Cont: Why James Webb Telescope rewrites/doesn't the laws of Physics/Redshifts (2)

[The Man]

The expanding universe wasn't working.

They "fixed it" by adding a nanosecond right at the beginning where 990 billion years of expansion takes place.

So it was working, as was your implication that you had to roll it back to when we knew less about the universe.

For your own edification "universe" is from old English ~1580's, from Française "univers" ~1200 and from latain "universum".

So, centuries before expansion theories.

Now they're trying to fix inflation.

How? Please try to be specific to just inflation and not just whatever aspects of modern physics you doubt.

The basis for my doubts is that expansion didn't work and adding exponential amounts of it is silly.

Again, your implication was that it did work, you had to claim a roll back of the theory and our knowledge to the 60's to assert it doesn’t work.

Again, your perception of silliness is no relevant mathematical factor in modern cosmology that I can recall.

So, our your doubts, you know, rational? If they aren’t there is no rational way to address them, hence my potential recommendation below.

I believed this stuff as a kid. I'm not a kid anymore.

Ok, beliefs again. Might have to recommend that this thread be moved to the Religion and Philosophy section where beliefs are actually a relevant factor.

I think a reasonable person would have serious doubts about the status of cosmology as a real science. No controlled experiments. The null hypothesis is always ignored. The subject matter itself is the creation of existence as we know it.

WTF, who ignores the “null hypothesis”? You do understand that observational evidence makes the hypothetical theoretical environment not just “null”. Models have to match observations as best they can, doubts and all.

All experiments are controlled to the degree permissible. How exactly does one have a control of the universe? You do understand that would require a perception into some kind of multi-verse? Rather than expansion requiring a multi-verse, as you fallaciously asserted before, you seem to need one (ETA: to point at other universal ****).

I don't blame my childhood self for believing in it. But at some point we're gonna have to circle back to reality.

I do, as you seem not to have grown past that. I studied relativity because I didn't believe it. Special relativity is just a no fixed time frame extension of Galilean. It’s General I'm talking about, that must subsume special. Beliefs are ****, I have to do things in my job I don't believe in. Management and customers make those determinations. I’ve had to do solutions I didn’t believe in that worked and became standard practice. I likewise do things I don't believe in to just put the question concerned to bed.

That's the thing, if you are going to oppose some theory, solution, answer or something you have to know it so well, while doing it open and honestly, as to make that affront valid. Otherwise all you do is invalidate your own attempt at invalidation.

 

[The Man]

I used to believe that too.

Always the beliefs, I'm getting even closer to the R&P recommendation.

You seem to be under the impression the steady state universe wasn't an expanding one.

That's wrong.

You seem to be under the impression that "steady state" doesn't preclude expansion, let alone accelerating expansion.

However, let's just go with that for the sake of argument. So, if the universe can expand and evidence indicates it is expanding faster now then just recently before. Expanding faster in the past ain't no problem. As long as it is consistent with other current theories, like General Relativity.


Remember those expansion graphics you posted and I gave a self-referential reply to when you said “That’s not how “relative” works” my reply being “That’s how relativity works”. We have no external universe reference, like you seeing and pointing at ****. Without a perceptible multi-verse reference frame, which we don’t have and might not even be possible, if such did exist. So, universal self-reference is all we can work with. Those expansion graphics you posted were done from a reference frame co-moving with the expansion. Referencing how the universe was at time T to how it was at time T-X or T+X, as you prefer. Got anything like that to explain the “steady state” “expanding” universe you just asserted, that is consistent with observational evidence? (ETA: and i don't just mean 1960's observational evidence).

 

[Mike Helland]

Always the beliefs, I'm getting even closer to the R&P recommendation.

Ok. I hear ya, loud and clear. I don't want to go in that direction.

 

[Mike Helland]

http://latex.codecogs.com/gif.latex?\frac{c - Hd}{c} = 1- \frac{Hd}{c}

Working on this a little more with some other people, and it seems to be the right line element is:

http://latex.codecogs.com/gif.latex? ds^2 = -(1-\frac{rH}{c})^{2} c^2 dt^2 + dr^2 + r^2 d\theta^2 + r^2\sin^2 (\theta) d\psi^2​

That makes sense intuitively that a tiny change in time multiplied 1/(1+z) equals the tiny change in space. The tiny change in space times (1+z) equals the tiny change in time.

Which is nice because that simplifies:

http://latex.codecogs.com/gif.latex?g_{tt} = -(1-\frac{rH}{c})^{2} c^2

http://latex.codecogs.com/gif.latex?g_{tt} = -\left(\frac{c- rH}{c}\right)^{2} c^2

http://latex.codecogs.com/gif.latex?g_{tt} = -(c- rH)^2

 

[W.D.Clinger]

The Helland universe (part 1 of 4)

This is the first of a 4-part series.


[size=+1]Abstract.[/size]

The Helland universe analyzed herein is defined by the metric tensor field Mike Helland proposed on 24 October 2023 and confirmed as the best of his several guesses on 26 October. Using seven different but equivalent coordinate systems, the Ricci curvature tensor calculated by Mike Helland, and the field equations of general relativity, this note derives the following consequences of the Helland metric:

• The Helland universe is static in these three senses:
  1. Its metric tensor field does not change over time.
  2. Its curvature tensor field does not change over time.
  3. Its distribution of mass-energy does not change over time.
  The second and third of those are of course immediate consequences of the first.
  
  
• The Helland universe is geocentric in the following sense: Its spatial slices are spherically symmetric around a central point whose physical properties distinguish it from all other points of space.
  
  
• The Helland universe has a curvature singularity at that central point.
  
  
• The mass-energy density at that central point is negative infinity.
  
  
• If the cosmological constant is zero, then the mass-energy density is negative throughout the Helland universe.
  
  
• The static universe implied by the Helland metric is not consistent with Einstein's general theory of relativity, because a universe with the Helland universe's distribution of mass-energy would not be static.

[size=+1]Disclaimer.[/size]

The author and sole proponent of the Helland universe has disputed several of the conclusions derived below. For the moment, it is enough to note that he has been arguing against mathematical consequences of a metric form he proposed. His only recourse is to abandon his advocacy of that metric form and the Helland universe it implies, just as he has abandoned all of the other metric forms he has proposed since 20 March. We should therefore anticipate that the Helland metric and Helland universe discussed below will be superseded by yet another Helland metric and Helland universe.

Indeed, Mike Helland came up with yet another Helland metric form just minutes before I submitted this analysis of the metric form he adopted just one week ago.


[size=+1]Outline.[/size]

• Abstract.
• Disclaimer.
• Outline.
• Manifolds.
• Spacetime manifolds.
• The Helland universe.
• The Helland universe in spherical coordinates.
  • Radial coordinate r
  • Radial coordinate q
  • Radial coordinate ρ
  • Coordinate singularities
  • Coordinate speed of light
  • Ricci scalar
• The Helland universe in Cartesian coordinates.
  • With origin at center of the universe
  • Translation in time
  • Translation in space
  • Observers in motion
• Distribution of mass/energy in the Helland universe.
• Summary of conclusions.

Readers who already know about spacetime manifolds should skip the next two sections.


[size=+1]Manifolds.[/size]

Euclidean plane geometry is about 2-dimensional Euclidean space. That 2-dimensional Euclidean space is a manifold.

Euclidean solid geometry is about 3-dimensional Euclidean space. That 3-dimensional Euclidean space is another example of a manifold.

Those familiar examples can be generalized to n-dimensional Euclidean space. The points of an n-dimensional Euclidean space can be named using Cartesian coordinates, in which each point is represented by an ordered list of n real numbers. Alternatively, those points can be named using spherical coordinates, in which each point is once again represented by an ordered list of n real numbers, but one of those numbers represents the point's distance from an origin and the other numbers represent angles.

Not all manifolds are Euclidean.

The surface of the earth is an example of a 2-dimensional manifold that is not Euclidean, but sufficiently small pieces of the earth's surface are approximately Euclidean. What that means is that, when examining a sufficiently small piece of the earth's surface, you can pretend it is a fragment of the 2-dimensional Euclidean space that is the subject of plane geometry, and you can construct flat maps of those pieces in which the error introduced by pretending they are Euclidean is not so great as to make the maps altogether useless.

Using the mathematics of calculus in n dimensions, that concept of "approximately Euclidean" can be made precise. That concept can be extended to say it is possible to apply theorems of calculus to small fragments without introducing too much error into the results, with enough overlap between fragments to allow larger calculations to be constructed by piecing small calculations together.

A Riemannian manifold is an n-dimensional space that is locally approximately Euclidean as described above, and has certain other nice properties as well. One of those other properties is the existence of a metric that allows the distance between two points along some path to be defined and calculated.


[size=+1]Spacetime manifolds.[/size]

General relativity and cosmology use mathematical models as idealized simplifications of a real or hypothesized physical universe. These models are used to understand the physical consequences of assumptions (or hypotheses) we might make about such a universe.

Following Einstein, most cosmological models of the entire universe are known as spacetime manifolds. The Helland universe is an example of a spacetime manifold.

A spacetime manifold is a special kind of 4-dimensional model with one dimension of time and three dimensions of space.

Spacetime manifolds are not Euclidean. They are not even Riemannian. They are pseudo-Riemannian, which means the metric is not positive definite: The interval between two distinct points can be negative or zero. That interval has implications for causality.

The points of a spacetime manifold are said to be events. If the interval between two events is negative, then it is possible for a massive particle or object to have travelled from one event to the other, which means the two events might be causally connected. If the interval is negative zero, then it is not possible for a massive particle or object to go from one to the other, but it is possible for a massless particle such as a photon to go from one event to the other. If the interval is positive, then it is not possible for any particle or object to go from one to the other, because the two events are so far apart in space and so close together in time that you couldn't get from one to the other even if you were travelling at the speed of light.

The pseudo-Riemannian metric of a spacetime manifold doesn't tell us everything we might want to know about the manifold, but it does tell us enough to derive several important facts about the manifold.

The following sections show how the Helland metric form implies several interesting facts about the Helland universe.

 

[W.D.Clinger]

The Helland universe (part 2 of 4)

This is the second of a 4-part series that analyzes the Helland universe.


[size=+1]The Helland universe in spherical coordinates.[/size]

These three coordinate systems use spherical coordinates to describe spatial dimensions of the Helland universe. They differ only in their radial coordinates: r, q, or ρ.

Radial coordinate r

The Helland metric form, as stated by Mike Helland on 26 October, uses a radial coordinate r that is restricted to the range 0 ≤ r < 1.

ds² = − (1 − r)⁻² c² dt² + dr² + r² [ dθ² + sin² θ dφ² ]​

For any fixed value t₀ of the time coordinate t, the set of all spacetime events with coordinates of the form (t₀, r, θ, φ) is called a spatial slice of the spacetime manifold.

Using spherical coordinates for spatial dimensions emphasizes the spherical symmetry of spatial slices around the central event E₀. In this coordinate system, E₀ has coordinates (t₀, 0, 0, 0).

If t₀ = 0, that central event happens to be the origin of this particular coordinate system.

Whether there is anything special about that central event is an important question. That question can be answered by calculating the Ricci scalar field and comparing its value at the origin to its value at other events within the same spatial slice, but it is easier to answer the question by transforming the metric form into a Cartesian coordinate system as will be done in a later section.

In the metric form above, the coefficient of dt² is

g_tt = − (1 − r)⁻² c²​

The presence of that radial coordinate r within g_tt suggests the Helland universe is neither homogeneous nor isotropic, and that events at the origin of a spatial slice are indeed special. We need to be cautious when drawing such conclusions, however, because is possible for a coordinate system to disguise the homogeneity and isotropy of a manifold. The Rindler coordinate system is a standard example of this phenomenon.

Another important question is whether the Helland universe is static. Apart from a few very special exceptions, the FLRW models of mainstream cosmology are not static, because they describe universes that are always expanding or contracting.

The time coordinate t does not appear within the Helland metric form, which implies the metric tensor field is the same for all spatial slices, which means the Helland universe is static from the point of view provided by this coordinate system.

It is possible to devise a coordinate system in which the Helland universe appears to be nonstatic. Consider, for example, an observer in motion whose r coordinate increases with time. If we were to transform into a coordinate system in which that observer is at rest with respect to the spatial coordinates, then the Helland universe would not be static from the point of view provided by that coordinate system. As will be shown by the seventh coordinate system considered in this essay, that observer (considering himself to be at rest) would see the coordinates of the universe's center changing over time.

It is conventional, however, to say a universe is static when its most natural coordinate systems provide a description in which all spatial slices have the same metric tensor field. In the first six coordinate systems described within this essay, the Helland universe is static. The fact that those coordinate systems agree upon this point makes it reasonable to say the Helland universe is static.


Radial coordinate q

The origin of a coordinate system is essentially arbitrary. A coordinate transformation that simply moves the origin to some other place is called a translation.

With Cartesian coordinate systems, translations are easy because they are performed by adding a constant (possibly zero) to each coordinate.

With spherical coordinate systems, translations can be messy because (in general) the angular coordinates must be adjusted in nonlinear fashion.

As a sort of warmup exercise, consider the coordinate transformation that leaves the t, θ, and φ coordinates unchanged but replaces the radial coordinate r by a new coordinate q = r − ½, where −½ ≤ q < ½. In this new coordinate system, the central event E₀ has coordinates (0, −½, 0, 0). A different event E₁ has coordinates (0, 0, 0, 0).

Whether we regard E₀ or E₁ as the origin of this new coordinate system is a matter of taste, but it is important to understand that the angular coordinates θ and φ are still defined with respect to the spatial position of E₀.

Whether we regard q as a radial coordinate is another matter of taste. I will refer to q as a radial coordinate, but some may insist that a coordinate that can have negative values should not be regarded as a radial coordinate. Regardless, it is important to understand that using q instead of r is a legitimate coordinate transformation.

Finally, it is reasonable to question whether the q-coordinate system should be regarded as a spherical coordinate system at all. The fact that its angular coordinates are defined with respect to a point other than the origin argues against thinking of it as a spherical coordinate system. Regardless, it is important to understand that the q-coordinate system is a perfectly legitimate coordinate system.

When transformed to use q instead of r, the Helland metric form becomes

ds² = − (½ − q)⁻² c² dt² + dq² + (q + ½)² [ dθ² + sin² θ dφ² ]​

By substituting r − ½ for q throughout that metric form, it is easy to prove that this transformed metric form is equivalent to the Helland metric as originally stated by Mike Helland using the r-coordinate system.

Note, however, that the algebraic simplicity of that proof depends upon the fact that dq = dr. If the relationship between q and r were non-linear, we would have to use calculus to derive the relationship between dq and dr.


Radial coordinate ρ

[size=-1]Notation:
In this section, the Greek letter ρ (rho) is used as a radial coordinate. That radial coordinate has nothing to do with the mass-energy density corresponding to the stress-energy tensor's T_tt component, for which ρ is a common notation. To reduce confusion, I will write ρ_m for the mass-energy density when I discuss it in a later section.[/size]​

In the original r-coordinate system, the fact that r is restricted to r < 1 might cause some people to wonder whether the Helland universe is bounded by some sort of wall. Would an observer moving toward increasing values of r eventually run into that wall? Or fall off the edge of the Helland universe?

The answer to both of those questions is that it is impossible for an observer ever to get to r = 1, because there are no events of the Helland universe for which the r-coordinate is 1.

That is not a terribly satisfying answer. Some people might prefer a radial coordinate that ranges over all non-negative real numbers.

We can accommodate their aesthetic concern by transforming from the r-coordinate system to a system that leaves the t, θ, and φ coordinates unchanged but replaces the radial coordinate r by a new radial coordinate ρ = r/(1−r) = 1/(1−r) − 1.

From that relationship between ρ and r, it is easy to see that the possible values of ρ should include all non-negative reals.

When transformed to use ρ instead of r, the Helland metric form becomes

ds² = − (ρ + 1)² c² dt² + (ρ + 1)⁻⁴ dρ² + (ρ / (ρ + 1))² [ dθ² + sin² θ dφ² ]​

To prove this metric form is equivalent to the original Helland metric form, one must prove that

dr = (ρ + 1)⁻² dρ​

That's just a matter of calculating dρ/dr and applying some high school algebra.

Having proved that the three coordinate systems defined above yield equivalent metric forms, we know those three metric forms all describe exactly the same metric tensor field.

We can therefore use any of the three coordinate systems to reason about the Helland universe.

We have to be careful, however, because all three of those coordinate systems have a coordinate singularity at the central event E₀ and at all other events whose spatial coordinates are the same as those of E₀.


Coordinate singularities

Suppose some intrepid adventurer tells you his expedition plans to ski southward from McMurdo Station until they reach the South Pole. From there they will head north until they reach the Antarctic coast, where you are expected to pick them up.

Does that plan tell you where they expect you to pick them up? No, it does not.

At most places on earth, what we mean by "north" is well-defined. At the South Pole, however, "north" is not well-defined.

That is an example of a coordinate singularity.

Informally, a coordinate singularity is a place where something goes haywire, typically because some coordinate or direction or derivative is not well-defined.

Spherical coordinate systems have two distinct kinds of coordinate singularity. One kind of coordinate singularity occurs with the angular coordinates because an angle of 0° is the same as an angle of 360°. Those coordinate singularities are typically handled by restricting the range of an angular coordinate, as by saying 0 ≤ θ < 2π.

Even with that convention, the coordinate singularity at θ=0 remains problematic because the mapping from points to their θ coordinates is discontinuous at θ=0, which causes derivatives involving θ to become undefined.

That is why the mathematical definition of a Riemannian or pseudo-Riemannian manifold insists that coordinate charts be defined on open subsets of the manifold, and that the mapping from points in that open set to their coordinates be continuous and sufficiently differentiable.

That is also why the surface of the earth cannot be covered by a single chart. You can't flatten the earth's surface (which is homeomorphic to a 2-sphere) onto 2-dimensional Euclidean space without puncturing or tearing the 2-sphere.

The surface of the earth can be covered by two charts that overlap at their boundaries. That is the mathematical basis for the idea that "Well that's no problem, you can just rotate the spherical coordinates a little bit if you're trying to do calculations at that boundary."

As an example of that workaround, consider the g_φφ component of the Helland metric form in the r-coordinate system:

g_φφ = r² sin² θ dφ²​

That is the g_φφ component of the covariant metric tensor. When we calculate the Ricci scalar in the next subsection, we will need to calculate the g^φφ component of the contravariant metric tensor, which is related to g_φφ by

g^φφ = 1 / g_φφ​

When θ is zero, sin θ is zero as well, so g_φφ = 0 and our calculation of g^φφ has to divide by zero.

We can avoid that by taking care to use nonzero values of θ whenever we calculate the Ricci scalar. Because the Helland universe is spherically symmetric, the Ricci scalar will be the same for θ = π/2 as for θ = 0, but you can see how this coordinate singularity makes the Ricci scalar slightly harder to calculate, and we'd have a more serious problem if we were calculating the Ricci scalar for a spacetime manifold that isn't spherically symmetric.

Spherical coordinates have another coordinate singularity involving the radial coordinate, because the angular coordinates are not well-defined at r=0. That coordinate singularity is typically handled by a convention that says the angular coordinates are zero whenever r=0.

Even with that convention, the coordinate singularity at r=0 remains problematic because the mapping from points to their angular coordinates is discontinuous at r=0, which causes derivatives to be undefined at r=0.

And that coordinate singularity definitely impedes analysis of the Helland universe at r=0, which is a serious problem because one of the most important questions we want to answer is whether points whose r-coordinate is zero actually represent a special center of the Helland universe's spatial slices.

One way we might try to answer that question is to look at the behavior of the Ricci curvature tensor. Mike Helland has calculated the components of that tensor for his r-coordinate system, from which we see that the R_tt component of that tensor blows up (becomes infinite) at r=0. That suggests a point at r=0 occupies a special place in the Helland universe, and likely represents the center of the Helland universe.

But wait a minute! We ought to consider the possibility that the Ricci tensor blows up at r=0 because of the coordinate singularity at r=0, and that the components of that tensor at such points might be well-behaved if we were using some other coordinate system.

In short, coordinate singularities make it difficult to draw reliable conclusions at those singularities or in their vicinity.

To draw reliable conclusions, it is often best to transform into a coordinate system that doesn't have coordinate singularities in the region of interest.


Ricci scalar

The Ricci scalar R is the trace of the Ricci curvature tensor, and can be computed from the diagonal components of the Ricci tensor using the following equation, in which summation is implied by the Einstein summation convention.

R = gⁱⁱ R_ii​

The index i ranges over the four coordinates t, r, 0, and φ, the gⁱⁱ are components of the contravariant metric tensor, and the R_ii are components of the covariant Ricci tensor.

As was explained in the previous section, a coordinate singularity leads to division by zero in that equation unless we take care to work around the problem.

Unlike the coordinate-dependent components of the metric tensor and Ricci tensor, the Ricci scalar is an invariant that does not depend on the coordinate system. If we calculate the Ricci scalar correctly, avoiding coordinate singularities, it may tell us whether the Helland universe's curvature varies from point to point.

The following graph, which relies upon Mike Helland's calculation of the Ricci tensor, shows that the magnitude of the Ricci scalar does indeed vary as a function of the radial coordinate r.

The Ricci scalar is negative throughout the Helland universe, so that graph shows the logarithm of its absolute value. On the horizontal axis, lg r = -30 is less than one millimeter, while lg r = -1 is 1.38 billion light years.

As can be seen from the log-log graph, the Ricci scalar converges to negative infinity as r approaches zero.

That tells us the Helland universe is not homogeneous and is not isotropic.

That graph also tells us the Helland universe is geocentric in the sense of having a spatial center at r=0 at which the curvature is far greater than at any other point of space, and the spacetime curvature is spherically symmetric around that center.


Coordinate speed of light

We can use a metric form to calculate the coordinate speed of light at various points of a spacetime manifold.

The coordinate speed of light seldom tells you anything interesting about the physics, because it has more to do with the coordinate system than with physical reality, but the coordinate speed of light can tell you something about the coordinate system.

Consider a photon moving directly away from the r=0 origin of the r-coordinate system. Its angular coordinates don't change, so dθ = dφ = 0. Photons travel along null geodesics, so

ds² = 0 = − (1 − r)⁻² c² dt² + dr²(1 − r)⁻² c² dt² = dr²dr/dt = (1 − r)⁻¹ c
dr/dt = c/(1 − r)​

That tells us the coordinate speed of light depends upon the r-coordinate of the photon. The coordinate speed of light is c at r=0 and increases as the photon moves further away from r=0.

Points with r=0 are the only points of the Helland universe at which the coordinate speed of light is c. Does that mean points with r=0 are special?

Not necessarily. The fact that points with r=0 are the only points at which the coordinate speed of light is c might be a fairly meaningless artifact of the r-coordinate system.

Let's calculate the coordinate speed of light in the ρ-coordinate system:

ds² = 0 = − (ρ + 1)² c² dt² + (ρ + 1)⁻⁴ dρ²(ρ + 1)² c² dt² = (ρ + 1)⁻⁴ dρ²dρ/dt = (ρ + 1) c / (ρ + 1)⁻²dρ/dt = c (ρ + 1)³​

Points with ρ=0 are the only points of the Helland universe at which the ρ-coordinate speed of light is c. The ρ-coordinate speed of light increases as the photon moves further away from ρ=0.

At event E₀, the coordinate speed of light is c in both of those coordinate systems.

Let's calculate the coordinate speed of light at E₁, whose q-coordinates are (0, 0, 0, 0).

In the r-coordinate system, the coordinates of E₁ are (0, ½, 0, 0) so the r-coordinate speed of light at E₁ is c/(1 − ½) = 2c.

In the ρ-coordinate system, the coordinates of E₁ are (0, 1, 0, 0) so the ρ-coordinate speed of light at E₁ is c (1 + 1)³ = 8c.

E₁ is a point of the spacetime manifold, and that point doesn't change just because we decide to express its coordinates using a different coordinate system. In physical reality, the actual speed of light at E₁ has to be exactly the same as the actual speed of light at E₁.

But the coordinate speed of light at E₁ depends on your choice of coordinate system.

Using a coordinate speed of light to draw conclusions about physical reality is like using a Mercator map to compare the areas of Greenland and Africa.

It might be a good idea to consult more than one map.

A man with only one watch knows what time it is.

A man with two watches is never quite sure.

The man with two watches is less certain because he knows more than the man with only one watch.​

 

[Mike Helland]

This is the first of a 4-part series.


[size=+1]Abstract.[/size]

The Helland universe...

I have already objected to this characterization.

The "territory" of the metric is an observer's past light cone (ie, observed reality).

"The Helland universe" is a straw man.

 

[W.D.Clinger]

The Helland universe (part 3 of 4)

This is the third of a 4-part series that analyzes the Helland universe.


[size=+1]The Helland universe in Cartesian coordinates.[/size]

As explained in the subsection dealing with the q-coordinate system, translating the origin of a spherical coordinate system is quite messy. It is far easier to translate the origin of a Cartesian coordinate system.


With origin at center of the universe

We start with the well-known translation from spherical coordinates r, θ, φ to Cartesian coordinates x, y, z:

x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ​

The time coordinate is unchanged, so the Helland metric form becomes

ds² = − (1 − sqrt(x² + y² + z²))⁻² c² dt² + dx² + dy² + dz²​

Proving the Cartesian form of the spatial metric is equivalent to the spatial metric in spherical coordinates is a bit messy because it involves 9 partial derivatives:

∂x/∂r = sin θ cos φ
∂x/∂θ = r cos θ cos φ
∂x/∂φ = − r sin θ sin φ

∂y/∂r = sin θ sin φ
∂y/∂θ = r cos θ sin φ
∂y/∂φ = r sin θ cos φ

∂z/∂r = cos θ
∂z/∂θ = − r sin θ
∂z/∂φ = 0​

Readers who want to use those partial derivatives to prove the Cartesian form of the Helland metric is equivalent to the Helland metric in spherical coordinates are welcome to do so.

Other readers will not be making a mistake if they just trust the fact that this well-known transformation from spherical to Cartesian coordinates is known to be correct.

Translation in time

To shift the origin of the time coordinate from t = 0 to t = 0 + Δt, we introduce a new time coordinate τ = t − Δt. Because none of the metric components involve t, and dt = dτ, writing the Helland metric in the τ-coordinate system is easy:

ds² = − (1 − sqrt(x² + y² + z²))⁻² c² dτ ² + dx² + dy² + dz²​

The only change is highlighted in red.

Because the transformed metric form is exactly the same as the Cartesian metric form before the translation in time, apart from renaming the time coordinate from t to τ, we conclude the Helland universe is static in the sense of having a metric tensor field that does not change over time.


Translation in space

To shift the origin of the x coordinate from x = 0 to x = 0 + Δx, we introduce a new coordinate w = x − Δx. Because dx = dw, writing the Helland metric in the w-coordinate system is easy:

ds² = − (1 − sqrt((w + Δx) ² + y² + z²))⁻² c² dt² + dw ² + dy² + dz²​

The only changes are highlighted in red.

Translating the origin of a spatial coordinate involves more than a simple renaming of that coordinate, because the Helland universe is not homogeneous and is not isotropic.


Observers in motion

Suppose an observer is moving at some constant coordinate speed s along the x axis, so his coordinate velocity is v = s dx/dt. Suppose also that the observer passes through the Helland universe's central event E₀ at Cartesian coordinates (t=0,x=0,y=0,z=0).

That observer's world line is not a geodesic. If the observer's world line were a geodesic, its coordinate velocity would change in response to curvature of the Helland universe. Because the observer's world line is not a geodesic, the observer experiences a feeling of acceleration.

Although the observer will become painfully aware that his journey through the Helland universe is not inertial, let's suppose the observer stubbornly insists upon using a Cartesian coordinate system in which he resides forever at the spatial origin of that self-centered coordinate system.

To design a coordinate system with that property, we can replace the x coordinate by u = x − st.

Because u is a function of both x and t, du is not a simple function of dx. We have to calculate the partial derivatives

∂u/∂x = 1
∂u/∂t = − s​

to obtain the total differential

du = dx − s dt​

In that u-coordinate system, the Helland metric form becomes

ds² = − (1 − sqrt((u + st) ² + y² + z²))⁻² c² dt² + (du - s dt) ² + dy² + dz²​

which simplifies to

ds² = − [ (1 − sqrt((u + st)² + y² + z²))⁻² c² + s² ] dt² + 2s dt du + du² + dy² + dz²​

Although that metric form is equivalent to the metric form of the t, x, y, z coordinate system, the cross term for dt du makes this metric form more complicated. Calculations using the u-coordinate system would be a lot messier than calculations using the x-coordinate system.

An observer moving across the Helland universe might want to consider that extra complexity before he insists upon using a self-centered coordinate system.

 

[W.D.Clinger]

The Helland universe (part 4 of 4)

This is the fourth of a 4-part series that analyzes the Helland universe.


[size=+1]Distribution of mass/energy in the Helland universe.[/size]

As was noted in the subsection on the Ricci scalar, the Ricci scalar converges to negative infinity as r approaches zero.

When expressed using Mike Helland's original coordinates t, r, θ, and φ,
the R_tt component of the Ricci tensor converges to positive infinity as r approaches zero.

Einstein's field equations say

R_μν − ½ R g_μν + Λ g_μν = κ T_μν​

where κ = 8πG/c⁴ is Einstein's gravitational constant, with a numerical value of about κ = 2.077 × 10⁻⁴³ s²/(kg m).

Taking both μ and ν to be t, we get the equation for the mass-energy density T_tt = ρ_m:

R_tt − ½ R g_tt + Λ g_tt = κ T_tt​

The cosmological constant Λ is some fixed finite constant. As the radial coordinate r approaches zero, g_tt converges to −1, R_tt converges to positive infinity, and R converges even faster to negative infinity.

Here's what that means:

As we approach the center of the Helland universe,
the magnitude of the mass-energy density ρ_m increases without bound.​

The following graph shows how the mass-energy density ρ_m of the Helland universe varies as a function of the radial coordinate r.

The mass-energy density is negative throughout the Helland universe, coming closest to zero at r=½.

That statement, like the graph, assumes the cosmological constant is zero. With the r-coordinate system of the original Helland metric form, setting the cosmological constant to −4 has hardly any effect on the mass-energy density at small values of r, but allows the mass-energy density to reach zero at r=½ and to become positive for r > ½, increasing without bound as r approaches 1.

The negative mass-energy density of the Helland universe in the vicinity of its central point, and the infinitely negative mass-energy density at that central point, would seem to disqualify the Helland universe as a plausible model of reality.

Furthermore, a spacetime manifold that has the mass-energy density shown in the graph above at time t=0 would rapidly evolve into a rather different distribution of mass-energy density, contradicting the Helland metric form's implicit assertion that the Helland universe is static.

In other words, the Helland metric is not consistent with general relativity.


[size=+1]Summary of conclusions.[/size]

By using seven different but equivalent coordinate systems, the Ricci curvature tensor calculated by Mike Helland, and the field equations of general relativity, this note has derived the following consequences of the Helland metric:

• The Helland universe is static.
  
  
• The Helland universe is geocentric.
  
  
• The Helland universe has a curvature singularity at that central point.
  
  
• The mass-energy density at that central point is negative infinity.
  
  
• The mass-energy density is negative throughout the Helland universe, (unless the cosmological constant is negative).
  
  
• The static universe implied by the Helland metric is not consistent with Einstein's general theory of relativity, because a universe with the Helland universe's distribution of mass-energy would not be static.

 

[sphenisc]

"If the interval between two events is negative, then it is possible for a massive particle or object to have travelled from one event to the other, which means the two events might be causally connected. If the interval is negative, then it is not possible for a massive particle or object to go from one to the other, "

I assume the second "negative" was intended to be "zero"?

 

[W.D.Clinger]

I assume the second "negative" was intended to be "zero"?

Yes. Thank you for spotting this so quickly that I could correct the error.

 

[Mike Helland]

This is the fourth of a 4-part series that analyzes the Helland universe.

I appreciate the time and effort it took to write and format your very detailed, informative essay.

Here are some of the things that stand out.

The presence of that radial coordinate r within g_tt suggests the Helland universe is neither homogeneous nor isotropic, and that events at the origin of a spatial slice are indeed special.

Empirical reality is neither homogeneous nor isotropic. You can only see Andromeda in one direction.

The strategy behind cosmology seems to be define a manifold, and then add a light cone, and wherever they intersect, that's observed reality.

Here's the handy "de Sitter sight-seeing tour" I've referenced before.

http://www.bourbaphy.fr/moschella.pdf

Page 4 has this:


(Click to enlarge to read the caption)

My strategy seems to be different.

Let's use the light cone (specifically, the past light cone, ie, observed reality) as a manifold, and describe that. Presumably the light cone should be a submanifold of the greater "universe", if such a "metric of everything" does in fact exist. We have know a priori reason to suppose Einstein was right about all that.

In any case, the tip of the past light cone is special. It's actually in the present, unlike the rest of the cone. It's the at the horizon between the past and future.

Points with r=0 are the only points of the Helland universe at which the coordinate speed of light is c. Does that mean points with r=0 are special?

Yes.

Using a coordinate speed of light to draw conclusions about physical reality is like using a Mercator map to compare the areas of Greenland and Africa.

But if your coordinates are proper distance and cosmic time, then dx/dt is the real velocity.

This shows the real velocity of light as being c - H(z)d(z) in LCDM:

We all know that the light wants to go at c. And its the expansion of space that doesn't allow it to go at c.

But it's change in real distance over its change in real time is c - H(z)d(z).

[*]The mass-energy density is negative throughout the Helland universe, (unless the cosmological constant is negative).

That doesn't seem so bad.

 

[W.D.Clinger]

This is the first of a 4-part series.
[size=+1]Abstract.[/size]

The Helland universe...

I have already objected to this characterization.

When discussing the spacetime manifold defined by a metric form, it is convenient and traditional to refer to the manifold using the name of the person who proposed the metric, as in de Sitter space, anti-de Sitter space, de Sitter universe, Einstein universe, Gödel universe, etc.

The "territory" of the metric is an observer's past light cone (ie, observed reality).

Nonsense.

The Helland metric form you proposed on 24 October, confirmed on 26 October, and repudiated earlier this morning defines a spacetime manifold and implies many important properties of that manifold. My 4-part series analyzing the Helland universe explains how the Helland metric form implies those important consequences for the Helland universe.

That Helland universe is the territory described by all seven of the charts (aka maps, aka coordinate systems) defined in that 4-part series.

Although you have now abandoned the Helland metric form you had settled upon just one week ago, my 4-part analysis of that Helland universe stands as an example of how different maps describe exactly the same territory in different ways, of how the metric form looks different in different coordinate systems but continues to describe exactly the same metric tensor field, and of how that metric tensor field implies so many important properties of the spacetime manifold (aka universe).

Indeed, my 4-part series shows how to go about analyzing the new Helland universe you described earlier this morning by stating a new Helland metric form.

 

[Mike Helland]

When discussing the spacetime manifold defined by a metric form, it is convenient and traditional to refer to the manifold using the name of the person who proposed the metric, as in de Sitter space, anti-de Sitter space, de Sitter universe, Einstein universe, Gödel universe, etc.

You know that's not what I meant.

In this case the manifold is obviously not a universe.

It's a light cone. A submanifold of the universe.

If you want to call it the Helland light cone, be my guest.

If we want to describe the universe, maybe we should start with the light cone, and get that right first.

Here's the light cone (part of it anyways) of special relativity in blue and my light cone in red.

 

[The Man]

You know that's not what I meant.

In this case the manifold is obviously not a universe.

It's a light cone. A submanifold of the universe.

If you want to call it the Helland light cone, be my guest.

If we want to describe the universe, maybe we should start with the light cone, and get that right first.

Here's the light cone (part of it anyways) of special relativity in blue and my light cone in red.

[qimg]https://mikehelland.github.io/hubbles-law/img/xt0.png[/qimg]

Well there ya go. The implication being that some spectime events subsumed by the "special relativity in blue" "light cone" would not be perceptible by your "light cone in red". Provided a vertical orientation of your unlabeled chart. However if the orientation was meant to be horizontal the would reverse the situation. Some spectime events subsumed by your "light cone in red" would not be perceptible by the "special relativity in blue" "light cone".

Besides observations being consistent with special relativity all you've done is add a deliberately imperceptible change in the speed of light. Since whenever you measure the speed of light it would be at your origin of your light cone. An observer measuring in the past at the same location and recording for the future would report the same speed of light. The only time the speed changes (angle of the light cone boundary change) in the past is only when you observe it from the future. So events observable to an observer in the past at the same location becomes non-observable and not casually relatable to some observer in the future in spite of being within both observers "special relativity in blue" "light cone". This happens very quickly by just one of your time intervals (with time as the vertical axis) and effects past causality from -5 intervals on. This appears to improve for past observers with the discrepancies falling further and further into their past.

 

[Mike Helland]

Besides observations being consistent with special relativity all you've done is add a deliberately imperceptible change in the speed of light.

Well, observations are not consistent with special relativity. Cosmological redshift and cosmic time dilation are observed phenomena.

My system can be stated like this:

Postulate light always travels at c to those who observe it
Observed fact: clocks observed from great distance are seen running in the distant past as slower than our clocks

Then the it must be true, that to an observer there and then (far away and long ago) would still be measuring the speed of light as ~300,000 km/s according to their local and contemporary clocks, which are running at half speed to compared to ours (using z=1 as an example). Their second lasts as long as two of our seconds.

Therefore, the speed of light there and then has to be half of what it is here and now.

 

[The Man]

Well, observations are not consistent with special relativity. Cosmological redshift and cosmic time dilation are observed phenomena.

You do understand what the "special" in special relativity means, right? Oh, and you are the one who cited special relativity as the blue line on your graph with no mention of "Cosmological redshift and cosmic time dilation" in reference to that graphic. "Cosmological redshift and cosmic time dilation" are "observed phenomena" of the metric expansion of space which would fall under General Relativity.

My system can be stated like this:

Postulate light always travels at c to those who observe it
Observed fact: clocks observed from great distance are seen running in the distant past as slower than our clocks

Then the it must be true, that to an observer there and then (far away and long ago) would still be measuring the speed of light as ~300,000 km/s according to their local and contemporary clocks, which are running at half speed to compared to ours (using z=1 as an example). Their second lasts as long as two of our seconds.

Therefore, the speed of light there and then has to be half of what it is here and now.

No, it doesn't and can't be as you just asserted as "Postulate light always travels at c to those who observe it". You are mixing clocks and rulers (reference frames) using the past's clocks to say half the time and the present's rulers to say the same distance. The metric of space time has expanded, that's what makes the cosmological time dilation (events at cosmological distances seem to happen slower). Distances were shorter in the frame of the past's apparently slower clock. Half the time and half the distance results in the same speed. Again merely an effect of the metric expansion of space time.

Think of it this way as you are moving away from the source of a sound the wavelength gets longer. So a sound of say 10 wavelengths at the source is still 10 wavelengths at the listener but the wavelength is longer now and with the speed of sound still the same means the overall time of the sound is lengthened as well. Time changes, distance changes, wavelength and period changes but number of cycles and speed stay the same.

 

[Mike Helland]

"Cosmological redshift and cosmic time dilation" are "observed phenomena" of the metric expansion of space which would fall under General Relativity.
...

Again merely an effect of the metric expansion of space time.

According to the standard model of cosmology.

Basically, it says, our eyes deceive us. Time isn't actually running slower, everything is actually getting farther away.

What we observe is an effect of something we don't observe, the expansion of space.

How fast is space expanding, btw?

The lack of a clear answer to that question is part of the motivation behind finding an alternative to the standard model.

So I say, our eyes don't deceive us. What we observe to be real should be considered real. And if it is, the whole of cosmology as we know it has zero basis in reality.

 

[The Man]

According to the standard model of cosmology.

Basically, it says, our eyes deceive us. Time isn't actually running slower, everything is actually getting farther away.

No, it doesn't say anything about our eyes deceiving us, what it does say is that there is a metric expansion of space time and it appears to be accelerating.

What we observe is an effect of something we don't observe, the expansion of space.

We do observe it, gravity bound clusters are moving away from each other consistent with a metric expansion of space time.

How fast is space expanding, btw?

How fast is the universe expanding? New supernova data could help nail it down

Data from the CMB suggests that the universe is expanding at the rate of about 41.9 miles (67.5 kilometers) per second per megaparsec (a distance equivalent to 3.26 million light-years). In contrast, data from supernovas and Cepheids in the nearby universe suggests a rate of about 46 miles (74 km) per second per megaparsec.

I expect "in the nearby universe" should have been 'in the nearby galaxy'

The lack of a clear answer to that question is part of the motivation behind finding an alternative to the standard model.

What lack of a clear answer? You're taking a difference of 4 miles across 3.26 million light-years. Try comparing that to the accuracy of GPS.

So I say, our eyes don't deceive us. What we observe to be real should be considered real. And if it is, the whole of cosmology as we know it has zero basis in reality.

Actually you did in fact just say "our eyes deceive us" but merely tried to ascribe what you said to others. So you'll say it but just won't own it.

Meanwhile ignoring the bulk of the relevant facts in the post you are responding to as well as the explanations in that of the self-inconsistencies in your assertions.

 

[Mike Helland]

I expect "in the nearby universe" should have been 'in the nearby galaxy'

They meant "nearby universe."

But, nearby is pretty misleading. The supernovae data extends to 2/3rds of the observable universe.