also stated several times that you can quote in H in any units you like provided they have a dimension of T^{1}, but if you change the units then the numerical value will change.
Quote:
You haven't begun to explain why any of this leads one to conclude that LCDM is incompatible with the Einstein equivalence principle. Can you state in your own words the logic that might lead one to that conclusion? Can you state why you think that galaxies are not moving on geodesics in de Sitter space?
Quote:
Quote:

__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

26th June 2023, 08:07 AM  #293 
Philosopher
Join Date: Oct 2009
Posts: 5,497

Amusingly, that is what Mike Helland concludes from a 1917 paper in which de Sitter explicitly acknowledges two exact solutions (Einstein's static universe and Minkowski spacetime) for which g_{tt} = c^{2} holds everywhere.
In 1917, Einstein and de Sitter were borderline obsessed with the problem of reconciling the theory of general relativity with Mach's principle. They weren't yet completely free of the Newtonian legacy that led them to think of the coordinatedependent components g_{μν} as components of a gravitational field, and they were also limited by their understanding of absolute differential calculus, a branch of mathematics that was still evolving into modern differential geometry. That is why de Sitter's 1917 papers take g_{μν} = 0 at infinity (where they assumed there is no matter) as a boundary condition. With the modern understanding of pseudoRiemannian manifolds, g_{μν} = 0 is forbidden because the metric tensor must be nondegenerate. In effect, de Sitter was actually trying to build a coordinate singularity into his coordinate systems. This was of course several years before those pioneering relativists understood the distinction between coordinate singularities and genuine singularities. Their thinking evolved. In 1932, Einstein and de Sitter coauthored a twopage paper describing an FLRW model that came to be known as the Einsteinde Sitter universe. Their equation (1) is the standard FLRW metric form for an expanding universe with flat space, for which g_{tt} = c^{2} everywhere. For the next sixty years, that Einsteinde Sitter model reigned as one of the leading models of the universe, until it was dethroned in the 1990s by accumulation of new data and the Nobelprizewinning research of James Peebles. It is therefore quite amusing that Mike Helland rejects the FLRW metric form of Einstein and de Sitter's equation (1), and continues to insist that FLRW models are incompatible with relativity, ignoring derivations of the FLRW models directly from the field equations as found in standard textbooks and even within the archives of this forum. 
Last edited by W.D.Clinger; 26th June 2023 at 08:18 AM. Reason: added word in gray, corrected a typo 

26th June 2023, 09:16 AM  #294 
Illuminator
Join Date: Nov 2020
Posts: 4,240

Makes sense. As I noted, his cosmology's equation for luminosity distance is the same as mine, so they are equal fits to the supernovae data.
Quote:
"While this simplifies the metric coefficients greatly, it ignores the important fact that the metric coefficients depend on the chosen stressenergy tensor, which changes as the universe evolves through phases of deceleration (when matter and radiation dominate) and acceleration (when dominated by inflation and dark energy)."
Quote:

__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 09:22 AM  #295 
Penultimate Amazing
Join Date: Aug 2007
Location: The Antimemetics Division
Posts: 67,130


__________________
There is no Antimemetics Division. 

26th June 2023, 09:28 AM  #296 
Illuminator
Join Date: Nov 2020
Posts: 4,240

If you accept the expansion of the universe, and you accept that it rewinds back to a singularity, then a cosmic time is a simple, convenient, and "natural" choice of coordinates.
You also are admitting a time coordinate that is not completely relative. It seems your argument is that since this is how GR is applied in cosmology, it is therefore compatible with GR. That is to say, canonically, FLRW is compatible with GR. It is because we say it is. But cosmic time is not completely relative. The derivations of FLRW all have in common that you deliberately choose a time coordinate that's not entirely relative. That might be how GR is used. But that doesn't get around the fact that you've abandoned what de Sitter calls the "mathematical postulate of the relativity of inertia." 
__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 09:36 AM  #297 
Penultimate Amazing
Join Date: Jun 2003
Posts: 55,286


__________________
"As long as it is admitted that the law may be diverted from its true purpose  that it may violate property instead of protecting it  then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and allabsorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious."  Bastiat, The Law 

26th June 2023, 09:45 AM  #298 
Illuminator
Join Date: Nov 2020
Posts: 4,240

de Sitter:
Quote:
That's system A. He says that his system B is "entirely relative" and "completely relative" at various points.
Quote:
Quote:

__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 10:13 AM  #299 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696

He knows what he’s talking about and is able to fully make his arguments in mathematical terms. You, not at all. Don’t even dream of thinking your silly ideas are in any way equivalent to his.
Quote:
Quote:
Quote:

__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

26th June 2023, 10:23 AM  #300 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696

You seem to be ignoring the fact that FLRW models are derived from GR, and that both Einstein and de Sitter had no problem in proposing a flat expanding metric (which is an FLRW metric) in that 1930s paper WDC referenced. It’s clear that their thinking evolved and you’re quote mining an earlier and flawed interpretation, because that’s just what crackpots do. This is the very definition of quote mining  using quotes from authorities which appear to support you pov, either taken out of context or subsequently abandoned, and where the authority would certainly not agree with the crackpot’s proposition. Just like a YEC, and not for the first time.
What do you actually mean by the statement that “cosmic time is not relative”. Could you define what you mean precisely? ETA: or perhaps you think that by the early 1930s de Sitter and Einstein were geriatric drooling idiots. 
__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

26th June 2023, 10:28 AM  #301 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696

And while we’re about it could you do the converse and explain what a relative time would be in a homogeneous and isotropic universe?

__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

26th June 2023, 10:30 AM  #302 
Penultimate Amazing
Join Date: Jun 2003
Posts: 55,286

So what? Relativity requires that the laws of physics be invariant. It does not require that the universe itself be invariant, and it obviously cannot be. A reference frame where mass is on average stationary is unique, and can be differentiated from reference frames where mass is on average not stationary. This is unavoidable, but it is not the problem you imagine it to be.
Quote:

__________________
"As long as it is admitted that the law may be diverted from its true purpose  that it may violate property instead of protecting it  then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and allabsorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious."  Bastiat, The Law 

26th June 2023, 10:41 AM  #303 
Illuminator
Join Date: Nov 2020
Posts: 4,240

Well, you can read it yourself, and you tell me. I don't see where a time varying cosmological constant is proposed tho:
Quote:
As for his cosmology and my idea, of course they are not the same. But they have the same luminosity distance equation, which he argues is a better fit to the data along a different line of reasoning, involving the "cosmic distance duality relation": https://arxiv.org/pdf/1804.09906.pdf 
__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 10:45 AM  #304 
Illuminator
Join Date: Nov 2020
Posts: 4,240


__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 10:50 AM  #305 
Illuminator
Join Date: Nov 2020
Posts: 4,240

Wouldn't it be this:
? I think this covers a part of the manifold centered on an observer. So to the observer, I think you will argue it wouldn't be exactly homogeneous. But it's inhomogeneous in the same way our observation of the universe is inhomogeneous: the farther we look out, the more time dilation and redshift we see. 
__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 10:50 AM  #306 
Penultimate Amazing
Join Date: Jun 2003
Posts: 55,286

Again, so what?
Why is that a problem?
Quote:

__________________
"As long as it is admitted that the law may be diverted from its true purpose  that it may violate property instead of protecting it  then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and allabsorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious."  Bastiat, The Law 

26th June 2023, 10:56 AM  #307 
Illuminator
Join Date: Nov 2020
Posts: 4,240


__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 11:06 AM  #308 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696


__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

26th June 2023, 11:10 AM  #309 
Penultimate Amazing
Join Date: Jun 2003
Posts: 55,286


__________________
"As long as it is admitted that the law may be diverted from its true purpose  that it may violate property instead of protecting it  then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and allabsorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious."  Bastiat, The Law 

26th June 2023, 11:35 AM  #310 
Illuminator
Join Date: Nov 2020
Posts: 4,240

Yeah. I see that.
Although in the Schwarz metric, r > r_{s} and there is mass present. In this case r < a. That equation is from: https://physics.stackexchange.com/qu...timedilation
Quote:
W. de Sitter's line element (8B) is: I suppose through trigonometry they're equivalent? Haven't fully worked that part out yet. 
__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 11:41 AM  #311 
Penultimate Amazing
Join Date: Aug 2007
Location: The Antimemetics Division
Posts: 67,130


__________________
There is no Antimemetics Division. 

26th June 2023, 11:55 AM  #312 
Illuminator
Join Date: Nov 2020
Posts: 4,240

If you're comfortable making a special exception for nonrelative time in FLRW, that's cool.
I'm just pointing out, that de Sitter pointed out that in system B, time is completely relative. If you set aside cosmology, the story of the universe's beginning and such, and look at the rest of physics, where experiments take place, nonrelative time isn't necessary. If alien's ever showed up for dinner, and started going through our science textbooks, they would probably look at the laws of motion, gravity, electromagnetism, chemistry, GR, QM, and say "well, it looks like you're coming along." But when we get to cosmology, and tell them how old the universe is, are they going to be like "yeah, dude, 13.8 billion years!" Or will they be like "wut?" You could throw cosmology out in its entirety and it would have no affect on any other field of physics, the ones where controlled experiments take place. The one's that are useful. We make a special exception for the quasiabsolute cosmic time because it feels natural in a universe that has a beginning of time. I get that the description of the universe at large is a majorly active area in physics. I would suppose the combination of the human need for a creation story and the lack of any real controlled experiments is what promotes the robust amount of theories and interpretations. At some point, though, as interesting as all this is, one has to ask, is any of it real? Did galaxies appear a few hundred million years after the beginning of time fully formed? Maybe this cosmic time is not only unnecessary, but straight up wrong. 
__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 01:40 PM  #313 
Penultimate Amazing
Join Date: Jun 2003
Posts: 55,286

What exception do you imagine is required? I don't think you actually know what you're talking about.
Quote:
Quote:
Quote:
Quote:
Too bad you don't.
Quote:
You don't understand what you're talking about. You never have. 
__________________
"As long as it is admitted that the law may be diverted from its true purpose  that it may violate property instead of protecting it  then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and allabsorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious."  Bastiat, The Law 

26th June 2023, 03:39 PM  #314 
Illuminator
Join Date: Nov 2020
Posts: 4,240


__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 09:40 PM  #315 
Penultimate Amazing
Join Date: Jun 2003
Posts: 55,286


__________________
"As long as it is admitted that the law may be diverted from its true purpose  that it may violate property instead of protecting it  then everyone will want to participate in making the law, either to protect himself against plunder or to use it for plunder. Political questions will always be prejudicial, dominant, and allabsorbing. There will be fighting at the door of the Legislative Palace, and the struggle within will be no less furious."  Bastiat, The Law 

26th June 2023, 10:24 PM  #316 
Illuminator
Join Date: Nov 2020
Posts: 4,240

Interesting new paper:
https://iopscience.iop.org/article/1...418213/acdb49 Distance Duality Test: The Evolution of Radio Sources Mimics a Nonexpanding Universe Distance duality relation (DDR) marks a fundamental difference between expanding and nonexpanding universes, as an expanding metric causes angular diameter distance smaller than luminosity distance by an extra factor of (1 + z). Here we report a test of this relation using two independent samples of ultracompact radio sources observed at 2.29 GHz and 5.0 GHz. The test with radio sources involves only geometry, so it is independent of cosmological models. Since the observed radio luminosities systematically increase with redshift, we do not assume a constant source size. Instead, we start with assuming the intensive property, luminosity density, does not evolve with redshift and then infer its evolution from the resultant DDR. We make the same assumption for both samples, and find it results in the same angular size–redshift relation. Interestingly, the resultant DDR is fully consistent with a nonexpanding universe. Imposing the DDR predicted by the expanding universe, we infer the radio luminosity density evolves as ρL ∝ (1 + z)3. However, the perfect agreement with a nonexpanding universe under the assumption of constant luminosity densities poses a conspiracy and finetuning problem: the size and luminosity density of ultracompact radio sources evolve in the way that precisely mimics a nonexpanding universe. 
__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

26th June 2023, 10:39 PM  #317 
Philosopher
Join Date: Oct 2009
Posts: 5,497

Abstract
In 1917, Willem de SItter published the first description of an FLRW model for a universe that is expanding at an exponential rate. That FLRW model has come to be known as de Sitter space. In 1917, it was philosophically fashionable to assume the universe in which we live is static, so de Sitter described the space using a particular set of static coordinates. Those static coordinates present a misleading impression of the space, because the metric form of de Sitter's equation (8B) is not spatially homogeneous: the g_{tt} component of the metric tensor depends upon one of the spatial coordinates. The author and sole proponent of Helland physics has been citing and quoting two different metric forms for de Sitter space, without understanding how those two metric forms assume two distinct systems of static coordinates. He has also been failing to understand that the metric tensor described by those two metric forms is an FLRW metric. The homogeneity and isotropy of de Sitter space becomes obvious when that metric is written as an FLRW metric form. Warning What follows is mostly first year calculus. The author and sole proponent of Helland physics won't understand it, but scientifically literate readers who have been reading this thread for laughs might enjoy it. As for the sentence I highlighted: Yes, the theory of relativity allows you to select a coordinate system you prefer to use. That's why it's called the theory of relativity. As this thread and its predecessor have demonstrated again and again, some coordinate systems are less misleading than others. Although all admissible coordinate systems can be used to reason correctly about the physics, a poor choice of coordinate system makes the physics considerably harder to understand. If your knowledge of general relativity and cosmology is marginal to begin with, so you struggle to understand even the best choices of coordinate system, then a poor choice of coordinate system is going to make the physics even harder for you to understand, to such an extent that you will almost certainly make mistakes. Which explains why individuals whose hobby horses require them to make such mistakes are especially likely to prefer a poor choice of coordinate system. For example: No, that is not the Schwarzschild metric. It is a metric form for de Sitter space, expressed using a certain set of static coordinates that (misleadingly) makes the space appear to be inhomogeneous: the g_{tt} component depends upon the spatial coordinate r. Because de Sitter space really is homogeneous and isotropic, and that particular choice of coordinates really does make the progress of time depend on one's spatial location, Mike Helland actually gave a fairly good answer to hecd2's challenge. But Mike Helland came up with that metric form via his usual cargo cult process of searching the web for something he can misinterpret as support for his mistakes. He doesn't understand how that particular coordinate system and that particular metric form are related to the coordinate system and metric form in de Sitter's paper from 1917: I can help with that. Theorem Metric form 1 is expressed using de Sitter's static coordinates from his 1917 paper, and is equation (8B) of that paper. Its sign convention is (− − − +). The g_{tt} component depends on the spatial coordinate r, so these static coordinates obscure the homogeneity and isotropy of the model. Metric form 2 is expressed using a different set of static coordinates, a different sign convention (− + + +), and the abbreviationOnce again, the g_{tt} component depends on the spatial coordinate r, so this second set of static coordinates also obscures the homogeneity and isotropy of the model. Metric form 3 is expressed using spherical spatial coordinates in which the homogeneity and isotropy of the model become evident. Metric form 4 is expressed using Cartesian spatial coordinates and the (− + + +) sign convention. Once again, the homogeneity and isotropy of the model are evident. Metric forms 3 and 4 are instances of the familiar FLRW metric form. With metric form 3, the spatial slices are hyperspherical, with constant positive curvature. With metric form 4, the spatial slices are flat (Euclidean), with zero curvature. How can they be describing the same metric tensor? The Ricci and Riemann curvature tensors are invariants, independent of the coordinate system you choose to use, but their numerical components depend on that coordinate system. Furthermore, they are defined on the 4dimensional spacetime manifold. In relativity, the division of 4dimensional spacetime into one time dimension and 3 spatial dimensions is not absolute. (That's part of why we call it the theory of relativity.) In some cases, that limited choice of what counts as time and what counts as space allows your choice of coordinate system to push curvature into spatial dimensions if you so desire, or into the time dimension if that is what you prefer. I will now sketch a proof that metric form 1 (de Sitter's equation (8B) from 1917) describes the same metric tensor as metric form 2. (Mike Helland guessed that their equivalence could be proved using trigonometry, but he guessed wrong.) Their equivalence is proved by a change of variables, just as in my proofs that the (revised) Helland metric form and the TDP metric form are nothing more than obfuscations of the familiar Minkowski metric. Sketch of proofTo prove the rest of the theorem, I'm going to let Wikipedia do the work. Metric form 2 is the metric form of Wikipedia's static coordinates for de Sitter space. Wikipedia gives equations that define another set of coordinates x_{i} in terms of those static coordinates. (Wikipedia's presentation actually starts with the x_{i} coordinates and defines the static coordinates in terms of those x_{i}, but we can go in the other direction.) Metric form 3 is obtained from metric form 2 via transformation to Wikipedia's closed slicing coordinates, which are defined (implicitly) by equations that have the x_{i} on the left hand side. Metric form 4 is obtained from metric form 2 via transformation to Wikipedia's flat slicing coordinates, which are also defined (implicitly) by equations that have the x_{i} on the left hand side. Metric form 4 is the metric form that makes it easiest to see that de Sitter space describes a universe that is expanding at an exponentially increasing rate. It is also the metric form that makes it easiest to see that the relationship between redshift and distance in a de Sitter universe coincides with the relationship postulated by Helland physics, for an allegedly nonexpanding universe (with absolutely no attempt to explain how a nonexpanding universe could possibly give rise to the same cosmological redshift as the exponentially expanding de Sitter universe). 
27th June 2023, 12:03 AM  #318 
Philosopher
Join Date: Oct 2009
Posts: 5,497

I have a couple of questions about that paper. In section 4, the paragraph bracketed by equations (10) and (11) makes three references to proper distance. Consulting a Wikipedia article for definitions of the various distance measures, it looks to me as though all three of that paragraph's references to proper distances should have referred to comoving distances. It seems to me that is a matter of some importance, because (in an expanding universe, in the context of that paragraph) proper distances differ from comoving distances by a factor of 1+z. Can you comment on this? I'd also be interested to hear your comments with regard to Figure 3 and its caption, whose concluding sentence says "The predictions of the expanding and nonexpanding universes are presented as the solid and dashed lines, respectively." From the figure, it is rather obvious that the solid line (which is said to be the prediction for an expanding universe) fits the data better than the dashed line (which is said to be the prediction for a nonexpanding universe). 
27th June 2023, 02:15 AM  #319 
Graduate Poster
Join Date: Oct 2013
Posts: 1,696


__________________
Gulielmus Princeps Haroldum Principem in catino canino impulit 

27th June 2023, 02:17 AM  #320 
Illuminator
Join Date: Nov 2020
Posts: 4,240

Very interesting. Thanks for the detailed, informative response.
So in eq1 we have a 1 (well, 1) as the coefficient for dr^{2}, and in eq2, that Schwarzschildy looking coefficient for it. The steps you give to get from one to the other make sense. Though I'm still working toward an intuition what's going on in both. The stack exchange answer gives a reason for why eq2 has static coordinates, as opposed to "length contracted" coordinates, but I'm still working out why that is. The reason given has to do with that integral, which wolfram alpha says equals an arcsin() expression.
Quote:
The one that I mentioned when I first cited the de Sitter Sightseeing Tour is: http://www.bourbaphy.fr/moschella.pdf Figure 8: Here time is defined by relation: From this you get that static time coefficient and the dynamic space coefficients that satisfy the equation for the hyperboloid. As the caption says, covers half the manifold. So I'm thinking, how about this: When t=0, , and as t goes to infinity, goes to R. If the standard FLRW metric were derived from this, there would be a short period of contraction, and then an indefinite period of expansion. Not very useful for expanding space. But if we could turn it around so space is static and time is dynamic, then this describes a time dilated past and time contracted future. What about choosing the time coordinate as the dynamic one, and the space coordinates as static ones? Could you still satisfy the equation for the hyperboloid? Does it help if you flip the hyperboloid on its side? I think that means you're in antide Sitter territory: ? Still learning about all that and working that out. One thing (out of many things) I find fundamentally confusing is whether it makes more sense to state as we have been, or instead makes more sense. The scale factor in FLRW takes t as the cosmic time. But the time scale factor in TDP () takes t as the measured time. The slicings are "hypersurfaces of equal cosmic time". So if the slicings should be of equal amounts, is it naive to think that means something like t=0, t=1, t=2, t=3? Because if that's what's up, it seems that should be using the latter, the ln() expression in the metric form. That probably might not make a ton of sense. Basically the time we measure (t) should be: Here I suppose takes the place of "constant cosmic time" in the slicings, but that doesn't end up being t in the resulting 4D spacetime. 
__________________
I'm not entirely sure what I'm talking about, but based on what little I know, the above seemed like a reasonable thing to say. Thank you in advance for any corrections. 

Thread Tools  

