The gist is that the manifold implied by the metric is really only good for one observer at r=0 and t=0.
And supposedly that isn't a "habitable" place.
I'm not sure why that is, unless you're making the perfect fluid assumption to get there.
But here's the thing... our current place in spacetime isn't habitable. That's congruent with our experience of physical reality. Something keeps propelling us into the future. In my metric, only at r=0 is the present, which is closer to the future than the other points. It should feel the "arrow of time" more sharply than other events in spacetime.
The idea is that the reality we observe isn't universe-shaped. The reality we observe is shaped like a 4D light cone. We live on the tip.
It's a consequence of thermodynamics that if you start with a low entropy state it will evolve toward a higher entropy state. But I don't think that justifies the statement that thermodynamics is propelling us toward the future.
That's an odd metaphysical claim whose meaning I can't even really parse here.
It's a consequence of thermodynamics that if you start with a low entropy state it will evolve toward a higher entropy state. But I don't think that justifies the statement that thermodynamics is propelling us toward the future.
That's an odd metaphysical claim whose meaning I can't even really parse here.
The physicist Sean Carroll explains in his book about the arrow of time that it appears as a result of thermodynamics. If there is no matter, there is no arrow of time.
The physicist Sean Carroll explains in his book about the arrow of time that it appears as a result of thermodynamics. If there is no matter, there is no arrow of time.
There is no such assumption. The impossibility of an observer at r=0 and t=0 surviving for a single microsecond is a mathematical consequence of Mike Helland's stated metric form and the Einstein field equations of general relativity.
That has been explained in considerable detail, but Mike Helland doesn't understand differential equations, so he doesn't understand the Einstein field equations, so he doesn't understand general relativity, so he doesn't understand modern cosmology, so he can only guess, which is what he did when he wrote "unless you're making the perfect fluid assumption to get there."
Mike Helland's guesses are often quite comical. Here's a typical example:
But here's the thing... our current place in spacetime isn't habitable. That's congruent with our experience of physical reality. Something keeps propelling us into the future. In my metric, only at r=0 is the present, which is closer to the future than the other points. It should feel the "arrow of time" more sharply than other events in spacetime.
The idea is that the reality we observe isn't universe-shaped. The reality we observe is shaped like a 4D light cone. We live on the tip.
There is no such assumption. The impossibility of an observer at r=0 and t=0 surviving for a single microsecond is a mathematical consequence of Mike Helland's stated metric form and the Einstein field equations of general relativity.
That has been explained in considerable detail, but Mike Helland doesn't understand differential equations, so he doesn't understand the Einstein field equations, so he doesn't understand general relativity, so he doesn't understand modern cosmology, so he can only guess, which is what he did when he wrote "unless you're making the perfect fluid assumption to get there."
Mike Helland's guesses are often quite comical. Here's a typical example:
There is no such assumption. The impossibility of an observer at r=0 and t=0 surviving for a single microsecond is a mathematical consequence of Mike Helland's stated metric form and the Einstein field equations of general relativity.
That has been explained in considerable detail, but Mike Helland doesn't understand differential equations, so he doesn't understand the Einstein field equations, so he doesn't understand general relativity, so he doesn't understand modern cosmology, so he can only guess, which is what he did when he wrote "unless you're making the perfect fluid assumption to get there."
The perfect fluid assumption gives us the following form of the energy-momentum tensor Tμν:
There are various parameters here but most strikingly the tensor is diagonal – it only has entries along its leading diagonal.
This is a manifestation of the isotropy we mentioned earlier – any off-diagonal elements would describe directional preferences of the pressure and energy density in terms of shear stresses, which would violate our isotropic assumption!
The first thing one does to understand modern cosmology is assume the universe is homogeneous and isotropic, thus creating a fantasy land which is obviously incongruent with observed reality.
There are no atoms, stars, galaxies, or empty space in the FLRW universe. Just some kind of indistinct jello.
Practitioners of this mythos are not likely to come back from their fantasy land, because once you've discarded empirical reality in favor of simplified equations, why would you?
The universe, as we observe it, is not homogeneous and it is not isotropic. The earth is below you and the sky is above you. Inhomogeneous and anisotropic. The sun isn't observed in all directions. Neither is Andromeda, or any other galaxy.
The FLRW model does not describe observed reality, and deliberately so.
The web page continues:
This is the interpretation for all energy-momentum tensors, not just our perfect fluid tensor.
In case you’re interested in understanding this in more detail, I cover the physical interpretations of each term in the energy-momentum tensor here.
In most cases, these shear stress components will be zero (since it’s common to study objects that could be considered as “perfect fluids“, which don’t have any internal stresses).
Whatever conclusions you've reached do seem to be based on a diagonal energy-momentum tensor, which seems to come from the perfect fluid assumption.
In any case, back to the first page:
The effect of this is “absorbing” the cosmological constant as an additional contribution to the energy density and pressure – we’re effectively just rescaling our variables by a constant amount.
We can see that it takes away from the pressure p(t), hence counting as a negative pressure!
In my metric, we stay in observable reality, which is neither homogeneous nor isotropic. We don't invoke a fantasy jello.
And instead of space "absorbing" the cosmological constant, it should probably be time that does that. If that makes any sense. I'm sure you'll say it doesn't. Only the jello makes sense to you.
The first thing one does to understand modern cosmology is assume the universe is homogeneous and isotropic, thus creating a fantasy land which is obviously incongruent with observed reality.
There are no atoms, stars, galaxies, or empty space in the FLRW universe. Just some kind of indistinct jello.
Seriously, you don't even understand the consequences of your own metric. You don't know what it would do. And you sure as hell can't compare it to observable reality. That's beyond your capabilities. So you just assume it agrees with reality, even though you've got nothing to indicate that this is so.
That's just crackpottery.
And instead of space "absorbing" the cosmological constant, it should probably be time that does that.
The first thing one does to understand modern cosmology is assume the universe is homogeneous and isotropic, thus creating a fantasy land which is obviously incongruent with observed reality.
Not at all. My conclusions were entirely based upon applying the most basic equations of general relativity to Mike Helland's metric form of 2 November.
Mike Helland's metric form is diagonal. That is why the stress-energy-momentum tensor implied by Mike Helland's metric form is diagonal.
That stress-energy tensor shows that the Helland universe described by that metric is not a perfect fluid. The pressure components of that tensor vary from point to point, because the Helland universe is not homogeneous. With a perfect fluid, the pressure components would be the same at all events within a spatial slice, as is true of FLRW models.
Mike Helland is profoundly ignorant of the facts stated above, leading him to write this:
Yes, Mike Helland's fantasy jello is neither homogeneous nor isotropic. That is why its associated stress-energy tensor does not resemble that of a perfect fluid.
Mike Helland's ignorance is so profound that he didn't realize my conclusions say the very opposite of the assumption he is falsely accusing me of making.
It is a lot easier to interpret the components of a stress-energy tensor when using Cartesian coordinates instead of spherical coordinates, which is why my analysis of the Helland universe considered only the Ttt and Trr components of that tensor.
Although I set out to calculate the Ricci tensor's components in Cartesian coordinates, I gave up when I realized (1) the necessary derivatives were not actually defined at the Helland universe's central point, (2) the curvature blows up at that central point, (3) the calculations were extremely messy, and (4) there wasn't a whole lot of point to completing those calculations after (1) and (2) had become obvious.
It would have been even more helpful if Mike Helland's screen shot had shown all four components of the Ricci tensor, instead of showing only the two in its leftmost column.
It should be noted that Mike Helland's Cartesian coordinates are not the same as mine, because Mike Helland did not transform the time coordinate to τ = ct. That doesn't change things very much, but it does make it more difficult for me to compare the EinsteinPy output against my own calculations.
For reasons explained below, I am pretty sure the Ricci scalar is undefined at x=0, which would mean the EinsteinPy software erred by reporting that scalar as zero.
The unnecessary complexity of the symbolic expressions reported by the EinsteinPy software is rather comical. That complexity resulted from some serious shortcomings of its simplifier, and I rather suspect those shortcomings are responsible for the simplifier's belief that the Ricci scalar is zero at x=0.
With Cartesian coordinates (t, x, y, z), all four of those coordinates are real numbers. Hence re(x)=x and im(x)=0. This spoiler lists those simplifications and a number of others that the EinsteinPy simplifier failed to apply.
re (x) = x
im = 0
d/dx re (x) = 1
d/dx im (x) = 0
d2/dx2 re (x) = 0
d2/dx2 im (x) = 0
|x2| = x2|x|2 = x2(H2 |x|2 − 2 Hc |x| + c2) = (H |x| − c)2 = (c − H |x|)2
Et cetera.
I applied those simplifications and a number of others to obtain much simpler expressions for Rtt and the Ricci scalar R. The following spoiler shows the simplified expression I got for Rtt.
Rtt = H ( − H + (H |x| − c) |x| + H |x| − (H |x| − c) ( |x| / x2 ) )
Doing that simplification was so tedious that I would not be at all surprised to learn I made some mistakes, but the really important part of that expression is the ( |x| / x2 ) term, and I checked that part multiple times using several different techniques.
Furthermore, I had identified the probable presence of such a term before I even began the simplification process. The reason I knew to look for it was the presence of similar terms within both my own partially completed calculation using Cartesian coordinates and the EinsteinPy output for spherical coordinates that Mike Helland posted almost a month ago.
If you try to evaluate that expression for Rtt at x=0 you will encounter a division by zero.
Using first-semester calculus, however, we can try to calculate the limit of Rtt as x approaches zero. That attempt fails because the limit does not exist. The magnitude of Rtt increases without bound as x approaches zero, because the magnitude of Rtt is roughly proportional to |1/x|.
EinsteinPy's output for the Ricci scalar is even more complicated than its output for Rtt, so there was more opportunity for error in my simplification of that output. If I got it right, however, the magnitude of the Ricci scalar R also increases without bound as x approaches zero.
The first thing one does to understand modern cosmology is assume the universe is homogeneous and isotropic, thus creating a fantasy land which is obviously incongruent with observed reality.
Wrong. It is precisely because we observe that the universe is homogeneous and isotropic, that we make these assumptions.
But it seems that you missed the bit that it is only at large scales that these assumptions are valid. Everywhere we look, the universe looks the same. Galaxies are not bunched together in one part of the sky, and the CMB is almost perfectly uniform.
Wrong. It is precisely because we observe that the universe is homogeneous and isotropic, that we make these assumptions.
But it seems that you missed the bit that it is only at large scales that these assumptions are valid. Everywhere we look, the universe looks the same. Galaxies are not bunched together in one part of the sky, and the CMB is almost perfectly uniform.
If you examine Mike Helland's history of whining about the assumption of homogeneity, you will see almost all of his complaints involve the discontinuous distribution of matter as seen in objects such as planets, stars, and galaxies.
In short, he is complaining about the continuity of the FLRW metric.
He hasn't yet noticed that his Helland metric is continuous as well, so almost all of his complaints about FLRW models apply equally well to his Helland models.
But he doesn't know enough calculus to notice such things.
The standard model of cosmology is based on the existence of homogeneous surfaces as the background arena for structure formation. Homogeneity underpins both general relativistic and modified gravity models and is central to the way in which we interpret observations of the CMB and the galaxy distribution. However, homogeneity cannot be directly observed in the galaxy distribution or CMB, even with perfect observations, since we observe on the past lightcone and not on spatial surfaces. We can directly observe and test for isotropy, but to link this to homogeneity, we need to assume the Copernican Principle. First, we discuss the link between isotropic observations on the past lightcone and isotropic spacetime geometry: what observations do we need to be isotropic in order to deduce spacetime isotropy? Second, we discuss what we can say with the Copernican assumption. The most powerful result is based on the CMB: the vanishing of the dipole, quadrupole and octupole of the CMB is sufficient to impose homogeneity. Real observations lead to near-isotropy on large scales - does this lead to near-homogeneity? There are important partial results, and we discuss why this remains a difficult open question. Thus we are currently unable to prove homogeneity of the Universe on large-scales, even with the Copernican Principle. However we can use observations of the CMB, galaxies and clusters to test homogeneity itself.
Is the observable Universe consistent with the cosmological principle?
The cosmological principle (CP)—the notion that the Universe is spatially isotropic and homogeneous on large scales—underlies a century of progress in cosmology. It is conventionally formulated through the Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies as the spacetime metric, and culminates in the successful and highly predictive Λ-Cold-Dark-Matter (ΛCDM) model. Yet, tensions have emerged within the ΛCDM model, most notably a statistically significant discrepancy in the value of the Hubble constant, H0. Since the notion of cosmic expansion determined by a single parameter is intimately tied to the CP, implications of the H0 tension may extend beyond ΛCDM to the CP itself. This review surveys current observational hints for deviations from the expectations of the CP, highlighting synergies and disagreements that warrant further study. Setting aside the debate about individual large structures, potential deviations from the CP include variations of cosmological parameters on the sky, discrepancies in the cosmic dipoles, and mysterious alignments in quasar polarizations and galaxy spins. While it is possible that a host of observational systematics are impacting results, it is equally plausible that precision cosmology may have outgrown the FLRW paradigm, an extremely pragmatic but non-fundamental symmetry assumption.
Everybody on the side of the homogeneous, isotropic universe seems to be silent on this.
That's one hell of an ass pull. Why would you think that? Do you even know why you think that?
The speed of light in the past was lower than it is today.
Either that's because of changing distances, or changing clock rates. So whatever would cause space to distort, should also be capable of causing time to distort.
The divergences from homogeneity and isotropism become smaller as the scale gets larger, for basically the same reason that the more coins you flip the more the ratio of heads to tails will tend to 1:1.
The observable universe is a very large place, and those divergences at large scales become very small. Do you have reason to think that these incredibly small divergences impact the evolution of the universe in a meaningful way?
The divergences from homogeneity and isotropism become smaller as the scale gets larger, for basically the same reason that the more coins you flip the more the ratio of heads to tails will tend to 1:1.
The observable universe is a very large place, and those divergences at large scales become very small. Do you have reason to think that these incredibly small divergences impact the evolution of the universe in a meaningful way?
The observable universe is, according to LCDM, 92 billion light years across.
There's a Giant Arc that's 3 billion light years long.
So out of 92 pixels, 3 would represent that thing.
That's not homogeneous.
How far do you have to zoom out for the assumptions to (maybe) become true?
Well beyond the observable universe.
It seems to me science should be in the business of talking about observable reality. The only thing FLRW cosmology talks about technically lies outside of that.
So, we can conclude the universe had a hot, dense birth if we assume that, beyond the observable universe, things are a certain way, namely, that everything we see and know exists is absent.
That's very clearly fiction.
My model is one of the observable universe. It has a cosmological horizon.
FLRW cosmology starts somewhere after my model ends.
I'm not saying Einstein's dream of "knowing the mind of God" or whatever is a bad one.
But the path to that should address observable reality somewhere along the line. It seems that's skipped right over.
Denser than if it wasn't filled with galaxies, I presume.
The Giant Arc is a large-scale structure discovered in June 2021 that spans 3.3 billion light years.[1] The structure of galaxies exceeds the 1.2 billion light year threshold, challenging the cosmological principle that at large enough scales the universe is considered to be the same in every place (homogeneous) and in every direction (isotropic). The Giant Arc consists of galaxies, galactic clusters, as well as gas and dust. It is located 9.2 billion light-years away and stretches across roughly a 15th of the radius of the observable universe.[2]
The bigger point is that if the cosmological principle doesn't "kick in" until beyond observable reality... what are we really even talking about here?
Obviously. The question is how far from homogeneity and isotropy is the observable universe on the largest scales. A large scale structure like the Giant Arc shows that there is some divergence from that, but not the degree of divergence, because you haven't given a value for how far that structure diverges from the average. And that's the meat of the point that you're trying to make.
The universe is clearly approximately homogenous and isotropic. But how good of an approximation is that? Your point is that it's not good enough for what we're trying to do with it, but if you want to make that point it's a quantitative point.
The bigger point is that if the cosmological principle doesn't "kick in" until beyond observable reality... what are we really even talking about here?
It "kicks in" at much lower scales than that, but only approximately. Again, the question is whether or not that approximation is good enough for whatever conclusions we're trying to draw. So far you haven't made a case that it isn't.
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